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Christian Franz,1, 3 Jan Spallek,1 Rudolf +Sch¨onmann,1 Michal Stekiel,1 Astrid Schneidewind,3 Marc Wilde,1, 2 and C. Pfleiderer1, 2, 4 +1Physik-Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany +2Zentrum f¨ur QuantumEngineering (ZQE), Technische Universit¨at M¨unchen, D-85748 Garching, Germany +3J¨ulich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), D-85748 Garching, Germany +4Munich Center for Quantum Science and Technology (MCQST), +Technische Universit¨at M¨unchen, D-85748 Garching, Germany +(Dated: January 23, 2023) +We report the magnetization, ac susceptibility, and specific heat of optically float-zoned single +crystals of CePdAl3. In comparison to the properties of polycrystalline CePdAl3 reported in the +literature, which displays a tetragonal crystal structure and no long-range magnetic order, our single +crystals exhibit an orthorhombic structure (Cmcm) and order antiferromagnetically below a N´eel +temperature TN = 5.6 K. The specific heat at zero-field shows a clear λ-type anomaly with a broad +shoulder at TN. +A conservative estimate of the Sommerfeld coefficient of the electronic specific +heat, γ = 121 mJ K−2 mol−1, indicates a moderately enhanced heavy-fermion ground state. A twin +microstructure evolves in the family of planes spanned by the basal plane lattice vectors ao and co, +with the magnetic hard axis bo common to all twins. The antiferromagnetic state is characterized +by a strong magnetic anisotropy and a spin-flop transition induced under magnetic field along the +easy direction, resulting in a complex magnetic phase diagram. Taken together our results reveal a +high sensitivity of the magnetic and electronic properties of CePdAl3 to its structural modifications. +I. +INTRODUCTION +Cerium-based intermetallic compounds exhibit a vari- +ety of ground states and various underlying exotic phys- +ical phenomena, such as unconventional superconductiv- +ity [1–8], heavy-fermion states [9, 10], non-Fermi liquid +behavior [11], vibronic hybrid excitations [12–16], and +complex magnetic order [17–24]. On the phenomenologi- +cal level, the origin of this remarkable diversity of ground +states has been attributed to the competition of narrow f- +electron bands and strong electronic correlations together +with spin-orbit interaction, crystal electric field (CEF) ef- +fects, and strong magneto-elastic coupling. An overarch- +ing theme connecting much of the research in f-electron +compounds concerns the condition of the emergence of +magnetic order. +A class of compounds with the general formula CeTX 3 +(T is a transition metal and X is a p-block element) +crystallizing in subgroups of the BaAl4-type (I4/mmm) +tetragonal structure has received special attention [3, 6, +22–42]. In these compounds, a large number of struc- +tural variants and diverse magnetic and electrical proper- +ties can be obtained by changing the transition metal T. +Many members of this class such as CeRhGe3, CeAuAl3, +CeCuAl3, and CeCoGe3 adopt a non-centrosymmetric +tetragonal structure (BaNiSn3-type I4mm) and exhibit +antiferromagnetic behavior [25–33]. Other members such +as CeAgAl3 display ferromagnetism with a centrosym- +metric orthorhombic crystal structure [34, 35]. A spin- +glass state was reported in non-centrosymmetric tetrago- +nal CePtAl3 below 0.8 K [32]. Complex magnetic phases +∗ vivek.kumar@tum.de +have been observed in antiferromagnetic CeNiGe3[22, +24], CeCoGe3 [23, 36–38] and CePtSi3 [39]. The discov- +ery of pressure-induced unconventional superconductiv- +ity in the non-centrosymmetric tetragonal heavy-fermion +antiferromagnets CeRhSi3, CeIrSi3, CeCoGe3, CeIrGe3, +and CeRhGe3 even suggests a new direction in condensed +matter physics [3, 6, 40–42]. +An important aspect is the structural stability of these +systems and the emergence of different electronic ground +states. As one of the first examples, CePd2Al2 [13, 15], +which is closely related to the class of CeTAl3 of ma- +terials, was found to undergo a structural phase trans- +formation from a tetragonal to an orthorhombic lat- +tice at 13.5 K. An inelastic neutron scattering study re- +vealed three magnetic excitations in the paramagnetic +phase. +However, according to Kramer’s theorem, only +two CEF excitations are expected due to the splitting of +ground state J = 5/2 of the Ce3+ ion into three doublets +in tetragonal/orthorhombic point symmetry suggesting +strong coupling between the crystal fields and the crystal +structure. Later, Adroja et al. found a similar anomaly +in CeCuAl3 [14], where a structural instability manifests +itself in terms of a drastic change in lattice parameters +of the tetragonal structure around 300 ◦C [29]. +These +anomalous excitations have been interpreted by means +of Thalmeier and Fulde’s model of bound states be- +tween phonons and CEF excitations as generalized to the +tetragonal point symmetry. Recently, ˇCerm´ak et al. con- +firmed related hybrid CEF-phonon excitations even for +weak magnetoelastic coupling in isostructural CeAuAl3 +[43]. Moreover, CePd2Al2, CeCuAl3 and CeAuAl3 order +antiferromagnetically at low temperatures and exhibit +incommensurate amplitude-modulated magnetic struc- +tures [15, 44–46]. The presence of multi-step magnetism +and complex magnetic phase diagrams suggests the pos- +arXiv:2301.08617v1 [cond-mat.str-el] 20 Jan 2023 + +2 +sible existence of topologically non-trivial multi-k struc- +tures akin to skyrmion lattices [47]. This raises the ques- +tion, if and how the formation of magnetic order depends +on the stabilization of specific crystal structure. +In this paper we focus on CePdAl3. A study of as- +cast polycrystalline CePdAl3 by Schank et al. in 1994 +revealed a tetragonal I4mm structure with lattice con- +stants a = 4.343 ˚A and c = 10.578 ˚A [48], where the +heat treatment at high temperature results in a struc- +tural phase transformation with an antiferromagnetic or- +der below TN ≃ 6 K. In contrast, no magnetic order was +found down to 0.1 K in a recent investigation by Franz +et al. on single crystalline tetragonal CePdAl3 grown by +optical float zoning with a growth rate of 6 mm/h [49]. +For the work reported in the following, a single crystal +was prepared by optical float zoning using a much lower +growth rate of 1 mm/h. Under these conditions we found +that CePdAl3 crystallizes in an orthorhombic as opposed +to a tetragonal structure [50]. In this paper, we report +comprehensive magnetization, ac susceptibility, and spe- +cific heat measurements on single crystalline orthorhom- +bic CePdAl3. +As our main result we find the charac- +teristics of antiferromagnetic order below TN = 5.6 K. We +determine the magnetic phase diagram upto 14 T, where +we find the emergence of complex magnetic phases un- +der magnetic fields applied along the easy direction. The +presence of different structural and magnetic configura- +tions of CePdAl3 identifies a new example of a material in +which to search for hybrid excitations and new magnetic +phases in the future. +Our paper is organized as follows. After a brief account +of the experimental methods in Sec. II, we present our ex- +perimental results in Sec. III. We start with the structural +properties and notation in Sec. III A, followed by the spe- +cific heat results in Sec. III B and magnetic susceptibility +data in Sec. III C. The temperature- and field-dependence +of the magnetization is presented in Sec. III D. We find +that the magnetic field-driven transitions for fields ap- +plied along the easy direction are consistent with the +specific heat as a function of temperature as presented +in Sec. III E. In Sec. III F, we examine the magnetic tran- +sitions in more detail by analyzing the hysteresis of the +field-dependent magnetic susceptibility. Comprehensive +datasets allow to infer the magnetic phase diagram pre- +sented in Sec. III G. The conclusions are summarized in +Sec. IV. +II. +EXPERIMENTAL METHODS +A single-crystal of CePdAl3 was grown using the op- +tical floating-zone technique following a process similar +that described in Ref. [49, 51, 52]. As the main difference, +the growth rate was reduced from 6 mm/h [49] to 1 mm/h +which resulted in the formation of an orthorhombic crys- +tal. +The crystal structure of CePdAl3 was determined by +means of single-crystal x-ray diffraction (SCXRD). A +platelet-shaped crystal with dimensions 50 µm × 40 µm × +10 µm was cleaved of the CePdAl3 crystal as grown. The +platelet was investigated at a Rigaku XtaLAB Synergy-S +diffractometer, using a Mo x-ray source with λ = 0.71 ˚A +and a two-dimensional HyPix-Arc 150◦ detector. Bragg +reflections were indexed using CrysAlisP ro [53] as inte- +grated with the diffractometer. +The single crystals were oriented by Laue x-ray diffrac- +tion and a cuboidal sample was cut with orientations a⋆ +o, +c⋆ +o and bo as introduced below for the measurement of +the bulk properties. +The ac susceptibility, magnetiza- +tion, and specific heat were measured in a Quantum De- +sign physical property measurement system (PPMS) at +temperatures down to 2 K under magnetic fields up to +14 T. In order to determine the temperature dependence +of the bulk properties, the sample was first cooled from a +high temperature, well above TN, to the lowest attainable +temperature in the absence of a magnetic field. Subse- +quently, the field was set to the desired value and data +were collected for increasing temperature. This protocol +was repeated for different target magnetic fields. The ac +susceptibility was measured at an excitation amplitude of +1 mT and an excitation frequency of 911 Hz. The specific +heat was measured down to 2 K using a large heat-pulse +method [54]. For temperatures between 0.08 K and 4 K +the specific heat was measured in a Dryogenic adiabatic +demagnetization refrigerator using a conventional heat- +pulse method. +The field dependence of the magnetization and the ac +susceptibility was measured using the following temper- +ature versus field protocol. First, the sample was cooled +from a high temperature well above TN to the target tem- +perature in the absence of a magnetic field. Second, data +as a function of magnetic field were recorded in a se- +quence of field sweeps from zero-field to 14 T, 14 T to +-14 T, and -14 T to 14 T. +The bulk properties recorded on different pieces cut +from the large single crystal ingot were consistent. The +temperature and field dependent features along a⋆ +o and +c⋆ +o were qualitatively identical. +Therefore, comprehen- +sive data focused on one of these directions, c⋆ +o, were +recorded. Summarizing the key result of our study, the +magnetic phase diagrams of CePdAl3 were inferred. Sig- +natures detected in measurements as a function of tem- +perature and magnetic field are labelled as Tj and Hj, +respectively. For clarity, the same subscript j is assigned +to the transitions corresponding to the same line in the +phase diagram. +III. +EXPERIMENTAL RESULTS +A. +Crystal structure and twinning +Different crystal growth conditions favor a tetrago- +nal (I4mm) [49] or orthorhombic crystal structures of +CePdAl3. By means of single crystal x-ray diffraction, +we determined that the orthorhombic lattice stabilizes + +3 +in the Cmcm space group. +The lattice parameters at +room temperature are ao = 6.379 ˚A, bo = 10.407 ˚A and +co = 5.975 ˚A. The orthorhombic phase exhibits a pseudo- +tetragonal twinning in the basal plane, evident, for in- +stance, by the splitting of the Bragg reflections shown in +Fig. 1(a). The twinning law was determined by index- +ing all measured reflections with components of the four +twins presented in Fig. 1(b). An illustration of the twin +orientation is shown in Figs. 1(c) and (d). The three per- +pendicular cartesian directions of twins for i = 1, 2, 3, 4 +are denoted by ai +o, bi +o and ci +o, where ai +o and ci +o construct +an effective basal plane and bi +o mutually represents the +long axis. The volume fraction of the four twins labelled +i = 1, 2, 3, and 4 are 0.38, 0.26, 0.23, and 0.13, respec- +tively. The mismatch angle between the twins numbered +1 and 2, as well as 3 and 4, are around 3◦. +Measurements on different pieces cleaved of the sin- +gle crystalline ingot demonstrate the same twinning +scheme with minor differences in twin fractions of differ- +ent twins. An attempt to detwin the crystals by means +of high-temperature treatment, etching, or cleaving of +micrometer-sized crystals neither affected the twinning +as such nor the twinning fractions. +In turn, measurements in any direction in the effective +basal plane reflect effectively an admixture of ai +o and ci +o +directions due to the four twins. We define, therefore, +two mutually perpendicular effective sample directions +a⋆ +o and c⋆ +o, explicitly taking into account the volume frac- +tions of the four twins. This definition is schematically +depicted in Figs. 1(c) and (d) where a⋆ +o is nearly aligned +along a1,2 +o +and c3,4 +o , while c⋆ +o is aligned to that of c1,2 +o +and a3,4 +o . The third crystal direction, corresponding to +the long axis bo, remains unaffected by the twin defor- +mations. +B. +Temperature-dependence of the specific heat +The temperature dependence of the specific heat C(T) +of single-crystalline tetragonal (I4mm) and orthorhom- +bic (Cmcm) CePdAl3, as well as nonmagnetic polycrys- +talline tetragonal (I4mm) LaPdAl3 measured in the ab- +sence of a magnetic field are shown in Fig. 2. +No evi- +dence suggesting magnetic order was observed in tetrag- +onal CePdAl3 [49]. In orthorhombic CePdAl3, a λ-type +anomaly comprising a peak at 5.4 K followed by a shoul- +der closely above the transition temperature TN = 5.6 K +is observed, where the magnetization is characteristic of +antiferromagnetism as reported below. The behavior ob- +served is consistent with a previous study of polycrys- +talline CePdAl3 [48]. Moreover, the properties are rem- +iniscent of the commensurate to incommensurate mag- +netic transition reported of other strongly correlated sys- +tems [55, 56]. +A pronounced shoulder in the specific heat has also +been seen in other systems, notably, the chiral cubic mag- +net MnSi [54, 57], where it reflects a change of char- +acter of the critical spin-fluctuations when approach- +FIG. 1. +Twin scheme in the basal plane of orthorhombic +CePdAl3 as derived from single crystal x-ray diffraction. (a) +X-ray scattering intensity reconstructed in the H0L plane. +The splitting of the reflections is characteristic of twin for- +mation. (b) Indexed reflections of panel (a) with the colors +corresponding to different twin domains. Schematics of the +lattice vectors ai +o and ci +o of twin i in the basal plane H0L of +the orthorhombic crystal are depicted in the lower panels (c) +and (d). Four twins labelled i = 1, 2, 3, and 4 were identi- +fied. a⋆ +o and c⋆ +o are defined as mutual perpendicular sample +directions comprising the admixtures of twin lattice vectors. +ing long-range helimagnetic order and a concomitant +fluctuation-induced first-order transition. Details of the +low-temperature specific heat of orthorhombic CePdAl3 +at zero-field are presented in Sec. III E below, which also +includes data collected at different magnetic fields. +Above TN, the expression C/T = γ + βT 2, where +γ and β are the electronic and phononic contributions +to the specific heat, respectively, has been fitted to the +specific heat data in the range ∼18 to ∼23 K of or- +thorhombic CePdAl3. The values obtained for γ and β +are 234 mJ mol−1 K−2 and 3.437 × 10−4 J mol−1 K−4, +respectively. +The Debye temperature, ΘD = 305 K, +associated with β may be derived using the relation +β = (12/5)π4nR/Θ3 +D, where n is the number of atoms +per formula unit and R is the gas constant. The phonon +contribution to the specific heat in the Debye model [or- +ange line in Fig. 2] is given by +Cph,Debye = 9nR +� T +ΘD +�3 � xD +0 +x4ex +(ex − 1)2 dx +(1) +where xD = ΘD/T. At high temperatures the experimen- +tal data of tetragonal LaPdAl3 and CePdAl3, as well as +orthorhombic CePdAl3 approach the Dulong-Petit limit, +3nR = 15R = 124.7 J mol−1 K−1, where n = 5. + +(a) +(b) +-2 +2 +0 +1it +0 +2 +-2 +0 +2 +-2 +0 +2 +H +H +(c) +(d) +Twin volumes +* +* +1: 38 % +a.ttas +ch +t c? +2: 26 % +3: 23 % +4: 13 % +3 +* +a +.4 +The large value of γ = 234 mJ mol−1 K−2 obtained +from the low-temperature specific heat above TN is typ- +ical for a heavy-fermion system. It has to be borne in +mind, however, that evaluating γ at the relatively high- +temperature range above TN is associated with substan- +tial uncertainties. A lower bound of γ, fitting the exper- +imental data in the antiferromagnetic state at tempera- +tures between ∼0.9 K and ∼3.7 K, yields a value of γ = +121 mJ mol−1 K−2 still characteristic of heavy-fermion +behaviour. +At high temperatures (T > 100 K), the specific heat of +all three compounds exhibits essentially the same tem- +perature dependence. However, the specific heat of or- +thorhombic CePdAl3 is slightly smaller than for tetrago- +nal CePdAl3, suggesting reduced electronic and phononic +contributions associated with the reduced crystal sym- +metry. Compared to nonmagnetic LaPdAl3, the specific +heat of orthorhombic CePdAl3 is also slightly smaller, +yet within the experimental error of experiment. Indeed, +a multiplication with a fraction of 0.99 to the total signal +of LaPdAl3 fully superimposes the data of CePdAl3 as +shown in Fig. 3(a) of C/T vs T. The corresponding dif- +ference in specific heats may be attributed to the mag- +netic contribution of the specific heat of orthorhombic +CePdAl3. +Shown in Fig. 3(b) is a sharp peak at T = 5.4 K in +the magnetic contribution to the specific heat following +subtraction of the phonon contribution signaling an an- +tiferromagnetic transition. +In addition, a broad maxi- +mum around 30 K may be discerned as characteristic of +a Schottky anomaly due to crystal electric field contribu- +tions. +In the tetragonal as well as the orthorhombic symme- +try of the lattice, the degeneracy of the sixfold ground +state multiplet of the Ce3+ ion splits into three doublet +states. These lift the first and second excited state with +respect to the ground state resulting in a contribution to +the specific heat which can be expressed as [58] +CCEF =R +Z +2 +� +l=0 +gl +� El +kT +�2 +exp +� +− El +kT +� +− R +Z2 +� 2 +� +l=0 +El +kT glexp +� +− El +kT +��2 +(2) +where +Z = +2 +� +l=0 +glexp +� +− El +kT +� +(3) +is the partition function, and l = 0, 1 and 2 denote the +ground, first and second excited states, respectively. The +degeneracy of the three doublet states is g0 = g1 = g2 = +2. +The energy difference E1−E0 = ∆1 and E2−E0 = ∆2 +represent the levels of the first and the second excited +states, respectively. A fit of the data to Eqn. (2) between +20 K and 100 K yields ∆1 = 25.4 K and ∆2 = 76.0 K, +0 +50 +100 +150 +200 +0 +50 +100 +150 +0 +5 +10 +15 +20 +25 +0 +8 +16 +Specific heat C (J mol-1 K-1) +CePdAl3 +Cmcm +CePdAl3 +I4mm +ΘD = 305 K +Dulong-Petit limit +LaPdAl3 +I4mm +Temperature T (K) +(a) +Temperature T (K) +(b) +TN +FIG. 2. +(a) Zero-field specific heat of single-crystalline or- +thorhombic (black) and tetragonal (blue) [49] CePdAl3 as +a function of temperature. Data of orthorhombic CePdAl3 +were measured in a Dryogenic system between 0.08 K and 4 K, +and in a PPMS between 2 K and 200 K. Also shown are the +specific heat of nonmagnetic polycrystalline LaPdAl3 (Gray +line) and the Debye fit (orange line) calculated from the low- +temperature specific heat of the Cmcm structure. The Debye +temperature is ΘD = 305 K. The Dulong-Petit limit for all +three compounds, 15R = 124.7 J mol−1 K−1 is depicted by +a dashed line. (b) The low-temperature part of the specific +heat of orthorhombic CePdAl3 shows a pronounced λ-type +anomaly with a broad shoulder at the magnetic transition at +TN. +respectively. Note that, the normalized subtraction of the +LaPdAl3 signal may introduce systematic errors in the +determination of the precise values of the excited states. +For instance, subtraction of the signal of LaPdAl3 after +multiplication with a fraction of 0.98 yields ∆1 = 28.6 K +and ∆2 = 95.5 K. +Furthermore, we have calculated the magnetic entropy +S = +� +(C/T)dT presented in Fig. 3(c). At the magnetic +transition temperature, the entropy reaches the theoret- +ical value of Rln 2 for a doublet ground state expected +of Ce3+ ions. When increasing the temperature, the en- +tropy increases and reaches Rln4 around 30 K, approach- +ing saturation above 100 K consistent with the scheme of +crystal electric field levels. + +5 +0 +8 +16 +0 +1 +2 +3 +0 +50 +100 +150 +200 +0 +6 +12 +0 +4 +8 +0 +6 +∆C (J mol-1 K-1) +(b) +∆1 = 25.4 K +∆2 = 76.0 K + Cmag + Fit +C/T (J mol-1 K-2) +(a) + CePdAl3 (Cmcm) + LaPdAl3 (99 %) + Difference +Entropy S (J mol-1 K-1) +Temperature T (K) +RLn2 +RLn4 +(c) +S (J mol-1 K-1) +T (K) +RLn2 +FIG. 3. Magnetic contribution to the specific heat and crys- +tal electric field levels. (a) Specific heat per unit tempera- +ture, C/T, of orthorhombic CePdAl3 and tetragonal LaPdAl3 +(with a multiplication of a fraction of 0.99) as well as their +difference. (b) Magnetic specific heat, Cmag, and the fit to +the expression for the crystal electric field contribution to the +specific heat yields ∆1 = 25.4 K and ∆2 = 76.0 K (c) Magnetic +contribution to the entropy. The inset shows the entropy at +low temperatures. +C. +Temperature-dependence of the magnetic +susceptibility +The real part of the ac susceptibility, Re χac of or- +thorhombic CePdAl3 as a function of temperature is +shown in Fig. 4(a) for a⋆ +o, c⋆ +o and bo. A clear magnetic +transition is observed at TN = 5.6 K in the low temper- +ature range, characteristic of the onset of antiferromag- +netic order as indicated by arrows in the inset. Namely, +0 +0.5 +1 +1.5 +0 +4 +8 +0 +0.5 +1 +1.5 +0 +100 +200 +300 +0 +1 +2 +0 +20 +40 +0 +4 +Re� ac (10-2) + Hac || a +� +o + Hac || c +� +o + Hac || bo +(a) +Re� ac (10-2) +T (K) +TN +H/M (102) +H/M (103) +Temperature T (K) + H || a +� +o + H || c +� +o + H || bo +� 0H = 0.1 T +(b) +T (K) +FIG. 4. (a) Temperature dependence of the real part of the +ac susceptibility, Re χac of orthorhombic CePdAl3 measured +along a⋆ +o, c⋆ +o and bo at an excitation amplitude of 1 mT and +a frequency of 911 Hz. The inset shows the low-temperature +part of Re χ ac, reflecting the characteristics of an antiferro- +magnetic transition at TN = 5.6 K. (b) Susceptibility, H /M, +as a function of temperature for H ∥ a⋆ +o, H ∥ c⋆ +o and H ∥ bo +measured in a field of 0.1 T. Gray lines are Curie-Weiss fits. +The inset shows the data for temperatures below 50 K. +below TN, Re χac monotonically decreases along a⋆ +o and +c⋆ +o with decreasing temperature, while slightly increas- +ing along bo. The magnitude of R eχac along different +axes differs significantly for T <100 K, indicating size- +able magnetic anisotropy. +Figure 4(b) shows the normalized susceptibility, H /M, +as a function of temperature in a field of 0.1 T for H +∥ a⋆ +o, H ∥ c⋆ +o and H ∥ bo. In the paramagnetic state +well above TN, a Curie-Weiss dependence is observed. +A linear fit to the data above 100 K yields Weiss tem- +peratures Θa⋆ +W = -0.8 K, Θc⋆ +W = -13.5 K and Θb +W = -33.0 K +for H ∥ a⋆ +o, H ∥ c⋆ +o and H ∥ bo, respectively, char- +acteristic of an antiferromagnetic coupling. +Moreover, +the effective moments of 2.39, 2.49 and 2.44 µB per ion +obtained under magnetic field along a⋆ +o, c⋆ +o and bo, re- +spectively, are close to the value of 2.54 µB expected for +a free Ce3+ ion. This might suggest a localized nature +of the Ce moments in CePdAl3. The deviation of H / +M from the Curie-Weiss dependence for TN < T < 100 K + +6 +shown in the inset of Fig. 4(b) may be related to CEF +effects and electronic correlations. Furthermore, despite +the twin deformations, a significant difference between +the susceptibilities along a⋆ +o and c⋆ +o in the paramagnetic +state indicates a large anisotropy in the basal plane, char- +acteristic of an easy-axis system. +D. +Magnetization +The magnetic field dependence of the isothermal mag- +netization of orthorhombic CePdAl3 at 2 K for H ∥ a⋆ +o, H +∥ c⋆ +o and H ∥ bo is shown in Fig. 5(a). No hysteresis is ob- +served. The magnetization varies linearly in the low-field +region up to 1 T as shown in the inset of Fig. 5(a) con- +sistent with antiferromagnetic order. For fields along a⋆ +o +and c⋆ +o, an S-shaped rise is observed in the magnetization +when further increasing field. A kink around 5.5 T sug- +gests a field-driven transition. The magnetization values +at 5.5 T are 0.85 µB for H ∥ a⋆ +o, 0.44 µB for H ∥ c⋆ +o, and +0.18 µB for H ∥ bo. The magnetization increases mono- +tonically above this transition where the moments along +a⋆ +o and c⋆ +o at 14 T, the highest field strength of studied, +are 1.3 and 0.7 µB per Ce atom, respectively. In compar- +ison, the magnetization increases linearly with field for +H ∥ bo. The moment at 14 T is 0.4 µB per Ce atom. +Keeping in mind the twinned crystal structure, the +magnetization along a⋆ +o and c⋆ +o represent a mixture of +the crystallographic ao and co axes. A large quantita- +tive difference in the magnetization at 5.5 T along a⋆ +o and +c⋆ +o makes it unlikely, that a metamagnetic transition oc- +curs at the same field value in the ao and co directions +in a single twin domain. Instead, it appears most likely +that the increase in the magnetization corresponds to a +spin-flop in the ai +o easy direction of each twin only. +Shown in Fig. 5(b), (c) and (d) are the isothermal mag- +netization at various temperatures for H ∥ a⋆ +o, H ∥ c⋆ +o +and H ∥ bo, respectively. +The spin-flop transition in +M (H ) for a⋆ +o and c⋆ +o shifts to lower fields under increas- +ing temperature and vanishes above TN. In contrast, the +variation in M (H ) along bo is essentially temperature +independent at and above TN. +In order to trace the field-driven magnetic transition, +we have calculated the differential susceptibility, dM /dH +from the isothermal magnetization at various tempera- +tures presented in Fig. 6(a), (b), and (c) for H ∥ a⋆ +o, H +∥ c⋆ +o and H ∥ bo, respectively. For fields along a⋆ +o and +c⋆ +o, the transition is characterized by a broad peak at +∼5.2 T at 2 K, which resolves into two peaks at elevated +temperatures. +These peaks exist below TN as marked +by arrows at the transition fields H3 and H4, following +the labelling scheme described in Sec. II. With increasing +temperature the field range between the peaks increases +and both peaks shift to lower field values. No indication +exists of a field-induced transition in dM /dH for field +along bo. +The evolution of the field-induced transitions may be +traced in further detail by the temperature dependence +-12 +-6 +0 +6 +12 +-1.2 +-0.6 +0 +0.6 +1.2 +0 +6 +12 +0 +0.6 +1.2 +0 +6 +12 0 +6 +12 +-1 +0 +1 +-0.05 +0 +0.05 +H || bo +Magnetization M (� B f.u.-1) +Magnetic field�� 0H (T) +T = 2 K +Magnetic field�� 0H (T) +H || a +� +o +H || c +� +o +(a) +(b) +H || a +� +o +(c) +M (� B f.u.-1) +H || c +� +o + 2 + 3 + 4 + 5 + 6 + 10 +(d) +T (K) +H || bo +M (� B f.u.-1) +� 0H (T) +FIG. 5. +(a) Isothermal magnetization of orthorhombic +CePdAl3 at 2 K measured in a field along a⋆ +o, c⋆ +o and bo up +to 14 T. The arrow indicates the direction of increasing mag- +netic field. The inset shows the linear variation of the mag- +netization below 1 T. Typical field dependence of isothermal +magnetization at various temperatures for (b) H ∥ a⋆ +o, (c) H +∥ c⋆ +o and (d) H ∥ bo. A field-driven spin-flop transition at +∼5.5 T is observed below TN for H ∥ a⋆ +o (blue arrow) and H +∥ c⋆ +o (red arrow). +0 +2 +4 +6 +0 +2 +4 +6 +0 +0.1 +0.2 +0 +2 +4 +6 ++0.025 +H || a +� +o +H || c +� +o +H || bo +(c) +10 K +6 K +5.5 K +5 K +4.5 K +4 K +3.5 K +3 K +2.5 K +2 K +Magnetic field�� 0H (T) ++0.025 +H4 +H3 +(a) +10 K +6 K +5.5 K +5 K +4.5 K +4 K +3.5 K +3 K +2.5 K +2 K ++0.025 +H4 H3 +(b) +2 K +2.5 K +10 K +3 K +3.5 K +4 K +4.5 K +5 K +5.5 K +6 K +Susceptibility dM/dH +FIG. 6. (a) Susceptibility, dM /dH, calculated from the mea- +sured magnetization of orthorhombic CePdAl3 for (a) H ∥ a⋆ +o, +(b) H ∥ c⋆ +o and (c) H ∥ bo. Data are shifted by 0.025 for clar- +ity. Peaks correspond to field-induced transitions marked by +arrows at H3 (pink) and H4 (sky blue). The peaks disappear +above TN. + +7 +0 +0.4 +0.8 +0 +0.4 +0.8 +1.2 +0 +0.2 +0.4 +2 +4 +6 +8 10 +0 +2 +4 +2 +4 +6 +8 10 +0 +3 +6 +2 +4 +6 +8 10 +0.4 +0.5 +0.6 +0.7 +T3 +T1 +(b) +M (� B f.u.-1) +T4 +T1 +(a) +� 0H (T) +T3 +T1 +(c) +T (K) + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 +H || c +� +o +T1 +T3 ++0.3 +T4 +(e) +Re� ac (10-2) +T (K) +H || a +� +o +T1 +T4 ++0.4 +T3 +(d) +T (K) +H || bo +T1 ++0.3 +(f) +FIG. 7. +Temperature dependence of magnetization, M (T) +and real part of ac susceptibility, Reχac(T) of orthorhombic +CePdAl3 in magnetic fields up to 14 T. M (T) is shown in +panels (a), (b) and (c), and Reχac(T) in (d), (e) and (f) for +H ∥ a⋆ +o, H ∥ c⋆ +o and H ∥ bo, respectively. Reχac(T) is shifted +for clarity. Magnetic transitions are marked by vertical lines +at temperatures T1 (red), T3 (blue) and T4 (green). A complex +behavior with multiple transitions is present for field along a⋆ +o +and c⋆ +o between 2 T and 6 T. +of the magnetization M (T) and the ac susceptibility +Reχac(T). Shown in Fig. 7 is M (T) and Reχac(T) at +various fields up to 14 T. By decreasing the temperature, +orthorhombic CePdAl3 undergoes a phase transforma- +tion from paramagnetism to antiferromagnetic order at +a transition temperature T1 (marked by red lines). This +transition is visible in Reχac(T) in all crystallographic +directions. The transition at T1 shifts to lower temper- +atures under increasing field but does not vanish upto +the highest field of 14 T studied. +In the intermediate +field range from 2 T to 6 T, clear changes in M (T) and +Reχac(T) for field along a⋆ +o [Fig. 7(a) and (d)]) and +c⋆ +o [Fig. 7(b) and (e)] point to two additional phase +transitions at temperatures denoted T3 (blue line) and +T4 (green line). +These transitions disappear at fields +above 6 T. For H ∥ bo, only the first transition at T1 is +observed in M (T) and Reχac(T) [Fig. 7(c) and (f)]. +E. +Field-dependence of the specific heat +The specific heat of orthorhombic CePdAl3 as a +function of temperature at different magnetic fields for +H ∥ c⋆ +o is presented in Fig. 8. +At zero magnetic field +0 +8 +16 +0 +8 +16 +0 +2 +4 +6 +0 +8 +16 +0 +2 +4 +6 0 +2 +4 +6 +0 +1 +2 +3 +4 +5 +6 +0 +8 +16 +� 0H = 0 +T1 +T2 +(b) +1 T +T1 +T2 +T4 +(c) +2 T +T1 +T2 +T4 +(d) +Specific heat C (J K-1 mol-1) +3 T +T2 +T1 +T3 +T4 +(e) +4 T +T1 +T2 +T3 +T4 +(f) +5 T +T1 +T2 +T4 +T3 +(g) +6 T +T1 +T2 +T3 +(h) +Temperature T (K) +9 T +T1 +T2 +(i) +14 T +T1 +T2 +(j) +Specific heat C (J K-1 mol-1) +Temperature T (K) + 0 + 1 + 2 + 3 + 4 + 5 + 6 +� 0H (T) +H || c +� +o +(a) +FIG. 8. +Specific heat of orthorhombic CePdAl3 as a func- +tion of temperature under selected magnetic fields up to 14 T +applied along the c⋆ +o axis. Data measured in the Dryogenic +system between 0.08 K and 4 K are combined with data mea- +sured in the PPMS above 2 K. At H = 0 the magnetic transi- +tion displays a peak at T2 preceded by a broad shoulder with +a point of inflection at T1. Additional peaks emerge at T3 and +T4 for magnetic fields between 2 T and 6 T. +[Fig. 8(b)], a broad shoulder with a point of inflection +is observed at T1 followed by a sharp peak at T2. +Increasing the applied field results in a broadening of the +peak at T2 [Fig. 8(c)] and a splitting with an additional +peak emerging at a lower temperature T4. +For even +higher fields up to 6 T [Fig. 8(d) to (h)], the position +of T4 continues to shift to lower temperatures with +the emergence of another peak at T3. +The emergence +of the peaks at T3 and T4 in the specific heat in the +intermediate field range from 2 to 6 T is consistent with +the phase transitions deduced from the magnetization +and the ac susceptibility (see Figs. 6 and 7). For fields +above 6 T, a noticeable shift of T1 and T2 to lower +temperatures is observed. +F. +Field-dependence of the magnetic susceptibility +In order to investigate the qualitative difference be- +tween the transitions labelled as H3 and H4 in dM /dH + +8 +(see Fig. 6), we have measured the magnetic susceptibility +as a function of magnetic field between 0 and 14 T. Fig- +ure 9 shows the real part of the ac susceptibility, Reχac, +and the susceptibility calculated from the magnetization, +dM /dH, as a function of increasing and decreasing field. +At 2 K, dM /dH exhibits two peaks under increasing field, +first, a pronounced peak at 5.15 T, followed by a second +broad peak at 5.3 T for both H ∥ c⋆ +o [Fig. 9(a) and (d)] +and H ∥ a⋆ +o [Fig. 9(c) and (f)]. +The first peak shifts +to 5 T resulting in a hysteresis, while the second peak +remains at the same field value under decreasing field. +Similar effects exists in Reχac where the first peak be- +comes less pronounced with a smaller hysteresis and a +slightly lower field of 5.05 T. Also, the value of Reχac +is slightly lower around the transition. At higher tem- +peratures, both peaks are shifted to lower field values. +The hysteresis in Reχac decreases significantly and drops +below the noise level at 5 K [Fig. 9(b) and (e)]. +Here, +the magnitude of Reχac matches well with dM /dH ex- +cept around the first peak. The difference in character +of the transitions labelled as H3 and H4 suggest their +intrinsic origin rather than being related to the twinned +microstructure. +On the one hand, the hysteresis observed in dM /dH +and Reχac is reminiscent of changes of population of mul- +tidomain states. On the other hand, the smaller ampli- +tude of Reχac as compared to dM /dH indicates the pres- +ence of slow relaxation processes around the phase tran- +sition. Similar features are known to trace spin textures +like helimagnetic disclination or skyrmions in magnetic +materials [59–61]. +Further experimental investigations +are needed to explore such a possibility in orthorhombic +CePdAl3. +G. +Magnetic phase diagram +Combining the features detected in the magnetization +and the specific heat, we infer the magnetic phase dia- +grams for field parallel to c⋆ +o and bo shown in Fig. 10(a) +and (b), respectively. Due to the twinned microstructure, +the response of the magnetization, specific heat, and ac +susceptibility are qualitatively alike for H ∥ a⋆ +o and H ∥ +c⋆ +o. In addition, the enhanced signal observed along a⋆ +o +as compared to c⋆ +o indicates that a⋆ +o reflects a larger frac- +tion of the easy axis, ao. Therefore, the transitions along +both a⋆ +o and c⋆ +o reflect equally the phenomenon belonging +to the easy ao axis of the untwinned single domain. +Four magnetic regions may be distinguished for field +along c⋆ +o, denoted AF-I, AF-II, AF-III and AF-IV. At +low temperature and zero-field, the ground state is de- +noted as AF-I. With increase temperature, AF-II appears +at 5.4 K before entering in the paramagnetic (PM) state +above 5.6 K. Signatures of the AF-II region are detected +only in the specific heat. The application of a magnetic +field at low temperature drives a spin-flop transition from +AF-I to AF-IV with an intermediate region AF-III in a +narrow field range only. For finite field applied along the +0 +6 +12 +0 +2 +4 +0 +6 +12 +0 +0.6 +1.2 +2.0 +2.5 +3.0 +0.9 +1.0 +0 +6 +12 +0 +4 +8 +4.8 +5.1 +5.4 +2.0 +2.4 +2.8 +4.8 +5.1 +5.4 +4 +6 +H || c +� +o +T = 2 K +Magnetic field�� 0H (T) +(a) +Re� ac, dM/dH (10-2) +H || c +� +o +T = 5 K +Magnetic field�� 0H (T) +Re� ac, dM/dH (10-2) +(b) +1 +Magnetic field�� 0H (T) +(e) +Magnetic field�� 0H (T) +H || a +� +o +T = 2 K +(c) +dM/dH +Re� ac +Magnetic field�� 0H (T) +(d) +H3 +H4 +Magnetic field�� 0H (T) +(f) +FIG. 9. +Details of the magnetic transitions labelled as +H3 and H4. +Shown are the real part of ac susceptibility, +Reχac, and the susceptibility calculated from the magneti- +zation, dM /dH of orthorhombic CePdAl3 as a function of +increasing and decreasing field for (a) H ∥ c⋆ +o at 2 K, (b) H +∥ c⋆ +o at 5 K, and (c) H ∥ a⋆ +o at 2 K. (d), (e) and (f) show the +magnetic transition regions corresponding to the blue rectan- +gles in (a), (b) and (c), respectively. Colors denote dM /dH +for increasing (orange) and decreasing (green) magnetic field. +Black circles correspond to Reχac for increasing (filled sym- +bols) and decreasing (open symbols) field, respectively. dM / +dH was calculated after smoothing the data. +hard axis, i.e., H ∥ bo [Fig. 10(b)] only the AF-I and PM +phases were observed, possibly due to the lack of specific +heat data for finite fields along the hard axis. However, +the AF-II transition was observed in zero field and the +AF-II regime is shown in the phase diagram in Fig. 10(b) +for consistency. +While the magnetization suggests a collinear antifer- +romagnetic structure along ao in the AF-I phase, and +AF-IV shows the characterisics of a spin-flop phase, +the nature of AF-II and AF-III remain completely un- +known. Neutron scattering studies under magnetic field +are needed to determine the nature of the four antifer- +romagnetic phases we observed in orthorhombic single +crystal CePdAl3. + +9 +0 +2 +4 +6 +0 +4 +8 +12 +0 +4 +8 +12 +Magnetic field � 0H (T) +Temperature T (K) +PM +H || bo +AF-I +AF-II +(b) +1 +2 +Magnetic field � 0H (T) +H || c +� +o +PM +AF-II +AF-IV +AF-I +(a) +1 +2 +4 +3 + C(T) + M(T) +M(H) + T1 + T1 + T2 + T3 + T3 + H3 + T4 + T4 + H4 +AF-IIl +FIG. 10. +Magnetic phase diagram of orthorhombic CePdAl3 +for (a) H ∥ c⋆ +o and (b) H ∥ bo as inferred from the magneti- +zation and specific heat. Due to crystal twinning, the phase +diagram for H ∥ a⋆ +o qualitatively resembles the phase diagram +for H ∥ c⋆ +o shown in (a). Phase transitions are guided by the +lines which are denoted by numerals j = 1, 2, 3, and 4. The +associated temperature and field values are labelled as Tj and +Hj, respectively. Four magnetically ordered phases may be +distinguished as discussed in the text. +IV. +CONCLUSIONS +In summary, we measured the magnetization, ac +susceptibility, and specific heat of a single crystal of +CePdAl3 grown by optical float-zoning. +A highly +anisotropic behavior with a twinned orthorhombic crystal +symmetry was observed. Antiferromagnetic order with +TN = 5.6 K was observed in terms of transitions in the +ac susceptibility and specific heat. +The magnetization +is characteristic of antiferromagnetic order with an easy +ao direction in the basal plane. Field-driven transitions +were detected in the magnetization along the easy di- +rection, consistent with the ac susceptibility and specific +heat. Taken together, our study reveals a strong inter- +play of electronic correlations, complex magnetic order +and structural modifications in CePdAl3. +ACKNOWLEDGMENTS +We wish to thank A. Engelhardt, S. Mayr, and W. +Simeth for fruitful discussions and assistance with the +experiments. We thank T. E. Schrader on measurements +with the Rigaku single-crystal diffractometer in the x-ray +labs of the J¨ulich Centre for Neutron Science (JCNS). +This study has been funded by the Deutsche Forschungs- +gemeinschaft (DFG, German Research Foundation) un- +der TRR80 (From Electronic Correlations to Function- +ality, Project No. 107745057, Project E1), SPP2137 +(Skyrmionics, Project No. 403191981, Grant PF393/19), +and the excellence cluster MCQST under Germany’s Ex- +cellence Strategy EXC-2111 (Project No. 390814868). +Financial support by the European Research Council +(ERC) through Advanced Grants No. 291079 (TOPFIT) +and No. 788031 (ExQuiSid) is gratefully acknowledged. +[1] E. Bauer, G. Hilscher, H. Michor, C. Paul, E.-W. Scheidt, +A. Gribanov, Y. Seropegin, H. No¨el, M. Sigrist, and +P. 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Pfleiderer, and M. Garst, Symmetry break- +ing, slow relaxation dynamics, and topological defects at +the field-induced helix reorientation in MnSi, Physical +Review B 95, 024429 (2017). +[61] Y. +Tokunaga, +X. +Yu, +J. +White, +H. +M. +Rønnow, +D. Morikawa, Y. Taguchi, and Y. Tokura, A new class +of chiral materials hosting magnetic skyrmions beyond +room temperature, Nature Communications 6, 7638 +(2015). + diff --git a/3tFAT4oBgHgl3EQflR3K/content/tmp_files/load_file.txt b/3tFAT4oBgHgl3EQflR3K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20daefd505c91c98d5dd30c4d9493157ddcd2c1e --- /dev/null +++ b/3tFAT4oBgHgl3EQflR3K/content/tmp_files/load_file.txt @@ -0,0 +1,982 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf,len=981 +page_content='Low-temperature antiferromagnetic order in orthorhombic CePdAl3 Vivek Kumar,1, ∗ Andreas Bauer,1, 2 Christian Franz,1, 3 Jan Spallek,1 Rudolf Sch¨onmann,1 Michal Stekiel,1 Astrid Schneidewind,3 Marc Wilde,1, 2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Pfleiderer1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 4 1Physik-Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Germany 2Zentrum f¨ur QuantumEngineering (ZQE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Germany 3J¨ulich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Germany 4Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Germany (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 2023) We report the magnetization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' ac susceptibility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' and specific heat of optically float-zoned single crystals of CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In comparison to the properties of polycrystalline CePdAl3 reported in the literature, which displays a tetragonal crystal structure and no long-range magnetic order, our single crystals exhibit an orthorhombic structure (Cmcm) and order antiferromagnetically below a N´eel temperature TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The specific heat at zero-field shows a clear λ-type anomaly with a broad shoulder at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A conservative estimate of the Sommerfeld coefficient of the electronic specific heat, γ = 121 mJ K−2 mol−1, indicates a moderately enhanced heavy-fermion ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A twin microstructure evolves in the family of planes spanned by the basal plane lattice vectors ao and co, with the magnetic hard axis bo common to all twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The antiferromagnetic state is characterized by a strong magnetic anisotropy and a spin-flop transition induced under magnetic field along the easy direction, resulting in a complex magnetic phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Taken together our results reveal a high sensitivity of the magnetic and electronic properties of CePdAl3 to its structural modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' INTRODUCTION Cerium-based intermetallic compounds exhibit a vari- ety of ground states and various underlying exotic phys- ical phenomena, such as unconventional superconductiv- ity [1–8], heavy-fermion states [9, 10], non-Fermi liquid behavior [11], vibronic hybrid excitations [12–16], and complex magnetic order [17–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' On the phenomenologi- cal level, the origin of this remarkable diversity of ground states has been attributed to the competition of narrow f- electron bands and strong electronic correlations together with spin-orbit interaction, crystal electric field (CEF) ef- fects, and strong magneto-elastic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' An overarch- ing theme connecting much of the research in f-electron compounds concerns the condition of the emergence of magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A class of compounds with the general formula CeTX 3 (T is a transition metal and X is a p-block element) crystallizing in subgroups of the BaAl4-type (I4/mmm) tetragonal structure has received special attention [3, 6, 22–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In these compounds, a large number of struc- tural variants and diverse magnetic and electrical proper- ties can be obtained by changing the transition metal T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Many members of this class such as CeRhGe3, CeAuAl3, CeCuAl3, and CeCoGe3 adopt a non-centrosymmetric tetragonal structure (BaNiSn3-type I4mm) and exhibit antiferromagnetic behavior [25–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Other members such as CeAgAl3 display ferromagnetism with a centrosym- metric orthorhombic crystal structure [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A spin- glass state was reported in non-centrosymmetric tetrago- nal CePtAl3 below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 K [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Complex magnetic phases ∗ vivek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='kumar@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='de have been observed in antiferromagnetic CeNiGe3[22, 24], CeCoGe3 [23, 36–38] and CePtSi3 [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The discov- ery of pressure-induced unconventional superconductiv- ity in the non-centrosymmetric tetragonal heavy-fermion antiferromagnets CeRhSi3, CeIrSi3, CeCoGe3, CeIrGe3, and CeRhGe3 even suggests a new direction in condensed matter physics [3, 6, 40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' An important aspect is the structural stability of these systems and the emergence of different electronic ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' As one of the first examples, CePd2Al2 [13, 15], which is closely related to the class of CeTAl3 of ma- terials, was found to undergo a structural phase trans- formation from a tetragonal to an orthorhombic lat- tice at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' An inelastic neutron scattering study re- vealed three magnetic excitations in the paramagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' However, according to Kramer’s theorem, only two CEF excitations are expected due to the splitting of ground state J = 5/2 of the Ce3+ ion into three doublets in tetragonal/orthorhombic point symmetry suggesting strong coupling between the crystal fields and the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Later, Adroja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' found a similar anomaly in CeCuAl3 [14], where a structural instability manifests itself in terms of a drastic change in lattice parameters of the tetragonal structure around 300 ◦C [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' These anomalous excitations have been interpreted by means of Thalmeier and Fulde’s model of bound states be- tween phonons and CEF excitations as generalized to the tetragonal point symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Recently, ˇCerm´ak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' con- firmed related hybrid CEF-phonon excitations even for weak magnetoelastic coupling in isostructural CeAuAl3 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Moreover, CePd2Al2, CeCuAl3 and CeAuAl3 order antiferromagnetically at low temperatures and exhibit incommensurate amplitude-modulated magnetic struc- tures [15, 44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The presence of multi-step magnetism and complex magnetic phase diagrams suggests the pos- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='08617v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='str-el] 20 Jan 2023 2 sible existence of topologically non-trivial multi-k struc- tures akin to skyrmion lattices [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' This raises the ques- tion, if and how the formation of magnetic order depends on the stabilization of specific crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In this paper we focus on CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A study of as- cast polycrystalline CePdAl3 by Schank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' in 1994 revealed a tetragonal I4mm structure with lattice con- stants a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='343 ˚A and c = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='578 ˚A [48], where the heat treatment at high temperature results in a struc- tural phase transformation with an antiferromagnetic or- der below TN ≃ 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In contrast, no magnetic order was found down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 K in a recent investigation by Franz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' on single crystalline tetragonal CePdAl3 grown by optical float zoning with a growth rate of 6 mm/h [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For the work reported in the following, a single crystal was prepared by optical float zoning using a much lower growth rate of 1 mm/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Under these conditions we found that CePdAl3 crystallizes in an orthorhombic as opposed to a tetragonal structure [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In this paper, we report comprehensive magnetization, ac susceptibility, and spe- cific heat measurements on single crystalline orthorhom- bic CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' As our main result we find the charac- teristics of antiferromagnetic order below TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' We determine the magnetic phase diagram upto 14 T, where we find the emergence of complex magnetic phases un- der magnetic fields applied along the easy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The presence of different structural and magnetic configura- tions of CePdAl3 identifies a new example of a material in which to search for hybrid excitations and new magnetic phases in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' After a brief account of the experimental methods in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' II, we present our ex- perimental results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' We start with the structural properties and notation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III A, followed by the spe- cific heat results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III B and magnetic susceptibility data in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The temperature- and field-dependence of the magnetization is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' We find that the magnetic field-driven transitions for fields ap- plied along the easy direction are consistent with the specific heat as a function of temperature as presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III F, we examine the magnetic tran- sitions in more detail by analyzing the hysteresis of the field-dependent magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Comprehensive datasets allow to infer the magnetic phase diagram pre- sented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The conclusions are summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' EXPERIMENTAL METHODS A single-crystal of CePdAl3 was grown using the op- tical floating-zone technique following a process similar that described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' [49, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' As the main difference, the growth rate was reduced from 6 mm/h [49] to 1 mm/h which resulted in the formation of an orthorhombic crys- tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The crystal structure of CePdAl3 was determined by means of single-crystal x-ray diffraction (SCXRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A platelet-shaped crystal with dimensions 50 µm × 40 µm × 10 µm was cleaved of the CePdAl3 crystal as grown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The platelet was investigated at a Rigaku XtaLAB Synergy-S diffractometer, using a Mo x-ray source with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='71 ˚A and a two-dimensional HyPix-Arc 150◦ detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Bragg reflections were indexed using CrysAlisP ro [53] as inte- grated with the diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The single crystals were oriented by Laue x-ray diffrac- tion and a cuboidal sample was cut with orientations a⋆ o, c⋆ o and bo as introduced below for the measurement of the bulk properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The ac susceptibility, magnetiza- tion, and specific heat were measured in a Quantum De- sign physical property measurement system (PPMS) at temperatures down to 2 K under magnetic fields up to 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In order to determine the temperature dependence of the bulk properties, the sample was first cooled from a high temperature, well above TN, to the lowest attainable temperature in the absence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Subse- quently, the field was set to the desired value and data were collected for increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' This protocol was repeated for different target magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The ac susceptibility was measured at an excitation amplitude of 1 mT and an excitation frequency of 911 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The specific heat was measured down to 2 K using a large heat-pulse method [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For temperatures between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='08 K and 4 K the specific heat was measured in a Dryogenic adiabatic demagnetization refrigerator using a conventional heat- pulse method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The field dependence of the magnetization and the ac susceptibility was measured using the following temper- ature versus field protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' First, the sample was cooled from a high temperature well above TN to the target tem- perature in the absence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Second, data as a function of magnetic field were recorded in a se- quence of field sweeps from zero-field to 14 T, 14 T to 14 T, and -14 T to 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The bulk properties recorded on different pieces cut from the large single crystal ingot were consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The temperature and field dependent features along a⋆ o and c⋆ o were qualitatively identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Therefore, comprehen- sive data focused on one of these directions, c⋆ o, were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Summarizing the key result of our study, the magnetic phase diagrams of CePdAl3 were inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Sig- natures detected in measurements as a function of tem- perature and magnetic field are labelled as Tj and Hj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For clarity, the same subscript j is assigned to the transitions corresponding to the same line in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' EXPERIMENTAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Crystal structure and twinning Different crystal growth conditions favor a tetrago- nal (I4mm) [49] or orthorhombic crystal structures of CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' By means of single crystal x-ray diffraction, we determined that the orthorhombic lattice stabilizes 3 in the Cmcm space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The lattice parameters at room temperature are ao = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='379 ˚A, bo = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='407 ˚A and co = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='975 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The orthorhombic phase exhibits a pseudo- tetragonal twinning in the basal plane, evident, for in- stance, by the splitting of the Bragg reflections shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The twinning law was determined by index- ing all measured reflections with components of the four twins presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' An illustration of the twin orientation is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 1(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The three per- pendicular cartesian directions of twins for i = 1, 2, 3, 4 are denoted by ai o, bi o and ci o, where ai o and ci o construct an effective basal plane and bi o mutually represents the long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The volume fraction of the four twins labelled i = 1, 2, 3, and 4 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='38, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='26, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='23, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='13, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The mismatch angle between the twins numbered 1 and 2, as well as 3 and 4, are around 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Measurements on different pieces cleaved of the sin- gle crystalline ingot demonstrate the same twinning scheme with minor differences in twin fractions of differ- ent twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' An attempt to detwin the crystals by means of high-temperature treatment, etching, or cleaving of micrometer-sized crystals neither affected the twinning as such nor the twinning fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In turn, measurements in any direction in the effective basal plane reflect effectively an admixture of ai o and ci o directions due to the four twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' We define, therefore, two mutually perpendicular effective sample directions a⋆ o and c⋆ o, explicitly taking into account the volume frac- tions of the four twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' This definition is schematically depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 1(c) and (d) where a⋆ o is nearly aligned along a1,2 o and c3,4 o , while c⋆ o is aligned to that of c1,2 o and a3,4 o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The third crystal direction, corresponding to the long axis bo, remains unaffected by the twin defor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Temperature-dependence of the specific heat The temperature dependence of the specific heat C(T) of single-crystalline tetragonal (I4mm) and orthorhom- bic (Cmcm) CePdAl3, as well as nonmagnetic polycrys- talline tetragonal (I4mm) LaPdAl3 measured in the ab- sence of a magnetic field are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' No evi- dence suggesting magnetic order was observed in tetrag- onal CePdAl3 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In orthorhombic CePdAl3, a λ-type anomaly comprising a peak at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 K followed by a shoul- der closely above the transition temperature TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K is observed, where the magnetization is characteristic of antiferromagnetism as reported below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The behavior ob- served is consistent with a previous study of polycrys- talline CePdAl3 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Moreover, the properties are rem- iniscent of the commensurate to incommensurate mag- netic transition reported of other strongly correlated sys- tems [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A pronounced shoulder in the specific heat has also been seen in other systems, notably, the chiral cubic mag- net MnSi [54, 57], where it reflects a change of char- acter of the critical spin-fluctuations when approach- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Twin scheme in the basal plane of orthorhombic CePdAl3 as derived from single crystal x-ray diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) X-ray scattering intensity reconstructed in the H0L plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The splitting of the reflections is characteristic of twin for- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (b) Indexed reflections of panel (a) with the colors corresponding to different twin domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Schematics of the lattice vectors ai o and ci o of twin i in the basal plane H0L of the orthorhombic crystal are depicted in the lower panels (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Four twins labelled i = 1, 2, 3, and 4 were identi- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' a⋆ o and c⋆ o are defined as mutual perpendicular sample directions comprising the admixtures of twin lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' ing long-range helimagnetic order and a concomitant fluctuation-induced first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Details of the low-temperature specific heat of orthorhombic CePdAl3 at zero-field are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' III E below, which also includes data collected at different magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Above TN, the expression C/T = γ + βT 2, where γ and β are the electronic and phononic contributions to the specific heat, respectively, has been fitted to the specific heat data in the range ∼18 to ∼23 K of or- thorhombic CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The values obtained for γ and β are 234 mJ mol−1 K−2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='437 × 10−4 J mol−1 K−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The Debye temperature, ΘD = 305 K, associated with β may be derived using the relation β = (12/5)π4nR/Θ3 D, where n is the number of atoms per formula unit and R is the gas constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The phonon contribution to the specific heat in the Debye model [or- ange line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 2] is given by Cph,Debye = 9nR � T ΘD �3 � xD 0 x4ex (ex − 1)2 dx (1) where xD = ΘD/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At high temperatures the experimen- tal data of tetragonal LaPdAl3 and CePdAl3, as well as orthorhombic CePdAl3 approach the Dulong-Petit limit, 3nR = 15R = 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='7 J mol−1 K−1, where n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) (b) 2 2 0 1it 0 2 2 0 2 2 0 2 H H (c) (d) Twin volumes 1: 38 % a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='ttas ch +t c?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 2: 26 % 3: 23 % 4: 13 % 3 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 The large value of γ = 234 mJ mol−1 K−2 obtained from the low-temperature specific heat above TN is typ- ical for a heavy-fermion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' It has to be borne in mind, however, that evaluating γ at the relatively high- temperature range above TN is associated with substan- tial uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A lower bound of γ, fitting the exper- imental data in the antiferromagnetic state at tempera- tures between ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='9 K and ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='7 K, yields a value of γ = 121 mJ mol−1 K−2 still characteristic of heavy-fermion behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At high temperatures (T > 100 K), the specific heat of all three compounds exhibits essentially the same tem- perature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' However, the specific heat of or- thorhombic CePdAl3 is slightly smaller than for tetrago- nal CePdAl3, suggesting reduced electronic and phononic contributions associated with the reduced crystal sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Compared to nonmagnetic LaPdAl3, the specific heat of orthorhombic CePdAl3 is also slightly smaller, yet within the experimental error of experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Indeed, a multiplication with a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='99 to the total signal of LaPdAl3 fully superimposes the data of CePdAl3 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 3(a) of C/T vs T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The corresponding dif- ference in specific heats may be attributed to the mag- netic contribution of the specific heat of orthorhombic CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 3(b) is a sharp peak at T = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 K in the magnetic contribution to the specific heat following subtraction of the phonon contribution signaling an an- tiferromagnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In addition, a broad maxi- mum around 30 K may be discerned as characteristic of a Schottky anomaly due to crystal electric field contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In the tetragonal as well as the orthorhombic symme- try of the lattice, the degeneracy of the sixfold ground state multiplet of the Ce3+ ion splits into three doublet states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' These lift the first and second excited state with respect to the ground state resulting in a contribution to the specific heat which can be expressed as [58] CCEF =R Z 2 � l=0 gl � El kT �2 exp � − El kT � − R Z2 � 2 � l=0 El kT glexp � − El kT ��2 (2) where Z = 2 � l=0 glexp � − El kT � (3) is the partition function, and l = 0, 1 and 2 denote the ground, first and second excited states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The degeneracy of the three doublet states is g0 = g1 = g2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The energy difference E1−E0 = ∆1 and E2−E0 = ∆2 represent the levels of the first and the second excited states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A fit of the data to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (2) between 20 K and 100 K yields ∆1 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 K and ∆2 = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 K, 0 50 100 150 200 0 50 100 150 0 5 10 15 20 25 0 8 16 Specific heat C (J mol-1 K-1) CePdAl3 Cmcm CePdAl3 I4mm ΘD = 305 K Dulong-Petit limit LaPdAl3 I4mm Temperature T (K) (a) Temperature T (K) (b) TN FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) Zero-field specific heat of single-crystalline or- thorhombic (black) and tetragonal (blue) [49] CePdAl3 as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Data of orthorhombic CePdAl3 were measured in a Dryogenic system between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='08 K and 4 K, and in a PPMS between 2 K and 200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Also shown are the specific heat of nonmagnetic polycrystalline LaPdAl3 (Gray line) and the Debye fit (orange line) calculated from the low- temperature specific heat of the Cmcm structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The Debye temperature is ΘD = 305 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The Dulong-Petit limit for all three compounds, 15R = 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='7 J mol−1 K−1 is depicted by a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (b) The low-temperature part of the specific heat of orthorhombic CePdAl3 shows a pronounced λ-type anomaly with a broad shoulder at the magnetic transition at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Note that, the normalized subtraction of the LaPdAl3 signal may introduce systematic errors in the determination of the precise values of the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For instance, subtraction of the signal of LaPdAl3 after multiplication with a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='98 yields ∆1 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K and ∆2 = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Furthermore, we have calculated the magnetic entropy S = � (C/T)dT presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At the magnetic transition temperature, the entropy reaches the theoret- ical value of Rln 2 for a doublet ground state expected of Ce3+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' When increasing the temperature, the en- tropy increases and reaches Rln4 around 30 K, approach- ing saturation above 100 K consistent with the scheme of crystal electric field levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 5 0 8 16 0 1 2 3 0 50 100 150 200 0 6 12 0 4 8 0 6 ∆C (J mol-1 K-1) (b) ∆1 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 K ∆2 = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 K Cmag Fit C/T (J mol-1 K-2) (a) CePdAl3 (Cmcm) LaPdAl3 (99 %) Difference Entropy S (J mol-1 K-1) Temperature T (K) RLn2 RLn4 (c) S (J mol-1 K-1) T (K) RLn2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Magnetic contribution to the specific heat and crys- tal electric field levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) Specific heat per unit tempera- ture, C/T, of orthorhombic CePdAl3 and tetragonal LaPdAl3 (with a multiplication of a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='99) as well as their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (b) Magnetic specific heat, Cmag, and the fit to the expression for the crystal electric field contribution to the specific heat yields ∆1 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 K and ∆2 = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 K (c) Magnetic contribution to the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The inset shows the entropy at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Temperature-dependence of the magnetic susceptibility The real part of the ac susceptibility, Re χac of or- thorhombic CePdAl3 as a function of temperature is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 4(a) for a⋆ o, c⋆ o and bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A clear magnetic transition is observed at TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K in the low temper- ature range, characteristic of the onset of antiferromag- netic order as indicated by arrows in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Namely, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 0 4 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 0 100 200 300 0 1 2 0 20 40 0 4 Re� ac (10-2) Hac || a � o Hac || c � o Hac || bo (a) Re� ac (10-2) T (K) TN H/M (102) H/M (103) Temperature T (K) H || a � o H || c � o H || bo � 0H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 T (b) T (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) Temperature dependence of the real part of the ac susceptibility, Re χac of orthorhombic CePdAl3 measured along a⋆ o, c⋆ o and bo at an excitation amplitude of 1 mT and a frequency of 911 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The inset shows the low-temperature part of Re χ ac, reflecting the characteristics of an antiferro- magnetic transition at TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (b) Susceptibility, H /M, as a function of temperature for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo measured in a field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Gray lines are Curie-Weiss fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The inset shows the data for temperatures below 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' below TN, Re χac monotonically decreases along a⋆ o and c⋆ o with decreasing temperature, while slightly increas- ing along bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The magnitude of R eχac along different axes differs significantly for T <100 K, indicating size- able magnetic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Figure 4(b) shows the normalized susceptibility, H /M, as a function of temperature in a field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 T for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In the paramagnetic state well above TN, a Curie-Weiss dependence is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A linear fit to the data above 100 K yields Weiss tem- peratures Θa⋆ W = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 K, Θc⋆ W = -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K and Θb W = -33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 K for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo, respectively, char- acteristic of an antiferromagnetic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Moreover, the effective moments of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='39, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='49 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='44 µB per ion obtained under magnetic field along a⋆ o, c⋆ o and bo, re- spectively, are close to the value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='54 µB expected for a free Ce3+ ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' This might suggest a localized nature of the Ce moments in CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The deviation of H / M from the Curie-Weiss dependence for TN < T < 100 K 6 shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 4(b) may be related to CEF effects and electronic correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Furthermore, despite the twin deformations, a significant difference between the susceptibilities along a⋆ o and c⋆ o in the paramagnetic state indicates a large anisotropy in the basal plane, char- acteristic of an easy-axis system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Magnetization The magnetic field dependence of the isothermal mag- netization of orthorhombic CePdAl3 at 2 K for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' No hysteresis is ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The magnetization varies linearly in the low-field region up to 1 T as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 5(a) con- sistent with antiferromagnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For fields along a⋆ o and c⋆ o, an S-shaped rise is observed in the magnetization when further increasing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A kink around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 T sug- gests a field-driven transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The magnetization values at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 T are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='85 µB for H ∥ a⋆ o, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='44 µB for H ∥ c⋆ o, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='18 µB for H ∥ bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The magnetization increases mono- tonically above this transition where the moments along a⋆ o and c⋆ o at 14 T, the highest field strength of studied, are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='7 µB per Ce atom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In compar- ison, the magnetization increases linearly with field for H ∥ bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The moment at 14 T is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 µB per Ce atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Keeping in mind the twinned crystal structure, the magnetization along a⋆ o and c⋆ o represent a mixture of the crystallographic ao and co axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A large quantita- tive difference in the magnetization at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 T along a⋆ o and c⋆ o makes it unlikely, that a metamagnetic transition oc- curs at the same field value in the ao and co directions in a single twin domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Instead, it appears most likely that the increase in the magnetization corresponds to a spin-flop in the ai o easy direction of each twin only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 5(b), (c) and (d) are the isothermal mag- netization at various temperatures for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The spin-flop transition in M (H ) for a⋆ o and c⋆ o shifts to lower fields under increas- ing temperature and vanishes above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In contrast, the variation in M (H ) along bo is essentially temperature independent at and above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In order to trace the field-driven magnetic transition, we have calculated the differential susceptibility, dM /dH from the isothermal magnetization at various tempera- tures presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 6(a), (b), and (c) for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For fields along a⋆ o and c⋆ o, the transition is characterized by a broad peak at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 T at 2 K, which resolves into two peaks at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' These peaks exist below TN as marked by arrows at the transition fields H3 and H4, following the labelling scheme described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' With increasing temperature the field range between the peaks increases and both peaks shift to lower field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' No indication exists of a field-induced transition in dM /dH for field along bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The evolution of the field-induced transitions may be traced in further detail by the temperature dependence 12 6 0 6 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 0 6 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 0 6 12 0 6 12 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='05 H || bo Magnetization M (� B f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='-1) Magnetic field�� 0H (T) T = 2 K Magnetic field�� 0H (T) H || a � o H || c � o (a) (b) H || a � o (c) M (� B f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='-1) H || c � o 2 3 4 5 6 10 (d) T (K) H || bo M (� B f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='-1) � 0H (T) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) Isothermal magnetization of orthorhombic CePdAl3 at 2 K measured in a field along a⋆ o, c⋆ o and bo up to 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The arrow indicates the direction of increasing mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The inset shows the linear variation of the mag- netization below 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Typical field dependence of isothermal magnetization at various temperatures for (b) H ∥ a⋆ o, (c) H ∥ c⋆ o and (d) H ∥ bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A field-driven spin-flop transition at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 T is observed below TN for H ∥ a⋆ o (blue arrow) and H ∥ c⋆ o (red arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 0 2 4 6 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 0 2 4 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='025 H || a � o H || c � o H || bo (c) 10 K 6 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 5 K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 4 K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 3 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 2 K Magnetic field�� 0H (T) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='025 H4 H3 (a) 10 K 6 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 5 K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 4 K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 3 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 2 K +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='025 H4 H3 (b) 2 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 10 K 3 K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 4 K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 5 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 K 6 K Susceptibility dM/dH FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (a) Susceptibility, dM /dH, calculated from the mea- sured magnetization of orthorhombic CePdAl3 for (a) H ∥ a⋆ o, (b) H ∥ c⋆ o and (c) H ∥ bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Data are shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='025 for clar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Peaks correspond to field-induced transitions marked by arrows at H3 (pink) and H4 (sky blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The peaks disappear above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 2 4 6 8 10 0 2 4 2 4 6 8 10 0 3 6 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='7 T3 T1 (b) M (� B f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='-1) T4 T1 (a) � 0H (T) T3 T1 (c) T (K) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 H || c � o T1 T3 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='3 T4 (e) Re� ac (10-2) T (K) H || a � o T1 T4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 T3 (d) T (K) H || bo T1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='3 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Temperature dependence of magnetization, M (T) and real part of ac susceptibility, Reχac(T) of orthorhombic CePdAl3 in magnetic fields up to 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' M (T) is shown in panels (a), (b) and (c), and Reχac(T) in (d), (e) and (f) for H ∥ a⋆ o, H ∥ c⋆ o and H ∥ bo, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Reχac(T) is shifted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Magnetic transitions are marked by vertical lines at temperatures T1 (red), T3 (blue) and T4 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A complex behavior with multiple transitions is present for field along a⋆ o and c⋆ o between 2 T and 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' of the magnetization M (T) and the ac susceptibility Reχac(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 7 is M (T) and Reχac(T) at various fields up to 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' By decreasing the temperature, orthorhombic CePdAl3 undergoes a phase transforma- tion from paramagnetism to antiferromagnetic order at a transition temperature T1 (marked by red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' This transition is visible in Reχac(T) in all crystallographic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The transition at T1 shifts to lower temper- atures under increasing field but does not vanish upto the highest field of 14 T studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In the intermediate field range from 2 T to 6 T, clear changes in M (T) and Reχac(T) for field along a⋆ o [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 7(a) and (d)]) and c⋆ o [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 7(b) and (e)] point to two additional phase transitions at temperatures denoted T3 (blue line) and T4 (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' These transitions disappear at fields above 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For H ∥ bo, only the first transition at T1 is observed in M (T) and Reχac(T) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 7(c) and (f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Field-dependence of the specific heat The specific heat of orthorhombic CePdAl3 as a function of temperature at different magnetic fields for H ∥ c⋆ o is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At zero magnetic field 0 8 16 0 8 16 0 2 4 6 0 8 16 0 2 4 6 0 2 4 6 0 1 2 3 4 5 6 0 8 16 � 0H = 0 T1 T2 (b) 1 T T1 T2 T4 (c) 2 T T1 T2 T4 (d) Specific heat C (J K-1 mol-1) 3 T T2 T1 T3 T4 (e) 4 T T1 T2 T3 T4 (f) 5 T T1 T2 T4 T3 (g) 6 T T1 T2 T3 (h) Temperature T (K) 9 T T1 T2 (i) 14 T T1 T2 (j) Specific heat C (J K-1 mol-1) Temperature T (K) 0 1 2 3 4 5 6 � 0H (T) H || c � o (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Specific heat of orthorhombic CePdAl3 as a func- tion of temperature under selected magnetic fields up to 14 T applied along the c⋆ o axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Data measured in the Dryogenic system between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='08 K and 4 K are combined with data mea- sured in the PPMS above 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At H = 0 the magnetic transi- tion displays a peak at T2 preceded by a broad shoulder with a point of inflection at T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Additional peaks emerge at T3 and T4 for magnetic fields between 2 T and 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 8(b)], a broad shoulder with a point of inflection is observed at T1 followed by a sharp peak at T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Increasing the applied field results in a broadening of the peak at T2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 8(c)] and a splitting with an additional peak emerging at a lower temperature T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For even higher fields up to 6 T [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 8(d) to (h)], the position of T4 continues to shift to lower temperatures with the emergence of another peak at T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The emergence of the peaks at T3 and T4 in the specific heat in the intermediate field range from 2 to 6 T is consistent with the phase transitions deduced from the magnetization and the ac susceptibility (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For fields above 6 T, a noticeable shift of T1 and T2 to lower temperatures is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Field-dependence of the magnetic susceptibility In order to investigate the qualitative difference be- tween the transitions labelled as H3 and H4 in dM /dH 8 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 6), we have measured the magnetic susceptibility as a function of magnetic field between 0 and 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Fig- ure 9 shows the real part of the ac susceptibility, Reχac, and the susceptibility calculated from the magnetization, dM /dH, as a function of increasing and decreasing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At 2 K, dM /dH exhibits two peaks under increasing field, first, a pronounced peak at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='15 T, followed by a second broad peak at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='3 T for both H ∥ c⋆ o [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 9(a) and (d)] and H ∥ a⋆ o [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 9(c) and (f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The first peak shifts to 5 T resulting in a hysteresis, while the second peak remains at the same field value under decreasing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Similar effects exists in Reχac where the first peak be- comes less pronounced with a smaller hysteresis and a slightly lower field of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='05 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Also, the value of Reχac is slightly lower around the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At higher tem- peratures, both peaks are shifted to lower field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The hysteresis in Reχac decreases significantly and drops below the noise level at 5 K [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 9(b) and (e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Here, the magnitude of Reχac matches well with dM /dH ex- cept around the first peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The difference in character of the transitions labelled as H3 and H4 suggest their intrinsic origin rather than being related to the twinned microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' On the one hand, the hysteresis observed in dM /dH and Reχac is reminiscent of changes of population of mul- tidomain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' On the other hand, the smaller ampli- tude of Reχac as compared to dM /dH indicates the pres- ence of slow relaxation processes around the phase tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Similar features are known to trace spin textures like helimagnetic disclination or skyrmions in magnetic materials [59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Further experimental investigations are needed to explore such a possibility in orthorhombic CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Magnetic phase diagram Combining the features detected in the magnetization and the specific heat, we infer the magnetic phase dia- grams for field parallel to c⋆ o and bo shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 10(a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Due to the twinned microstructure, the response of the magnetization, specific heat, and ac susceptibility are qualitatively alike for H ∥ a⋆ o and H ∥ c⋆ o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' In addition, the enhanced signal observed along a⋆ o as compared to c⋆ o indicates that a⋆ o reflects a larger frac- tion of the easy axis, ao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Therefore, the transitions along both a⋆ o and c⋆ o reflect equally the phenomenon belonging to the easy ao axis of the untwinned single domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Four magnetic regions may be distinguished for field along c⋆ o, denoted AF-I, AF-II, AF-III and AF-IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' At low temperature and zero-field, the ground state is de- noted as AF-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' With increase temperature, AF-II appears at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 K before entering in the paramagnetic (PM) state above 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Signatures of the AF-II region are detected only in the specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The application of a magnetic field at low temperature drives a spin-flop transition from AF-I to AF-IV with an intermediate region AF-III in a narrow field range only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' For finite field applied along the 0 6 12 0 2 4 0 6 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 0 6 12 0 4 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='4 4 6 H || c � o T = 2 K Magnetic field�� 0H (T) (a) Re� ac, dM/dH (10-2) H || c � o T = 5 K Magnetic field�� 0H (T) Re� ac, dM/dH (10-2) (b) 1 Magnetic field�� 0H (T) (e) Magnetic field�� 0H (T) H || a � o T = 2 K (c) dM/dH Re� ac Magnetic field�� 0H (T) (d) H3 H4 Magnetic field�� 0H (T) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Details of the magnetic transitions labelled as H3 and H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Shown are the real part of ac susceptibility, Reχac, and the susceptibility calculated from the magneti- zation, dM /dH of orthorhombic CePdAl3 as a function of increasing and decreasing field for (a) H ∥ c⋆ o at 2 K, (b) H ∥ c⋆ o at 5 K, and (c) H ∥ a⋆ o at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' (d), (e) and (f) show the magnetic transition regions corresponding to the blue rectan- gles in (a), (b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Colors denote dM /dH for increasing (orange) and decreasing (green) magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Black circles correspond to Reχac for increasing (filled sym- bols) and decreasing (open symbols) field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' dM / dH was calculated after smoothing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' hard axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=', H ∥ bo [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 10(b)] only the AF-I and PM phases were observed, possibly due to the lack of specific heat data for finite fields along the hard axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' However, the AF-II transition was observed in zero field and the AF-II regime is shown in the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 10(b) for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' While the magnetization suggests a collinear antifer- romagnetic structure along ao in the AF-I phase, and AF-IV shows the characterisics of a spin-flop phase, the nature of AF-II and AF-III remain completely un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Neutron scattering studies under magnetic field are needed to determine the nature of the four antifer- romagnetic phases we observed in orthorhombic single crystal CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 9 0 2 4 6 0 4 8 12 0 4 8 12 Magnetic field � 0H (T) Temperature T (K) PM H || bo AF-I AF-II (b) 1 2 Magnetic field � 0H (T) H || c � o PM AF-II AF-IV AF-I (a) 1 2 4 3 C(T) M(T) M(H) T1 T1 T2 T3 T3 H3 T4 T4 H4 AF-IIl FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Magnetic phase diagram of orthorhombic CePdAl3 for (a) H ∥ c⋆ o and (b) H ∥ bo as inferred from the magneti- zation and specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Due to crystal twinning, the phase diagram for H ∥ a⋆ o qualitatively resembles the phase diagram for H ∥ c⋆ o shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Phase transitions are guided by the lines which are denoted by numerals j = 1, 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The associated temperature and field values are labelled as Tj and Hj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Four magnetically ordered phases may be distinguished as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' CONCLUSIONS In summary, we measured the magnetization, ac susceptibility, and specific heat of a single crystal of CePdAl3 grown by optical float-zoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' A highly anisotropic behavior with a twinned orthorhombic crystal symmetry was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Antiferromagnetic order with TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content='6 K was observed in terms of transitions in the ac susceptibility and specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' The magnetization is characteristic of antiferromagnetic order with an easy ao direction in the basal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Field-driven transitions were detected in the magnetization along the easy di- rection, consistent with the ac susceptibility and specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Taken together, our study reveals a strong inter- play of electronic correlations, complex magnetic order and structural modifications in CePdAl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' ACKNOWLEDGMENTS We wish to thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Engelhardt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Mayr, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Simeth for fruitful discussions and assistance with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' We thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Schrader on measurements with the Rigaku single-crystal diffractometer in the x-ray labs of the J¨ulich Centre for Neutron Science (JCNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' This study has been funded by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) un- der TRR80 (From Electronic Correlations to Function- ality, Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 107745057, Project E1), SPP2137 (Skyrmionics, Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 403191981, Grant PF393/19), and the excellence cluster MCQST under Germany’s Ex- cellence Strategy EXC-2111 (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 390814868).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Financial support by the European Research Council (ERC) through Advanced Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 291079 (TOPFIT) and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' 788031 (ExQuiSid) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Bauer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQflR3K/content/2301.08617v1.pdf'} +page_content=' Hilscher, H.' metadata={'source': 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@@ +arXiv:2301.01698v1 [hep-th] 4 Jan 2023 +Quantum Energy Inequalities along stationary worldlines +Christopher J. Fewster +1,∗ and Jacob Thompson +1,2,† +1 Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom. +2 School of Mathematics and Statistics, The University of Sheffield, Hicks Building, Hounsfield Road, +Sheffield S3 7RH, United Kingdom. +January 5, 2023 +Abstract +Quantum energy inequalities (QEIs) are lower bounds on the averaged energy density of a +quantum field. They have been proved for various field theories in general curved spacetimes but +the explicit lower bound is not easily calculated in closed form. In this paper we study QEIs for the +massless minimally coupled scalar field in four-dimensional Minkowski spacetime along stationary +worldlines – curves whose velocity evolves under a 1-parameter Lorentz subgroup – and find closed +expressions for the QEI bound, in terms of curvature invariants. Our general results are illustrated +by specific computations for the six protoypical stationary worldlines. When the averaging period +is taken to infinity, the QEI bound is consistent with a constant energy density along the worldline. +For inertial and uniformly linearly accelerated worldlines, this constant value is attained by the +Minkowski and Rindler vacuums respectively. It is an open question as to whether the bounds for +other stationary worldlines are attained by other states of interest. +1 +Introduction +Even if a classical field theory obeys local energy conditions, such as positivity of energy density, the +corresponding quantum field theory (QFT) will fail to do so, as a result of a general theorem [7]. +In fact, it is typical that the expectation value of energy density at any given point can be made +arbitrarily negative by a suitable choice of the quantum state [9]. Nonetheless, in many QFT models, +local averages of the expected energy density are bounded below by Quantum Energy Inequalities +(QEIs), independent of the state. +Starting with results of Ford and Roman [17, 18, 19] QEIs have been derived for a variety of +quantum fields in flat and curved spacetimes. References and discussion may be found in the recent +reviews [11, 32]. For example, consider the real scalar field of mass m ≥ 0 in any globally hyperbolic +spacetime (M, g), recalling that global hyperbolicity demands only that the spacetime possesses a +global Cauchy surface. Let γ(s) be any smooth timelike curve, parameterised by proper time. It was +shown in [8] that the energy density of the quantum field along γ obeys the QEI +� ∞ +−∞ +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +� ∞ +0 +dα +π +� +g ⊗ gT(−α, α) > −∞, +(1.1) +which holds for all real-valued compactly supported smooth test functions g, and all Hadamard states +ω of the field. Here, the hat denotes a Fourier transform, defined according to the convention ˆg(α) = +� ∞ +−∞ ds eiαsg(s), and we employ units where ℏ = c = 1, which will be in force throughout this paper. On +the left-hand side, the normal ordering is conducted with respect to an arbitrary Hadamard reference +state ω0, whose two-point function is used to construct the distribution T(s, s′) that appears on the +right-hand side. Recall also that the Hadamard states form a large class of physically reasonable states, +determined by their short-distance structure [29, 37]. The two most important features of the QEI (1.1) +∗chris.fewster@york.ac.uk +†jthompson16@sheffield.ac.uk +1 + +are that the right-hand side is completely independent of the state ω, and that the bound is finite – +which is proved using the microlocal properties of Hadamard states uncovered by Radzikowski [39]. +Discussion of QEIs for other QFTs, including non-free models, may be found in [11, 32]; see [21] for +a very recent development. +Although the lower bound in (1.1) is explicit and rigorous, it is not easy to compute in closed +form except in special cases. +To the best of our knowledge this has only been achieved when T +exhibits translational invariance T(s + r, s′ + r) = T(s, s′) which occurs, for instance, when (M, g) +is a stationary spacetime, γ is a timelike Killing orbit and ω0 is stationary. Translational invariance +allows us to write, with an abuse of notation, T(s, s′) = T(s − s′), from which one easily finds that +the QEI (1.1) simplifies to +� ∞ +−∞ +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +� ∞ +−∞ +dα|ˆg(α)|2Q(α), +(1.2) +where +Q(α) = +1 +2π2 +� α +−∞ +du ˆT(u); +(1.3) +the QEI (1.2) is also valid for complex-valued g. Taking the massless free field as an example, averaging +along an inertial worldline in Minkowski space and using the Minkowski vacuum as the reference state +ω0, this results in Q(α) = α4Θ(α)/(16π3). Using the evenness of |ˆg|2 together with Parseval’s theorem +then yields +� ∞ +−∞ +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +1 +16π2 +� ∞ +−∞ +ds|g′′(s)|2. +(1.4) +Similar expressions are known for massive fields and in Minkowski spacetime of general dimension [12]; +for some curved spacetime examples see [15]. Another explicit example arises where γ is a uniformly +linearly accelerated worldline in four-dimensional Minkowski spacetime with proper acceleration a, in +which case the QEI (1.2) becomes [13] +� ∞ +−∞ +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +1 +16π2 +� ∞ +−∞ +ds +� +|g′′(s)|2 + 2a2|g′(s)|2 + 11 +30a4|g(s)|2 +� +, +(1.5) +and is again valid for all Hadamard states ω and complex-valued test functions g. +Using such expressions the scaling behaviour of the bound is easily understood and phenomena +such as ‘quantum interest’ may be explored [20, 16, 11]. For example, let gλ(s) = λ−1/2g(s/λ), where +g is normalised so that +� ∞ +−∞ ds|g(s)|2 = 1. Then (1.5) implies +lim inf +λ−→∞ +� ∞ +−∞ +ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 11a4 +480π2 , +(1.6) +reducing to the Averaged Weak Energy Condition (AWEC) +lim inf +λ−→∞ +� ∞ +−∞ +dτ|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ 0 +(1.7) +in the limit a → 0, which can also be obtained directly from (1.4). An interesting observation is that +the lower bound in (1.6) is exactly the constant energy density of the Rindler vacuum state along the +accelerated worldline, while the lower bound in (1.7) is the energy density of the Minkowski vacuum. +As closed form expressions for QEI bounds are relatively few in number, it is of interest to find +others. The purpose of this paper is to present a calculation of the QEI bound for a massless scalar +field along any stationary worldline in 4-dimensional Minkowski spacetime. By a stationary worldline, +we mean any timelike curve γ(s), parameterised by proper time s, whose velocity vector evolves under +a 1-parameter subgroup of the Lorentz group: ˙γ(s) = exp(sM) ˙γ(0) for some fixed M ∈ so(1, 3) and +future-pointing unit timelike ˙γ(0). +Stationary worldlines have a long history. Kottler [33], Synge [41] and Letaw [34] (see also [36]) all +obtained them as the solutions to four-dimensional Frenet-Serret equations subject to constancy of the +curvature invariants; the name ‘stationary worldlines’ is due to Letaw. The three curvature invariants +2 + +are the curvature, which measures the proper acceleration, and the torsion and hypertorsion, which +specify its proper angular velocity. More details are given in Section 2. Stationary worldlines are +equivalently described as the orbits of timelike Killing vector fields in Minkowski spacetime. There +are also overlaps with the theory of rigid motions in special relativity that goes back to Born [1] and +Herglotz [25]; in particular, any rotational rigid motion is the flow of a timelike Killing vector by the +Herglotz–Noether theorem, although the same theorem allows any C2 timelike curve to be a flow line +of an irrotational rigid motion. See [22] for discussion and references. +By a Poincar´e transformation, any stationary worldline can be reduced to one of six prototypes: +the inertial, uniformly linearly accelerated, and uniformly rotating worldlines are all familiar, while +the three remaining ones have spatial projections corresponding to a semicubical parabola, a catenary +or a helix. We will give more detail as we discuss each case separately. +The main result of this paper is that the QEI (1.2) along any stationary worldline in Minkowski +spacetime may be given explicitly as +� +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +1 +16π2 +� ∞ +−∞ +ds +� +|g′′(s)|2 + 2A|g′(s)|2 + B|g(s)|2� +, +(1.8) +where A and B are expressed in terms of the curvature κ, torsion τ, and hypertorsion υ as +A = κ2 + τ 2 + υ2 +(1.9) +and +B = 1 +90 +� +3κ4 + 62κ2τ 2 + 30(κ2 + τ 2 + υ2)2� +, +(1.10) +and the inequality (1.8) holds for all Hadamard states ω and all smooth compactly supported test +functions g. +To interpret the QEI (1.8), it is useful to consider its scaling behaviour. As before, we take a test +function gλ which is just a scaled version of the test function g, namely gλ(s) = λ−1/2g(s/λ), so the +support width of gλ is proportional to λ. Observing that g(k) +λ (s) = λ−k−1/2g(k)(s/λ), we find +� +ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − ∥g′′∥2 +16π2λ4 − A∥g′∥2 +8π2λ2 − Treg(0)∥g∥2, +(1.11) +where the norms are those of L2(R), i.e., ∥g∥2 = +� ∞ +−∞ ds|g(s)|2. +Here we have written Treg(0) = +B/(16π2) for reasons that will become clear later – see, for example, equation (1.13) and the ar- +guments presented in Section 3. +For sampling times shorter than the curvature scales, i.e., λ ≪ +min{κ−1, τ −1, υ−1}, the leading term dominates, reflecting the fact that any worldline is approxi- +mately inertial on short enough timescales. At intermediate and long timescales relative to curvature +scales, the bound will receive corrections from, and eventually be dominated by the last two terms +in (1.11), showing that the QEI is sensitive to the curvature invariants of the worldline γ. In the limit +λ → +∞, and with g normalised so that ∥g∥ = 1, we obtain the remarkably simple formula +lim inf +λ−→∞ +� ∞ +−∞ +ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −Treg(0), +(1.12) +which bounds the average energy density along the entire trajectory. In particular, the QEI is con- +sistent with the existence of a constant renormalised energy density −Treg(0) along γ, and this is the +most negative value that any constant energy density could take. An intriguing question is whether +or not this value is attained by some Hadamard state, or a sequence of Hadamard states in a limiting +sense, which we will address in Section 6. +The derivation of (1.8) requires a number of innovations. Although the point-split energy density +can be obtained easily enough for any given stationary worldline, its Fourier transform does not have +a closed form – as far as we know – for three of the six prototypes. In Section 3, we develop a new +method for computing the QEI bound for massless fields in four-dimensions that avoids the use of +Fourier transforms. The result is that the QEI will take the form (1.8) provided that the point-split +energy density takes the form +T(s, s′) = lim +ǫ→0+ +� +3 +2π2(s − s′ − iǫ)4 − +A +4π2(s − s′ − iǫ)2 +� ++ Treg(s − s′), +(1.13) +3 + +where the regular part Treg must satisfy various conditions, whereupon the coefficient B is given by B = +16π2Treg(0) as before. In Section 4, we apply these ideas to stationary worldlines, resulting in formulae +for the point-split energy density in terms of functions easily computed from the Lorentzian distance +between two points on the curve and a tetrad that is adapted to it, in a manner we describe. Most of the +required conditions on Treg follow directly from this analysis, and the values A and B are identified +in terms of Taylor coefficients of these functions. Appendix A gives more detail on our methods, +while in Appendix B the relevant Taylor coefficients are evaluated in terms of curvature invariants +thus establishing (1.9) and (1.10). In Section 5, we work through each prototype in turn, providing +explicit formulae for the point-split energy density that allow the remaining technical condition to +be verified, and also as a check on our Taylor series calculations. In three cases, (inertial worldlines, +linearly accelerated worldlines and the semicubical parabola), a closed form may be found for ˆT, +and we can also check our calculations by using (1.2) and (1.3). Finally, in Section 6, we discuss +the physical significance of our results and some open problems. Two further appendices contain +additional computations: Appendix C computes a quantum inequality for the Wick square along +stationary worldlines following the same general method of the main text, while Appendix D records +the calculation of the minimally coupled stress-energy tensor in the Rindler vacuum and Rindler +thermal states, which is needed for our discussion. +2 +Stationary worldlines +Throughout this paper we work on 4-dimensional Minkowski spacetime, with metric η = dt2 − dx2 − +dy2 − dz2, and we employ the inertial coordinates (t, x, y, z) except where otherwise specified. +A +stationary worldline is any smooth curve γ : R → R4, whose velocity vector ˙γ is a future-pointing unit +timelike vector evolving under a 1-parameter subgroup of the Lorentz group SO(1, 3), i.e., +˙γµ(s) = exp(sM)µ +ν ˙γν(0), +(2.1) +where M is any fixed element of so(1, 3) (which requires precisely that Mµν is antisymmetric). As +every component of exp(sM) is analytic in s, it follows that the Cartesian components of ˙γ(s) and, in- +tegrating, the Cartesian coordinates of γ(s) are also s-analytic. An equivalent definition of a stationary +worldline is that γ is an orbit of a future-pointing timelike Killing vector field +ξµ(x) = Mµ +ν(xν − γ(0)ν) + ˙γµ(0), +(2.2) +which is necessarily timelike in a neighbourhood of γ and future-pointing unit vector on γ. +Finally, stationary worldlines can also be described as the solutions to the Frenet-Serret equations +with constant curvatures [33, 41, 34]. Here, the curvature invariants of a general timelike curve γ(s), +parameterised by proper time, are defined as follows. Suppose a right-handed tetrad eµ +a has been +chosen along γ so that +γ(k+1)(s) ∈ span{e0(s), . . . , ek(s)} +(0 ≤ k ≤ 3), +and +˙γ(s) = e0(s), +(2.3) +in which case we say that eµ +a is adapted to γ. If the tetrad also satisfies +e1(s)µ¨γ(s)µ ≤ 0, +e2(s)µ...γ (s)µ ≤ 0, +(2.4) +then it will be called a Frenet–Serret tetrad. If the tetrad is defined by ea(s) = exp(sM)ea(0), then it +is adapted (respectively, Frenet–Serret) if and only if (2.3) holds at s = 0 (resp., (2.3) and (2.4) hold +at s = 0). Explicit formulae resulting from a Gram–Schmidt procedure are given in [34]. Expanding +the derivatives of the tetrad vectors in terms of the tetrad, one obtains the generalized Frenet–Serret +equations +˙eµ +a = K b +a eµ +b , +(2.5) +where Kab is antisymmetric and tridiagonal (due to (2.3)). Thus it takes the form +K••(s) = + + + + +0 +−κ(s) +0 +0 +κ(s) +0 +−τ(s) +0 +0 +τ(s) +0 +−υ(s) +0 +0 +υ(s) +0 + + + + , +(2.6) +4 + +which defines the curvature κ, torsion τ and hypertorsion υ. Here, and elsewhere in this paper, bullets +are used to indicate tensorial type, when displaying tensorial components in vector or matrix form. +Explicitly, one has +κ = e0µ ˙eµ +1 = −e1µ ˙eµ +0, +τ = e1µ ˙eµ +2 = −e2µ ˙eµ +1, +υ = e2µ ˙eµ +3 = −e3µ ˙eµ +2. +(2.7) +The choices made when specifying the Frenet–Serret tetrad ensure that κ and τ are nonnegative, while +υ can take any real value. +As the curvature invariants are constant along stationary worldlines, it is easy to compute higher +derivatives of the tetrad, +dk +dsk eµ +a = (Kk) b +a eµ +b , +(Kk) b +a = K c1 +a +K +c2 +c1 +· · · Kck−1 +b. +(2.8) +For example, the first three derivatives of the velocity u = ˙γ may be computed as +˙uµ = ˙eµ +0 = κeµ +1, +¨uµ = κ2eµ +0 + κτeµ +2, +...u µ = κ(κ2 − τ 2)eµ +1 + κτυeµ +3. +(2.9) +It is also possible to give a general formula for γ(s) in terms of M, γ(0) and ˙γ(0). As M• +• is +antisymmetric with respect to η, there is a unique decomposition +˙γ(0)µ = Mµ +νvν + kµ, +(2.10) +where Mµ +νkν = 0. One then has +γ(s)µ = exp(sM)µ +νvν + skµ + γ(0)µ − vµ. +(2.11) +Any stationary worldline γ may be related to one of six basic types by a proper orthochronous +Poincar´e transformation. Note that γ(s) is determined by the initial position, γ(0) ∈ R4, the initial +four-velocity ˙γ(0) and the element M ∈ so(1, 3) that fixes the evolution ˙γ(s) = exp(sM) ˙γ(0). Under +a Poincar´e transformation x �→ Λx + w, γ is mapped to ˜γ(s) = Λγ(s) + w, whose velocity evolves +according to the 1-parameter Lorentz subgroup exp +� +sΛMΛ−1� +and which is therefore also a stationary +worldline. As the Lorentz transformation maps a Frenet–Serret tetrad for γ to a Frenet–Serret tetrad +for ˜γ, it follows from (2.7) that the curvature invariants of ˜γ are identical to those of γ. Using the +classification of conjugacy classes in so(1, 3) [40], we may choose Λ in such a way that ˜ +M = ΛMΛ−1 +is one of five possible types: (a) the zero element, generating the trivial subgroup of SO(1, 3), (b) a +generator of boosts in the tx-plane, corresponding to a hyperbolic subgroup of SO(1, 3), (c) a generator +of rotations in the yz-plane, corresponding to an elliptic subgroup of SO(1, 3), (d) a generator of a null +rotation that fixes the null vector ∂t + ∂x but acts nontrivially on all other null vectors, corresponding +to a parabolic subgroup of SO(1, 3); (e) the sum of a generator of boosts in the tx-plane and a +generator of rotations in the yz plane, corresponding to a loxodromic subgroup of SO(1, 3). In each +case, Lorentz transformations that commute with the 1-parameter subgroup in question can be used +to arrange that ˙˜γ(0) takes a convenient form. +Taking these possibilities in turn: in case (a), all Lorentz transformations commute with the trivial +subgroup, so we may without loss assume that ˜γ(s) = (s, 0, 0, 0). In case (b), the subgroup of boosts +parallel to the x-axis commutes with itself and the subgroup of rotations in the yz-plane. Thus, we +may arrange that ˙˜γ(0) = cosh χ∂t + sinh χ∂y for some χ ∈ R,1 leading to two subcases: χ = 0, in +which case (after possible translation) +˜γ(s) = (a−1 sinh as, a−1 cosh as, 0, 0) +(2.12) +is a uniformly linearly accelerated worldline with a ̸= 0, or χ ̸= 0, in which case (up to translations) +˜γ(s) = (a−1 cosh χ sinh as, a−1 cosh χ cosh as, −s sinh χ, 0) +(2.13) +is a catenary. The curvature invariants (in either subcase) are κ = |a| cosh χ and τ = |a sinh χ|, while +the hypertorsion is υ = 0. For convenience, the curvature invariants for all six prototypes are tabulated +in Table 1, in agreement with [36]. +1We could even arrange that χ ≥ 0, but it is convenient not to insist on this. +5 + +Inertial +Linear Acc. +Catenary +Parabolic +Elliptic +Loxodromic +κ = τ = υ = 0 +κ > 0 +κ > τ > 0 +κ = τ > 0 +τ > κ > 0 +κ, τ > 0 +τ = υ = 0 +υ = 0 +υ = 0 +υ = 0 +υ ̸= 0 +κ +0 +|a| +|a| cosh χ +|a| +rω2 +√ +C2a2 + V 2ω2 +τ +0 +0 +|a sinh χ| +|a| +|ω| +� +1 + (rω)2 +(a2 + ω2)C|V |/κ +υ +0 +0 +0 +0 +0 +aω/κ +Table 1: Curvature invariants for the stationary worldlines. +In case (c), the 1-parameter parabolic subgroup takes the form +P • +•(s) = + + + + +1 + (as)2/2 +−(as)2/2 +0 +as +(as)2/2 +1 − (as)2/2 +0 +as +0 +0 +1 +0 +as +−as +0 +1 + + + + = exp + + + + +0 +0 +0 +as +0 +0 +0 +as +0 +0 +0 +0 +as +−as +0 +0 + + + + +(2.14) +for some constant nonzero a ∈ R, and commutes with Lorentz transformations of the form +Λ• +• = + + + + +1 + r2/2 +−r2/2 +r cos θ +r sin θ +r2/2 +1 − r2/2 +r cos θ +r sin θ +r cos θ +−r cos θ +1 +0 +r sin θ +−r sin θ +0 +1 + + + + +(2.15) +which can be used to bring the initial velocity into the form ˙˜γ(0) = cosh χ∂t+sinh χ∂x for some χ ∈ R. +Conjugating P • +•(s) with a boost in the tx-plane results in P • +•(λs) for some λ > 0; in other words +effectively rescaling a. Therefore there is no loss of generality in assuming that the initial 4-velocity +is ˙˜γ(0) = ∂t, in which case the worldline (up to translation) is the semicubical parabola, +˜γ(s) = +� +s + 1 +6a2s3, 1 +6a2s3, 0, 1 +2as2 +� +. +(2.16) +Next, the elliptic subgroup in case (d) commutes with boosts in the tx-plane and rotations in +the yz-plane. Accordingly, we may arrange the initial velocity to be ˙˜γ(0) = cosh χ∂t + sinh χ∂z for +some χ ∈ R; the special case χ = 0 corresponds to inertial motion and may be discarded. Up to a +translation, this results in the uniformly rotating worldline +˜γ•(s) = (s cosh χ, 0, r cos ωs, r sin ωs) , +(2.17) +where the radius r > 0 and proper angular velocity ω ̸= 0 are related to the initial rapidity by +rω = sinh χ. The proper acceleration is κ = rω2, while the torsion is τ = |ω| +� +1 + (rω)2 and the +hypertorsion vanishes. +Lastly, in case (e), the loxodromic subgroup is generated by a linear combination of a tx-boost +generator and a yz-rotation generator. +As it commutes with tx-boosts and yz-rotations, we may +assume without loss that the initial velocity is ˙˜γ(0) = cosh χ∂t + sinh χ∂z for χ ∈ R \ {0}; the +possibility χ = 0 corresponds to a hyperbolic worldline and is rejected. Up to a translation, this +results in the worldline +γ•(s) = (Ca−1 sinh(as), Ca−1 cosh(as), V ω−1 cos(ωs), V ω−1 sin(ωs)), +(2.18) +where C = cosh χ and V = sinh χ, which undergoes both rotation in the yz-plane at constant proper +angular velocity ω ̸= 0 and constant distance |V/ω| from the x-axis, while undergoing uniform acceler- +ation in the x-direction controlled by a ̸= 0 (the cases where one or both of a or ω vanish are already +covered under (a), (b) and (d)). The curvature invariants for this worldline are +κ = +� +C2a2 + V 2ω2, +τ = (a2 + ω2)C|V |/κ, +υ = aω/κ. +(2.19) +6 + +3 +Reformulation of the QEI bound +We study the massless minimally coupled scalar field in 4-dimensional Minkowski spacetime, with field +equation □φ = ηµν∇µ∇νφ = 0 and classical stress-energy tensor +Tµν = (∇µφ)∇νφ − 1 +2ηµνηαβ(∇αφ)∇βφ. +(3.1) +Consider an observer following a timelike curve γ, parameterised by proper time, with 4-velocity +uµ = ˙γµ. This observer sees energy density +Tµνuµuν = 1 +2 +3 +� +a=0 +(eµ +a∇µφ)2, +(3.2) +where eµ +a (0 ≤ a ≤ 3) is a tetrad defined around γ with eµ +0|γ = uµ. +In quantum field theory, the stress-energy tensor requires renormalisation. Let +G(x, x′) = ⟨φ(x)φ(x′)⟩ω +(3.3) +be the Wightman function of the field in a state ω. The Wick square has expectation value +⟨:φ2(x):⟩ω = (G − G0)(x, x), +(3.4) +where +G0(x, x′) = lim +ǫ→0+ +−1 +4π2((t − t′ − iǫ)2 − ∥x − x′∥2) +(3.5) +is the Wightman function of the Poincar´e invariant vacuum ω0. This expression makes sense if (like +ω0) ω is a Hadamard state [29, 37], because the difference G−G0 is then a smooth function. Similarly, +the renormalised stress-energy tensor has expectation value +⟨:Tµν(x):⟩ω = Dµν(x) − 1 +2ηµνηαβDαβ(x), +(3.6) +where +Dµν(x) = [[(∇ ⊗ ∇)(G − G0)]]µν (x) +(3.7) +and the double square brackets denote a coincidence limit. +Although the classical energy density (3.2) is everywhere nonnegative, the quantised energy density +may assume negative expectation values. The QEIs provide lower bounds on averaged expectation +values, for which a prototype is a lower bound on the following expression +� +ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)), +(3.8) +where Q is a partial differential operator with smooth real coefficients and g ∈ C∞ +0 (R) is a smooth +real-valued test function. In the case where Q is the identity, (3.8) is an averaged Wick square, while +by considering a sum of similar terms for Qa = 2−1/2eµ +a∇µ for 0 ≤ a ≤ 3, we can bound averages of +the energy density along γ. +A lower bound on (3.8) was established in [8] – in fact the bound applies to general timelike curves +in arbitrary globally hyperbolic spacetimes for massive as well as massless fields. In our case it asserts +that +� ∞ +−∞ +ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ − +� ∞ +0 +dα +π +÷ +g ⊗ gT (−α, α) > −∞ +(3.9) +holds for all real-valued compactly supported smooth test functions g, and all Hadamard states ω, +where +T(s, s′) = ⟨Qφ(γ(s))Qφ(γ(s′))⟩ω0 = ((Q ⊗ Q)G0)(γ(s), γ(s′)). +(3.10) +Here, the vacuum two-point function enters because normal ordering is performed relative to the +vacuum; the general results of [8] also allow for any Hadamard state to be used as the reference state +for this purpose. At a more formal level, T is the pull-back of the distribution (Q ⊗ Q)G0 by the map +7 + +(s, s′) �→ (γ(s), γ(s′)), and its existence is owed to the special properties of the Hadamard condition +and the fact that γ is timelike – see [8] for full details and rigorous proofs. +As already mentioned, a QEI for the energy density involves a sum of such bounds, leading to (1.1) +with +T(s, s′) = 1 +2 +3 +� +a=0 +((∇ea ⊗ ∇ea)G0)(γ(s), γ(s′)). +(3.11) +While it is usually not hard to obtain the distribution T for a given timelike curve in Minkowski +spacetime, assuming that G0 is given, it is not usually possible to find the Fourier transform required +to compute the QEI bound (3.9) in closed form. +The situation is somewhat simplified if T(s, s′) is translationally invariant, in which case one has +the bound given by (1.2) and (1.3). This can be taken a little further, on observing that |ˆg(α)|2 is +even, so only the even part Qeven(α) = 1 +2(Q(α) + Q(−α)) of Q contributes to (1.2), resulting in the +bound +� +ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ − +� ∞ +−∞ +dα|ˆg(α)|2Qeven(α), +(3.12) +which is the final form of our prototypical quantum inequality. +A convenient expression for Qeven may be found by manipulating equation (1.3) in the following +way: +Qeven(α) = +1 +4π2 +�� α +−∞ +ˆT(u) du + +� −α +−∞ +ˆT(u) du +� += +1 +4π2 +� +2 +� 0 +−∞ +ˆT(u) du + +� α +0 +ˆT(u) du − +� α +0 +ˆT(−u) du +� += +1 +2π2 +�� 0 +−∞ +ˆT(u) du + +� α +0 +ˆTodd(u) du +� +, +(3.13) +where ˆTodd(u) = 1 +2( ˆT(u) − ˆT(−u)). In the above calculation, ˆT is assumed to be continuous, as is the +case for the examples we will study. +Evaluating Qeven from (3.13) requires several steps. Computing T is a tedious but straightforward +calculation best handled using computer algebra. In the simplest cases, the transform may be eval- +uated in closed form, which (as will be seen later) is the case for the inertial, uniformly accelerated +and semicubical parabola worldlines, but is not possible (to our knowledge) in the case of the other +stationary worldlines. However, this obstacle can be circumvented, as we now describe. +Using the Minkowski vacuum as the reference state, we will show in Section 4 that the point-split +energy density along a stationary worldline may be written in the form +T(s, s′) = Tsing(s − s′) + Treg(s − s′), +(3.14) +where Tsing is given by the distributional limit +Tsing(s) = lim +ǫ→0+ +� +3 +2π2(s − iǫ)4 − +A +4π2(s − iǫ)2 +� +(3.15) +for some constant A (the sign is chosen for later convenience) and Treg is smooth, real and even, and +decaying as O(s−2) as |s| → ∞. In particular, Treg is absolutely integrable and has a well-defined +Fourier transform that is continuous, real and even. Therefore it does not contribute to ˆTodd. Turning +to Tsing, its leading singularity is universal, essentially because all stationary worldlines resemble +inertial worldlines on sufficiently short timescales. The specific coefficient is fixed by the Hadamard +form and the definition of the energy density along the curve. Meanwhile the coefficient A carries +information about the specific curve at hand. The Fourier transform of Tsing, in our convention, is +ˆTsing(u) = 1 +2π(u3 + Au)Θ(u), +(3.16) +8 + +where Θ is the Heaviside distribution. Evidently Tsing does not contribute to the first term in (3.13), +while the odd part of ˆT is +ˆTodd(u) = 1 +4π (u3 + Au), +(3.17) +recalling that ˆTreg is even. We now have Qeven in the form +Qeven(α) = +1 +2π2 +�� 0 +−∞ +du ˆTreg(u) + 1 +4π +� α +0 +du(u3 + Au) +� += +1 +32π3 (α4 + 2Aα2) + Treg(0) +2π +, +(3.18) +where we have again used the evenness of ˆTreg and the Fourier inversion formula. Inserting (3.18) +into (3.12) and using Parseval’s theorem gives the QEI bound +� +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +1 +16π2 +� ∞ +−∞ +ds +� +|g′′(s)|2 + 2A|g′(s)|2 + B|g(s)|2� +, +(3.19) +where B = 16π2Treg(0). +The upshot of this analysis is a direct route to the QEI once the point-split expression T is obtained; +all that is needed is to isolate the appropriate values of A and Treg(0), avoiding the need to compute ˆT +explicitly. This apparent royal road is made possible because of the special structure of the Minkowski +vacuum two-point function for the massless scalar field in four dimensions – closely related to Huygens’ +principle. A similar analysis for a QI on the Wick square can be found in Appendix C. +4 +Computation of the point-split energy density +In this section we establish that the point-split energy density along stationary worldlines obeys +equations (3.14) and (3.15), and also that Tsing and Treg have the properties mentioned above, with +one exception that will be treated by examining the six prototypical cases in Section 5. +Let γ be any stationary worldline with ˙γ(s) = exp(sM)˙γ(0) and ˙γ(0) a future-pointing unit timelike +vector. Suppose that +ea(s) = exp(sM)ea(0) +(0 ≤ a ≤ 3) +(4.1) +is an adapted frame on γ satisfying (2.3). In general there may be many possible adapted tetrads of +this type. However, if ˜ea(s) is any other then it is related to ea(s) by a rigid rotation, i.e., ˜e0(s) = e0(0) +and ˜ei(0) = R j +i ej(0) (summing j over 1, 2, 3), where δimR j +i R +n +m = δmn, det R = 1. This must be true +for some R at s = 0, and extends to all s as both tetrads evolve under exp(sM). +Next, recall that the vacuum 2-point function may be given as a distributional limit +G0(x, x′) = lim +ǫ→0+ F(σǫ(x, x′)) +(4.2) +where F(z) = 1/(4π2z) and +σǫ(x, x′) = −ηµν(x − x′ − iǫ∂t)µ(x − x′ − iǫ∂t)ν +(4.3) +is the regulated signed squared geodesic separation of x and x′. As usual, we have identified Minkowski +spacetime with its tangent spaces at all points. +Distributional derivatives may be taken under the limit in (4.2), giving +1 +2(∇µ ⊗ 1)G0(x, x′) = − lim +ǫ→0+ F ′(σǫ(x, x′))(x − x′ − iǫ∂t)µ +(4.4) +and +1 +2(∇µ ⊗ ∇ν)G0(x, x′) = lim +ǫ→0+ +� +F ′(σǫ(x, x′))ηµν − 2F ′′(σǫ(x, x′))(x − x′ − iǫ∂t)µ(x − x′ − iǫ∂t)ν +� +. +(4.5) +9 + +Contracting with ea(x)µea(x′)ν (without summing on a) and pulling back to the worldline, we find +1 +2((∇ea ⊗ ∇ea)G0)(γ(s), γ(s′)) = lim +ǫ→0+ F ′(σǫ(γ(s), γ(s′)))Ca(s, s′) ++ lim +ǫ→0+ 2F ′′(σǫ(γ(s), γ(s′)))Da(s, s′)Da(s′, s). +(4.6) +(note the order of variables in the last two factors in the second term) where +Ca(s, s′) = ηµνeµ +a(s)eν +a(s′), +Da(s, s′) = (γ(s) − γ(s′))µeµ +a(s). +(4.7) +Under a change of frame from ea to ˜ea as described above, one has ˜C0 = C0, ˜D0 = D0, while +˜Di = R j +i Dj and ˜Ci(s, s′) = R j +i R k +i ηµνeµ +j (s)eν +k(s′). By orthogonality, this implies that �3 +i=1 ˜Ci(s, s′) = +�3 +i=1 Ci(s, s′) and �3 +i=1 ˜Da(s, s′) ˜Da(s′, s) = �3 +i=1 Da(s, s′)Da(s′, s). +In Appendix A, we give some further details to justify the above distributional manipulations and +prove the following result, where κ, τ and υ are the curvature invariants of γ as described in Section 2. +Lemma. (a) With the choice of tetrad just described, Ca(s, s′) and Da(s, s′) are translationally in- +variant, depending only on s − s′. There are entire analytic functions Ga and Ha such that +Ca(s, s′) = Ga(κ2(s − s′)2), +Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2), +(4.8) +where, in the limit z → 0, +3 +� +a=0 +Ga(z) = −2 + τ 2 + υ2 +κ2 +z + (κτ)2 − (τ 2 + υ2)2 +κ4 +z2 + O(z3), +(4.9) +and +3 +� +a=0 +Ha(z) = 1 + z +12 + κ2 + 19τ 2 +360κ2 +z2 + O(z3). +(4.10) +(b) The signed square geodesic separation of points along γ obeys +σ0(γ(s), γ(s′)) = −(s − s′)2Υ(κ2(s − s′)2), +(4.11) +where Υ is entire analytic with +Υ(z) = 1 + 1 +12z + κ2 − τ 2 +360κ2 z2 + O(z3) +(4.12) +as z → 0. Furthermore, for z ∈ [0, ∞), Υ(z) is real with Υ(z) ≥ 1. +The Lemma now allows us to compute the point-split energy density by evaluating the right-hand +side of (4.6) and summing over a. We use the fact (explained in Appendix A) that +lim +ǫ→0+ +(s − s′)2j +σǫ(γ(s), γ(s′))k = +(−1)k +Υ(κ2(s − s′)2)k lim +ǫ→0+ +1 +(s − s′ − iǫ)2(k−j) , +(4.13) +where the limits are taken in the sense of distributions, as is the multiplication by a smooth prefactor +on the right-hand side. If j = k, the distributional limit on the right-hand side may be replaced +by unity. In particular, when calculating T(s, s′) from (4.6), the factor (s − s′)2 in Da(s, s′)Da(s′, s) +cancels a factor of (s − s′ − iǫ)2 in the denominator, as ǫ → 0+. The upshot is that +T(s, s′) = − 1 +4π2 lim +ǫ→0+ +K(κ2(s − s′)2) +(s − s′ − iǫ)4 , +where +K(z) = +3 +� +a=0 +�Ga(z) +Υ(z)2 − 4Ha(z) +Υ(z)3 +� +(4.14) +is a meromorphic function that is analytic in a neighbourhood of the positive real axis (on which Υ is +bounded away from zero). +10 + +The singular part is easily isolated by splitting off the first two terms of the Taylor series for K +from the remainder, which carries a leading factor of (s − s′)4 that cancels the denominator in the +limit ǫ → 0+. Similarly, the O(z) part of the Taylor series partly cancels the denominator. Thus, +T(s, s′) = Tsing(s − s′) + Treg(s − s′) with +Tsing(s) = − 1 +4π2 lim +ǫ→0+ +K(0) +(s − iǫ)4 − +1 +4π2 lim +ǫ→0+ +κ2K′(0) +(s − iǫ)2 , +(4.15) +and +Treg(s) = − κ4 +4π2 J((κs)2), +where +J(z) = K(z) − K(0) − K′(0)z +z2 +(4.16) +is analytic on a neighbourhood of the positive real axis, so J((κs)2) is smooth for s ∈ R. +Using the Lemma, we may read off that K(0) = −6, thus establishing (3.15), with A = κ2K′(0). +Meanwhile, Treg(s) is smooth, even, and real-valued for s ∈ R. Provided that K(z) = O(z) as z → ∞ +on the real axis, we find that Treg(s) = O(s−2) as s → ∞, which completes the properties needed in +Section 3. Furthermore, +Treg(0) = −J(0)κ4 +4π2 += −K′′(0)κ4 +8π2 +. +(4.17) +Note that if we had used the tetrad ˜e instead, the function K would be unchanged, owing to the +remarks before the Lemma. Thus the QEIs obtained from ˜ea and ea are identical. +These results now provide a calculational method to determine the QEI along stationary worldlines. +Starting from the generator M ∈ so(1, 3) and the initial 4-velocity u(0), choose a tetrad as described +at the start of this section, and compute the proper acceleration κ = +� +−η(Mu(0), Mu(0)). The +translational invariance of Ca and Da means that they can be calculated conveniently as +Ca(s, s′) = ηµνeµ +a(s − s′)eν +a(0), +Da(s, s′) = −(γ(s′ − s) − γ(0))µeµ +a(0), +(4.18) +from which Ga and Ha are easily obtained. The function Υ is computed directly from the Lorentz +interval between γ(0) and γ(s). Then construct K(z) according to (4.14) and check that K(z) = O(z) +as z → ∞. Then the QEI along γ is given by (3.19), with constants +A = κ2K′(0), +B = −2κ4K′′(0). +(4.19) +The constants A and B can be computed from the first few terms of the Taylor expansions of +� +a Ga, � +a Ha and Υ, given in (4.9), (4.10) and (4.12) respectively. After a calculation, one finds +K(z) = −6 + z κ2 + τ 2 + υ2 +κ2 +− z2 +1 +360κ4 +� +3κ4 + 62κ2τ 2 + 30(κ2 + τ 2 + υ2)2� ++ O(z3), +(4.20) +from which the formulae (1.9) and (1.10) follow immediately. Nonetheless, this is perhaps not the +most illuminating calculation and also does not provide a check that K(z) = O(z) for large real z, +which was assumed above. For these reasons, and their own intrinsic interest, we will also provide +explicit calculations in Section 5 that together cover all possible stationary worldlines. +5 +QEIs for the prototypical stationary worldlines +We have now established the general QEI for stationary worldlines in Minkowski spacetime, assuming +a technical condition on the growth of K. In this section, we reduce the problem of computing the +QEI for a general stationary worldline to six prototypical cases, which will be treated in turn. These +calculations follow the method of Section 4 and result in explicit formulae for K. In this way it is seen +that the growth condition holds in all cases and we also obtain a check on the Taylor series calculations +in Appendix B. +We have already discussed the fact that any stationary worldline may be brought into one of the +six standard forms by a Poincar´e transformation, without changing the curvature invariants. Owing +to Poincar´e invariance of the vacuum state, and because Poincar´e invariance maps an adapted tetrad +of the form ea(s) = exp(sM)ea(0) along the original curve to a tetrad with the same properties on +11 + +the new one, the point-split energy density obtained by the method of Section 4 is exactly the same +for the two worldlines, which accordingly share the same QEI bound. +The QEIs for the prototypical stationary worldlines are now given in turn. Most of the computa- +tions that follow were conducted using the computer algebra system Maple. +5.1 +Trivial subgroup: inertial motion +For the inertial worldline γ(s) = (s, 0, 0, 0), we employ the adapted tetrad ∂t, ∂x, ∂y, ∂z, which is +constant along γ, leading immediately to the relations C0(s, s′) = 1, Ci(s, s′) = −1 for i = 1, 2, 3, while +D0(s, s′) = s−s′, Di(s, s′) = 0 for all s, s′. It follows that G0 = H0 ≡ 1, Gi ≡ −1, Hi ≡ 0. Furthermore, +Υ ≡ 1 because σ0(γ(s), γ(s′)) = −(s − s′)2. Hence K ≡ −6 and one finds T(s, s′) = Tsing(s − s′) where +Tsing(s) = lim +ǫ→0+ +3 +2π2(s − iǫ)4 . +(5.1) +Consequently Treg vanishes identically, and we may read off immediately that A = B = 0, reproducing +QEI (1.4) by substituting into (3.19), and in agreement with (1.9) and (1.10). Of course these results +are easily obtained by direct differentiation of the two-point function; our purpose here is to show how +they follow from formulae in Section 4. +Alternatively, we may proceed by taking the Fourier transform +ˆTsing(u) = u3Θ(u)/(2π), +(5.2) +from which we obtain Q(α) = α4Θ(α)/(16π3) by (1.3), leading to (1.4) as discussed in the introduction. +5.2 +Hyperbolic subgroups: linear acceleration +We consider a uniformly linearly accelerated worldline +γ(s) = (a−1 sinh as, a−1 cosh as, 0, 0), +(5.3) +whose velocity evolves under the 1-parameter group of tx-boosts ˙γµ(s) = Hµ +ν(s)˙γν(0), where +H• +•(s) = + + + + +cosh as +sinh as +0 +0 +sinh as +cosh as +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 + + + + = exp + + + + +0 +as +0 +0 +as +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + + + +(5.4) +and 0 ̸= a ∈ R is fixed. Noting that the initial velocity and its first two derivatives are ˙γ(0) = ∂t, +¨γ(0) = a∂x, ¨γ(0) = a2∂t, we obtain an adapted tetrad by choosing the tetrad ∂t, ∂x, ∂y, ∂z at s = 0, +and applying the prescription eµ +a(s) = Hµ +ν(s)eν +a(0) to find +e0(s) = cosh as∂t + sinh as∂x, +e1(s) = sinh as∂t + cosh as∂x, +e2(s) = ∂y, +e3(s) = ∂z. +(5.5) +Straightforward calculation, following the method of Section 4, gives +K(a2s2) = − +3(as)4 +8 sinh4(as/2) +(5.6) +and hence +T(s, s′) = lim +ǫ→0+ +3a4(s − s′)4 cosech4(a(s − s′)/2) +32π2(s − s′ − iǫ)4 +, +(5.7) +which may be simplified to +T(s, s′) = lim +ǫ→0+ +3a4 +32π2 cosech4 � +a(s − s′ − iǫ)/2 +� +. +(5.8) +Here, we have used the general fact that limǫ→0+ g(x)f(x − iǫ) = limǫ→0+ g(x − iǫ)f(x − iǫ) in the +sense of distributions, when f is analytic in a strip Z = {x − iy : x ∈ R, 0 < y < y0} ⊂ C with +supz∈Z |f(z)(Im z)N| < ∞ for some N > 0 and g is analytic on Z and continuous on Z ∪ R. +12 + +As the function K(z) evidently decays rapidly as z → ∞ on the real axis, the method of Section 4 +allows us to read off the QEI from the derivatives of K(z) at z = 0 according to (4.19). Using +K(z) = +3z2 +8 sinh4(√z/2) = −6 + z − 11 +120z2 + O(z3), +(5.9) +we find A = a2 and B = 11a4/30, in agreement with (1.9) and (1.10) using the invariants from Table 1 +and reproducing the result (1.5) from [13]. In that reference, the point-split energy density (5.8) was +found by a direct calculation. Writing T(s, s′) = T(s − s′), the Fourier transform yields +ˆT(u) = +u3 − a2u +2π(1 − e−2πu/a) +(5.10) +and by using the last expression in (3.13), a calculation gives +Qeven(α) = +1 +32π3 +� +α4 + 2a2α2 + 11 +30a4 +� +, +(5.11) +from which (1.5) follows on inserting the above expression into (3.12) and using Parseval’s theorem. +5.3 +Hyperbolic subgroups: the catenary +Now consider the catenary +γ(s) = (a−1 cosh χ sinh as, a−1 cosh χ cosh as, −s sinh χ, 0), +(5.12) +for constant a ̸= 0, with initial velocity +˙γ•(0) = (cosh χ, 0, − sinh χ, 0), +(5.13) +and second and third derivatives +¨γ•(0) = (0, a cosh χ, 0, 0), +...γ •(0) = (a2 cosh χ, 0, 0, 0). +(5.14) +The velocity evolves under the hyperbolic subgroup (5.4). Writing C = cosh χ and V = sinh χ, the +tetrad +e• +0(s) = (C cosh as, C sinh as, −V, 0), +e• +1(s) = (sinh as, cosh as, 0, 0), +e• +2(s) = (−V cosh as, −V sinh as, C, 0), +e• +3(s) = (0, 0, 0, 1) +(5.15) +is adapted to γ with eµ +a(s) = Hµ +ν(s)eν +a(0). A calculation results in the formula +K(z) = −4V 2(sinhc2(r) + v2) sinh2(r) + 2(4C2 − 1) sinhc2(r) − 16V 2 sinhc(2r) + 2v2(4C2 − 3) +C4(sinhc2(r) − v2)3 +(5.16) +where v = tanh χ, r = √z/(2 cosh χ) and sinhc(x) = sinh(x)/x is the hyperbolic version of the sinc +function. Note that we need not specify a branch for the square root as it always appears in the +argument of an even entire function, and also that K(z) → 0 as z → ∞ in R. The series expansion is +K(z) = −6 + 2C2 − 1 +C2 +z − 185C4 − 182C2 + 30 +360C4 +z2 + O(z3) +(5.17) +and as κ = aC we may read off A = a2(2C2 − 1) = a2 cosh 2χ and B = (185C4 − 182C2 + 30)a4/90. +It is straightforward that these values agree with (1.9) and (1.10) using the curvature invariants for +this case. In particular, the resulting QEI is compatible with a constant negative energy density of +− Treg(0) = −(185 cosh4 χ − 182 cosh2 χ + 30)a4 +1440π2 +(5.18) +along the worldline (5.12). As would be expected, the QEI for linear acceleration is obtained in the +limit χ → 0, but for χ ̸= 0, we have −Treg(0) < −11a4/480π2, and the QEI bound is consistent with +a strictly more negative constant energy density than is the case for the linearly accelerated worldline +with the same value of a. +13 + +5.4 +Parabolic subgroups: the semicubical parabola +We now consider the semicubical parabola +γ(s) = +� +s + 1 +6a2s3, 1 +6a2s3, 0, 1 +2as2 +� +, +(5.19) +for constant a ̸= 0, whose velocity evolves as ˙γµ(s) = P µ +ν(s)˙γ(0) with ˙γ(0) = ∂t, where P µ +ν was +defined in (2.14). From the initial derivatives ˙γ(0) = ∂t, ¨γ(0) = a∂z, ...γ (0) = a2(∂t + ∂x) one sees that +the initial tetrad e0(0) = ∂t, e1(0) = ∂z, e2(0) = ∂x, e3(0) = ∂y determines an adapted tetrad +e• +0(s) = +� +1 + 1 +2(as)2, 1 +2(as)2, 0, as +� +, +e• +1(s) = (as, as, 0, 1) , +e• +2(s) = +� +− 1 +2(as)2, 1 − 1 +2(as)2, 0, −as +� +, +e• +3(s) = (0, 0, 1, 0), +(5.20) +at general proper time s obeying eµ +a(s) = P µ +ν(s)eν +a(0). +Straightforward calculation now gives +K(z) = −6 − z/2 + 5z2/36 +(1 + z/12)3 +, +(5.21) +with +K(z) = −6 + 2z − 37 +72z2 + O(z3) +(5.22) +as z → 0 and K(z) = O(z−1) for z → ∞. Thus, the point-split energy density is +T(s, s′) = lim +ǫ→0+ +3 − a2(s − s′)2/4 + 5a4(s − s′)4/72 +π2(s − s′ − iǫ)4(1 + a2(s − s′)2/12)3 +(5.23) +and (4.19) gives A = 2a2 and B = 37a4/18, in agreement with (1.9) and (1.10). Thus the QEI along +a semicubical parabola is +� +ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +1 +16π2 +� ∞ +−∞ +ds +� +|g′′(s)|2 + 4a2|g′(s)|2 + 37 +18a4|g(s)|2 +� +, +(5.24) +for any Hadamard state ω. The long-time scaling limit of the above QEI is then +lim inf +λ−→∞ +� ∞ +−∞ +ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − +37 +288π2 a4, +(5.25) +where as usual we choose g with unit L2-norm. The QEI is therefore compatible with a constant +negative energy density −37a4/(288π2) along the semicubical parabola. As one would expect, the +QEI reduces to the inertial case as a → 0. +In fact the QEI (5.24) can also be obtained by a different method. Writing T(s, s′) = T(s − s′), +the Fourier transform may be computed by contour methods as +ˆT(u) = 1 +2π +�� 2u2 +√ +12 + 7|u| +8 a + +15 +8 +√ +12a2 +� +ae−|u| +√ +12/a + +� +u3 + 2ua2� +Θ(u) +� +. +(5.26) +The calculation is considerably simplified if one first replaces powers of s − s′ in (5.23) by powers of +s − s′ − iǫ. To find Qeven(α), we note that ˆTodd(u) = (u3 + 2ua2)/(4π), and also that the integral of ˆT +over (−∞, 0] may be evaluated in terms of Γ-functions. After manipulation, the formula (3.13) gives +Qeven(α) = +1 +4π3 +� 0 +−∞ +� 2u2 +√ +12 + 7|u| +8 a + +15 +8 +√ +12a2 +� +ae−|u| +√ +12/a du + +1 +8π3 +� α +0 +� +u3 + 2ua2� +du += +1 +32π3 α4 + a2 +8π3 α2 + 37a4 +576π3 . +(5.27) +Inserting this expression in (3.12) and using Parseval’s theorem we reproduce (5.24). +14 + +5.5 +Elliptic subgroups: uniform rotation +Next, consider the uniformly rotating worldline +γ(s) = (s cosh χ, 0, r cos ωs, r sin ωs) , +(5.28) +where the radius r > 0 and proper angular velocity ω ̸= 0 together fix the rapidity χ = sinh−1(rω). +In this case, the velocity evolves under rotations in the yz-plane as ˙γµ(s) = Rµ +ν(s)˙γν(0), where +R• +•(s) = + + + + +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +cos ωs +− sin ωs +0 +0 +sin ωs +cos ωs + + + + = exp + + + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−ωs +0 +0 +ωs +0 + + + + . +(5.29) +Meanwhile, the initial velocity and its first two derivatives are +˙γ•(0) = (C, 0, 0, V ) +¨γ•(0) = (0, 0, −V ω, 0) +...γ •(0) = +� +0, 0, 0, −V ω2� +, +where we have written C = cosh χ and V = rω = sinh χ. +Then e• +0(0) = (C, 0, 0, V ), e• +1(0) = +(0, 0, −1, 0), e• +2(0) = (−V, 0, 0, −C), e• +3(0) = (0, 1, 0, 0), defines an adapted tetrad at s = 0, which +can be extended along γ by eµ +a(s) = Rµ +ν(ωs)eν +a(0) to give +e• +0(s) = (C, 0, −V sin ωs, V cos ωs), +e• +1(s) = (0, 0, − cos ωs, − sin ωs), +e• +2(s) = (−V, 0, C sin ωs, −C cos ωs), +e• +3(s) = (0, 1, 0, 0). +(5.30) +A calculation gives +K(z) = 4C2 sin2(θ)(1 + v2 sinc2(θ)) − 2(4C2 − 3)v2 sinc2(θ) + 16V 2 sinc(2θ) + 2(4C2 − 1) +C4(1 − v2 sinc2(θ))3 +, +(5.31) +where θ = √z/(2 sinh(χ)), with series expansion +K(z) = −6 + 2 cosh2 χ − 1 +sinh2 χ +z − 185 cosh4 χ − 188 cosh2 χ + 33 +360 sinh4 χ +z2 + O(z3). +(5.32) +As κ = rω2 = ω sinh χ we read off A = ω2 cosh(2χ) = (2(rω)2 + 1)ω2 and +B = ω4(185 cosh4 χ − 188 cosh2 χ + 33) +90 += ω4(30 + 182(rω)2 + 185(rω)4) +90 +, +(5.33) +which may be substituted into (1.8) to obtain the QEI in this case. In particular, the QEI is compatible +with a constant negative energy density of +− Treg(0) = −ω4(30 + 182(rω)2 + 185(rω)4) +1440π2 +(5.34) +along the worldline. While the point-split energy density may be written down in terms of K, we do +not know of any closed-form expression for its transform. Thus the method of Sections 3 and 4 is the +only available way to compute this QEI. +Note that the QEI reduces to the inertial case if ω → 0 with r fixed – indeed, even if r = o(ω−2). +One might initially be surprised that it does not reduce in the same way when r → 0+ with ω fixed. +The explanation is that the torsion of the curve does not vanish in this limit, even though the curvature +κ does. This neatly illustrates the influence of higher curvature invariants on the QEI bound. +15 + +5.6 +Loxodromic subgroups +Finally, we study the loxodromic worldline +γ•(s) = (Ca−1 sinh(as), Ca−1 cosh(as), V ω−1 cos(ωs), V ω−1 sin(ωs)), +(5.35) +where C = cosh χ, V = sinh χ for fixed χ ̸= 0, a ̸= 0 and ω ̸= 0. This worldline undergoes both rotation +in the yz-plane at constant proper angular velocity ω and constant distance |V/ω| from the x-axis, while +undergoing uniform acceleration in the x-direction. The velocity evolves as ˙γµ(s) = La,ω +µ +ν(s)˙γν(0), +where +La,ω• +•(s) = + + + + +cosh as +sinh as +0 +0 +sinh as +cosh as +0 +0 +0 +0 +cos ωs +− sin ωs +0 +0 +sin ωs +cos ωs + + + + = exp + + + + +0 +as +0 +0 +as +0 +0 +0 +0 +0 +0 +−ωs +0 +0 +ωs +0 + + + + . +(5.36) +It can be checked that +e• +0(s) = (C cosh as, C sinh as, −V sin ωs, V cos ωs), +e• +1(s) = (Caκ−1 sinh as, Caκ−1 cosh as, −V ωκ−1 cos ωs, −V ωκ−1 sin ωs), +e• +2(s) = (−V cosh as, −V sinh as, C sin ωs, −C cos ωs), +e• +3(s) = (V ωκ−1 sinh as, V ωκ−1 cosh as, Caκ−1 cos ωs, Caκ−1 sin ωs) +(5.37) +defines an adapted tetrad for γ, obeying eµ +a(s) = La,ω +µ +ν(s)eν +a(0), while the calculation of K by computer +algebra produces +K(z) = +1 +(C2 sinhc2(ar) − V 2 sinc2(ωr))3 +� +16C2V 2 sinc(2ωr) sinhc(2ar) ++4(C2 sin2(ωr) − V 2 sinh2(ar))(V 2 sinc2(ωr) + C2 sinhc2(ar)) +−2V 2(C2 + 3V 2) sinc2(ωr) − 2C2(3C2 + V 2) sinhc2(ar) +� +, +(5.38) +where r = √z/(2 +√ +C2a2 + V 2ω2). For large real z, it is easily seen that +K(z) ∼ −4V 2(ar)2/(C4 sinhc2(ar)) → 0 +(5.39) +as z → ∞ in R. Meanwhile, the Taylor expansion about z = 0 reads +K(z) = −6 + (a2 + ω2)(C2 + V 2) +C2a2 + V 2ω2 +z − +z2 +360(C2a2 + V 2ω2)2 +� +(33a4 + 60a2ω2 + 30ω4)C4 ++ (122a4 + 250a2ω2 + 122ω4)(CV )2 + (33ω4 + 60a2ω2 + 30a4)V 4� ++ O(z3) +so A = (C2 + V 2)(a2 + ω2), while B is given by +90B = (3a4 + 30(a2 + ω2)2)C4 + (3ω4 + (30(a2 + ω2)2)V 4 + (122(a2 + ω2)2 + 6a2ω2)C2V 2 += 3(C2a2 + V 2ω2)2 + 62(a2 + ω2)2(CV )2 + 30(a2 + ω2)2(C2 + V 2)2, +(5.40) +in which the last term is 30A2. These values are easily expressed in terms of curvature invariants. +Using (2.19) and C2 − V 2 = 1 one has +κ2(τ 2 + υ2) = (a2 + ω2)2(CV )2 + (aω)2 = (V 2a2 + C2ω2)(C2a2 + V 2ω2) = κ2(V 2a2 + C2ω2), (5.41) +from which the identity +κ2 + τ 2 + υ2 = (a2 + ω2)(C2 + V 2) = A +(5.42) +follows directly, in agreement with (1.9). Using this in (5.40) together with (2.19) we see that B +takes the form (1.10). We see that the QEI is compatible with a constant negative energy density of +−Treg(0) along the worldline (2.18), where +Treg(0) = 185(a2 + ω2)2C4 − (182a4 + 370a2ω2 + 188ω4)C2 + 33ω4 + 60a2ω2 + 30a4 +1440π2 +(5.43) +16 + +and we have used V 2 = C2 − 1. Note that the QEI does not reduce to the hyperbolic QEI in the limit +χ → 0 with a and ω fixed. This is because the hypertorsion has a nonzero limit υ → sgn(a)ω, even +though the torsion vanishes and the curvature tends to a. Nonetheless, it is easily seen from (5.40) that +90B ≥ 33a4 and hence that −Treg(0) < −11a4/(480π2), so that the QEI for loxodromic worldlines can +be consistent with a more negative constant value of the energy density than the linearly accelerated +worldline with the same value of a. +6 +Summary and discussion +In this paper we have succeeded in giving an exact closed form expression (1.8)–(1.10) for the QEI +for the massless scalar field on any stationary worldline in four-dimensional Minkowski spacetime. +This was achieved by a novel method that circumvented the need to take Fourier transforms of the +point-split energy density along the worldline, and which reduced the problem to the computation of +certain Taylor coefficients of functions determined by a tetrad adapted to the worldline. In addition, +we have given explicit calculations for the six prototypical classes of stationary trajectory, obtaining +agreement with our general result (and also verifying a technical condition needed for the general +analysis). The resulting QEI bound depends only on the curvature, torsion and hypertorsion of the +worldline. We have also conducted – in Appendix C – a parallel exercise for a quantum inequality +on the Wick square. A scaling analysis (see (1.11)) shows how these bounds take a universal form on +timescales short in relation to the curvature scales, from which they then deviate at longer timescales. +In the infinite time limit, they would all allow the field to exhibit a constant negative energy density +(or zero in the inertial case). +Our results complement those of Kontou and Olum [30, 31], who computed an absolute QEI [14] in +an approximation of spacetimes where the curvature was weak. There, the worldline was taken to be +a geodesic. Our present results indicate the corrections that should enter at leading order when that +assumption is dropped. (We reemphasise that our results are exact for massless fields in Minkowski +spacetime on stationary trajectories.) +To conclude, we first mention various potential extensions of our work and then return to the +question of whether the long-time limits of the QEI are saturated by physical states of the field. +Starting with extensions, we expect that our general method would extend fairly directly to station- +ary worldlines in any even-dimensional Minkowski spacetimes, leading to closed form results in terms +of the appropriate curvature invariants. In odd dimensions, the vacuum two-point function involves +noninteger powers of the geodesic separation, which adds an extra complication. It would be interest- +ing to investigate this case in more detail. (For higher-dimensional treatment of the Unruh detector +response in higher dimensions, which would be related to the Wick QI in these cases, see [26], and for +specific calculations relating to the detailed balance definition of Unruh temperature along stationary +worldlines in 4-dimensions, see [23].) Next, massive fields typically have QEI bounds that are expo- +nentially suppressed relative to the massless ones. Here, we do not expect that our method would +easily produce closed-form results, but again, it would be worth investigating, as would the situation +for higher spin fields. +Finally, we consider the extent to which the long term average bounds can be attained. In the case +of inertial worldlines this is obvious: the long-term average value of zero is attained in the Minkowski +vacuum state. For uniformly accelerated curves it was noted in [13] that the bound (1.6) is attained +by the Rindler vacuum for the right wedge x > |t| in Minkowski spacetime. It is useful to put this +in a broader context. Adopting coordinates t = ξ sinh χ, x = ξ cosh χ, the Rindler wedge x > |t| of +Minkowski spacetime has metric ξ2 dχ2 − dξ2 − dy2 − dz2, and any curve χ �→ (aχ, 1/a, y0, z0) with +a > 0 is a curve of proper acceleration a in proper time parameterisation. +Moreover, the energy +density measured by an observer moving on a curve of constant ξ, in the thermal state of temperature +β−1 with respect to the coordinate χ, is +⟨:Tµνuµuν:⟩β = (4π2 − β2)(33β2 + 12π2) +1440π2β4ξ4 +, +(6.1) +17 + +β +ρ +−11 +0 +2π +4π +6π +8π +10π +12π +14π +Figure 1: Plot of ρ = (480π2ξ4)⟨:Tµνuµuν:⟩β on a curve of constant ξ, against β. The dotted line +corresponds to the QEI bound (1.6), which is attained as β → ∞, corresponding to the Rindler +ground state. +reducing to +⟨:Tµνuµuν:⟩∞ = − +11 +480π2ξ4 +(6.2) +for the Rindler ground state. At β = 2π, the thermal state on Rindler spacetime is precisely the +restriction of the Minkowski vacuum to the right wedge, which is why the energy density vanishes. +Because most references (e.g., [5, 6, 2]) only discuss the conformally coupled stress-energy tensor (the +‘new improved’ stress tensor) and [13] only considered the ground state without giving details, the +relevant calculations are briefly reviewed in Appendix D. On restriction to the curve ξ = 1/a we see +that all these states have constant energy density consistent with (1.6) (see Fig. 1) and that this bound +is attained by the Rindler ground state. +One should note that the Rindler ground state (and indeed all the β-KMS states other than the +special case β = 2π) is not defined on all of Minkowski, but just on the wedge x > |t|. The obvious +divergence of the stress-energy tensor as ξ → 0+ shows that the state cannot be extended as a +Hadamard state beyond the wedge. The reason they satisfy the Minkowski QEI is because this QEI +is local and covariant – see [13] for a discussion and many similar calculations, and [10] for a more +abstract viewpoint inspired by [3]. Nonetheless, it remains open as to whether equality in (1.6) can be +attained by a Hadamard state defined on all of Minkowski; our conjecture is that one can find global +Hadamard states that approximate the Rindler ground state sufficiently well that the bound (1.6) is +satisfied in a limiting sense. These issues will be addressed elsewhere. +Turning to the remaining stationary worldlines, the QEI is again consistent with a constant strictly +negative energy density and we can again ask whether the bound is attained in any sense. Letaw +and Pfautsch [35] considered the problem of quantising the field in coordinates associated with the +various stationary worldlines and seeking an appropriate ground state. For the inertial, uniformly +rotating, and semicubical parabolic worldlines, they concluded that the resulting state was precisely +the Minkowski vacuum state. This means that we have no obvious candidate state associated with the +uniformly rotating and semicubical parabolic worldlines with negative energy density. On the other +hand, the catenary (5.12) and loxodromic worldlines (2.18) both result in a Rindler vacuum state +on the x > |t| wedge, which is the causal hull of the worldline in question. One may compute the +energy density along these curves in the Rindler vacuum, using the renormalised stress energy tensor +given in Appendix D, yielding constant energy densities −(14 cosh2 χ+19)a4/(1440π2 cosh4 χ) in each +18 + +case. This value is strictly greater than −11a4/(480π2) for χ ̸= 0, which is greater than the most +negative constant energy density consistent with the QEIs in these cases (see the remarks at the end +of sections 5.3 and 5.6). Thus they are are consistent with the QEIs but do not saturate them. +It therefore remains an open and intriguing question, whether there are (sequences of) Hadamard +states that attain these QEI bounds (in a limiting sense). Resolving this question, and its analogues in +2+1 dimensions, may have relevance to proposed experiments to detect the Unruh effect using a laser +beam whose intersection with a Bose-Einstein condensate follows a uniformly rotating worldline [24]. +Acknowledgements CJF thanks Alexander Strohmaier and Valter Moretti for useful conversations +concerning the H¨ormander pseudo-topologies, and Aron Wall for posing an interesting direction for +further study. The work of JT was in part funded by an EPSRC studentship at the University of +Sheffield and a summer studentship from the University of York. We thank Elizabeth Winstanley for +a reading of the manuscript and some helpful suggestions. +A +Details on the method +We give further details on the method described in Section 4 and prove the Lemma stated there. Some +aspects are treated using techniques of microlocal analysis – we will be rather brief on those details, +referring the reader to appropriate literature, while indicating the structure of the argument. +To start, we observe that, for ǫ > 0, F(σǫ(x, 0)) can be written +F(σǫ(x, 0)) = +� +d3k +(2π)3 +e−∥k∥ǫ−ik·x +2∥k∥ +, +(A.1) +where k• = (∥k∥, k), x• = (t, x). Thus for any ϕ ∈ C∞ +0 (R4), the distribution uǫ(x) = ϕ(x)F(σǫ(x, 0)) +has Fourier transform +ˆuǫ(k′) = +� +d3k +(2π)3 +e−ǫ∥k∥ +2∥k∥ ˆϕ(k′ − k). +(A.2) +As ˆϕ decays faster than inverse polynomials and k ∈ N +, where N +/− is the bundle of future/past- +pointing null covectors, it may be shown that F(σǫ(x, 0)) converges in D′ +N +(R4) with respect to the +H¨ormander pseudo-topology [28]. It follows from this that the vacuum 2-point function G0(x, x′) is +the limit of F(σǫ(x, x′)) = F(σǫ(x − x′, 0)) in D′ +N +×N −(R4 × R4) and has wavefront set WF(G0) ⊂ +N + × N −, as is also known on general grounds because the state is Hadamard [39]. +These facts have various consequences. First, the pull-back of (any derivative operator acting on) +G0 by ϕ : (s, s′) �→ (γ(s), γ(s′)) is well-defined because the set of normals to ϕ does not intersect +WF(G0), essentially because timelike and null vectors cannot be orthogonal – see [8] for details. +Consequently the pull-back is well-defined by standard results explained in Chapter 8 of [28] and has +wavefront set contained in ϕ∗ WF(G0) ⊂ ϕ∗(N + × N −) = Γ × (−Γ), where Γ = R × (0, ∞) ⊂ T ∗R. +Moreover, ϕ∗G0 is the limit in D′ +Γ×(−Γ)(R × R) of ϕ∗Fǫ ◦ σǫ as ǫ → 0+, which justifies taking the +pull-back under the ǫ → 0+ limits in (4.6). Similar arguments apply to the convergence of F ′(σǫ(x, x′)) +and F ′′(σǫ(x, x′)) as ǫ → 0+. +Next, recall that the stationary worldline γ has velocity u = ˙γ evolving according to u(s) = +exp(sM)u(0), for M ∈ so(1, 3) with dimensions of inverse time, and that the right-handed tetrad ea(s) +obeys ea(s) = exp(sM)ea(0), with u(s) = e0(s), ˙u(s) ∈ span{e1(s)}, and ¨u(s) ∈ span{e0(s), e1(s), +e2(s)}. The Cartesian coordinates of γ(s), and components of ea(s) are evidently real analytic in s. +We extend ea to a smooth tetrad in a neighbourhood of γ in an arbitrary fashion. Recall that the +functions Ca and Da are defined, in index-free notation, by +Ca(s, s′) = η(ea(s), ea(s′)), +Da(s, s′) = η(γ(s) − γ(s′), ea(s)). +(A.3) +We now prove the lemma needed in Section 4, which we restate for convenience. +Lemma. (a) With the choice of tetrad just described, Ca(s, s′) and Da(s, s′) are translationally in- +variant, depending only on s − s′. There are entire analytic functions Ga and Ha such that +Ca(s, s′) = Ga(κ2(s − s′)2), +Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2), +(A.4) +19 + +where in the limit z → 0, +3 +� +a=0 +Ga(z) = −2 + τ 2 + υ2 +κ2 +z + (κτ)2 − (τ 2 + υ2)2 +κ4 +z2 + O(z3), +(A.5) +and +3 +� +a=0 +Ha(z) = 1 + z +12 + κ2 + 19τ 2 +360κ2 +z2 + O(z3). +(A.6) +(b) The signed square geodesic separation of points along γ obeys +σ0(γ(s), γ(s′)) = −(s − s′)2Υ(κ2(s − s′)2), +(A.7) +where Υ is entire analytic with Υ(z) = 1 + 1 +12z + +1 +360(1 − τ 2/κ2)z2 + O(z3) as z → 0. Furthermore, +for z ∈ [0, ∞), Υ(z) is real with Υ(z) ≥ 1. +Proof. (a) For inertial worldlines, ea(s) is constant and the result holds trivially with G0(z) ≡ 1, +Gi(z) ≡ 1, H0(z) ≡ −1, Hi(z) ≡ 0. From now on we may assume that κ is nonzero. +It follows from (4.1) that +Ca(s, s′) = η(exp +� +s′M +� +ea(0), exp(sM)ea(0)) = η(ea(0), exp +� +(s − s′)M +� +ea(0)), +(A.8) +so Ca depends only on s − s′. As every component of the matrix exp(sM) is analytic, and because +Ca(s, s′) = Ca(s′, s), we deduce that Ca(s, s′) = Ga(κ2(s − s′)2) for dimensionless entire analytic +functions Ga. +Next, observe that +∂ +∂s′ Da(s, s′) = −η(e0(s′), ea(s)) = −η(e0(0), exp +� +(s − s′)M +� +ea(0)). +(A.9) +Integrating with respect to s′ and using Da(s, s) = 0 we may deduce that κDa(s, s′) is a dimensionless +entire analytic function of (s − s′)κ. Again using Da(s, s) = 0 and because (A.9) gives ∂D0/∂s′|s′=s = +−1 and ∂Di/∂s′|s′=s = 0 for i = 1, 2, 3, we have +D0(s, s′) = (s − s′) +� +1 + O((κ(s − s′))2) +� +, +Di(s, s′) = κ−1O((κ(s − s′))2), +(A.10) +where we have also used the fact that D0(s, s′) = −D0(s′, s) as a consequence of (A.9). Because +Da(s, s′)Da(s′, s) is invariant under interchange of s and s′, we now have +Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2) +(A.11) +for dimensionless entire analytic functions Ha. +The Taylor series of Ga, Ha and their sums, are +computed up to second order in Appendix B. +(b) Next, we study the geodesic separation between γ(s) and γ(s′). We note that +∂ +∂sσ0(γ(s), γ(s′)) = −2D0(s, s′) +(A.12) +depends only on s − s′, so σ0(γ(s), γ(s′)) = Σ(s − s′) + f(s′) and on considering s = s′ we find that f +is constant and may be absorbed into Σ, which is also seen to be even. The first terms in its Taylor +expansion are easily found: Σ(0) = 0, while +Σ′′(s − s′) = −2η(u(s), u(s′)), +Σ(4)(s − s′) = 2η( ˙u(s), ˙u(s′)), +Σ(6)(s − s′) = −2η(¨u(s), ¨u(s′)) +(A.13) +giving +Σ′′(0) = −2, +Σ(4)(0) = −2κ2, +Σ(6)(0) = −2κ2(κ2 − τ 2) +(A.14) +using (2.9). Accordingly, we have established (A.7), the analyticity of Υ, and also the expansion +Υ(z) = 1 + z +12 + κ2 − τ 2 +360κ2 z2 + O(z3) +(A.15) +as z → 0. Finally, as γ(0) and γ(s) are connected by a smooth timelike curve, the timelike geodesic +that connects them maximises proper time. Thus −σ0(γ(s), γ(0)) ≥ s2 for all s ∈ R and consequently, +Υ(z) ≥ 1 for z ∈ [0, ∞), which concludes the proof. +20 + +Finally, we explain how the identity (4.13) may be proved. First note that +σǫ(γ(s), γ(s′)) = σ0(γ(s), γ(s′)) + 2iǫ(γ0(s) − γ0(s′)) + ǫ2 += −(s − s′)2Υ(κ2(s − s′)2) + 2iǫ(γ0(s) − γ0(s′)) + ǫ2 += −(s − s′ − iǫ)2Υ(κ2(s − s′)2) + ǫΨ(s, s′) + ǫ2Ξ(s, s′) +for smooth (indeed analytic) functions Ψ and Ξ. Let S be the difference between the distribution on +the left-hand side of (4.13) and the distribution on the right-hand side. Then, using the fact that Υ +is nonvanishing on the real axis, S takes the form +S(s, s′) = lim +ǫ→0+ +2k +� +r=1 +ǫrSr(s, s′) +σǫ(γ(s), γ(s′))k(s − s′ − iǫ)2k +(A.16) +for smooth functions Sr ∈ C∞(R2) (1 ≤ r ≤ 2k). +All that is needed now is to show that the +distributional limit +lim +ǫ→0+ +1 +σǫ(γ(s), γ(s′))k(s − s′ − iǫ)2k +(A.17) +exists, whereupon S must vanish due to the strictly positive powers of ǫ in (A.16). The required result +now follows from the sequential continuity of the distributional product with respect to the H¨ormander +pseudo-topology (Theorem 2.5.10 in [27]), and the fact that both 1/σǫ(γ(s), γ(s′)) and +1 +s − s′ − iǫ = i +� ∞ +0 +dk e−ik(s−s′−iǫ) +(A.18) +have limits as ǫ → 0+ in D′ +Γ×(−Γ)(R2), where, as before, Γ = R × (0, ∞) ⊂ ˙T ∗R. +B +Taylor series calculation +We compute the Taylor series of both Ga and Ha up to second order, using equations (4.7), (4.8) and +(4.18). Recalling that Ca(s, s′) = Ga(κ2(s − s′)2), one can expand the right hand side into a Taylor +series in s − s′ about the point s − s′ = 0 and then differentiate to yield +− 1 +2κ2 +∂2Ca +∂s∂s′ = G′ +a(0) + 3κ2(s − s′)2G′′ +a(0) + O((s − s′)4) +(B.1) +1 +12κ4 +∂4Ca +∂2s∂2s′ = G′′ +a(0) + O((s − s′)2) +(B.2) +as s − s′ → 0. Differentiating equation (4.7) and setting s = s′ = 0, one easily finds +G′ +a(0) = −η( ˙ea(0), ˙ea(0)) +2κ2 +, +G′′ +a(0) = η(¨ea(0), ¨ea(0)) +12κ4 +(B.3) +by equating powers of s − s′. The derivatives of the ea can be read off from the generalized Frenet- +Serret equations (2.5) and its derivatives (2.8), allowing us to express G′ +a(0) and G′′ +a(0) in terms of +curvature invariants. +An easy computation shows that +G′ +a(0) = 1 +2η0a + κ2 − τ 2 +2κ2 +η1a − τ 2 + υ2 +2κ2 +η2a − υ2 +2κ2 η3a +(B.4) +and +G′′ +a(0) = κ2 − τ 2 +12κ2 η0a + τ 2υ2 + (κ2 − τ 2)2 +12κ4 +η1a − κ2τ 2 − (τ 2 + υ2)2 +12κ4 +η2a + υ2 τ 2 + υ2 +12κ4 η3a, +(B.5) +21 + +where η(ea(0), eb(0)) = ηab by orthogonality of the tetrad field. Reconstructing Ga using a Taylor +series therefore yields +Ga(z) = ηaa + +1 +2κ2 z +� +η0aκ2 − η1a(τ 2 − κ2) − η2a(υ2 + τ 2) − η3aυ2� ++ +z2 +24κ4 +� +η0aκ2(κ2 − τ 2) + η1a(τ 2υ2 + (κ2 − τ 2)2) − η2a(κ2τ 2 − (τ 2 + υ2)2) + η3aυ2(τ 2 + υ2) +� ++ O(z3). +(B.6) +Summing, we obtain +3 +� +a=0 +Ga(z) = −2 + τ 2 + υ2 +κ2 +z + (κτ)2 − (τ 2 + υ2)2 +κ4 +z2 + O(z3) +(B.7) +as z → 0. +Applying exactly the same methodology to Ha, one writes Ea(s, s′) = Da(s, s′)Da(s′, s) so that +Ea(s, s′) = −(s − s′)2Ha(κ2(s − s′)2) += −(s − s′)2Ha(0) − κ2(s − s′)4H′ +a(0) − 1 +2κ4(s − s′)6H′′ +a(0) + O((s − s′)8). +(B.8) +Differentiation yields +∂2Ea +∂s∂s′ = 2Ha(0) + 12κ2(s − s′)2H′ +a(0) + 15κ4(s − s′)4H′′ +a(0) + O((s − s′)6) +(B.9) +∂4Ea +∂2s∂2s′ = −24κ2H′ +a(0) − 180κ4(s − s′)2H′′ +a(0) + O((s − s′)4) +(B.10) +∂6Ea +∂3s∂3s′ = 360κ4H′′ +a(0) + O((s − s′)4), +(B.11) +from which Ha(0), H′ +a(0) and H′′ +a(0) can be obtained differentiating equation (4.18) using Leibniz’ +rule and subsequently setting s = s′ = 0. It is easily verifiable that this yields +Ha(0) = [η(˙γ(0), ea(0))]2 = [η(e0(0), ea(0))]2 , +(B.12) +H′ +a(0) = − 1 +4κ2 [η(¨γ(0), ea(0))]2 + +1 +3κ2 η(˙γ(0), ea(0))η(...γ (0), ea(0)), +(B.13) +H′′ +a(0) = +1 +18κ4 [η(...γ (0), ea(0))]2 − +1 +12κ4 η(¨γ(0), ea(0))η(γ(4)(0), ea(0)) ++ +1 +30κ4 η(˙γ(0), ea(0))η(γ(5)(0), ea(0)), +(B.14) +and after some straightforward computation, +Ha(0) = η0a +(B.15) +H′ +a(0) = 1 +3η0a + 1 +4η1a +(B.16) +H′′ +a(0) = (η0a)2 +� 1 +18 + κ2 − τ 2 +30κ2 +� +− κ2 − τ 2 +12κ2 (η1a)2 + +τ 2 +18κ2 (η2a)2 += η0a +� 1 +18 + κ2 − τ 2 +30κ2 +� ++ κ2 − τ 2 +12κ2 η1a − +τ 2 +18κ2 η2a +(B.17) +using the fact that (η0a)2 = η0a and (ηia)2 = −ηia for i = 1, 2, 3, as can be explicitly seen in the +calculation of H′′ +a(0). Reconstructing Ha using a Taylor series, one obtains +Ha(z) = η0a + 1 +12z (4η0a + 3η1a) ++ +1 +360κ2 z2 � +η0a(10κ2 + 6(κ2 − τ 2)) + 15η1a(κ2 − τ 2) − 10η2aτ 2� ++ O(z3), +(B.18) +and summing, +3 +� +a=0 +Ha(z) = 1 + z +12 + κ2 + 19τ 2 +360κ2 +z2 + O(z3). +(B.19) +22 + +C +Wick square +In this Appendix we show how a quantum inequality for the Wick square can be obtained along +stationary trajectories. This is a simpler calculation than the one used for the energy density and we +shall be relatively brief. +Recall that the general QEI involves a (sum of) pull-backs of a suitable differential operator acting +on the two-point function, +T(s, s′) = ⟨Qφ(γ(s))Qφ(γ(s′))⟩ω0 = ((Q ⊗ Q)G0)(γ(s), γ(s′)). +(C.1) +For a quantum inequality on the Wick square, the operator Q can be simply identified as the identity, +so T(s, s′) can be written in this case as +T(s, s′) = G0(γ(s), γ(s′)). +(C.2) +Using the results of Section 4 and in particular, equation (4.13), the two-point function can be neatly +expressed as +T(s, s′) = lim +ǫ→0+ +1 +4π2σǫ(γ(s), γ(s′)) = − lim +ǫ→0+ +1 +4π2(s − s′ − iǫ)2 +� +Υ +� +κ2(s − s′)2��−1 . +(C.3) +As Υ(κ2s2) ≥ 1 for s ∈ R by the Lemma, the entire function Υ(z) is nonvanishing on the real axis, +and Υ(z)−1 is therefore analytic in a neighbourhood of the real axis. +Using (4.12) we may write +Υ(z)−1 = 1 + zJ(z), where J is also analytic in a neighbourhood of the real axis, with J(0) = −1/12. +Because 0 < 1 + zJ(z) ≤ 1 for z ≥ 0, we may deduce that 0 ≤ −J(z) < 1/z for z > 0. +We can now split the pulled back two-point function into its singular and regular parts as T(s, s′) = +Tsing(s − s′) + Treg(s − s′), where +Tsing(s) = − 1 +4π2 lim +ǫ→0+ +1 +(s − iǫ)2 , +(C.4) +and +Treg(s) = −J(κ2s2) +4π2 +lim +ǫ→0+ +κ2s2 +(s − iǫ)2 = −κ2J(κ2s2) +4π2 +, +(C.5) +with Treg(0) = κ2/(48π2). Here we used the identity limǫ→0+ x2/(x−iǫ)2 = limǫ→0+(x−iǫ)2/(x−iǫ)2 = +1 of distributional limits, because g(z) = z2 is entire, while f(z) = z−2 is analytic in the open lower +half-plane Z ⊂ C and obeys supz∈Z|f(z)(Im z)2| = 1 (see the argument below equation (5.8)). +Observing that the two-point function given above is translationally invariant, we can use the +bound given by (1.2) and (1.3) and thus write +� +ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ − +� ∞ +−∞ +dα|ˆg(α)|2Qeven(α) +(C.6) +where +Qeven(α) = +1 +2π2 +�� 0 +−∞ +ˆT(u) du + +� α +0 +ˆTodd(u) du +� +. +(C.7) +The Fourier transform of Tsing is easily shown to be ˆTsing(u) = +u +2πΘ(u). Again, Treg is smooth, real +and even on R, decaying like O(s−2) as |s| → ∞ because of the decay of J. Evidently Treg does not +contribute to ˆTodd as Treg is absolutely integrable and has a well defined, continuous, real and even +Fourier transform. In this case, Tsing is actually universal; the information relating to the specific +worldline is encoded in Treg, as can also be seen below in Eq. (C.10). Clearly, ˆTsing does not contribute +to the first term in (C.7) and, recalling that Treg is even, the odd part of ˆT is +ˆTodd(u) = u +4π, +(C.8) +and so Qeven is given in the form +Qeven(α) = +1 +2π2 +�� 0 +−∞ +du ˆTreg(u) + 1 +4π +� α +0 +du u +� += +1 +16π3 α2 + Treg(0) +2π +. +(C.9) +23 + +In direct analogy to the analysis of the energy density, the evenness of Treg and the Fourier inversion +formula have been used. Inserting this into (C.6) gives the QI bound +� +ds|g(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − 1 +8π2 +� ∞ +−∞ +ds +� +|g′(s)|2 + C|g(s)|2� +. +(C.10) +where C = 8π2Treg(0) = κ2/6. +Considering the scaling behaviour, using the same test function gλ(s) = λ−1/2g(λ/s) as in the case +for the QEI (1.11), one can easily verify that +� +ds|gλ(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − ∥g′∥2 +8π2λ2 − κ2∥g∥2 +48π2 , +(C.11) +where again ∥g∥2 denotes the L2-norm of the function g. Taking the limit λ → ∞ yields the following +formula, +lim inf +λ−→∞ +� ∞ +−∞ +ds|gλ(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − κ2 +48π2 +(C.12) +when considering the functions g such that ∥g∥2 = 1. Physically, since one can interpret 12⟨:φ2:⟩ as +the square of a local temperature [4], states with negative expected Wick square are regarded as being +locally out of equilibrium. The above bound therefore quantifies the extent to which the thermal +interpretation may fail uniformly along these worldlines, in terms of their proper acceleration. This +raises an intriguing question as to whether there are states that would saturate this bound – something +quite relevant to the Unruh experiments discussed in Section 6. +In relation to the Unruh effect, a study of the detailed balance temperature obtained from the +excitation of an Unruh-DeWitt detector carried along stationary worldlines can be found in [23]. Here +the quantum field is assumed to be in the vacuum state, and the temperature depends not only on +the curvature invariants but also on the energy gap of the detector. Although this is a different focus +from our results, which concern averages of the Wick square in arbitrary Hadamard states, there are +technical similarities, because the pulled back vacuum Wightman function plays a key role in both. +It would be interesting to understand whether some of the methods described here can be used to +corroborate the numerical results of [23]. +D +Computation of the renormalised stress-tensor for thermal and +ground states on Rindler spacetime +The Feynman propagator for a thermal state at inverse temperature β of the massless scalar field +in Minkowski spacetime was given by Dowker [6] and the Wightman functions (including for higher +spin) can be found in [38]. Adopting coordinates t = ξ sinh χ, x = ξ cosh χ, the Rindler wedge x > |t| +of Minkowski spacetime has metric ξ2 dχ2 − dξ2 − dy2 − dz2, and any curve χ �→ (aχ, 1/a, y0, z0) +with a > 0 is a curve of proper acceleration a in proper time parameterisation. Given two points +x = (χ, ξ, y, z) and x′ = (χ′, ξ′, y′, z′), write +α(x, x′) = cosh−1 +�ξ2 + (ξ′)2 + (y − y′)2 + (z − z′)2 +2ξξ′ +� +, +(D.1) +whereupon the Wightman function Gβ(x, x′) = ⟨φ(x)φ(x′)⟩β for the temperature β−1 KMS state with +respect to the coordinate χ is +Gβ(x, x′) = +1 +4πβξξ′ sinh α(x, x′) +� +sinh(2πα(x, x′)/β) +cosh(2πα(x, x′)/β) − cosh(2π(χ − χ′ − iǫ)/β) +� +. +(D.2) +The β = 2π case coincides with the restriction of the Minkowski vacuum state to the wedge, while the +zero temperature limit has Wightman function +G∞(x, x′) = − +α(x, x′) +4π2ξξ′ sinh α(x, x′)(α(x, x′)2 − (χ − χ′ − iǫ)2). +(D.3) +24 + +To obtain the renormalised (minimally coupled) stress-energy tensor, we first apply suitable derivatives +to Gβ − G2π and take the limit x′ → x, obtaining +⟨:(∇µφ)(x)(∇νφ)(x):⟩β = +4π2 − β2 +1440π2β4ξ4 +� +(16π2 + 14β2)ˆuµˆuν + 30β2ˆaµˆaν − (4π2 + 11β2)ηµν +� +, +(D.4) +where, at spacetime position x, ˆuµ = ξ−1(∂χ)µ is the 4-velocity of the curve through x with constant +ξ, y and z, and ˆaµ = (∂ξ)µ is the unit spacelike vector parallel to the 4-acceleration of this curve. +Consequently, +⟨:Tµν:⟩β = +4π2 − β2 +1440π2β4ξ4 +� +(16π2 + 14β2)ˆuµˆuν + 30β2ˆaµˆaν − (4π2 − 19β2)ηµν +� +(D.5) +and the result for Rindler ground state is obtained by taking β → ∞, giving +⟨:Tµν:⟩∞ = − +1 +1440π2ξ4 (14ˆuµˆuν + 30ˆaµˆaν + 19ηµν) . +(D.6) +Computing the energy density on curves of constant ξ yields (6.1). +Bibliography +[1] Born, M.: +Die Theorie des starren Elektrons in der Kinematik des Relativit¨atsprinzips. +Annalen +der +Physik +335(11), +1–56 +(1909). +DOI +10.1002/andp.19093351102. +URL +https://doi.org/10.1002/andp.19093351102 +[2] Brown, M.R., Ottewill, A.C., Page, D.N.: Conformally invariant quantum field theory in static +Einstein space-times. 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URL http://www.jstor.org/stable/20488646 +27 + diff --git a/49AzT4oBgHgl3EQfu_0d/content/tmp_files/load_file.txt b/49AzT4oBgHgl3EQfu_0d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dfb66d916d54b46e7c0655b818c186065936df55 --- /dev/null +++ b/49AzT4oBgHgl3EQfu_0d/content/tmp_files/load_file.txt @@ -0,0 +1,1304 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf,len=1303 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='01698v1 [hep-th] 4 Jan 2023 Quantum Energy Inequalities along stationary worldlines Christopher J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Fewster 1,∗ and Jacob Thompson 1,2,† 1 Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 2 School of Mathematics and Statistics, The University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' January 5, 2023 Abstract Quantum energy inequalities (QEIs) are lower bounds on the averaged energy density of a quantum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' They have been proved for various field theories in general curved spacetimes but the explicit lower bound is not easily calculated in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In this paper we study QEIs for the massless minimally coupled scalar field in four-dimensional Minkowski spacetime along stationary worldlines – curves whose velocity evolves under a 1-parameter Lorentz subgroup – and find closed expressions for the QEI bound, in terms of curvature invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Our general results are illustrated by specific computations for the six protoypical stationary worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' When the averaging period is taken to infinity, the QEI bound is consistent with a constant energy density along the worldline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For inertial and uniformly linearly accelerated worldlines, this constant value is attained by the Minkowski and Rindler vacuums respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It is an open question as to whether the bounds for other stationary worldlines are attained by other states of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 1 Introduction Even if a classical field theory obeys local energy conditions, such as positivity of energy density, the corresponding quantum field theory (QFT) will fail to do so, as a result of a general theorem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In fact, it is typical that the expectation value of energy density at any given point can be made arbitrarily negative by a suitable choice of the quantum state [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Nonetheless, in many QFT models, local averages of the expected energy density are bounded below by Quantum Energy Inequalities (QEIs), independent of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Starting with results of Ford and Roman [17, 18, 19] QEIs have been derived for a variety of quantum fields in flat and curved spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' References and discussion may be found in the recent reviews [11, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For example, consider the real scalar field of mass m ≥ 0 in any globally hyperbolic spacetime (M, g), recalling that global hyperbolicity demands only that the spacetime possesses a global Cauchy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Let γ(s) be any smooth timelike curve, parameterised by proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It was shown in [8] that the energy density of the quantum field along γ obeys the QEI � ∞ −∞ ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − � ∞ 0 dα π � g ⊗ gT(−α, α) > −∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) which holds for all real-valued compactly supported smooth test functions g, and all Hadamard states ω of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here, the hat denotes a Fourier transform, defined according to the convention ˆg(α) = � ∞ −∞ ds eiαsg(s), and we employ units where ℏ = c = 1, which will be in force throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' On the left-hand side, the normal ordering is conducted with respect to an arbitrary Hadamard reference state ω0, whose two-point function is used to construct the distribution T(s, s′) that appears on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Recall also that the Hadamard states form a large class of physically reasonable states, determined by their short-distance structure [29, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The two most important features of the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) ∗chris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='fewster@york.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='uk †jthompson16@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='uk 1 are that the right-hand side is completely independent of the state ω, and that the bound is finite – which is proved using the microlocal properties of Hadamard states uncovered by Radzikowski [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Discussion of QEIs for other QFTs, including non-free models, may be found in [11, 32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' see [21] for a very recent development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Although the lower bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) is explicit and rigorous, it is not easy to compute in closed form except in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' To the best of our knowledge this has only been achieved when T exhibits translational invariance T(s + r, s′ + r) = T(s, s′) which occurs, for instance, when (M, g) is a stationary spacetime, γ is a timelike Killing orbit and ω0 is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Translational invariance allows us to write, with an abuse of notation, T(s, s′) = T(s − s′), from which one easily finds that the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) simplifies to � ∞ −∞ ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − � ∞ −∞ dα|ˆg(α)|2Q(α), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) where Q(α) = 1 2π2 � α −∞ du ˆT(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) is also valid for complex-valued g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Taking the massless free field as an example, averaging along an inertial worldline in Minkowski space and using the Minkowski vacuum as the reference state ω0, this results in Q(α) = α4Θ(α)/(16π3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using the evenness of |ˆg|2 together with Parseval’s theorem then yields � ∞ −∞ ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 1 16π2 � ∞ −∞ ds|g′′(s)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) Similar expressions are known for massive fields and in Minkowski spacetime of general dimension [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' for some curved spacetime examples see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Another explicit example arises where γ is a uniformly linearly accelerated worldline in four-dimensional Minkowski spacetime with proper acceleration a, in which case the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) becomes [13] � ∞ −∞ ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 1 16π2 � ∞ −∞ ds � |g′′(s)|2 + 2a2|g′(s)|2 + 11 30a4|g(s)|2 � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) and is again valid for all Hadamard states ω and complex-valued test functions g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using such expressions the scaling behaviour of the bound is easily understood and phenomena such as ‘quantum interest’ may be explored [20, 16, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For example, let gλ(s) = λ−1/2g(s/λ), where g is normalised so that � ∞ −∞ ds|g(s)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) implies lim inf λ−→∞ � ∞ −∞ ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 11a4 480π2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) reducing to the Averaged Weak Energy Condition (AWEC) lim inf λ−→∞ � ∞ −∞ dτ|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) in the limit a → 0, which can also be obtained directly from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' An interesting observation is that the lower bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) is exactly the constant energy density of the Rindler vacuum state along the accelerated worldline, while the lower bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) is the energy density of the Minkowski vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As closed form expressions for QEI bounds are relatively few in number, it is of interest to find others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The purpose of this paper is to present a calculation of the QEI bound for a massless scalar field along any stationary worldline in 4-dimensional Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' By a stationary worldline, we mean any timelike curve γ(s), parameterised by proper time s, whose velocity vector evolves under a 1-parameter subgroup of the Lorentz group: ˙γ(s) = exp(sM) ˙γ(0) for some fixed M ∈ so(1, 3) and future-pointing unit timelike ˙γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Stationary worldlines have a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Kottler [33], Synge [41] and Letaw [34] (see also [36]) all obtained them as the solutions to four-dimensional Frenet-Serret equations subject to constancy of the curvature invariants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' the name ‘stationary worldlines’ is due to Letaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The three curvature invariants 2 are the curvature, which measures the proper acceleration, and the torsion and hypertorsion, which specify its proper angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' More details are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Stationary worldlines are equivalently described as the orbits of timelike Killing vector fields in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' There are also overlaps with the theory of rigid motions in special relativity that goes back to Born [1] and Herglotz [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' in particular, any rotational rigid motion is the flow of a timelike Killing vector by the Herglotz–Noether theorem, although the same theorem allows any C2 timelike curve to be a flow line of an irrotational rigid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' See [22] for discussion and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' By a Poincar´e transformation, any stationary worldline can be reduced to one of six prototypes: the inertial, uniformly linearly accelerated, and uniformly rotating worldlines are all familiar, while the three remaining ones have spatial projections corresponding to a semicubical parabola, a catenary or a helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We will give more detail as we discuss each case separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The main result of this paper is that the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) along any stationary worldline in Minkowski spacetime may be given explicitly as � ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 1 16π2 � ∞ −∞ ds � |g′′(s)|2 + 2A|g′(s)|2 + B|g(s)|2� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) where A and B are expressed in terms of the curvature κ, torsion τ, and hypertorsion υ as A = κ2 + τ 2 + υ2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and B = 1 90 � 3κ4 + 62κ2τ 2 + 30(κ2 + τ 2 + υ2)2� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) and the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) holds for all Hadamard states ω and all smooth compactly supported test functions g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' To interpret the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8), it is useful to consider its scaling behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As before, we take a test function gλ which is just a scaled version of the test function g, namely gλ(s) = λ−1/2g(s/λ), so the support width of gλ is proportional to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Observing that g(k) λ (s) = λ−k−1/2g(k)(s/λ), we find � ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − ∥g′′∥2 16π2λ4 − A∥g′∥2 8π2λ2 − Treg(0)∥g∥2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) where the norms are those of L2(R), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', ∥g∥2 = � ∞ −∞ ds|g(s)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here we have written Treg(0) = B/(16π2) for reasons that will become clear later – see, for example, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) and the ar- guments presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For sampling times shorter than the curvature scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', λ ≪ min{κ−1, τ −1, υ−1}, the leading term dominates, reflecting the fact that any worldline is approxi- mately inertial on short enough timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' At intermediate and long timescales relative to curvature scales, the bound will receive corrections from, and eventually be dominated by the last two terms in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11), showing that the QEI is sensitive to the curvature invariants of the worldline γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In the limit λ → +∞, and with g normalised so that ∥g∥ = 1, we obtain the remarkably simple formula lim inf λ−→∞ � ∞ −∞ ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −Treg(0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) which bounds the average energy density along the entire trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In particular, the QEI is con- sistent with the existence of a constant renormalised energy density −Treg(0) along γ, and this is the most negative value that any constant energy density could take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' An intriguing question is whether or not this value is attained by some Hadamard state, or a sequence of Hadamard states in a limiting sense, which we will address in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The derivation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) requires a number of innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Although the point-split energy density can be obtained easily enough for any given stationary worldline, its Fourier transform does not have a closed form – as far as we know – for three of the six prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In Section 3, we develop a new method for computing the QEI bound for massless fields in four-dimensions that avoids the use of Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The result is that the QEI will take the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) provided that the point-split energy density takes the form T(s, s′) = lim ǫ→0+ � 3 2π2(s − s′ − iǫ)4 − A 4π2(s − s′ − iǫ)2 � + Treg(s − s′), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) 3 where the regular part Treg must satisfy various conditions, whereupon the coefficient B is given by B = 16π2Treg(0) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In Section 4, we apply these ideas to stationary worldlines, resulting in formulae for the point-split energy density in terms of functions easily computed from the Lorentzian distance between two points on the curve and a tetrad that is adapted to it, in a manner we describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Most of the required conditions on Treg follow directly from this analysis, and the values A and B are identified in terms of Taylor coefficients of these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Appendix A gives more detail on our methods, while in Appendix B the relevant Taylor coefficients are evaluated in terms of curvature invariants thus establishing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In Section 5, we work through each prototype in turn, providing explicit formulae for the point-split energy density that allow the remaining technical condition to be verified, and also as a check on our Taylor series calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In three cases, (inertial worldlines, linearly accelerated worldlines and the semicubical parabola), a closed form may be found for ˆT, and we can also check our calculations by using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Finally, in Section 6, we discuss the physical significance of our results and some open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Two further appendices contain additional computations: Appendix C computes a quantum inequality for the Wick square along stationary worldlines following the same general method of the main text, while Appendix D records the calculation of the minimally coupled stress-energy tensor in the Rindler vacuum and Rindler thermal states, which is needed for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 2 Stationary worldlines Throughout this paper we work on 4-dimensional Minkowski spacetime, with metric η = dt2 − dx2 − dy2 − dz2, and we employ the inertial coordinates (t, x, y, z) except where otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A stationary worldline is any smooth curve γ : R → R4, whose velocity vector ˙γ is a future-pointing unit timelike vector evolving under a 1-parameter subgroup of the Lorentz group SO(1, 3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', ˙γµ(s) = exp(sM)µ ν ˙γν(0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) where M is any fixed element of so(1, 3) (which requires precisely that Mµν is antisymmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As every component of exp(sM) is analytic in s, it follows that the Cartesian components of ˙γ(s) and, in- tegrating, the Cartesian coordinates of γ(s) are also s-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' An equivalent definition of a stationary worldline is that γ is an orbit of a future-pointing timelike Killing vector field ξµ(x) = Mµ ν(xν − γ(0)ν) + ˙γµ(0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) which is necessarily timelike in a neighbourhood of γ and future-pointing unit vector on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Finally, stationary worldlines can also be described as the solutions to the Frenet-Serret equations with constant curvatures [33, 41, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here, the curvature invariants of a general timelike curve γ(s), parameterised by proper time, are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Suppose a right-handed tetrad eµ a has been chosen along γ so that γ(k+1)(s) ∈ span{e0(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' , ek(s)} (0 ≤ k ≤ 3), and ˙γ(s) = e0(s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) in which case we say that eµ a is adapted to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' If the tetrad also satisfies e1(s)µ¨γ(s)µ ≤ 0, e2(s)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='γ (s)µ ≤ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) then it will be called a Frenet–Serret tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' If the tetrad is defined by ea(s) = exp(sM)ea(0), then it is adapted (respectively, Frenet–Serret) if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) holds at s = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) hold at s = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Explicit formulae resulting from a Gram–Schmidt procedure are given in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Expanding the derivatives of the tetrad vectors in terms of the tetrad, one obtains the generalized Frenet–Serret equations ˙eµ a = K b a eµ b , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) where Kab is antisymmetric and tridiagonal (due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus it takes the form K••(s) = \uf8eb \uf8ec \uf8ec \uf8ed 0 −κ(s) 0 0 κ(s) 0 −τ(s) 0 0 τ(s) 0 −υ(s) 0 0 υ(s) 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) 4 which defines the curvature κ, torsion τ and hypertorsion υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here, and elsewhere in this paper, bullets are used to indicate tensorial type, when displaying tensorial components in vector or matrix form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Explicitly, one has κ = e0µ ˙eµ 1 = −e1µ ˙eµ 0, τ = e1µ ˙eµ 2 = −e2µ ˙eµ 1, υ = e2µ ˙eµ 3 = −e3µ ˙eµ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) The choices made when specifying the Frenet–Serret tetrad ensure that κ and τ are nonnegative, while υ can take any real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As the curvature invariants are constant along stationary worldlines, it is easy to compute higher derivatives of the tetrad, dk dsk eµ a = (Kk) b a eµ b , (Kk) b a = K c1 a K c2 c1 · · Kck−1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) For example, the first three derivatives of the velocity u = ˙γ may be computed as ˙uµ = ˙eµ 0 = κeµ 1, ¨uµ = κ2eµ 0 + κτeµ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='u µ = κ(κ2 − τ 2)eµ 1 + κτυeµ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) It is also possible to give a general formula for γ(s) in terms of M, γ(0) and ˙γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As M• is antisymmetric with respect to η, there is a unique decomposition ˙γ(0)µ = Mµ νvν + kµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) where Mµ νkν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' One then has γ(s)µ = exp(sM)µ νvν + skµ + γ(0)µ − vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) Any stationary worldline γ may be related to one of six basic types by a proper orthochronous Poincar´e transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Note that γ(s) is determined by the initial position, γ(0) ∈ R4, the initial four-velocity ˙γ(0) and the element M ∈ so(1, 3) that fixes the evolution ˙γ(s) = exp(sM) ˙γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Under a Poincar´e transformation x �→ Λx + w, γ is mapped to ˜γ(s) = Λγ(s) + w, whose velocity evolves according to the 1-parameter Lorentz subgroup exp � sΛMΛ−1� and which is therefore also a stationary worldline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As the Lorentz transformation maps a Frenet–Serret tetrad for γ to a Frenet–Serret tetrad for ˜γ, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) that the curvature invariants of ˜γ are identical to those of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using the classification of conjugacy classes in so(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 3) [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' we may choose Λ in such a way that ˜ M = ΛMΛ−1 is one of five possible types: (a) the zero element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' generating the trivial subgroup of SO(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (b) a generator of boosts in the tx-plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' corresponding to a hyperbolic subgroup of SO(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (c) a generator of rotations in the yz-plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' corresponding to an elliptic subgroup of SO(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (d) a generator of a null rotation that fixes the null vector ∂t + ∂x but acts nontrivially on all other null vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' corresponding to a parabolic subgroup of SO(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (e) the sum of a generator of boosts in the tx-plane and a generator of rotations in the yz plane, corresponding to a loxodromic subgroup of SO(1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In each case, Lorentz transformations that commute with the 1-parameter subgroup in question can be used to arrange that ˙˜γ(0) takes a convenient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Taking these possibilities in turn: in case (a), all Lorentz transformations commute with the trivial subgroup, so we may without loss assume that ˜γ(s) = (s, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In case (b), the subgroup of boosts parallel to the x-axis commutes with itself and the subgroup of rotations in the yz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus, we may arrange that ˙˜γ(0) = cosh χ∂t + sinh χ∂y for some χ ∈ R,1 leading to two subcases: χ = 0, in which case (after possible translation) ˜γ(s) = (a−1 sinh as, a−1 cosh as, 0, 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) is a uniformly linearly accelerated worldline with a ̸= 0, or χ ̸= 0, in which case (up to translations) ˜γ(s) = (a−1 cosh χ sinh as, a−1 cosh χ cosh as, −s sinh χ, 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) is a catenary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The curvature invariants (in either subcase) are κ = |a| cosh χ and τ = |a sinh χ|, while the hypertorsion is υ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For convenience, the curvature invariants for all six prototypes are tabulated in Table 1, in agreement with [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 1We could even arrange that χ ≥ 0, but it is convenient not to insist on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 5 Inertial Linear Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Catenary Parabolic Elliptic Loxodromic κ = τ = υ = 0 κ > 0 κ > τ > 0 κ = τ > 0 τ > κ > 0 κ, τ > 0 τ = υ = 0 υ = 0 υ = 0 υ = 0 υ ̸= 0 κ 0 |a| |a| cosh χ |a| rω2 √ C2a2 + V 2ω2 τ 0 0 |a sinh χ| |a| |ω| � 1 + (rω)2 (a2 + ω2)C|V |/κ υ 0 0 0 0 0 aω/κ Table 1: Curvature invariants for the stationary worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In case (c), the 1-parameter parabolic subgroup takes the form P • (s) = \uf8eb \uf8ec \uf8ec \uf8ed 1 + (as)2/2 −(as)2/2 0 as (as)2/2 1 − (as)2/2 0 as 0 0 1 0 as −as 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = exp \uf8eb \uf8ec \uf8ec \uf8ed 0 0 0 as 0 0 0 as 0 0 0 0 as −as 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) for some constant nonzero a ∈ R, and commutes with Lorentz transformations of the form Λ• = \uf8eb \uf8ec \uf8ec \uf8ed 1 + r2/2 −r2/2 r cos θ r sin θ r2/2 1 − r2/2 r cos θ r sin θ r cos θ −r cos θ 1 0 r sin θ −r sin θ 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15) which can be used to bring the initial velocity into the form ˙˜γ(0) = cosh χ∂t+sinh χ∂x for some χ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Conjugating P • (s) with a boost in the tx-plane results in P • (λs) for some λ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' in other words effectively rescaling a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Therefore there is no loss of generality in assuming that the initial 4-velocity is ˙˜γ(0) = ∂t, in which case the worldline (up to translation) is the semicubical parabola, ˜γ(s) = � s + 1 6a2s3, 1 6a2s3, 0, 1 2as2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16) Next, the elliptic subgroup in case (d) commutes with boosts in the tx-plane and rotations in the yz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Accordingly, we may arrange the initial velocity to be ˙˜γ(0) = cosh χ∂t + sinh χ∂z for some χ ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' the special case χ = 0 corresponds to inertial motion and may be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Up to a translation, this results in the uniformly rotating worldline ˜γ•(s) = (s cosh χ, 0, r cos ωs, r sin ωs) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='17) where the radius r > 0 and proper angular velocity ω ̸= 0 are related to the initial rapidity by rω = sinh χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The proper acceleration is κ = rω2, while the torsion is τ = |ω| � 1 + (rω)2 and the hypertorsion vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Lastly, in case (e), the loxodromic subgroup is generated by a linear combination of a tx-boost generator and a yz-rotation generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As it commutes with tx-boosts and yz-rotations, we may assume without loss that the initial velocity is ˙˜γ(0) = cosh χ∂t + sinh χ∂z for χ ∈ R \\ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' the possibility χ = 0 corresponds to a hyperbolic worldline and is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Up to a translation, this results in the worldline γ•(s) = (Ca−1 sinh(as), Ca−1 cosh(as), V ω−1 cos(ωs), V ω−1 sin(ωs)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) where C = cosh χ and V = sinh χ, which undergoes both rotation in the yz-plane at constant proper angular velocity ω ̸= 0 and constant distance |V/ω| from the x-axis, while undergoing uniform acceler- ation in the x-direction controlled by a ̸= 0 (the cases where one or both of a or ω vanish are already covered under (a), (b) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The curvature invariants for this worldline are κ = � C2a2 + V 2ω2, τ = (a2 + ω2)C|V |/κ, υ = aω/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) 6 3 Reformulation of the QEI bound We study the massless minimally coupled scalar field in 4-dimensional Minkowski spacetime, with field equation □φ = ηµν∇µ∇νφ = 0 and classical stress-energy tensor Tµν = (∇µφ)∇νφ − 1 2ηµνηαβ(∇αφ)∇βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) Consider an observer following a timelike curve γ, parameterised by proper time, with 4-velocity uµ = ˙γµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This observer sees energy density Tµνuµuν = 1 2 3 � a=0 (eµ a∇µφ)2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) where eµ a (0 ≤ a ≤ 3) is a tetrad defined around γ with eµ 0|γ = uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In quantum field theory, the stress-energy tensor requires renormalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Let G(x, x′) = ⟨φ(x)φ(x′)⟩ω (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) be the Wightman function of the field in a state ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The Wick square has expectation value ⟨:φ2(x):⟩ω = (G − G0)(x, x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) where G0(x, x′) = lim ǫ→0+ −1 4π2((t − t′ − iǫ)2 − ∥x − x′∥2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) is the Wightman function of the Poincar´e invariant vacuum ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This expression makes sense if (like ω0) ω is a Hadamard state [29, 37], because the difference G−G0 is then a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Similarly, the renormalised stress-energy tensor has expectation value ⟨:Tµν(x):⟩ω = Dµν(x) − 1 2ηµνηαβDαβ(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) where Dµν(x) = [[(∇ ⊗ ∇)(G − G0)]]µν (x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) and the double square brackets denote a coincidence limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Although the classical energy density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) is everywhere nonnegative, the quantised energy density may assume negative expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The QEIs provide lower bounds on averaged expectation values, for which a prototype is a lower bound on the following expression � ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) where Q is a partial differential operator with smooth real coefficients and g ∈ C∞ 0 (R) is a smooth real-valued test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In the case where Q is the identity, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) is an averaged Wick square, while by considering a sum of similar terms for Qa = 2−1/2eµ a∇µ for 0 ≤ a ≤ 3, we can bound averages of the energy density along γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A lower bound on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) was established in [8] – in fact the bound applies to general timelike curves in arbitrary globally hyperbolic spacetimes for massive as well as massless fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In our case it asserts that � ∞ −∞ ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ − � ∞ 0 dα π ÷ g ⊗ gT (−α, α) > −∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) holds for all real-valued compactly supported smooth test functions g, and all Hadamard states ω, where T(s, s′) = ⟨Qφ(γ(s))Qφ(γ(s′))⟩ω0 = ((Q ⊗ Q)G0)(γ(s), γ(s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) Here, the vacuum two-point function enters because normal ordering is performed relative to the vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' the general results of [8] also allow for any Hadamard state to be used as the reference state for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' At a more formal level, T is the pull-back of the distribution (Q ⊗ Q)G0 by the map 7 (s, s′) �→ (γ(s), γ(s′)), and its existence is owed to the special properties of the Hadamard condition and the fact that γ is timelike – see [8] for full details and rigorous proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As already mentioned, a QEI for the energy density involves a sum of such bounds, leading to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) with T(s, s′) = 1 2 3 � a=0 ((∇ea ⊗ ∇ea)G0)(γ(s), γ(s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) While it is usually not hard to obtain the distribution T for a given timelike curve in Minkowski spacetime, assuming that G0 is given, it is not usually possible to find the Fourier transform required to compute the QEI bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The situation is somewhat simplified if T(s, s′) is translationally invariant, in which case one has the bound given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This can be taken a little further, on observing that |ˆg(α)|2 is even, so only the even part Qeven(α) = 1 2(Q(α) + Q(−α)) of Q contributes to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2), resulting in the bound � ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ − � ∞ −∞ dα|ˆg(α)|2Qeven(α), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) which is the final form of our prototypical quantum inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A convenient expression for Qeven may be found by manipulating equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) in the following way: Qeven(α) = 1 4π2 �� α −∞ ˆT(u) du + � −α −∞ ˆT(u) du � = 1 4π2 � 2 � 0 −∞ ˆT(u) du + � α 0 ˆT(u) du − � α 0 ˆT(−u) du � = 1 2π2 �� 0 −∞ ˆT(u) du + � α 0 ˆTodd(u) du � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) where ˆTodd(u) = 1 2( ˆT(u) − ˆT(−u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In the above calculation, ˆT is assumed to be continuous, as is the case for the examples we will study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Evaluating Qeven from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) requires several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Computing T is a tedious but straightforward calculation best handled using computer algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In the simplest cases, the transform may be eval- uated in closed form, which (as will be seen later) is the case for the inertial, uniformly accelerated and semicubical parabola worldlines, but is not possible (to our knowledge) in the case of the other stationary worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' However, this obstacle can be circumvented, as we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using the Minkowski vacuum as the reference state, we will show in Section 4 that the point-split energy density along a stationary worldline may be written in the form T(s, s′) = Tsing(s − s′) + Treg(s − s′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) where Tsing is given by the distributional limit Tsing(s) = lim ǫ→0+ � 3 2π2(s − iǫ)4 − A 4π2(s − iǫ)2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15) for some constant A (the sign is chosen for later convenience) and Treg is smooth, real and even, and decaying as O(s−2) as |s| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In particular, Treg is absolutely integrable and has a well-defined Fourier transform that is continuous, real and even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Therefore it does not contribute to ˆTodd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Turning to Tsing, its leading singularity is universal, essentially because all stationary worldlines resemble inertial worldlines on sufficiently short timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The specific coefficient is fixed by the Hadamard form and the definition of the energy density along the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Meanwhile the coefficient A carries information about the specific curve at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The Fourier transform of Tsing, in our convention, is ˆTsing(u) = 1 2π(u3 + Au)Θ(u), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16) 8 where Θ is the Heaviside distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Evidently Tsing does not contribute to the first term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13), while the odd part of ˆT is ˆTodd(u) = 1 4π (u3 + Au), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='17) recalling that ˆTreg is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We now have Qeven in the form Qeven(α) = 1 2π2 �� 0 −∞ du ˆTreg(u) + 1 4π � α 0 du(u3 + Au) � = 1 32π3 (α4 + 2Aα2) + Treg(0) 2π , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) where we have again used the evenness of ˆTreg and the Fourier inversion formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Inserting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) and using Parseval’s theorem gives the QEI bound � ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 1 16π2 � ∞ −∞ ds � |g′′(s)|2 + 2A|g′(s)|2 + B|g(s)|2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) where B = 16π2Treg(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The upshot of this analysis is a direct route to the QEI once the point-split expression T is obtained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' all that is needed is to isolate the appropriate values of A and Treg(0), avoiding the need to compute ˆT explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This apparent royal road is made possible because of the special structure of the Minkowski vacuum two-point function for the massless scalar field in four dimensions – closely related to Huygens’ principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A similar analysis for a QI on the Wick square can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 4 Computation of the point-split energy density In this section we establish that the point-split energy density along stationary worldlines obeys equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15), and also that Tsing and Treg have the properties mentioned above, with one exception that will be treated by examining the six prototypical cases in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Let γ be any stationary worldline with ˙γ(s) = exp(sM)˙γ(0) and ˙γ(0) a future-pointing unit timelike vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Suppose that ea(s) = exp(sM)ea(0) (0 ≤ a ≤ 3) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) is an adapted frame on γ satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In general there may be many possible adapted tetrads of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' However, if ˜ea(s) is any other then it is related to ea(s) by a rigid rotation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', ˜e0(s) = e0(0) and ˜ei(0) = R j i ej(0) (summing j over 1, 2, 3), where δimR j i R n m = δmn, det R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This must be true for some R at s = 0, and extends to all s as both tetrads evolve under exp(sM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Next, recall that the vacuum 2-point function may be given as a distributional limit G0(x, x′) = lim ǫ→0+ F(σǫ(x, x′)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) where F(z) = 1/(4π2z) and σǫ(x, x′) = −ηµν(x − x′ − iǫ∂t)µ(x − x′ − iǫ∂t)ν (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) is the regulated signed squared geodesic separation of x and x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As usual, we have identified Minkowski spacetime with its tangent spaces at all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Distributional derivatives may be taken under the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2), giving 1 2(∇µ ⊗ 1)G0(x, x′) = − lim ǫ→0+ F ′(σǫ(x, x′))(x − x′ − iǫ∂t)µ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) and 1 2(∇µ ⊗ ∇ν)G0(x, x′) = lim ǫ→0+ � F ′(σǫ(x, x′))ηµν − 2F ′′(σǫ(x, x′))(x − x′ − iǫ∂t)µ(x − x′ − iǫ∂t)ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) 9 Contracting with ea(x)µea(x′)ν (without summing on a) and pulling back to the worldline, we find 1 2((∇ea ⊗ ∇ea)G0)(γ(s), γ(s′)) = lim ǫ→0+ F ′(σǫ(γ(s), γ(s′)))Ca(s, s′) + lim ǫ→0+ 2F ′′(σǫ(γ(s), γ(s′)))Da(s, s′)Da(s′, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) (note the order of variables in the last two factors in the second term) where Ca(s, s′) = ηµνeµ a(s)eν a(s′), Da(s, s′) = (γ(s) − γ(s′))µeµ a(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) Under a change of frame from ea to ˜ea as described above, one has ˜C0 = C0, ˜D0 = D0, while ˜Di = R j i Dj and ˜Ci(s, s′) = R j i R k i ηµνeµ j (s)eν k(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' By orthogonality, this implies that �3 i=1 ˜Ci(s, s′) = �3 i=1 Ci(s, s′) and �3 i=1 ˜Da(s, s′) ˜Da(s′, s) = �3 i=1 Da(s, s′)Da(s′, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In Appendix A, we give some further details to justify the above distributional manipulations and prove the following result, where κ, τ and υ are the curvature invariants of γ as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (a) With the choice of tetrad just described, Ca(s, s′) and Da(s, s′) are translationally in- variant, depending only on s − s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' There are entire analytic functions Ga and Ha such that Ca(s, s′) = Ga(κ2(s − s′)2), Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) where, in the limit z → 0, 3 � a=0 Ga(z) = −2 + τ 2 + υ2 κ2 z + (κτ)2 − (τ 2 + υ2)2 κ4 z2 + O(z3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and 3 � a=0 Ha(z) = 1 + z 12 + κ2 + 19τ 2 360κ2 z2 + O(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) (b) The signed square geodesic separation of points along γ obeys σ0(γ(s), γ(s′)) = −(s − s′)2Υ(κ2(s − s′)2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) where Υ is entire analytic with Υ(z) = 1 + 1 12z + κ2 − τ 2 360κ2 z2 + O(z3) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Furthermore, for z ∈ [0, ∞), Υ(z) is real with Υ(z) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The Lemma now allows us to compute the point-split energy density by evaluating the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) and summing over a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We use the fact (explained in Appendix A) that lim ǫ→0+ (s − s′)2j σǫ(γ(s), γ(s′))k = (−1)k Υ(κ2(s − s′)2)k lim ǫ→0+ 1 (s − s′ − iǫ)2(k−j) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) where the limits are taken in the sense of distributions, as is the multiplication by a smooth prefactor on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' If j = k, the distributional limit on the right-hand side may be replaced by unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In particular, when calculating T(s, s′) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6), the factor (s − s′)2 in Da(s, s′)Da(s′, s) cancels a factor of (s − s′ − iǫ)2 in the denominator, as ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The upshot is that T(s, s′) = − 1 4π2 lim ǫ→0+ K(κ2(s − s′)2) (s − s′ − iǫ)4 , where K(z) = 3 � a=0 �Ga(z) Υ(z)2 − 4Ha(z) Υ(z)3 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) is a meromorphic function that is analytic in a neighbourhood of the positive real axis (on which Υ is bounded away from zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 10 The singular part is easily isolated by splitting off the first two terms of the Taylor series for K from the remainder, which carries a leading factor of (s − s′)4 that cancels the denominator in the limit ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Similarly, the O(z) part of the Taylor series partly cancels the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus, T(s, s′) = Tsing(s − s′) + Treg(s − s′) with Tsing(s) = − 1 4π2 lim ǫ→0+ K(0) (s − iǫ)4 − 1 4π2 lim ǫ→0+ κ2K′(0) (s − iǫ)2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15) and Treg(s) = − κ4 4π2 J((κs)2), where J(z) = K(z) − K(0) − K′(0)z z2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16) is analytic on a neighbourhood of the positive real axis, so J((κs)2) is smooth for s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using the Lemma, we may read off that K(0) = −6, thus establishing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15), with A = κ2K′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Meanwhile, Treg(s) is smooth, even, and real-valued for s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Provided that K(z) = O(z) as z → ∞ on the real axis, we find that Treg(s) = O(s−2) as s → ∞, which completes the properties needed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Furthermore, Treg(0) = −J(0)κ4 4π2 = −K′′(0)κ4 8π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='17) Note that if we had used the tetrad ˜e instead, the function K would be unchanged, owing to the remarks before the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus the QEIs obtained from ˜ea and ea are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' These results now provide a calculational method to determine the QEI along stationary worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Starting from the generator M ∈ so(1, 3) and the initial 4-velocity u(0), choose a tetrad as described at the start of this section, and compute the proper acceleration κ = � −η(Mu(0), Mu(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The translational invariance of Ca and Da means that they can be calculated conveniently as Ca(s, s′) = ηµνeµ a(s − s′)eν a(0), Da(s, s′) = −(γ(s′ − s) − γ(0))µeµ a(0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) from which Ga and Ha are easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The function Υ is computed directly from the Lorentz interval between γ(0) and γ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Then construct K(z) according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) and check that K(z) = O(z) as z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Then the QEI along γ is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19), with constants A = κ2K′(0), B = −2κ4K′′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) The constants A and B can be computed from the first few terms of the Taylor expansions of � a Ga, � a Ha and Υ, given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' After a calculation, one finds K(z) = −6 + z κ2 + τ 2 + υ2 κ2 − z2 1 360κ4 � 3κ4 + 62κ2τ 2 + 30(κ2 + τ 2 + υ2)2� + O(z3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='20) from which the formulae (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) follow immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Nonetheless, this is perhaps not the most illuminating calculation and also does not provide a check that K(z) = O(z) for large real z, which was assumed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For these reasons, and their own intrinsic interest, we will also provide explicit calculations in Section 5 that together cover all possible stationary worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 5 QEIs for the prototypical stationary worldlines We have now established the general QEI for stationary worldlines in Minkowski spacetime, assuming a technical condition on the growth of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In this section, we reduce the problem of computing the QEI for a general stationary worldline to six prototypical cases, which will be treated in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' These calculations follow the method of Section 4 and result in explicit formulae for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In this way it is seen that the growth condition holds in all cases and we also obtain a check on the Taylor series calculations in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We have already discussed the fact that any stationary worldline may be brought into one of the six standard forms by a Poincar´e transformation, without changing the curvature invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Owing to Poincar´e invariance of the vacuum state, and because Poincar´e invariance maps an adapted tetrad of the form ea(s) = exp(sM)ea(0) along the original curve to a tetrad with the same properties on 11 the new one, the point-split energy density obtained by the method of Section 4 is exactly the same for the two worldlines, which accordingly share the same QEI bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The QEIs for the prototypical stationary worldlines are now given in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Most of the computa- tions that follow were conducted using the computer algebra system Maple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1 Trivial subgroup: inertial motion For the inertial worldline γ(s) = (s, 0, 0, 0), we employ the adapted tetrad ∂t, ∂x, ∂y, ∂z, which is constant along γ, leading immediately to the relations C0(s, s′) = 1, Ci(s, s′) = −1 for i = 1, 2, 3, while D0(s, s′) = s−s′, Di(s, s′) = 0 for all s, s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It follows that G0 = H0 ≡ 1, Gi ≡ −1, Hi ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Furthermore, Υ ≡ 1 because σ0(γ(s), γ(s′)) = −(s − s′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Hence K ≡ −6 and one finds T(s, s′) = Tsing(s − s′) where Tsing(s) = lim ǫ→0+ 3 2π2(s − iǫ)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) Consequently Treg vanishes identically, and we may read off immediately that A = B = 0, reproducing QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) by substituting into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19), and in agreement with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Of course these results are easily obtained by direct differentiation of the two-point function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' our purpose here is to show how they follow from formulae in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Alternatively, we may proceed by taking the Fourier transform ˆTsing(u) = u3Θ(u)/(2π), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) from which we obtain Q(α) = α4Θ(α)/(16π3) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3), leading to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) as discussed in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2 Hyperbolic subgroups: linear acceleration We consider a uniformly linearly accelerated worldline γ(s) = (a−1 sinh as, a−1 cosh as, 0, 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) whose velocity evolves under the 1-parameter group of tx-boosts ˙γµ(s) = Hµ ν(s)˙γν(0), where H• (s) = \uf8eb \uf8ec \uf8ec \uf8ed cosh as sinh as 0 0 sinh as cosh as 0 0 0 0 1 0 0 0 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = exp \uf8eb \uf8ec \uf8ec \uf8ed 0 as 0 0 as 0 0 0 0 0 0 0 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) and 0 ̸= a ∈ R is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Noting that the initial velocity and its first two derivatives are ˙γ(0) = ∂t, ¨γ(0) = a∂x, ¨γ(0) = a2∂t, we obtain an adapted tetrad by choosing the tetrad ∂t, ∂x, ∂y, ∂z at s = 0, and applying the prescription eµ a(s) = Hµ ν(s)eν a(0) to find e0(s) = cosh as∂t + sinh as∂x, e1(s) = sinh as∂t + cosh as∂x, e2(s) = ∂y, e3(s) = ∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) Straightforward calculation, following the method of Section 4, gives K(a2s2) = − 3(as)4 8 sinh4(as/2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) and hence T(s, s′) = lim ǫ→0+ 3a4(s − s′)4 cosech4(a(s − s′)/2) 32π2(s − s′ − iǫ)4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) which may be simplified to T(s, s′) = lim ǫ→0+ 3a4 32π2 cosech4 � a(s − s′ − iǫ)/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) Here, we have used the general fact that limǫ→0+ g(x)f(x − iǫ) = limǫ→0+ g(x − iǫ)f(x − iǫ) in the sense of distributions, when f is analytic in a strip Z = {x − iy : x ∈ R, 0 < y < y0} ⊂ C with supz∈Z |f(z)(Im z)N| < ∞ for some N > 0 and g is analytic on Z and continuous on Z ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 12 As the function K(z) evidently decays rapidly as z → ∞ on the real axis, the method of Section 4 allows us to read off the QEI from the derivatives of K(z) at z = 0 according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using K(z) = 3z2 8 sinh4(√z/2) = −6 + z − 11 120z2 + O(z3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) we find A = a2 and B = 11a4/30, in agreement with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) using the invariants from Table 1 and reproducing the result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In that reference, the point-split energy density (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) was found by a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Writing T(s, s′) = T(s − s′), the Fourier transform yields ˆT(u) = u3 − a2u 2π(1 − e−2πu/a) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) and by using the last expression in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13), a calculation gives Qeven(α) = 1 32π3 � α4 + 2a2α2 + 11 30a4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) from which (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) follows on inserting the above expression into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) and using Parseval’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3 Hyperbolic subgroups: the catenary Now consider the catenary γ(s) = (a−1 cosh χ sinh as, a−1 cosh χ cosh as, −s sinh χ, 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) for constant a ̸= 0, with initial velocity ˙γ•(0) = (cosh χ, 0, − sinh χ, 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) and second and third derivatives ¨γ•(0) = (0, a cosh χ, 0, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='γ •(0) = (a2 cosh χ, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) The velocity evolves under the hyperbolic subgroup (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Writing C = cosh χ and V = sinh χ, the tetrad e• 0(s) = (C cosh as, C sinh as, −V, 0), e• 1(s) = (sinh as, cosh as, 0, 0), e• 2(s) = (−V cosh as, −V sinh as, C, 0), e• 3(s) = (0, 0, 0, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15) is adapted to γ with eµ a(s) = Hµ ν(s)eν a(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A calculation results in the formula K(z) = −4V 2(sinhc2(r) + v2) sinh2(r) + 2(4C2 − 1) sinhc2(r) − 16V 2 sinhc(2r) + 2v2(4C2 − 3) C4(sinhc2(r) − v2)3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16) where v = tanh χ, r = √z/(2 cosh χ) and sinhc(x) = sinh(x)/x is the hyperbolic version of the sinc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Note that we need not specify a branch for the square root as it always appears in the argument of an even entire function, and also that K(z) → 0 as z → ∞ in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The series expansion is K(z) = −6 + 2C2 − 1 C2 z − 185C4 − 182C2 + 30 360C4 z2 + O(z3) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='17) and as κ = aC we may read off A = a2(2C2 − 1) = a2 cosh 2χ and B = (185C4 − 182C2 + 30)a4/90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It is straightforward that these values agree with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) using the curvature invariants for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In particular, the resulting QEI is compatible with a constant negative energy density of − Treg(0) = −(185 cosh4 χ − 182 cosh2 χ + 30)a4 1440π2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) along the worldline (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As would be expected, the QEI for linear acceleration is obtained in the limit χ → 0, but for χ ̸= 0, we have −Treg(0) < −11a4/480π2, and the QEI bound is consistent with a strictly more negative constant energy density than is the case for the linearly accelerated worldline with the same value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4 Parabolic subgroups: the semicubical parabola We now consider the semicubical parabola γ(s) = � s + 1 6a2s3, 1 6a2s3, 0, 1 2as2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) for constant a ̸= 0, whose velocity evolves as ˙γµ(s) = P µ ν(s)˙γ(0) with ˙γ(0) = ∂t, where P µ ν was defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' From the initial derivatives ˙γ(0) = ∂t, ¨γ(0) = a∂z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='γ (0) = a2(∂t + ∂x) one sees that the initial tetrad e0(0) = ∂t, e1(0) = ∂z, e2(0) = ∂x, e3(0) = ∂y determines an adapted tetrad e• 0(s) = � 1 + 1 2(as)2, 1 2(as)2, 0, as � , e• 1(s) = (as, as, 0, 1) , e• 2(s) = � − 1 2(as)2, 1 − 1 2(as)2, 0, −as � , e• 3(s) = (0, 0, 1, 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='20) at general proper time s obeying eµ a(s) = P µ ν(s)eν a(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Straightforward calculation now gives K(z) = −6 − z/2 + 5z2/36 (1 + z/12)3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='21) with K(z) = −6 + 2z − 37 72z2 + O(z3) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='22) as z → 0 and K(z) = O(z−1) for z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus, the point-split energy density is T(s, s′) = lim ǫ→0+ 3 − a2(s − s′)2/4 + 5a4(s − s′)4/72 π2(s − s′ − iǫ)4(1 + a2(s − s′)2/12)3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) gives A = 2a2 and B = 37a4/18, in agreement with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus the QEI along a semicubical parabola is � ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 1 16π2 � ∞ −∞ ds � |g′′(s)|2 + 4a2|g′(s)|2 + 37 18a4|g(s)|2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='24) for any Hadamard state ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The long-time scaling limit of the above QEI is then lim inf λ−→∞ � ∞ −∞ ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 37 288π2 a4, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='25) where as usual we choose g with unit L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The QEI is therefore compatible with a constant negative energy density −37a4/(288π2) along the semicubical parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As one would expect, the QEI reduces to the inertial case as a → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In fact the QEI (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='24) can also be obtained by a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Writing T(s, s′) = T(s − s′), the Fourier transform may be computed by contour methods as ˆT(u) = 1 2π �� 2u2 √ 12 + 7|u| 8 a + 15 8 √ 12a2 � ae−|u| √ 12/a + � u3 + 2ua2� Θ(u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='26) The calculation is considerably simplified if one first replaces powers of s − s′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='23) by powers of s − s′ − iǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' To find Qeven(α), we note that ˆTodd(u) = (u3 + 2ua2)/(4π), and also that the integral of ˆT over (−∞, 0] may be evaluated in terms of Γ-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' After manipulation, the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) gives Qeven(α) = 1 4π3 � 0 −∞ � 2u2 √ 12 + 7|u| 8 a + 15 8 √ 12a2 � ae−|u| √ 12/a du + 1 8π3 � α 0 � u3 + 2ua2� du = 1 32π3 α4 + a2 8π3 α2 + 37a4 576π3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='27) Inserting this expression in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) and using Parseval’s theorem we reproduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5 Elliptic subgroups: uniform rotation Next, consider the uniformly rotating worldline γ(s) = (s cosh χ, 0, r cos ωs, r sin ωs) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='28) where the radius r > 0 and proper angular velocity ω ̸= 0 together fix the rapidity χ = sinh−1(rω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In this case, the velocity evolves under rotations in the yz-plane as ˙γµ(s) = Rµ ν(s)˙γν(0), where R• (s) = \uf8eb \uf8ec \uf8ec \uf8ed 1 0 0 0 0 1 0 0 0 0 cos ωs − sin ωs 0 0 sin ωs cos ωs \uf8f6 \uf8f7 \uf8f7 \uf8f8 = exp \uf8eb \uf8ec \uf8ec \uf8ed 0 0 0 0 0 0 0 0 0 0 0 −ωs 0 0 ωs 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='29) Meanwhile, the initial velocity and its first two derivatives are ˙γ•(0) = (C, 0, 0, V ) ¨γ•(0) = (0, 0, −V ω, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='γ •(0) = � 0, 0, 0, −V ω2� , where we have written C = cosh χ and V = rω = sinh χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Then e• 0(0) = (C, 0, 0, V ), e• 1(0) = (0, 0, −1, 0), e• 2(0) = (−V, 0, 0, −C), e• 3(0) = (0, 1, 0, 0), defines an adapted tetrad at s = 0, which can be extended along γ by eµ a(s) = Rµ ν(ωs)eν a(0) to give e• 0(s) = (C, 0, −V sin ωs, V cos ωs), e• 1(s) = (0, 0, − cos ωs, − sin ωs), e• 2(s) = (−V, 0, C sin ωs, −C cos ωs), e• 3(s) = (0, 1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='30) A calculation gives K(z) = 4C2 sin2(θ)(1 + v2 sinc2(θ)) − 2(4C2 − 3)v2 sinc2(θ) + 16V 2 sinc(2θ) + 2(4C2 − 1) C4(1 − v2 sinc2(θ))3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='31) where θ = √z/(2 sinh(χ)), with series expansion K(z) = −6 + 2 cosh2 χ − 1 sinh2 χ z − 185 cosh4 χ − 188 cosh2 χ + 33 360 sinh4 χ z2 + O(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='32) As κ = rω2 = ω sinh χ we read off A = ω2 cosh(2χ) = (2(rω)2 + 1)ω2 and B = ω4(185 cosh4 χ − 188 cosh2 χ + 33) 90 = ω4(30 + 182(rω)2 + 185(rω)4) 90 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='33) which may be substituted into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) to obtain the QEI in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In particular, the QEI is compatible with a constant negative energy density of − Treg(0) = −ω4(30 + 182(rω)2 + 185(rω)4) 1440π2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='34) along the worldline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' While the point-split energy density may be written down in terms of K, we do not know of any closed-form expression for its transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus the method of Sections 3 and 4 is the only available way to compute this QEI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Note that the QEI reduces to the inertial case if ω → 0 with r fixed – indeed, even if r = o(ω−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' One might initially be surprised that it does not reduce in the same way when r → 0+ with ω fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The explanation is that the torsion of the curve does not vanish in this limit, even though the curvature κ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This neatly illustrates the influence of higher curvature invariants on the QEI bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6 Loxodromic subgroups Finally, we study the loxodromic worldline γ•(s) = (Ca−1 sinh(as), Ca−1 cosh(as), V ω−1 cos(ωs), V ω−1 sin(ωs)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='35) where C = cosh χ, V = sinh χ for fixed χ ̸= 0, a ̸= 0 and ω ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This worldline undergoes both rotation in the yz-plane at constant proper angular velocity ω and constant distance |V/ω| from the x-axis, while undergoing uniform acceleration in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The velocity evolves as ˙γµ(s) = La,ω µ ν(s)˙γν(0), where La,ω• (s) = \uf8eb \uf8ec \uf8ec \uf8ed cosh as sinh as 0 0 sinh as cosh as 0 0 0 0 cos ωs − sin ωs 0 0 sin ωs cos ωs \uf8f6 \uf8f7 \uf8f7 \uf8f8 = exp \uf8eb \uf8ec \uf8ec \uf8ed 0 as 0 0 as 0 0 0 0 0 0 −ωs 0 0 ωs 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='36) It can be checked that e• 0(s) = (C cosh as, C sinh as, −V sin ωs, V cos ωs), e• 1(s) = (Caκ−1 sinh as, Caκ−1 cosh as, −V ωκ−1 cos ωs, −V ωκ−1 sin ωs), e• 2(s) = (−V cosh as, −V sinh as, C sin ωs, −C cos ωs), e• 3(s) = (V ωκ−1 sinh as, V ωκ−1 cosh as, Caκ−1 cos ωs, Caκ−1 sin ωs) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='37) defines an adapted tetrad for γ, obeying eµ a(s) = La,ω µ ν(s)eν a(0), while the calculation of K by computer algebra produces K(z) = 1 (C2 sinhc2(ar) − V 2 sinc2(ωr))3 � 16C2V 2 sinc(2ωr) sinhc(2ar) +4(C2 sin2(ωr) − V 2 sinh2(ar))(V 2 sinc2(ωr) + C2 sinhc2(ar)) −2V 2(C2 + 3V 2) sinc2(ωr) − 2C2(3C2 + V 2) sinhc2(ar) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='38) where r = √z/(2 √ C2a2 + V 2ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For large real z, it is easily seen that K(z) ∼ −4V 2(ar)2/(C4 sinhc2(ar)) → 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='39) as z → ∞ in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Meanwhile, the Taylor expansion about z = 0 reads K(z) = −6 + (a2 + ω2)(C2 + V 2) C2a2 + V 2ω2 z − z2 360(C2a2 + V 2ω2)2 � (33a4 + 60a2ω2 + 30ω4)C4 + (122a4 + 250a2ω2 + 122ω4)(CV )2 + (33ω4 + 60a2ω2 + 30a4)V 4� + O(z3) so A = (C2 + V 2)(a2 + ω2), while B is given by 90B = (3a4 + 30(a2 + ω2)2)C4 + (3ω4 + (30(a2 + ω2)2)V 4 + (122(a2 + ω2)2 + 6a2ω2)C2V 2 = 3(C2a2 + V 2ω2)2 + 62(a2 + ω2)2(CV )2 + 30(a2 + ω2)2(C2 + V 2)2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='40) in which the last term is 30A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' These values are easily expressed in terms of curvature invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) and C2 − V 2 = 1 one has κ2(τ 2 + υ2) = (a2 + ω2)2(CV )2 + (aω)2 = (V 2a2 + C2ω2)(C2a2 + V 2ω2) = κ2(V 2a2 + C2ω2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='41) from which the identity κ2 + τ 2 + υ2 = (a2 + ω2)(C2 + V 2) = A (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='42) follows directly, in agreement with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using this in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='40) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) we see that B takes the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We see that the QEI is compatible with a constant negative energy density of −Treg(0) along the worldline (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18), where Treg(0) = 185(a2 + ω2)2C4 − (182a4 + 370a2ω2 + 188ω4)C2 + 33ω4 + 60a2ω2 + 30a4 1440π2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='43) 16 and we have used V 2 = C2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Note that the QEI does not reduce to the hyperbolic QEI in the limit χ → 0 with a and ω fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This is because the hypertorsion has a nonzero limit υ → sgn(a)ω, even though the torsion vanishes and the curvature tends to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Nonetheless, it is easily seen from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='40) that 90B ≥ 33a4 and hence that −Treg(0) < −11a4/(480π2), so that the QEI for loxodromic worldlines can be consistent with a more negative constant value of the energy density than the linearly accelerated worldline with the same value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 6 Summary and discussion In this paper we have succeeded in giving an exact closed form expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) for the QEI for the massless scalar field on any stationary worldline in four-dimensional Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This was achieved by a novel method that circumvented the need to take Fourier transforms of the point-split energy density along the worldline, and which reduced the problem to the computation of certain Taylor coefficients of functions determined by a tetrad adapted to the worldline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In addition, we have given explicit calculations for the six prototypical classes of stationary trajectory, obtaining agreement with our general result (and also verifying a technical condition needed for the general analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The resulting QEI bound depends only on the curvature, torsion and hypertorsion of the worldline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We have also conducted – in Appendix C – a parallel exercise for a quantum inequality on the Wick square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A scaling analysis (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11)) shows how these bounds take a universal form on timescales short in relation to the curvature scales, from which they then deviate at longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In the infinite time limit, they would all allow the field to exhibit a constant negative energy density (or zero in the inertial case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Our results complement those of Kontou and Olum [30, 31], who computed an absolute QEI [14] in an approximation of spacetimes where the curvature was weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' There, the worldline was taken to be a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Our present results indicate the corrections that should enter at leading order when that assumption is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (We reemphasise that our results are exact for massless fields in Minkowski spacetime on stationary trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=') To conclude, we first mention various potential extensions of our work and then return to the question of whether the long-time limits of the QEI are saturated by physical states of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Starting with extensions, we expect that our general method would extend fairly directly to station- ary worldlines in any even-dimensional Minkowski spacetimes, leading to closed form results in terms of the appropriate curvature invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In odd dimensions, the vacuum two-point function involves noninteger powers of the geodesic separation, which adds an extra complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It would be interest- ing to investigate this case in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (For higher-dimensional treatment of the Unruh detector response in higher dimensions, which would be related to the Wick QI in these cases, see [26], and for specific calculations relating to the detailed balance definition of Unruh temperature along stationary worldlines in 4-dimensions, see [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=') Next, massive fields typically have QEI bounds that are expo- nentially suppressed relative to the massless ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here, we do not expect that our method would easily produce closed-form results, but again, it would be worth investigating, as would the situation for higher spin fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Finally, we consider the extent to which the long term average bounds can be attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In the case of inertial worldlines this is obvious: the long-term average value of zero is attained in the Minkowski vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For uniformly accelerated curves it was noted in [13] that the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) is attained by the Rindler vacuum for the right wedge x > |t| in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It is useful to put this in a broader context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Adopting coordinates t = ξ sinh χ, x = ξ cosh χ, the Rindler wedge x > |t| of Minkowski spacetime has metric ξ2 dχ2 − dξ2 − dy2 − dz2, and any curve χ �→ (aχ, 1/a, y0, z0) with a > 0 is a curve of proper acceleration a in proper time parameterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Moreover, the energy density measured by an observer moving on a curve of constant ξ, in the thermal state of temperature β−1 with respect to the coordinate χ, is ⟨:Tµνuµuν:⟩β = (4π2 − β2)(33β2 + 12π2) 1440π2β4ξ4 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) 17 β ρ −11 0 2π 4π 6π 8π 10π 12π 14π Figure 1: Plot of ρ = (480π2ξ4)⟨:Tµνuµuν:⟩β on a curve of constant ξ, against β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The dotted line corresponds to the QEI bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6), which is attained as β → ∞, corresponding to the Rindler ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' reducing to ⟨:Tµνuµuν:⟩∞ = − 11 480π2ξ4 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) for the Rindler ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' At β = 2π, the thermal state on Rindler spacetime is precisely the restriction of the Minkowski vacuum to the right wedge, which is why the energy density vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Because most references (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', [5, 6, 2]) only discuss the conformally coupled stress-energy tensor (the ‘new improved’ stress tensor) and [13] only considered the ground state without giving details, the relevant calculations are briefly reviewed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' On restriction to the curve ξ = 1/a we see that all these states have constant energy density consistent with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 1) and that this bound is attained by the Rindler ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' One should note that the Rindler ground state (and indeed all the β-KMS states other than the special case β = 2π) is not defined on all of Minkowski, but just on the wedge x > |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The obvious divergence of the stress-energy tensor as ξ → 0+ shows that the state cannot be extended as a Hadamard state beyond the wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The reason they satisfy the Minkowski QEI is because this QEI is local and covariant – see [13] for a discussion and many similar calculations, and [10] for a more abstract viewpoint inspired by [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Nonetheless, it remains open as to whether equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) can be attained by a Hadamard state defined on all of Minkowski;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' our conjecture is that one can find global Hadamard states that approximate the Rindler ground state sufficiently well that the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) is satisfied in a limiting sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' These issues will be addressed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Turning to the remaining stationary worldlines, the QEI is again consistent with a constant strictly negative energy density and we can again ask whether the bound is attained in any sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Letaw and Pfautsch [35] considered the problem of quantising the field in coordinates associated with the various stationary worldlines and seeking an appropriate ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' For the inertial, uniformly rotating, and semicubical parabolic worldlines, they concluded that the resulting state was precisely the Minkowski vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This means that we have no obvious candidate state associated with the uniformly rotating and semicubical parabolic worldlines with negative energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' On the other hand, the catenary (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) and loxodromic worldlines (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) both result in a Rindler vacuum state on the x > |t| wedge, which is the causal hull of the worldline in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' One may compute the energy density along these curves in the Rindler vacuum, using the renormalised stress energy tensor given in Appendix D, yielding constant energy densities −(14 cosh2 χ+19)a4/(1440π2 cosh4 χ) in each 18 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This value is strictly greater than −11a4/(480π2) for χ ̸= 0, which is greater than the most negative constant energy density consistent with the QEIs in these cases (see the remarks at the end of sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus they are are consistent with the QEIs but do not saturate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It therefore remains an open and intriguing question, whether there are (sequences of) Hadamard states that attain these QEI bounds (in a limiting sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Resolving this question, and its analogues in 2+1 dimensions, may have relevance to proposed experiments to detect the Unruh effect using a laser beam whose intersection with a Bose-Einstein condensate follows a uniformly rotating worldline [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Acknowledgements CJF thanks Alexander Strohmaier and Valter Moretti for useful conversations concerning the H¨ormander pseudo-topologies, and Aron Wall for posing an interesting direction for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The work of JT was in part funded by an EPSRC studentship at the University of Sheffield and a summer studentship from the University of York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We thank Elizabeth Winstanley for a reading of the manuscript and some helpful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A Details on the method We give further details on the method described in Section 4 and prove the Lemma stated there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Some aspects are treated using techniques of microlocal analysis – we will be rather brief on those details, referring the reader to appropriate literature, while indicating the structure of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' To start, we observe that, for ǫ > 0, F(σǫ(x, 0)) can be written F(σǫ(x, 0)) = � d3k (2π)3 e−∥k∥ǫ−ik·x 2∥k∥ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) where k• = (∥k∥, k), x• = (t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus for any ϕ ∈ C∞ 0 (R4), the distribution uǫ(x) = ϕ(x)F(σǫ(x, 0)) has Fourier transform ˆuǫ(k′) = � d3k (2π)3 e−ǫ∥k∥ 2∥k∥ ˆϕ(k′ − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) As ˆϕ decays faster than inverse polynomials and k ∈ N +, where N +/− is the bundle of future/past- pointing null covectors, it may be shown that F(σǫ(x, 0)) converges in D′ N +(R4) with respect to the H¨ormander pseudo-topology [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It follows from this that the vacuum 2-point function G0(x, x′) is the limit of F(σǫ(x, x′)) = F(σǫ(x − x′, 0)) in D′ N +×N −(R4 × R4) and has wavefront set WF(G0) ⊂ N + × N −, as is also known on general grounds because the state is Hadamard [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' These facts have various consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' First, the pull-back of (any derivative operator acting on) G0 by ϕ : (s, s′) �→ (γ(s), γ(s′)) is well-defined because the set of normals to ϕ does not intersect WF(G0), essentially because timelike and null vectors cannot be orthogonal – see [8] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Consequently the pull-back is well-defined by standard results explained in Chapter 8 of [28] and has wavefront set contained in ϕ∗ WF(G0) ⊂ ϕ∗(N + × N −) = Γ × (−Γ), where Γ = R × (0, ∞) ⊂ T ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Moreover, ϕ∗G0 is the limit in D′ Γ×(−Γ)(R × R) of ϕ∗Fǫ ◦ σǫ as ǫ → 0+, which justifies taking the pull-back under the ǫ → 0+ limits in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Similar arguments apply to the convergence of F ′(σǫ(x, x′)) and F ′′(σǫ(x, x′)) as ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Next, recall that the stationary worldline γ has velocity u = ˙γ evolving according to u(s) = exp(sM)u(0), for M ∈ so(1, 3) with dimensions of inverse time, and that the right-handed tetrad ea(s) obeys ea(s) = exp(sM)ea(0), with u(s) = e0(s), ˙u(s) ∈ span{e1(s)}, and ¨u(s) ∈ span{e0(s), e1(s), e2(s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The Cartesian coordinates of γ(s), and components of ea(s) are evidently real analytic in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We extend ea to a smooth tetrad in a neighbourhood of γ in an arbitrary fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Recall that the functions Ca and Da are defined, in index-free notation, by Ca(s, s′) = η(ea(s), ea(s′)), Da(s, s′) = η(γ(s) − γ(s′), ea(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) We now prove the lemma needed in Section 4, which we restate for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (a) With the choice of tetrad just described, Ca(s, s′) and Da(s, s′) are translationally in- variant, depending only on s − s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' There are entire analytic functions Ga and Ha such that Ca(s, s′) = Ga(κ2(s − s′)2), Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) 19 where in the limit z → 0, 3 � a=0 Ga(z) = −2 + τ 2 + υ2 κ2 z + (κτ)2 − (τ 2 + υ2)2 κ4 z2 + O(z3), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) and 3 � a=0 Ha(z) = 1 + z 12 + κ2 + 19τ 2 360κ2 z2 + O(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) (b) The signed square geodesic separation of points along γ obeys σ0(γ(s), γ(s′)) = −(s − s′)2Υ(κ2(s − s′)2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) where Υ is entire analytic with Υ(z) = 1 + 1 12z + 1 360(1 − τ 2/κ2)z2 + O(z3) as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Furthermore, for z ∈ [0, ∞), Υ(z) is real with Υ(z) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (a) For inertial worldlines, ea(s) is constant and the result holds trivially with G0(z) ≡ 1, Gi(z) ≡ 1, H0(z) ≡ −1, Hi(z) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' From now on we may assume that κ is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) that Ca(s, s′) = η(exp � s′M � ea(0), exp(sM)ea(0)) = η(ea(0), exp � (s − s′)M � ea(0)), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) so Ca depends only on s − s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' As every component of the matrix exp(sM) is analytic, and because Ca(s, s′) = Ca(s′, s), we deduce that Ca(s, s′) = Ga(κ2(s − s′)2) for dimensionless entire analytic functions Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Next, observe that ∂ ∂s′ Da(s, s′) = −η(e0(s′), ea(s)) = −η(e0(0), exp � (s − s′)M � ea(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) Integrating with respect to s′ and using Da(s, s) = 0 we may deduce that κDa(s, s′) is a dimensionless entire analytic function of (s − s′)κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Again using Da(s, s) = 0 and because (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) gives ∂D0/∂s′|s′=s = −1 and ∂Di/∂s′|s′=s = 0 for i = 1, 2, 3, we have D0(s, s′) = (s − s′) � 1 + O((κ(s − s′))2) � , Di(s, s′) = κ−1O((κ(s − s′))2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) where we have also used the fact that D0(s, s′) = −D0(s′, s) as a consequence of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Because Da(s, s′)Da(s′, s) is invariant under interchange of s and s′, we now have Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) for dimensionless entire analytic functions Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The Taylor series of Ga, Ha and their sums, are computed up to second order in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (b) Next, we study the geodesic separation between γ(s) and γ(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We note that ∂ ∂sσ0(γ(s), γ(s′)) = −2D0(s, s′) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) depends only on s − s′, so σ0(γ(s), γ(s′)) = Σ(s − s′) + f(s′) and on considering s = s′ we find that f is constant and may be absorbed into Σ, which is also seen to be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The first terms in its Taylor expansion are easily found: Σ(0) = 0, while Σ′′(s − s′) = −2η(u(s), u(s′)), Σ(4)(s − s′) = 2η( ˙u(s), ˙u(s′)), Σ(6)(s − s′) = −2η(¨u(s), ¨u(s′)) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) giving Σ′′(0) = −2, Σ(4)(0) = −2κ2, Σ(6)(0) = −2κ2(κ2 − τ 2) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Accordingly, we have established (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7), the analyticity of Υ, and also the expansion Υ(z) = 1 + z 12 + κ2 − τ 2 360κ2 z2 + O(z3) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15) as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Finally, as γ(0) and γ(s) are connected by a smooth timelike curve, the timelike geodesic that connects them maximises proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Thus −σ0(γ(s), γ(0)) ≥ s2 for all s ∈ R and consequently, Υ(z) ≥ 1 for z ∈ [0, ∞), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 20 Finally, we explain how the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) may be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' First note that σǫ(γ(s), γ(s′)) = σ0(γ(s), γ(s′)) + 2iǫ(γ0(s) − γ0(s′)) + ǫ2 = −(s − s′)2Υ(κ2(s − s′)2) + 2iǫ(γ0(s) − γ0(s′)) + ǫ2 = −(s − s′ − iǫ)2Υ(κ2(s − s′)2) + ǫΨ(s, s′) + ǫ2Ξ(s, s′) for smooth (indeed analytic) functions Ψ and Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Let S be the difference between the distribution on the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) and the distribution on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Then, using the fact that Υ is nonvanishing on the real axis, S takes the form S(s, s′) = lim ǫ→0+ 2k � r=1 ǫrSr(s, s′) σǫ(γ(s), γ(s′))k(s − s′ − iǫ)2k (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16) for smooth functions Sr ∈ C∞(R2) (1 ≤ r ≤ 2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' All that is needed now is to show that the distributional limit lim ǫ→0+ 1 σǫ(γ(s), γ(s′))k(s − s′ − iǫ)2k (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='17) exists, whereupon S must vanish due to the strictly positive powers of ǫ in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The required result now follows from the sequential continuity of the distributional product with respect to the H¨ormander pseudo-topology (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10 in [27]), and the fact that both 1/σǫ(γ(s), γ(s′)) and 1 s − s′ − iǫ = i � ∞ 0 dk e−ik(s−s′−iǫ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) have limits as ǫ → 0+ in D′ Γ×(−Γ)(R2), where, as before, Γ = R × (0, ∞) ⊂ ˙T ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' B Taylor series calculation We compute the Taylor series of both Ga and Ha up to second order, using equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Recalling that Ca(s, s′) = Ga(κ2(s − s′)2), one can expand the right hand side into a Taylor series in s − s′ about the point s − s′ = 0 and then differentiate to yield − 1 2κ2 ∂2Ca ∂s∂s′ = G′ a(0) + 3κ2(s − s′)2G′′ a(0) + O((s − s′)4) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) 1 12κ4 ∂4Ca ∂2s∂2s′ = G′′ a(0) + O((s − s′)2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) as s − s′ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Differentiating equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) and setting s = s′ = 0, one easily finds G′ a(0) = −η( ˙ea(0), ˙ea(0)) 2κ2 , G′′ a(0) = η(¨ea(0), ¨ea(0)) 12κ4 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) by equating powers of s − s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The derivatives of the ea can be read off from the generalized Frenet- Serret equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) and its derivatives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8), allowing us to express G′ a(0) and G′′ a(0) in terms of curvature invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' An easy computation shows that G′ a(0) = 1 2η0a + κ2 − τ 2 2κ2 η1a − τ 2 + υ2 2κ2 η2a − υ2 2κ2 η3a (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) and G′′ a(0) = κ2 − τ 2 12κ2 η0a + τ 2υ2 + (κ2 − τ 2)2 12κ4 η1a − κ2τ 2 − (τ 2 + υ2)2 12κ4 η2a + υ2 τ 2 + υ2 12κ4 η3a, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) 21 where η(ea(0), eb(0)) = ηab by orthogonality of the tetrad field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Reconstructing Ga using a Taylor series therefore yields Ga(z) = ηaa + 1 2κ2 z � η0aκ2 − η1a(τ 2 − κ2) − η2a(υ2 + τ 2) − η3aυ2� + z2 24κ4 � η0aκ2(κ2 − τ 2) + η1a(τ 2υ2 + (κ2 − τ 2)2) − η2a(κ2τ 2 − (τ 2 + υ2)2) + η3aυ2(τ 2 + υ2) � + O(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) Summing, we obtain 3 � a=0 Ga(z) = −2 + τ 2 + υ2 κ2 z + (κτ)2 − (τ 2 + υ2)2 κ4 z2 + O(z3) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Applying exactly the same methodology to Ha, one writes Ea(s, s′) = Da(s, s′)Da(s′, s) so that Ea(s, s′) = −(s − s′)2Ha(κ2(s − s′)2) = −(s − s′)2Ha(0) − κ2(s − s′)4H′ a(0) − 1 2κ4(s − s′)6H′′ a(0) + O((s − s′)8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) Differentiation yields ∂2Ea ∂s∂s′ = 2Ha(0) + 12κ2(s − s′)2H′ a(0) + 15κ4(s − s′)4H′′ a(0) + O((s − s′)6) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) ∂4Ea ∂2s∂2s′ = −24κ2H′ a(0) − 180κ4(s − s′)2H′′ a(0) + O((s − s′)4) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) ∂6Ea ∂3s∂3s′ = 360κ4H′′ a(0) + O((s − s′)4), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) from which Ha(0), H′ a(0) and H′′ a(0) can be obtained differentiating equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) using Leibniz’ rule and subsequently setting s = s′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It is easily verifiable that this yields Ha(0) = [η(˙γ(0), ea(0))]2 = [η(e0(0), ea(0))]2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) H′ a(0) = − 1 4κ2 [η(¨γ(0), ea(0))]2 + 1 3κ2 η(˙γ(0), ea(0))η(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='γ (0), ea(0)), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13) H′′ a(0) = 1 18κ4 [η(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='γ (0), ea(0))]2 − 1 12κ4 η(¨γ(0), ea(0))η(γ(4)(0), ea(0)) + 1 30κ4 η(˙γ(0), ea(0))η(γ(5)(0), ea(0)), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='14) and after some straightforward computation, Ha(0) = η0a (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='15) H′ a(0) = 1 3η0a + 1 4η1a (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='16) H′′ a(0) = (η0a)2 � 1 18 + κ2 − τ 2 30κ2 � − κ2 − τ 2 12κ2 (η1a)2 + τ 2 18κ2 (η2a)2 = η0a � 1 18 + κ2 − τ 2 30κ2 � + κ2 − τ 2 12κ2 η1a − τ 2 18κ2 η2a (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='17) using the fact that (η0a)2 = η0a and (ηia)2 = −ηia for i = 1, 2, 3, as can be explicitly seen in the calculation of H′′ a(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Reconstructing Ha using a Taylor series, one obtains Ha(z) = η0a + 1 12z (4η0a + 3η1a) + 1 360κ2 z2 � η0a(10κ2 + 6(κ2 − τ 2)) + 15η1a(κ2 − τ 2) − 10η2aτ 2� + O(z3), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18) and summing, 3 � a=0 Ha(z) = 1 + z 12 + κ2 + 19τ 2 360κ2 z2 + O(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19) 22 C Wick square In this Appendix we show how a quantum inequality for the Wick square can be obtained along stationary trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This is a simpler calculation than the one used for the energy density and we shall be relatively brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Recall that the general QEI involves a (sum of) pull-backs of a suitable differential operator acting on the two-point function, T(s, s′) = ⟨Qφ(γ(s))Qφ(γ(s′))⟩ω0 = ((Q ⊗ Q)G0)(γ(s), γ(s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) For a quantum inequality on the Wick square, the operator Q can be simply identified as the identity, so T(s, s′) can be written in this case as T(s, s′) = G0(γ(s), γ(s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) Using the results of Section 4 and in particular, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='13), the two-point function can be neatly expressed as T(s, s′) = lim ǫ→0+ 1 4π2σǫ(γ(s), γ(s′)) = − lim ǫ→0+ 1 4π2(s − s′ − iǫ)2 � Υ � κ2(s − s′)2��−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) As Υ(κ2s2) ≥ 1 for s ∈ R by the Lemma, the entire function Υ(z) is nonvanishing on the real axis, and Υ(z)−1 is therefore analytic in a neighbourhood of the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) we may write Υ(z)−1 = 1 + zJ(z), where J is also analytic in a neighbourhood of the real axis, with J(0) = −1/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Because 0 < 1 + zJ(z) ≤ 1 for z ≥ 0, we may deduce that 0 ≤ −J(z) < 1/z for z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' We can now split the pulled back two-point function into its singular and regular parts as T(s, s′) = Tsing(s − s′) + Treg(s − s′), where Tsing(s) = − 1 4π2 lim ǫ→0+ 1 (s − iǫ)2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) and Treg(s) = −J(κ2s2) 4π2 lim ǫ→0+ κ2s2 (s − iǫ)2 = −κ2J(κ2s2) 4π2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) with Treg(0) = κ2/(48π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here we used the identity limǫ→0+ x2/(x−iǫ)2 = limǫ→0+(x−iǫ)2/(x−iǫ)2 = 1 of distributional limits, because g(z) = z2 is entire, while f(z) = z−2 is analytic in the open lower half-plane Z ⊂ C and obeys supz∈Z|f(z)(Im z)2| = 1 (see the argument below equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Observing that the two-point function given above is translationally invariant, we can use the bound given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) and thus write � ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ − � ∞ −∞ dα|ˆg(α)|2Qeven(α) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) where Qeven(α) = 1 2π2 �� 0 −∞ ˆT(u) du + � α 0 ˆTodd(u) du � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) The Fourier transform of Tsing is easily shown to be ˆTsing(u) = u 2πΘ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Again, Treg is smooth, real and even on R, decaying like O(s−2) as |s| → ∞ because of the decay of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Evidently Treg does not contribute to ˆTodd as Treg is absolutely integrable and has a well defined, continuous, real and even Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In this case, Tsing is actually universal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' the information relating to the specific worldline is encoded in Treg, as can also be seen below in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Clearly, ˆTsing does not contribute to the first term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='7) and, recalling that Treg is even, the odd part of ˆT is ˆTodd(u) = u 4π, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='8) and so Qeven is given in the form Qeven(α) = 1 2π2 �� 0 −∞ du ˆTreg(u) + 1 4π � α 0 du u � = 1 16π3 α2 + Treg(0) 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='9) 23 In direct analogy to the analysis of the energy density, the evenness of Treg and the Fourier inversion formula have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Inserting this into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) gives the QI bound � ds|g(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − 1 8π2 � ∞ −∞ ds � |g′(s)|2 + C|g(s)|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='10) where C = 8π2Treg(0) = κ2/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Considering the scaling behaviour, using the same test function gλ(s) = λ−1/2g(λ/s) as in the case for the QEI (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11), one can easily verify that � ds|gλ(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − ∥g′∥2 8π2λ2 − κ2∥g∥2 48π2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='11) where again ∥g∥2 denotes the L2-norm of the function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Taking the limit λ → ∞ yields the following formula, lim inf λ−→∞ � ∞ −∞ ds|gλ(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − κ2 48π2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='12) when considering the functions g such that ∥g∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Physically, since one can interpret 12⟨:φ2:⟩ as the square of a local temperature [4], states with negative expected Wick square are regarded as being locally out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' The above bound therefore quantifies the extent to which the thermal interpretation may fail uniformly along these worldlines, in terms of their proper acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' This raises an intriguing question as to whether there are states that would saturate this bound – something quite relevant to the Unruh experiments discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' In relation to the Unruh effect, a study of the detailed balance temperature obtained from the excitation of an Unruh-DeWitt detector carried along stationary worldlines can be found in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Here the quantum field is assumed to be in the vacuum state, and the temperature depends not only on the curvature invariants but also on the energy gap of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Although this is a different focus from our results, which concern averages of the Wick square in arbitrary Hadamard states, there are technical similarities, because the pulled back vacuum Wightman function plays a key role in both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' It would be interesting to understand whether some of the methods described here can be used to corroborate the numerical results of [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' D Computation of the renormalised stress-tensor for thermal and ground states on Rindler spacetime The Feynman propagator for a thermal state at inverse temperature β of the massless scalar field in Minkowski spacetime was given by Dowker [6] and the Wightman functions (including for higher spin) can be found in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Adopting coordinates t = ξ sinh χ, x = ξ cosh χ, the Rindler wedge x > |t| of Minkowski spacetime has metric ξ2 dχ2 − dξ2 − dy2 − dz2, and any curve χ �→ (aχ, 1/a, y0, z0) with a > 0 is a curve of proper acceleration a in proper time parameterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Given two points x = (χ, ξ, y, z) and x′ = (χ′, ξ′, y′, z′), write α(x, x′) = cosh−1 �ξ2 + (ξ′)2 + (y − y′)2 + (z − z′)2 2ξξ′ � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1) whereupon the Wightman function Gβ(x, x′) = ⟨φ(x)φ(x′)⟩β for the temperature β−1 KMS state with respect to the coordinate χ is Gβ(x, x′) = 1 4πβξξ′ sinh α(x, x′) � sinh(2πα(x, x′)/β) cosh(2πα(x, x′)/β) − cosh(2π(χ − χ′ − iǫ)/β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2) The β = 2π case coincides with the restriction of the Minkowski vacuum state to the wedge, while the zero temperature limit has Wightman function G∞(x, x′) = − α(x, x′) 4π2ξξ′ sinh α(x, x′)(α(x, x′)2 − (χ − χ′ − iǫ)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='3) 24 To obtain the renormalised (minimally coupled) stress-energy tensor, we first apply suitable derivatives to Gβ − G2π and take the limit x′ → x, obtaining ⟨:(∇µφ)(x)(∇νφ)(x):⟩β = 4π2 − β2 1440π2β4ξ4 � (16π2 + 14β2)ˆuµˆuν + 30β2ˆaµˆaν − (4π2 + 11β2)ηµν � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4) where, at spacetime position x, ˆuµ = ξ−1(∂χ)µ is the 4-velocity of the curve through x with constant ξ, y and z, and ˆaµ = (∂ξ)µ is the unit spacelike vector parallel to the 4-acceleration of this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Consequently, ⟨:Tµν:⟩β = 4π2 − β2 1440π2β4ξ4 � (16π2 + 14β2)ˆuµˆuν + 30β2ˆaµˆaν − (4π2 − 19β2)ηµν � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='5) and the result for Rindler ground state is obtained by taking β → ∞, giving ⟨:Tµν:⟩∞ = − 1 1440π2ξ4 (14ˆuµˆuν + 30ˆaµˆaν + 19ηµν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='6) Computing the energy density on curves of constant ξ yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Bibliography [1] Born, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=': Die Theorie des starren Elektrons in der Kinematik des Relativit¨atsprinzips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Annalen der Physik 335(11), 1–56 (1909).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1002/andp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19093351102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1002/andp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='19093351102 [2] Brown, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Ottewill, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Page, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' : Conformally invariant quantum field theory in static Einstein space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' D (3) 33(10), 2840–2850 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' 2840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='2840 [3] Brunetti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Fredenhagen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Verch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=': The generally covariant locality principle: A new paradigm for local quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Commun.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1856 [7] Epstein, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Glaser, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Jaffe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=': Nonpositivity of the energy density in quantized field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Nuovo Cimento (10) 36, 1016–1022 (1965).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Springer International Publishing (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1007/978-3-319-55182-1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1007/978-3-319-55182-1_10 25 [12] Fewster, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='4277 [20] Ford, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=', Roman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' : The quantum interest conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='024031 [31] Kontou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1007/BF02100096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1007/BF02100096 [40] Shaw, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=': The subgroup structure of the homogeneous Lorentz group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Quart.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1093/qmath/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='101 [41] Synge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' : Timelike helices in flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Irish Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' A 65, 27–42 (1967) (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content=' URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} +page_content='org/stable/20488646 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfu_0d/content/2301.01698v1.pdf'} diff --git a/4tE3T4oBgHgl3EQfQQmX/vector_store/index.faiss b/4tE3T4oBgHgl3EQfQQmX/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..396a1486ca5239fa2c35b2bfc51d4e9377cd7e46 --- /dev/null +++ b/4tE3T4oBgHgl3EQfQQmX/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1906bda3d5ebd3b8bd484ee3202c9049f5f944184a8c008ed4636cd2d4dc1c79 +size 3670061 diff --git 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Nayak†, Avhishek Chatterjee, and Lav R. Varshney, Senior Member IEEE +Abstract—Existing lower bounds on redundancy in fault- +tolerant quantum circuits are applicable when both the input +and the intended output are quantum states. These bounds may +not necessarily hold, however, when the input and the intended +output are classical bits, as in the Deutsch-Jozsa, Grover, or Shor +algorithms. Here we show that indeed, noise thresholds obtained +from existing bounds do not apply to a simple fault-tolerant +implementation of the Deutsch-Jozsa algorithm. Then we obtain +the first lower bound on the minimum required redundancy for +fault-tolerant quantum circuits with classical inputs and outputs. +Recent results show that due to physical resource constraints +in quantum circuits, increasing redundancy can increase noise, +which in turn may render many fault-tolerance schemes useless. +So it is of both practical and theoretical interest to characterize +the effect of resource constraints on the fundamental limits of +fault-tolerant quantum circuits. Thus as an application of our +lower bound, we characterize the fundamental limit of fault- +tolerant quantum circuits with classical inputs and outputs under +resource constraint-induced noise models. +Keywords—fault-tolerant +computing, +redundancy, +resource +constraints +I. INTRODUCTION +Initial ideas [1], [2], and especially mathematical demon- +strations of advantages of quantum computing over classical +computing [3], [4], have spurred considerable interest. How- +ever, noise in quantum circuits heavily restricts the class of +problems that can be solved using quantum hardware. Indeed, +the formal term NISQ (Noisy Intermediate Scale Quantum) +has been introduced to describe the current era where quantum +processors are noise-limited [5]. +To limit the corruption of quantum states due to noise, +the pursuit of fault-tolerant quantum circuits has led to a +large literature in quantum error correction. Early papers +demonstrated that one can achieve arbitrary computational +accuracy when physical noise is below a certain threshold. +Achievability of any desired fault tolerance in these initial +works required a poly-logarithmic redundancy with respect to +the size of the quantum circuit [6]–[9], but more recent works +extend such threshold theorems to require only a constant +overhead [10], [11], reminiscent of work in classical fault- +tolerant computing [12], [13]. +† The student authors contributed equally. +Uthirakalyani. G and A. Chatterjee are with the Department of Electrical +Engineering, Indian Institute of Technology, Madras, Chennai 600036, India +(emails:{ee19d404@smail,avhishek@ee}.iitm.ac.in). +A. K. Nayak and L. R. Varshney are with Coordinated Science Labora- +tory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +(emails:{anujk4, varshney}@illinois.edu). +This work was supported in part by National Science Foundation grant +PHY-2112890. +In this direction, some works provide fundamental limits +(lower bounds) on redundancy for arbitrarily accurate compu- +tation [14]–[18]. However, all of these lower bounds are for +quantum input/output, rather than classical input/output which +is common for a large class of algorithms, such as those due +to Deutsch-Jozsa [4], Shor [19], and Grover [20]. Here, we +demonstrate by example that lower bounds obtained so far in +quantum fault tolerance are not applicable for quantum circuits +with classical input/output, and provide a general alternate +bound. As far as we know, this is the first lower bound on fault +tolerance for quantum circuits with classical input/output. +The effects of noise on computational accuracy of quantum +circuits are typically studied assuming the noise per physical +qubit is constant with respect to the the size of the circuit. Un- +fortunately, this is not true in many quantum devices today— +often due to limited physical resources such as energy [21], +volume [22], or available bandwidth [23]—that have physical +noise levels that grow as the quantum computer grows [24]. +Fellous-Asiani, et al. introduce physical models of such scale- +dependent noise and also aim to extend threshold theorems +to this setting. However, the characterization of computational +error (per logical qubit error) is empirical in nature, lacking +precise mathematical treatment. Moreover, the characterization +depends on specific implementation and is restricted to con- +catenated codes. Here, using our new redundancy lower bound +and tools from optimization theory, we characterize the limits +of scale-dependence on fault-tolerant quantum circuits with +classical input/output, agnostic to specific implementation and +error correction methods. +The two motivations for the present work are therefore to +obtain lower bounds on the required redundancy of a quantum +circuit for computation with classical input/output, and to +investigate the effect of resource constraints (like energy or +volume) on this bound. +The distance between the distributions of the output clas- +sical bit corresponding to two different quantum input states +vanishes exponentially with the depth of the circuit when noise +is above a threshold [16]. Similarly for the trace distance be- +tween the output quantum states [15]. These results, however, +do not apply when the depth of the circuit is small; they also +do not provide lower bounds on required redundancy for sub- +threshold noise when fault-tolerant computation is possible. +This work focuses on shallow quantum circuits whose +input and output are classical, aiming for converse results for +classical computation using quantum circuits that yield lower +bounds on required redundancy. In [14], [18], lower bounds on +required redundancy that also led to improved noise thresholds + +were obtained. However, the fault-tolerance criteria in [14], +[18] are not appropriate for our setting, as Section II argues +using the example of the well-known Deutsch-Jozsa algorithm. +The experimental finding that noise increases with more +redundancy under resource constraints implies that simple per +(logical) qubit redundancy cannot achieve arbitrary computa- +tional accuracy even if noise per physical qubit is below the +fault-tolerance threshold, in contrast to conventional threshold +theorems [24]. This limitation is due to two opposing forces: +improvement in accuracy due to increased redundancy and +worse overall noise with redundancy due to scale-dependence. +In this regard, we aim to find the sweet spot on redundancy +for a desired computational accuracy. +The remainder of the paper is organized as follows. Sec- +tion II motivates our work through a counterexample that illus- +trates the need for a new redundancy lower bound. Section III +then gives the mathematical models of computation, noise, and +resource constraints that form the basis of our analysis. Then, +the primary contributions follow: +• Section IV proves a converse bound on redundancy +required for classical computation on quantum circuits, +drawing on one-shot capacity of classical-quantum chan- +nels (Theorem 1). +• Section V through VII analyze the converse results on the +limits of scale-dependence for fault-tolerant computation, +including closed-form and numerical solutions for some +canonical quantum device models. +Finally, Section VIII concludes. +II. NEED FOR A NEW REDUNDANCY LOWER BOUND: +A COUNTEREXAMPLE +Consider a quantum circuit that suffers from erasure noise +(with erasure probability p) right before the final measurement. +From one of the best known noise threshold bounds [14] +and the capacity results for erasure channels, it follows that +for an erasure noise per physical qubit p > 1 +2, fault-tolerant +computation is not possible (i.e., the required redundancy is +not finite). However, we demonstrate that a simple adaptation +of the Deutsch-Jozsa algorithm on this quantum circuit (with +erasure noise before the final measurement) can have a prob- +ability of error less than any ǫ > 0 even if p > 1 +2. +Fig. 1. A schema of quantum circuit that implements Deutsch-Jozsa algorithm +with erasure noise, to demonstrate the need for a new redundancy bound for +fault-tolerant quantum computation with classical I/Os. +The Deutsch-Jozsa algorithm is used to determine if the +given function oracle, f +: {0, 1}n → {0, 1} is constant +(0 or 1 for all input strings) or balanced (0 for half the +input strings and 1 for the rest). From [25, Eq. 1.51], the +quantum state before measurement under the absence of noise +is ψ1 = � +z,x∈{0,1}n +(−1)x.z+f(x) +2n +|z⟩ |y⟩ , where |y⟩ = |0⟩−|1⟩ +√ +2 +. +Measuring the first n qubits yields either |0⟩⊗n if f(·) is +constant, or an n-qubit state from {|0⟩ , |1⟩}⊗n \ {|0⟩⊗n} if +f(·) is balanced. Suppose the quantum states are corrupted by +erasure right before measurement as in Fig. 1; then the state +of the circuit becomes +ψ2 = +� +z,x∈{0,1}n +(−1)x.z+f(x) +2n +|z⟩(e) |y⟩(e) , +where |z⟩(e) and |y⟩(e) are the corrupted (i.i.d. erased) versions +of |z⟩ and |y⟩, respectively. For example, if f(·) is constant, +|z⟩(e) = |e00e00 · · ·e0⟩, i.e., each qubit state |0⟩ is replaced +i.i.d. with probability p by qubit state |e⟩. Now, consider our +modified algorithm: +1) Run Deutsch-Jozsa algorithm T times. +2) If no |1⟩ state was measured in any run, declare function +oracle f(·) a constant. +When f(·) is balanced, the measurement in the no-erasure +case must have one or more |1⟩ states. Such an oracle can be +incorrectly declared as constant when all of these |1⟩ states +are erased. So the probability of error is: +Pe = P{f(·) is declared constant |f(·) is balanced}, += P + + + +� +j,t +|zj⟩(e) +t +̸= |1⟩ +���f(·) is balanced + + + , +≤ P +�� +t +|zj⟩(e) +t += |e⟩ +��� |zj⟩ = |1⟩ +� +, +for some j ∈ {1, 2, . . ., n}. Since erasures are independent, +Pe ≤ +T +� +t=1 +P +� +|zj⟩(e) +t += |e⟩ +��� |zj⟩ = |1⟩ +� += pT . +Choosing T +≥ +��� ln ǫ +ln p +���, one can achieve Pe ≤ ǫ for any +p ∈ [0, 1). This counterexample proves that the bound for +redundancy N ≥ +n +Q(N) proposed in [14] does not hold for +quantum computation with classical input/output, since for an +erasure channel, the quantum capacity Q(N) = max{0, 1 − +2p} = 0 as p > +1 +2. This motivates the need for a different +bound which holds for classical I/O. +Note that this does not imply the prior bounds are incorrect; +the apparent contradiction is due differences in the definition +of accuracy. Prior work [14] uses a notion of distance (or +similarity) between the output quantum states of noiseless +and noisy circuits to quantify accuracy. This requirement is +too stringent when the output bits are classical and error +probability is a more suitable performance criterion [26], [27]. +As such, we obtain a lower bound on the redundancy under the +error probability criterion and then study the effect of resource +constraints. + +III. MODEL +A. Model of Computation +Consider the computational model in Fig. 2, which is a +quantum circuit with classical inputs and classical outputs. +This is denoted by CQC : {0, 1}n → {0, 1}n or equivalently +CQC(x) for x ∈ {0, 1}n, where n is the input size. The goal +of the circuit is to realize a function f : {0, 1}n → {0, 1}n. +The circuit consists of l layers. The first layer takes n clas- +sical inputs (x) as orthogonal quantum states |0⟩ and |1⟩ along +with N − n ancillas. It maps the input to a density operator +of dimension 2N. Any subsequent layer i, for 2 ≤ i ≤ l − 1, +takes the output of the previous layer, layer i − 1 as input. +The output of any layer i, 1 ≤ i ≤ l − 1, is a density operator +of dimension 2N. The final layer, layer l, performs a POVM +measurement and obtains classical output CQC(x). +Each layer i, with i ∈ {1, 2, . . ., l − 1} is a noisy quantum +operation. This is modeled as a noiseless quantum operation +Li on density operators of dimensions 2N followed by N +i.i.d. quantum channels N (Fig. 2). Finally, the last layer, +layer l, performs a measurement (POVM), which yields a +classical output. Thus the quantum circuit can be represented +as a composition of quantum operations as CQC(x) = Ll ◦ +N ⊗N ◦ Ll−1 ◦ · · · ◦ L2 ◦ N ⊗N ◦ L1(x), where ◦ has the usual +meaning of function composition. +We use the notation QCl−1 to denote the combined opera- +tions of layers 2 to l, given by Ll◦N ⊗N ◦Ll−1◦· · ·◦N ⊗N ◦L2. +B. Noise models +Here, we consider only Holevo-additive channels charac- +terized by a single parameter p ∈ [0, 1] and whose Holevo +capacity is monotonically decreasing in p. We use the generic +notation Np for such a channel with parameter p. Examples +include erasure, depolarizing, and symmetric GAD channels. +a) Erasure Channel: +In a quantum erasure channel +(QEC), each qubit flips to |e⟩⟨e|, which is orthogonal to every +ρ ∈ L(Cd), with probability p. Therefore, whenever a qubit +gets corrupted, the location of corruption is known. +Np(ρ) = (1 − p)ρ + ρTr[ρ] |e⟩⟨e| . +The classical capacity is [28]: +χ(Np) = 1 − p. +(1) +b) Depolarizing Channel: When a qubit undergoes de- +polarizing noise, it is replaced by a maximally mixed state +I/2 with probability p [28]: +Np(ρ) = (1 − p)ρ + p +2I. +In contrast to the erasure channel, the receiver (or the decoder) +is not aware of the location of the error. The Holevo informa- +tion of the depolarizing channel is: +χ(Np) = 1 − h2 +� p +2 +� +, +(2) +where h2(·) is the binary entropy function. Note that the +Holevo information is similar to the capacity of a binary +symmetric channel with crossover probability p/2. +c) Generalized Amplitude Damping Channel (GADC): +Amplitude damping channels model the transformation of +an excited atom to ground state by spontaneous emission +of photons. The changes are expressed using |0⟩ for the +ground (no photon) state and |1⟩ for the excited state. If +the initial state of the environment |0⟩⟨0|, is replaced by the +state θµ ≜ (1 − µ) |0⟩⟨0| + µ |1⟩⟨1| , µ ∈ [0, 1] where, µ is +thermal noise, we get the generalized ADC described using +the following four Kraus operators [29]: +A1 = +� +1 − µ +�1 +0 +0 +√1 − p +� +, +A2 = +� +1 − µ +�0 +√p +0 +0 +� +, +A3 = √µ +�√1 − p +0 +0 +1 +� +, +A4 = √µ +� 0 +0 +√p +0 +� +. +GADC is not additive in general (for arbitrary µ). However, in +the special case of symmetric generalized amplitude damping, +i.e., generalized amplitude damping with µ = 1/2, it is a +Holevo additive channel. The classical capacity of symmetric +GADC (µ = 1/2) is [29]: +χ(Np) = 1 − h2 +� +1−√1−p +2 +� +, +(3) +where p is the probability an atom decays from excited to +ground state. +Remark 1. Note that we have used p to describe different +impairments in different channels, so p must be interpreted +appropriately based on context. +C. Resource Constraints and Scale-Dependent Noise +In [24], it was shown that resource constraints can lead +to an increase in noise with increase in redundancy, scale- +dependent noise. A few models of scale-dependent noise have +been studied in [24]. +Let k ≜ N/n ≥ 1 be the redundancy and p(k) be the noise +strength when the redundancy is k. (Recall that we consider +Holevo-additive noise models that can be characterized by +a single parameter 0 ≤ p ≤ 1.) In the polynomial model, +p(k) = min(p0(1 + α(k − 1))γ, 1) and in the exponential +model, p(k) = min(p0 exp(α(k − 1)γ), 1). Here, p0 ∈ [0, 1] +is the noise strength in the absence of any redundancy, i.e., +k = 1, and α and γ are positive parameters. +Motivated by practically useful noise models like erasure, +depolarization, and models for scale dependence in [24], we +consider the following generic scale-dependent noise model. +Definition 1. Noise Np is parameterized by a single parameter +p ∈ [0, 1] and the Holevo information χ(Np) is non-increasing +in p. The parameter p is a function of redundancy k, given +by min(p(k; p0, θ), 1), where θ is a tuple of non-negative +parameters from the set K, and +1) p0 = p(1; p0, θ) for all θ, +2) for any k ≥ 1, p(k; p0, θ) is non-decreasing in any +component of θ and in p0, given the other parameters +are fixed. +Here, p0 represents the noise without redundancy, i.e., the +initial noise without any resource constraint arising due to + +Fig. 2. CQC model of computation: classical input, quantum computation, and classical output. +redundancy. Clearly, the polynomial and exponential models +are special cases with θ = (α, γ) ∈ K = R2 +≥0. +The threshold for p0, i.e., the minimum p0 beyond which +reliable quantum computation is not possible, was studied in +[24] assuming concatenated codes for error correction. Here, +we obtain a universal threshold for all fault tolerance schemes. +IV. LOWER BOUND ON REQUIRED REDUNDANCY +We first define the accuracy criterion for computation using +quantum circuits with classical input/output. Then we show +how to convert the noisy computation problem to a communi- +cation problem over i.i.d. quantum channels. This finally leads +to the redundancy bound in Theorem 1, which we use to obtain +thresholds for p0 under resource constraints. +Definition 2. Suppose f(·) is a classical function realized by +a quantum circuit CQC(·) as defined in Sec. III. Then the +ǫ-accuracy is: +P{CQC(x) ̸= f(x)} < ǫ, for all x ∈ Zn +2 . +(4) +Eq. (4) holds for all x ∈ Zn +2 . Therefore, the ǫ-accuracy +condition holds for any subset of Zn +2. The following lemma +states a necessary condition for ǫ-accuracy. +Lemma 1. Consider x(1), x(2), . . . , x(R) ∈ Zn +2 s.t. |{f(x(i)) : +1 ≤ i ≤ R}| = R. Then a necessary condition for ǫ-accuracy +condition (4) to hold is +P{CQC(x(i)) ̸= f(x(i))} < ǫ, for all i = 1, 2, . . ., R. +Note that the domain is restricted to R inputs, such that +the restricted mapping is bijective. Using this bijectivity, we +obtain the following simpler lemma, which connects ǫ-accurate +computation with finite blocklength communication. +Lemma +2. +Suppose +there +exists +a +CQC(x) +s.t. +P{CQC(x(i)) +̸= +f(x(i))} +< +ǫ for all 1 +≤ +i +≤ +R. +Then there exists a classical circuit C(·) s.t. +P{C(CQC(x(i))) ̸= x(i)} < ǫ, +for all 1 ≤ i ≤ R. +(5) +Proof: Suppose ˆf(·) is a restriction of f(·) such that the +mapping ˆf : {x(i), 1 ≤ i ≤ R} → {f(x(i)), 1 ≤ i ≤ R}, is a +bijection. Then the inverse map ˆf −1(·) is unique. Choosing a +C(·) that implements ˆf −1(·), the probability of error can be +equivalently expressed as +P{C(CQC(x(i))) ̸= x(i)} < ǫ, +for all 1 ≤ i ≤ R. +We define ˆx(i) to be the output of C(CQC(x(i))). Then the +condition in (5) is equivalent to +max +x(i) P{x(i) ̸= ˆx(i)} < ǫ, +1 ≤ i ≤ R. +This implies that a necessary condition to satisfy accuracy +condition (4) is +inf +L1,C,QCl−1 max +x(i) P{x(i) ̸= ˆx(i)} ≤ max +x(i) P{x(i) ̸= ˆx(i)} < ǫ. +Note that L1 is an encoding of classical bits into a quantum +state, and QCl−1 followed by C(·) can be interpreted as the +decoding of the noisy version of the same quantum state +(depicted in Fig. 3). Hence, infL1,C,QCl−1 maxx(i) P{x(i) ̸= +ˆx(i)} is equivalent to the maximum probability of error for +transmitting message x(i), i ∈ {1, . . . , R} over the channel +N ⊗N. Using this reduction, we lower-bound the redundancy +for any classical computation using a quantum circuit. +Theorem 1. Let f : {0, 1}n → {0, 1}n be a classical function +and Rf = |{f(x) : x ∈ {0, 1}n}| be the cardinality of the +range of f. Then, for computing a classical function f with + +Fig. 3. Reduction of noisy computation model in Fig. 2 to noisy communi- +cation model. +ǫ-accuracy using a quantum circuit corrupted by i.i.d. Holevo- +additive noise, the required number of physical qubits N is +bounded as +N > (1 − ǫ) log2(Rf) − h2(ǫ) +χ(N) +for all ǫ ∈ [0, 1 +2]. +Proof: For any additive quantum channel N, an upper +bound for classical communication over a quantum channel +using an (M, N, ǫ) code is [28]: +log2(|M|) ≤ χ(N ⊗N) + h2(ǫ) +1 − ǫ +, +where M is the message alphabet. Assigning |M| = Rf +yields +ǫ > Pe ≥ 1 − χ(N ⊗N) + h2(Pe) +log2 Rf +, +(6) +≥ 1 − χ(N ⊗N) + h2(ǫ) +log2 Rf +. +The last inequality holds, since h2(·) is increasing in [0, 1 +2]. +Rearranging, we obtain +χ(N ⊗N) > (1 − ǫ) log2 Rf − h2(ǫ). +Noting that Holevo information is sub-additive, +Nχ(N) > (1 − ǫ) log2 Rf − h2(ǫ), +N > (1 − ǫ) log2 Rf − h2(ǫ) +χ(N) +. +(7) +The bound in Theorem 1 states that the number of quantum +buffers, N, needed for accurate computation of an n-bit +function f is lower bounded. As given in Sec. III, k ≜ +N +n +is the redundancy of the quantum circuit. Thus, Theorem 1 +can be seen as a lower bound on the required redundancy. +The quantity log2 Rf is the number of bits needed to encode +the output. We define η ≜ log2 Rf +n +as the compression factor of +f. To understand the impact of scale-dependent noise, we will +use the following corollary of Theorem 1 that gives a lower +bound on the redundancy k. +Corollary 1. The condition for ǫ-accuracy in Theorem 1 is +alternatively +k > c(ǫ, η, n) +χ(N) , +(8) +where c(ǫ, η, n) ≜ (1 − ǫ)η − h2(ǫ) +n . +Proof: Substituting log2 Rf = ηn and k = N/n in (7), +and rearranging we obtain +N > (1 − ǫ)ηn − h2(ǫ) +χ(N) +, +k > c(ǫ, η, n) +χ(N) . +V. SCALE-DEPENDENT NOISE: CONVERSE REGIONS +Let Np denote the channel parameterized by a noise in- +tensity term p. Examples include the probability of erasure, +p, for erasure channels; the probability of a quantum state +being replaced by a maximally mixed state, p, for depolarizing +channels; and the amplitude decay parameter, p, for symmetric +GAD channels. If the error per physical qubit p is a constant +w.r.t. k, then χ(Np) is constant. Therefore, one can ensure +that the necessary condition for ǫ-accuracy in (8) is satisfied +by sufficiently increasing redundancy (choosing large k). +On the other hand, if p scales (increases) with redundancy, +then χ(N) decreases with k (we denote the dependency on +p(k) as χ(Np(k))). Therefore, satisfying ǫ-accuracy condition +k > c(ǫ, η, n)/χ(Np(k)) is not guaranteed. In fact, Fig. 4 +plots the error probability lower bound (6) with both scale- +independent and scale-dependent erasure. Notice that when +the physical noise is independent of k, the Pe lower bound +rapidly decreases with an increase in redundancy, whereas +when the noise is scale dependent, the probability of error +initially decreases with increasing redundancy k, but then +grows beyond a certain optimum k. With this motivation, we +explore the limitations of ǫ-accurate computation under scale- +dependent noise. +We specifically aim to characterize the set of (p0, θ) for +which ǫ-accurate computation is not possible. This is equiv- +alent to the noise threshold in traditional models, with scale- +independent noise. The following corollary to Theorem 1 +provides a converse in terms of θ. +Corollary 2. Suppose we have, +¯Θ ≜ +� +(p0, θ) ∈ K +���min +k≥1 g(k, p0, θ, ǫ) ≥ 0 +� +, +where +g(k, p0, θ, ǫ) ≜ c(ǫ, η, n) +k +− χ(Np(k)). +Then ǫ-accurate computation is not possible for θ ∈ ¯Θ. Also, +if (p0, θ) ∈ ¯Θ then (p′ +0, θ′) ∈ ¯Θ if (p′ +0, θ′) ≥ (p0, θ) in a +component-wise sense. + +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +Redundancy (k) +10−2 +10−1 +100 +Pe (lower bound) +p0 = 0.15 +p0 = 0.20 +p0 = 0.30 +Fig. 4. Comparison between Pe lower bound with (solid lines) and without +(dashed-lines) scale-dependent physical noise for erasure channel. +Proof: From Definition 2, we must prove that if Pe < ǫ, +then (p0, θ) /∈ ¯Θ. Considering the scale-dependent noise in +Corollary 1, we have if Pe < ǫ, then +k > c(ǫ, η, n) +χ(Np((k)), +g(k, p0, θ, ǫ) = c(ǫ, η, n) +k +− χ(Np(k)) < 0. +(9) +For any θ ∈ K, (9) is satisfied only if +min +k≥1 g(k, p0, θ, ǫ) < 0. +In other words, (p0, θ) /∈ ¯Θ. +As χ(Np) is non-increasing in p and p(k; p0, θ) is +non-decreasing in each component, (p′ +0, θ′) ≥ (p0, θ) in +a component-wise sense implies (p′ +0, θ′) ∈ +¯Θ whenever +(p0, θ) ∈ ¯Θ. +We refer to ¯Θ as the converse region since ǫ-accurate +classical computation on quantum circuits is not possible if the +parameters of the scale-dependent noise are in ¯Θ. As any fault- +tolerant implementation has to avoid this region, characterizing +¯Θ is of particular interest. By Corollary 2, for characterizing +¯Θ, it is enough to find the minimum p0 for each θ such that +(p0, θ) ∈ ¯Θ. More precisely, for a fixed θ, the threshold pth(θ) +can be defined as: +pth(θ) := inf{p0 | (p0, θ) ∈ ¯Θ}. +(10) +The threshold pth(θ) (or pth for brevity) can be obtained +by solving the following optimization problem. +minimize p0 +s.t. +min +k≥1, 0≤p(k)≤1 gθ(k, p0) ≥ 0, +(11) +where, gθ(k, p0) := g(k, p0, θ, ǫ). +Consider the following optimization problem +PL : +min +k≥1, 0≤p(k)≤1 gθ(k, p0). +Clearly, (11) has the optimization problem PL, which we refer +to as the lower-level optimization problem, as a constraint. +Thus, (11) is a bi-level optimization problem. For a given set +of θ the solution to PL is a function of p0, which we denote +as g∗ +θ(p0). Thus, the bi-level optimization problem in (11) can +also be written as +min p0 +s.t. g∗ +θ(p0) ≥ 0. +(12) +In general, to compute the threshold pth one needs to solve +(11). However, for erasure noise and some special classes of +p(k; p0, θ), closed-form expressions for pth can be obtained. +Theorem 2. For erasure noise, thresholds are as follows. +1) If p(k; p0) = p0 (constant), then pth = 1. +2) If p(k; p0, α) = p0(1 + α(k − 1)), then +pth = + + + +1 − c, +if α ≥ +c +1−c, and +( +√cα−√cα−α+1) +2 +(α−1)2 +, +otherwise. +3) If p(k; p0, γ) = p0kγ, then +pth = + + + +1 − c, +if γ ≥ +c +1−c, and +( γ +c ) +γ +(γ+1)γ+1 , +otherwise. +Here c = c(ǫ, η, n), defined in Corollary 1. Note that θ = ∅, α +and γ in cases 1), 2) and 3), respectively. +Proof: Consider a procedure to find a closed-form expres- +sion for pth as follows. +1) Minimize gθ(k, p0) over k. Since p(k; p0, θ) is non- +decreasing in k, it is enough to minimize gθ(k, p0) over +[1, kmax], where kmax = max{k | p(k; p0, θ) ≤ 1} (see +Appendix B for more details). The minimum occurs at +either k = 1, k = kmax or a stationary point of gθ(k, p0) +in [1, kmax]. +2) Substitute the minimizer k into gθ(k, p0) ≥ 0, which +yields an equation in p0, θ. +3) Solving the equation for p0 yields a closed-form expres- +sion for pth. +The derivation of pth for corresponding p(k; p0, θ) is given in +Appendix A. +For a general p(k; p0, θ), however, a closed-form expression +for pth in terms of θ cannot be obtained, and therefore, pth +must be computed numerically. +We develop Algorithm 1 to obtain pth by solving bi-level +optimization problem (11). In Algorithm 1, we solve the +alternate formulation (12) using the bisection method, while +assuming access to an oracle that computes g∗ +θ(p0) for any p0. +Later, we also develop efficient algorithms that solve PL and +obtain g∗ +θ(p0) for any p0. +Algorithm 1 computes the threshold pth (up to an error of +δp0), for a pre-determined set of θ. First, a channel-specific +Lipschitz constant L is computed using Eqs. (15), (17), or +(19) for a given (p0, θ), which determines how quickly PL is +solved. Lines 7–17 describe the bisection method to compute +pth. Depending on whether PL is convex or non-convex, +Algorithm 2 or Algorithm 3 is used to compute g∗ +θ(p0), +respectively. + +The following theorem provides a proof of global conver- +gence of Algorithm 1, with only a monotonicity assumption +in θ (note that continuity in θ is not needed). +Theorem 3. Suppose a quantum circuit is corrupted by a scale- +dependent noise-per-physical qubit, p(k; θ) that is monotonic +in θ. Then for any given θ ∈ K, the sequence {p0i} generated +using Algorithm 1 converges to the threshold pth in (10). +Proof: Algorithm 1 generates a non-increasing sequence +{p+ +0i} and a non-decreasing sequence {p− +0i}, which at every +iteration yields g∗(p+ +0i) ≥ 0 and g∗(p− +0i) < 0, with p0i = +p+ +0i +p− +0i +2 +. Since the bisection method halves the difference +between p+ +0i and p− +0i at every iteration (i.e., p+ +0i+1 − p− +0i+1 = +p+ +0i −p− +0i +2 +), we have that for all ǫ > 0, there exists an i0 such +that for all i ≥ i0, we get p+ +0i −p− +0i < ǫ. Also, since both {p+ +0i} +and {p− +0i} are bounded, they converge, and since for all i ≥ i0, +p+ +0i −p− +0i < ǫ, they converge to a common limit point (say p∗). +Since, K is closed, (p∗ +0, α, γ) ∈ K. Due to the monotonicity +of g∗ +θ(p0) (non-decreasing with p0), the following inequality +holds: g∗ +θ(p− +0i) ≤ g∗ +θ(p∗ +0) ≤ g∗ +θ(p+ +0i). Therefore, g∗ +θ(p0) < 0, +for all p0 < p∗, and g∗ +θ(p0) ≥ 0, for all p0 > p∗, which is by +definition p∗ = pth. +Obtaining g∗ +θ(·) requires solving PL. Next, we present +efficient algorithms for solving PL for erasure, depolarizing, +and symmetric GAD channels, and numerically obtain the +converse surface for those noise models. +Algorithm 1 Algorithm to obtain pth/¯Θs numerically. +1: Initialize the set K′ ⊆ K. +2: Initialize max iters, δp0, δ, ¯Θs = {}, k ← 1. +3: for each θ ∈ K′ do +4: +Initialize i ← 0, ∆p0 ← 1, p0 ← 0.5, +5: +p− +0 ← 0, p+ +0 ← 1. +6: +% BISECTION METHOD +7: +while ∆p0 > δp0 and i 0 then p+ +0 ← p0, +13: +else p− +0 ← p0. +14: +end if +15: +∆p0 ← |p0 − p−1|. +16: +i ← i + 1. +17: +end while +18: +¯Θs = ¯Θs +�{(p0, α, γ)}. +19: end for +VI. CONVERSE REGION FOR ERASURE +In this section, we derive necessary conditions for ǫ-accurate +computation when the source of corruption of quantum states +is erasure. Substituting for the classical capacity of QEC from +(1) in (8) yields +g(k, p0, θ, ǫ) = gθ(k, p0) = c(ǫ, η, n) +k ++ p(k) − 1 < 0. (13) +Remark 2. One can equivalently solve PL by restricting the +range of k to [1, kmax], where kmax = max{k | p(k; p0, θ) ≤ +1}. Also, kmax is finite and hence [1, kmax] is compact, which +makes it convenient to solve (11). Therefore, one can replace +line 11 with g∗ +θ(p0) = mink∈[1,kmax] gθ(k, p0) to obtain the +same value of threshold pth. See Appendix B for more details. +A. Physical Noise p(k; p0, θ) is Convex in Redundancy k +For the erasure channel, if p(k; p0, θ) is convex, then +g(k, p0, θ, ǫ) is convex in [1, kmax], since the Holevo informa- +tion χ(Np) is affine in p. Therefore, the problem PL in (11), +is convex, and from Remark 2, the feasible set is compact. +A convex function over a compact set can be optimized +using a gradient projection method given in [30]. There are +many algorithms to solve general gradient projection prob- +lems such as sequential quadratic programming (SQP) and +augmented Lagrangian methods that can be directly applied +to solve PL. Since, our problem is a one-dimensional convex +problem (with only a Lipschitz gradient constraint) over a +finite range [1, kmax], we provide a simple constant step-size +gradient projection algorithm (Algorithm 2). +Algorithm 2 Projected gradient descent routine. +1: function PROJGD(gθ, p0, kin, kmax, L, ζ) +2: +Initialize j ← 0, kj ← kin, ξ = 1 +L, ∆g = 2ζ. +3: +while ∆g ≥ ζ do +4: +if gθ(·, p0) is convex then dj ← g′ +θ(kj, p0) +5: +else dj ← |g′ +θ(kj, p0)| +6: +end if +7: +kj+1 ← min{kmax, max{1, kj + ξdj}}. +8: +˜g ← gθ(kj+1, p0). +9: +∆g ← |gθ(kj, p0) − ˜g|. +10: +j ← j + 1. +11: +end while +12: +return ˜g, kj. +13: end function +Algorithm 2 solves PL optimally if step size (ξ) and +stopping criterion (ζ) are chosen appropriately. Sufficient +conditions for convergence are: 1) ξ ∈ (0, 1 +L], if g′ +θ(k, p0) ≜ +∂ +∂kgθ(k, p0) is L-Lipschitz over [1, kmax], and 2) stopping +criterion provided in Definition 3. In all our computations, +we choose ξ = 1 +L as the step size for fast convergence. +Definition 3. Stopping criterion 1: Let {kj} be the iterates +generated by the projected gradient descent algorithm (Algo- +rithm 2), we use the following stopping criterion for projected +gradient descent algorithm: +|gθ(kj, p0) − gθ(kj+1, p0)| < +δ2 +2Lk2max +=: ζ. +(14) +Then, it follows from the convexity and L-Lipschitz prop- +erty of gθ(·, p0) that stopping criterion (14) is a sufficient +condition for convergence, which is gθ(kj+1, p0)−g∗ +θ(p0) ≤ δ. +The following theorem provides proof of convergence of +Algorithm 2. For better readability, the associated lemmas used +in the proof are included in Appendix D. + +Theorem 4. Convergence of Algorithm 2: Suppose g∗ +θ(p0) = +min +k≥1 gθ(k, p0), which is convex in k. Then Algorithm 2 yields +˜g arbitrarily close to g∗ +θ(p0), i.e., for any pre-determined δ > 0, +|˜g − g∗ +θ(p0)| ≤ δ. +Proof: Let {1, . . . , kl} be a sequence generated by +projected gradient descent, PROJGD, where kl satisfies the +stopping criterion. Note that PROJGD does not cross any +stationary point if the step-size ξ ≤ +1 +L (from Lemma 8). +So, kl = 1 if and only if ˜g = g∗ +θ(p0) = gθ(1, p0), and +similarly kl = kmax if and only if ˜g = g∗ +θ(p0) = gθ(kmax, p0). +Otherwise kl ∈ (1, kmax) and gθ(1, p0) < 0, which implies +from Lemma 8 that gθ(kl, p0) ≤ 0. From Lemmas 6 and 7, +kl satisfying the stopping criterion in Def. 3 is sufficient for +convergence, i.e., ˜g = gθ(kl, p0) and |˜g − g∗ +θ(p0)| ≤ δ. +The following lemma shows g′ +θ(k, p0) is indeed L-Lipschitz +over [1, kmax] for a general polynomial noise model and gives +a closed-form expression for L. +Lemma 3. Computing Lipschitz constant L: g′ +θ(k, p0) is L- +Lipschitz over [1, kmax] for scale-dependent erasure noise +p(k; p0, θ) = p0(1 + α(k − 1))γ with γ ≥ 1 where, for +c = c(ǫ, η, n), +L = 2c + α2γ(γ − 1)p +1 +γ +0 . +(15) +Proof: The proof is given in Appendix C-A. +Next, we provide a closed-form upper bound for p0 for the +polynomial noise model; however, the bound is looser than +pth computed in closed-form in Corollary 2 and numerically +using Algorithm 1. +Claim 1. If p(k) = p0(1 + α(k − 1))γ for α > 0 and γ ≥ 1, +then for any p0 with +p0 ≥ max +�c(ǫ, η, n) +γα +, 1 − c(ǫ, η, n) +� +, +ǫ-accurate computation is not possible. +Proof: Since p(k) is convex, so is the following function: +gθ(k, p0) = p(k) − 1 + c(ǫ, η, n) +k +. +Notice that if the slope g′ +θ(k0, p0) = 0, for some k0 ≥ 1, +then gθ(k, p0) is increasing in k ≥ k0 due to its convexity +over k. Therefore, if gθ(k, p0) ≥ 0 and g′ +θ(k, p0) = 0 at +k = 1, then gθ(k, p0) ≥ 0 is always satisfied ∀k ≥ 1, and +such (p0, α, γ) ∈ ¯Θ. Solving the following +∂ +∂k +� +p(k) − 1 + c(ǫ, η, n) +k +����� +k=1 +≥ 0, +yields: +p0γα ≥ c(ǫ, η, n). +Noting that p0 ≥ 1 − c(ǫ, η, n) ≥ 0 (at k = 1), we obtain +p0 ≥ max +�c(ǫ, η, n) +γα +, 1 − c(ǫ, η, n) +� += pth. +■ +A comparison of the looser bound with that obtained by +Algorithm 1 is shown in Fig. 5. +Fig. 5. Comparison of converse surfaces, ¯Θs between numerical optimization +(Algorithm 1) and the derivative approach (Claim 1). Evidently, the converse +bound obtained using the algorithm is tighter. +B. Physical Noise p(k; p0, θ) is Non-convex in Redundancy k +Suppose p(k) is non-convex, then gθ(k; θ, ǫ) is also non- +convex. Hence, the lower-level problem PL cannot be solved +using Algorithm 2 (PROJGD). Therefore, we provide a line- +search algorithm (Algorithm 3) to compute solution for a non- +convex problem PL. +In Algorithm 3, the compact set [1, kmax] is traversed by +successive gradient descent (or ascent) and perturbation over a +one-dimensional non-convex function using an iterate starting +from k = 1 (w.l.o.g.) and moving in the positive k direction. +Lines 4–9 include one iteration of Algorithm 3, which contains +calls to PROJGD and PERTURB as subroutines. The variable +˜g keeps track of the minimum value of gθ(·, p0) encountered +thus far with an error of δ > 0. +In Algorithm 3 we reuse the PROJGD routine for gradient +ascent/descent but with a different (more relaxed) stopping +criterion than in Def. 3. +Definition 4. Stopping criterion 2: Let {kj} be the iterates +generated by the projected gradient descent algorithm (Algo- +rithm 2). We use the following stopping criterion for projected +gradient descent algorithm: +|gθ(kj, p0) − gθ(kj+1, p0)| < δ =: ζ. +Definition 5. Stopping criterion for PERTURB: Let {kj} be a +sequence generated by PERTURB routine. +|gθ(kj, p0) − gθ(k′ +j, p0)| ≥ δ +L. +where k′ +j += min{kmax, k + ξ|g′ +θ(kj, p0)|} in line 17 of +Algorithm 3, and g′ +θ(z, p0) = ∂g′ +θ(k,p0) +∂k +��� +k=z. +Remark 3. Note that the stopping criterion for PERTURB +is similar to Definition 4, but with the inequality reversed. +Since the stopping criteria of PROJGD and PERTURB are +complementary, only one of the routines will be active during +the execution of Algorithm 3. +Next, Theorem 5 proves convergence of Algorithm 3. Re- +quired lemmas are in Appendix E. + +ure Channel) +Optimization +Looser BoundConverse Regions (Eras +0.84 +3 +2 +20.6 +0 +p +0.4 +0.2 +0 +0 +2 +3 +1 +aAlgorithm 3 Line search algorithm to find mink≥1 gθ(k, p0), +when gθ(k, p0) is non-convex w.r.t. k. +1: function LINESEARCH(gθ, p0, kmax, L, δ) +2: +Initialize i ← 0, ki ← 1. +3: +˜g ← +min +k∈{1,kmax}gθ(k, p0). +4: +while i 0. +Proof: Suppose {. . . , ki, k− +i , ki+1, k− +i+1, . . .} is the se- +quence generated by Algorithm 3. From Lemma 8 there are +no stationary points in (ki, k− +i ). Then, the PERTURB routine +keeps track of the minimum value of gθ(·, p0) in [k− +i , ki+1] at +discrete increments: ˆgi = mink∈{k− +i ,k− +i +∆k,...,ki+1}gθ(k, p0). +This is followed by executing PROJGD again from ki+1 to +k− +i+1, and so on. In every call to the PERTURB routine, ˜g tracks +the minimum of ˆgi until the ith iteration. From Lemma 10, ˆgi +differs from mink∈[k− +i ,ki+1]gθ(k, p0) by at most δ. In line 3 +of Algorithm 3, ˜g is initialized with minimum at boundary +points k = {1, kmax}. Therefore, ˜g − g∗ ≤ δ. Finally, +from Lemma 11, Algorithm 3 terminates in finite steps when +kj = kmax or k− +j = kmax for some j ≥ i + 1. +VII. CONVERSE REGION FOR SYMMETRIC GAD AND +DEPOLARIZING CHANNELS +A. Converse Region for Symmetric GAD Channel +Let us compute converse regions when quantum states are +corrupted by GADCs. We only consider symmetric GADC +(with µ = 1/2), since its classical capacity is additive; for +µ ̸= 1/2, the additivity of classical capacity is not known. +Substituting classical capacity of symmetric GADC from (3) +in the necessary condition for ǫ-accuracy in (8) yields: +gθ(k, p0) = c(ǫ, η, n) +k +−1+h2 +� +1 − +� +1 − p(k) +2 +� +≤ 0. (16) +In (16), the last term is monotonic (increasing) in p, and +p(k; p0, θ) is monotonic (increasing) in θ. Therefore, Corol- +lary 2 also holds for symmetric GAD channel. Therefore, for +a given θ, the threshold pth can be computed by solving bi- +level optimization problem (11). However, we cannot obtain +closed-form expressions like for the erasure channel due to +the challenge from the binary entropy term in (16); therefore, +the threshold pth must be computed numerically. Since, sym- +metric GAD channel is additive, and scale-dependent noise +p(k; p0, θ) is monotonic in θ component-wise, the threshold +pth can be computed using Algorithm 1 (Theorem 3 holds). +However, since Holevo information of symmetric GADC +is concave in p, even if p(k; p0, θ) is convex in k, unlike +the erasure case, gθ(k, p0) is not convex in k. Therefore, the +lower-level problem PL can be solved using Algorithm 3 to +obtain the threshold pth for a given θ. For a polynomial noise +model described in Section III-C, we can compute Lipschitz +constant L in closed form for a given θ as follows. +Lemma 4. Computing Lipschitz constant L: g′ +θ(k, p0) is L- +Lipschitz over [1, kmax] for a polynomial scale-dependent +symmetric GAD noise p(k; p0, θ) = p0(1+α(k−1))γ, where +L := 2c+α2γ2p +min{1, 2 +γ } +0 +4(1 − p0) +� +1 +p0 ln 2 + 1 + 2(1 − p0) +2√1 − p0 +log2 P +� +, +(17) +with P = 1+√1−p0 +1−√1−p0 . +Proof: See Appendix C-B. +B. Converse Region for Depolarizing Channel +In this section, we compute the converse region when +computational states are corrupted by depolarizing noise. Sub- +stituting for the classical capacity of the depolarizing channel +from (2) in (8), we obtain +gθ(k, p0) = h2 +�p(k) +2 +� +− 1 + c(ǫ, η, n) +k +≤ 0. +(18) +Similar to the symmetric GAD channel, the first term is +increasing in p, and p(k; p0, θ) is non-decreasing in θ. There- +fore, Corollary 2 and computation of threshold pth by solving +bi-level optimization problem (11) also hold. Also, similar to +symmetric GADC, since obtaining closed-form expressions for +pth is not possible, it can be computed using Algorithm 1. +Since gθ(k, p0) is non-convex (due to h2(·) in (18) being +concave), the threshold pth can be computed using line-search +(Algorithm 3). Again, similar to the symmetric GAD channel, +Lipschitz constant L can be computed in closed form for a +given θ. + +Lemma 5. Computing Lipschitz constant L: g′ +θ(k, p0) is L- +Lipschitz over [1, kmax] for a polynomial scale-dependent +depolarizing noise p(k; p0, θ) = p0(1 + α(k − 1))γ, where +L = 2c + α2γ2p +min{1, 2 +γ } +0 +� +log2(e) +p0(2 − p0) + 1 +2 log2 +�2 − p0 +p0 +�� +. +(19) +Proof: Refer to Appendix C-C. +C. Comparing Converse Surfaces of Different Channels +Fig. 6 shows the converse surfaces ¯Θs when quantum +computation is affected by erasure, depolarizing, and general- +ized amplitude damping noise. For a given θ = (α, γ) the +thresholds are related as p(e) +th +≥ p(g) +th +≥ p(d) +th +(point-wise), +where the superscripts stand for erasure, symmetric GAD, and +depolarizing channels, respectively. This relation is expected +since Holevo information of the channels are related for a +given p ∈ (0, 1) as χ(e)(Np) ≥ χ(g)(Np) ≥ χ(d)(Np) (point- +wise). +Fig. 6. +Comparison of converse surfaces ¯Θs for erasure, depolarizing and +GADC with ǫ = 0.1, n = 128, and η = 1. The probability of error per +physical qubit is assumed to scale with redundancy k as p(k; p0, θ) = p0(1+ +α(k − 1))γ. +VIII. CONCLUSION +We considered a model of quantum circuits where inputs +and outputs are classical, which includes a large class of +quantum circuits that are used to efficiently solve classical +problems, such as algorithms due to Deutsch-Jozsa, Grover, +and Shor. We demonstrated that the currently best-known +redundancy bound for quantum computation is not applicable +for quantum circuits with classical input and output. We +considered the scenario where quantum states are corrupted +by i.i.d. additive quantum channels. We reduced the problem +of noisy computation to noisy classical communication over +a quantum channel and used one-shot classical capacity of +quantum channels to obtain a lower bound on redundancy. +We also considered a problem of practical interest, namely, +fault-tolerant quantum computation under resource constraints, +which results in physical noise per qubit being scale- +dependent. We cast determining limits of scale dependence +on computational accuracy as an optimization problem, and +derived closed-form expressions whenever possible, and for +other cases we solved the optimization problem numerically. +APPENDIX A +CLOSED-FORM EXPRESSIONS OF THRESHOLDS FOR +COMPUTATIONS CORRUPTED BY ERASURES +The Holevo capacity of erasure channel is χ(Np) = 1 − p. +Therefore, +g(k, p0, θ, ǫ) = c(ǫ, η, n) +k ++ p(k) − 1. +If θ ∈ ¯Θ, then from Corollary (2) the following holds: +g(k, p0, θ, ǫ) = c(ǫ, η, n) +k ++ p(k) − 1 ≥ 0, +∀k ≥ 1 +(20) +Differentiating w.r.t. k and equate to 0 (to find stationary +point), +g′(k, p0, θ, ǫ) = −c(ǫ, η, n) +k2 ++ p′(k; p0, θ) = 0 +(21) +Henceforth, we shall use c = c(ǫ, η, n) for brevity. For a fixed +θ = ∅, α and γ (respectively), the thresholds pth are derived +for some well-behaved p(k; p0, θ) as follows: +1) p(k; p0, θ) = p0 (scale-independent noise, θ = ∅): +In this case, g′(k, p0, θ, ǫ) = 0 as k → ∞. Substituting +in (20), we obtain +p0 − 1 ≥ 0 =⇒ pth = 1. +2) p(k; p0, θ) = p0(1 + α(k − 1)): +Suppose (20) holds for some p0, then: +g(k, p0, θ, ǫ) ≥ 0, +for k = 1, +c + p0 − 1 ≥ 0, +p0 ≥ 1 − c. +(22) +Note that kmax = 1 + α−1((p0)−1 − 1) (obtained by +solving for k in p(k; p0, θ) = 1). Also since p(·; p0, θ) +is linear, g(·, p0, θ, ǫ) is convex in [1, kmax]. Therefore, +g(k, p0, θ, ǫ) is minimized at any one of k = 1, k = +kmax or a stationary point in (1, kmax). Substituting +p′(k; p0, θ) = p0α in (21), the stationary point is: +k = +� c +p0α. +(23) +(a) Note that for k ∈ (1, kmax) to be the minimum, +g′(k, p0, θ, ǫ)|k=1 < 0. Also, noting that p0 ≥ 1 − c +(from Eq. (22)), we obtain +(1 − c)α ≤ p0α < c. +Therefore, α < +c +1−c. Substituting (23) in (20): +p0 ≥ +�√cα − √cα − α + 1 +�2 +(α − 1)2 +, +pth = +�√cα − √cα − α + 1 +�2 +(α − 1)2 +. + +onsConverse regi +Erasure +0.6 +GADC +Depolarizin +0.53 +2 +I +人0.4 +0.3 +0.2 +0.1 +0 +0 +I +2 +3 +aNote that since α < +c +1−c, the second term in the +numerator, cα − α + 1 = 1 − c ≥ 0. Therefore, the +threshold pth exists. +(b) If α ≥ +c +1−c, then +k = +� c +p0α ≥ +� +1 − c +p0 +≥ 1, +p0 ≤ 1 − c. +However, p0 ≥ 1−c from (23). Therefore, pth = 1−c. +3) p(k; p0, θ) = p0kγ: +Here, θ = γ. The value of k ranges from 1 ≤ k ≤ +� +1 +p0 +� 1 +γ . For the choice of pth, (20) must hold for all k in +this range. Similar to linear case, for this choice of p(k) +and range of k, (20) is convex. Hence for pth, +g(k, p0, θ, ǫ) |k=1 ≥ 0 +c + p0 − 1 ≥ 0 +p0 ≥ 1 − c. +Substituting p′(k; p0, θ) = p0γkγ−1 in (21), we obtain +the stationary point as: +k = +� c +p0γ +� +1 +γ+1 +. +(24) +Similar to linear p(k; p0, θ), there are two cases: +(a) If γ < +c +1−c, +Substituting the stationary point computed in (24) in +(20), the threshold pth can be computed as: +p0 ≥ +� γ +c +�γ +(γ + 1)γ+1 , +pth = +� γ +c +�γ +(γ + 1)γ+1 . +(b) If γ ≥ +c +1−c, then, +k = +� c +p0γ +� +1 +γ+1 +≥ +�1 − c +p0 +� +1 +γ+1 +≥ 1, +p0 ≤ 1 − c. +However, p0 ≥ 1−c from (23). Therefore, pth = 1−c. +APPENDIX B +RESTRICTION OF THE FEASIBLE SET OF PL TO [1, kmax] +Let kmax = max{k | p(k; p0, θ) ≤ 1}. If kmax = ∞, then +solving (11) yields pth = 1. Therefore, (11) is non-trivial only +if kmax is finite. Let g1(p0) = mink≥1 gθ(k, p0) and g2(p0) = +mink∈[1,kmax] gθ(k, p0). From (13), it can be observed that +g1(p0) = 0 whenever g2(p0) > 0, and g1(p0) = g2(p0) +whenever g2(p0) ≤ 0. Hence, the threshold pth obtained using +g1(·) and g2(·) as a solution to PL in (11) are identical. +Therefore, (11) can be equivalently solved by restricting the +domain of gθ(·, p0) in PL to [1, kmax]. In other words, one +can replace line 11 with g∗ +θ(p0) = mink∈[1,kmax] gθ(k, p0) +to obtain the same value of threshold pth. Additionally, this +restriction makes the feasible set compact. Moreover, notice +that the restriction and equivalence hold for all channels (not +just erasure) as long as χ(Np) = 0 whenever p = 1. +APPENDIX C +DERIVATION OF LIPSCHITZ CONSTANTS +A. Proof of Lemma 3: Erasure Channel +Let g′ +θ and g′′ +θ denote the partial derivatives +∂2 +∂k2 gθ(k, p0) +and +∂2 +∂k2 gθ(k, p0), respectively. The magnitude of the second +order partial derivative is bounded above as: +|g′′ +θ| ≤ max +k +���� +2c +k3 +���� + max +k +|p′′(k; p0, θ)| , +where the inequality follows from triangle inequality and +maximizing each summand. Observe that the first summand +is maximized when k = 1, and the second term is bounded +above as +p′′(k; p0, θ) ≤ +� +α2γ(γ − 1)p0, +1 ≤ γ < 2, k = 1, and +α2γ(γ − 1)p2/γ +0 +, +γ ≥ 2, k = kmax, +≤ α2γ2p +1 +γ +0 , +γ > 0, +where kmax = 1 + α−1(p−(1/γ) +0 +− 1). Therefore, +g′′ +θ ≤ 2c + α2γ2p +1 +γ +0 =: L. +B. Proof of Lemma 4: Symmetric GADC +Let q(p) = q(p(k)) = +1−√ +1−p(k) +2 +; the magnitude of the +second-order derivative of gθ(k, p0) is bounded above as: +|g′′ +θ| ≤ max +k +���� +2c +k3 +���� + max +k +|h′′ +2 (q(k))| , +≤ 2c + max +k {|h′′ +2(q)|q′(p)2p′(k; p0, θ)2 ++ h′ +2(q)q′′(p)p′(k; p0, θ)2 + h′ +2(q)q′(p)p′′(k; p0, θ)} +Noting that |h′′(q)| is maximized when k = 1, and remaining +the terms are maximized when k = kmax, we obtain: +|g′′ +θ| ≤ 2c+α2γ2p +min{1, 2 +γ } +0 +4(1 − p0) +� +1 +p0 ln 2 ++1 + 2(1 − p0) +2√1 − p0 +log2 P +� +=: L, +where P = 1+√1−p0 +1−√1−p0 . +C. Proof of Lemma 5: Depolarizing Channel +The second partial derivative of gθ(k, p0) is bounded above +as: +|g′′ +θ| ≤ max +k +���� +2c +k3 +���� + max +k +|h′′ +2(p(k)/2)| , +|g′′ +θ| ≤ 2c + max +k +�1 +4|h′′ +2(z)|p′(k; p0, θ)2 + 1 +2h′(z)p′′(k; p0, θ) +� +, + +where z = p(k; p0, θ)/2. +|g′′ +θ| ≤ 2c+ +α2γ2p2/γ +0 +p0(2 − p0) ln 2 ++α2γ(γ − 1)p +min{1, 2 +γ } +0 +2 +log2 +�2 − p0 +p0 +� +, +|g′′ +θ| ≤ 2c+α2γ2p +min{1, 2 +γ } +0 +� +log2 e +p0(2 − p0) + 1 +2 log2 +�2 − p0 +p0 +�� +=: L. +APPENDIX D +LEMMAS: PROJECTED GRADIENT DESCENT +Definition 6. In Appendices D and E, we consider g(·) to be +of the following form: g : [1, kmax] → +R : k �→ g(k), where +g′(·) is L-Lipschitz. +Lemma 6. Stopping criterion and bounded gradient: Suppose +a pair of iterates (kj, kj+1), which lie in the interior (1, kmax), +generated by PROJGD satisfy the stopping criterion g(kj) − +g(kj+1) < +δ2 +2Lk2max , then the first order derivative is bounded +above as |g′(kj)| < +δ +kmax . +Proof: Applying the descent lemma to kj, kj+1, we get +g(kj+1, p0) ≤ g(kj, p0) ++g′(kj, p0)(kj+1 − kj) + 1 +2L|kj+1 − kj|2. +(25) +Substituting kj+1 − kj = −ξgθ(kj) in (25): +ξ +� +1 − ξL +2 +� +|g′(kj)|2 ≤ g(kj) − g(kj+1). +Choosing ξ = 1 +L, we obtain: +1 +2L|g′(kj)|2 ≤ g(kj) − g(kj+1) < +δ2 +2Lk2max +. +Therefore, +|g′(kj)| < +δ +kmax +. +Lemma +7. +Suppose +g(·) +is +convex, +and +g∗ += +mink∈[1,kmax] +g(k). +If +|g′(kj)| +< +δ/kmax, +then +|g(kj) − g∗| ≤ δ, for any kj ∈ [1, kmax]. +Proof: From the convexity of g(·), we have g(k) ≥ +g′(kj)(k−kj)+g(kj), for any k, kj ∈ [1, kmax]. If g′(kj) < 0, +then: +g(k) − g(kj) ≥ g′(kj)(kmax − kj) ≥ g′(kj)(kmax − 1), ∀k. +Therefore, +g(kj) − g∗ ≤ |g′(kj)|(kmax − 1) ≤ |g′(kj)|kmax ≤ δ. +On the other hand, if g′(kj) ≥ 0, then +g(k) − g(kj) ≥ g′(kj)(k − kj) ≥ g′(kj)(1 − kj), ∀k. +g(kj) − g∗ ≤ g′(kj)(kmax − 1) ≤ g′(kj)kmax ≤ δ. +Therefore, combining both cases: if |g′(kj)| ≤ +δ +kmax , then +|g(kj) − g∗| ≤ δ. +Lemma 8. Projected gradient descent (PROJGD) does not +cross any stationary point: Let kj and kj+1 be the successive +iterates generated by PROJGD routine for g(·). Suppose, the +step-size ξ ∈ (0, 1 +L], then g′(kj)g′(kj+1) ≥ 0. +Proof: From the definition of Lipschitz gradient, we have +|g′(kj) − g′(kj+1)| ≤ L|kj − kj+1| = Lξ|g′(kj)|, where the +last equality holds, since kj+1 is generated from PROJGD +routine. Suppose, g′(kj) ≥ 0, then the following inequalities +hold: +−Lξg′(kj) ≤ g′(kj) − g′(kj+1) ≤ Lξg′(kj), +−Lξg′(kj) ≤ g′(kj+1) − g′(kj) ≤ Lξg′(kj), +(1 − Lξ)g′(kj) ≤ g′(kj+1) ≤ (1 + Lξ)g′(kj). +For ξ ≤ 1 +L, we obtain: +g′(kj+1) ≥ g′(kj)(1 − Lξ) ≥ 0. +Symmetrically, if g′(kj) ≤ 0, then g′(kj+1) ≤ 0. Combining +both cases, we obtain g′(kj)g′(kj+1) ≥ 0. +Lemma 9. Least difference between the successive iterates of +PROJGD: Let kj and kj+1 be the successive iterates generated +by PROJGD routine for g(·), with a step size ξ ∈ (0, 1 +L]. If +|g(kj) − g(kj+1)| > ζ, then |kj − kj+1| > +� +2ζ +L . +Proof: Using descent lemma [31] on g(·) at kj and kj+1, +we obtain +g(kj)−g(kj+1) ≤ g′(kj+1)(kj−kj+1)+ L +2 |kj−kj+1|2. (26) +Suppose we g′(kj) ≤ 0 and g(kj) − g(kj+1) > ζ, then +g′(kj+1) ≤ 0 (from Lemma 8), and noting that (kj−kj+1) ≥ 0 +we obtain: +ζ < L +2 |kj − kj+1|2, +(27) +Therefore, +|kj+1 − kj| > +�� +2ζ +L , +if kj+1 < kmax, +2 +L(g(kj) − g(kmax)), +if kj+1 = kmax +The second case should be handled separately since (27) may +not hold when kj+1 = kmax. +Corollary 3. Suppose PROJGD is used with stopping criterion +1 (Definition 3). If |kj − kj+1| ≤ +δ +Lkmax , then |g(kj) − +g(kj+1)| ≤ +δ2 +2Lk2max , when kj+1 < kmax. +Proof: The result follows by substituting ζ = +δ2 +2Lk2max in +Lemma 9. +Corollary 4. Suppose PROJGD is used with stopping criterion +2 (Definition 4). If |kj − kj+1| ≤ +� +2δ +L , then |g(kj) − +g(kj+1)| ≤ δ, when kj+1 < kmax. +Proof: The result follows by substituting ζ += δ in +Lemma 9. + +APPENDIX E +LEMMAS: PERTURBATION +Lemma 10. PERTURB routine does not miss stationary points: +Suppose kj meets the stopping criterion 2 (Definition 4). If the +perturbation ∆k ≤ +� +2δ +L , then the PERTURB routine does not +miss any stationary points with an error greater than δ. +Proof: Let kj, kj+1 be any two points in [1, kmax]. Since +g′(·) is L-Lipschitz, (26) holds. Let kj+1 be the closest +stationary point to kj, then: +|g(kj) − g(kj+1)| ≤ L +2 |kj − kj+1|2. +Therefore, the following condition is necessary for the stop- +ping criterion 2, i.e., |g(kj) − g(kj+1)| ≥ δ (Definition 4) to +hold: +∆k = |kj − kj+1| ≥ +� +2δ +L . +Lemma 11. Upper bound on the number of stationary points: +Consider a set of stationary points {ks} of g(·) in [1, kmax] +such that for every ks, the adjacent stationary point ks+1, +|g(ks) − g(ks+1)| ≥ δ. The number of such stationary points +is finite and bounded above as ⌈kmaxL/δ⌉. +Proof: From the proof of Lemma 10, it follows that if +|g(ks)−g(ks+1)| ≥ δ, then the stationary points are separated +by at least |ks − ks+1| ≥ +� +2δ +L . 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Wilde, “Principles of quantum communication +theory: A modern approach,” arXiv:2011.04672 [quant-ph]., Nov. 2020. +[29] S. Khatri, K. Sharma, and M. M. Wilde, “Information-theoretic aspects +of the generalized amplitude-damping channel,” Physical Review A, vol. +102, no. 1, p. 012401, 2020. +[30] S. Wright, J. Nocedal et al., “Numerical optimization,” Springer Science, +vol. 35, no. 67-68, p. 7, 1999. +[31] D. P. Bertsekas, “Nonlinear programming,” Journal of the Operational +Research Society, vol. 48, no. 3, pp. 334–334, 1997. + diff --git a/69A0T4oBgHgl3EQfOP-G/content/tmp_files/load_file.txt b/69A0T4oBgHgl3EQfOP-G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa7c84dd1db45772315cac11479dcc0cb9f0ff97 --- /dev/null +++ b/69A0T4oBgHgl3EQfOP-G/content/tmp_files/load_file.txt @@ -0,0 +1,911 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf,len=910 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='02158v1 [quant-ph] 5 Jan 2023 Limits of Fault-Tolerance on Resource-Constrained Quantum Circuits for Classical Problems Uthirakalyani G†, Anuj K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Nayak†, Avhishek Chatterjee, and Lav R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Varshney, Senior Member IEEE Abstract—Existing lower bounds on redundancy in fault- tolerant quantum circuits are applicable when both the input and the intended output are quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' These bounds may not necessarily hold, however, when the input and the intended output are classical bits, as in the Deutsch-Jozsa, Grover, or Shor algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here we show that indeed, noise thresholds obtained from existing bounds do not apply to a simple fault-tolerant implementation of the Deutsch-Jozsa algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then we obtain the first lower bound on the minimum required redundancy for fault-tolerant quantum circuits with classical inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Recent results show that due to physical resource constraints in quantum circuits, increasing redundancy can increase noise, which in turn may render many fault-tolerance schemes useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' So it is of both practical and theoretical interest to characterize the effect of resource constraints on the fundamental limits of fault-tolerant quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus as an application of our lower bound, we characterize the fundamental limit of fault- tolerant quantum circuits with classical inputs and outputs under resource constraint-induced noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Keywords—fault-tolerant computing, redundancy, resource constraints I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' INTRODUCTION Initial ideas [1], [2], and especially mathematical demon- strations of advantages of quantum computing over classical computing [3], [4], have spurred considerable interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' How- ever, noise in quantum circuits heavily restricts the class of problems that can be solved using quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Indeed, the formal term NISQ (Noisy Intermediate Scale Quantum) has been introduced to describe the current era where quantum processors are noise-limited [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' To limit the corruption of quantum states due to noise, the pursuit of fault-tolerant quantum circuits has led to a large literature in quantum error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Early papers demonstrated that one can achieve arbitrary computational accuracy when physical noise is below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Achievability of any desired fault tolerance in these initial works required a poly-logarithmic redundancy with respect to the size of the quantum circuit [6]–[9], but more recent works extend such threshold theorems to require only a constant overhead [10], [11], reminiscent of work in classical fault- tolerant computing [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' † The student authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Uthirakalyani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' G and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Chatterjee are with the Department of Electrical Engineering, Indian Institute of Technology, Madras, Chennai 600036, India (emails:{ee19d404@smail,avhishek@ee}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='iitm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Nayak and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Varshney are with Coordinated Science Labora- tory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (emails:{anujk4, varshney}@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This work was supported in part by National Science Foundation grant PHY-2112890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In this direction, some works provide fundamental limits (lower bounds) on redundancy for arbitrarily accurate compu- tation [14]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, all of these lower bounds are for quantum input/output, rather than classical input/output which is common for a large class of algorithms, such as those due to Deutsch-Jozsa [4], Shor [19], and Grover [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, we demonstrate by example that lower bounds obtained so far in quantum fault tolerance are not applicable for quantum circuits with classical input/output, and provide a general alternate bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As far as we know, this is the first lower bound on fault tolerance for quantum circuits with classical input/output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The effects of noise on computational accuracy of quantum circuits are typically studied assuming the noise per physical qubit is constant with respect to the the size of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Un- fortunately, this is not true in many quantum devices today— often due to limited physical resources such as energy [21], volume [22], or available bandwidth [23]—that have physical noise levels that grow as the quantum computer grows [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Fellous-Asiani, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' introduce physical models of such scale- dependent noise and also aim to extend threshold theorems to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, the characterization of computational error (per logical qubit error) is empirical in nature, lacking precise mathematical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Moreover, the characterization depends on specific implementation and is restricted to con- catenated codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, using our new redundancy lower bound and tools from optimization theory, we characterize the limits of scale-dependence on fault-tolerant quantum circuits with classical input/output, agnostic to specific implementation and error correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The two motivations for the present work are therefore to obtain lower bounds on the required redundancy of a quantum circuit for computation with classical input/output, and to investigate the effect of resource constraints (like energy or volume) on this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The distance between the distributions of the output clas- sical bit corresponding to two different quantum input states vanishes exponentially with the depth of the circuit when noise is above a threshold [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Similarly for the trace distance be- tween the output quantum states [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' These results, however, do not apply when the depth of the circuit is small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' they also do not provide lower bounds on required redundancy for sub- threshold noise when fault-tolerant computation is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This work focuses on shallow quantum circuits whose input and output are classical, aiming for converse results for classical computation using quantum circuits that yield lower bounds on required redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In [14], [18], lower bounds on required redundancy that also led to improved noise thresholds were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, the fault-tolerance criteria in [14], [18] are not appropriate for our setting, as Section II argues using the example of the well-known Deutsch-Jozsa algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The experimental finding that noise increases with more redundancy under resource constraints implies that simple per (logical) qubit redundancy cannot achieve arbitrary computa- tional accuracy even if noise per physical qubit is below the fault-tolerance threshold, in contrast to conventional threshold theorems [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This limitation is due to two opposing forces: improvement in accuracy due to increased redundancy and worse overall noise with redundancy due to scale-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In this regard, we aim to find the sweet spot on redundancy for a desired computational accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Sec- tion II motivates our work through a counterexample that illus- trates the need for a new redundancy lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Section III then gives the mathematical models of computation, noise, and resource constraints that form the basis of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then, the primary contributions follow: Section IV proves a converse bound on redundancy required for classical computation on quantum circuits, drawing on one-shot capacity of classical-quantum chan- nels (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Section V through VII analyze the converse results on the limits of scale-dependence for fault-tolerant computation, including closed-form and numerical solutions for some canonical quantum device models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Finally, Section VIII concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' NEED FOR A NEW REDUNDANCY LOWER BOUND: A COUNTEREXAMPLE Consider a quantum circuit that suffers from erasure noise (with erasure probability p) right before the final measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' From one of the best known noise threshold bounds [14] and the capacity results for erasure channels, it follows that for an erasure noise per physical qubit p > 1 2, fault-tolerant computation is not possible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', the required redundancy is not finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, we demonstrate that a simple adaptation of the Deutsch-Jozsa algorithm on this quantum circuit (with erasure noise before the final measurement) can have a prob- ability of error less than any ǫ > 0 even if p > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' A schema of quantum circuit that implements Deutsch-Jozsa algorithm with erasure noise, to demonstrate the need for a new redundancy bound for fault-tolerant quantum computation with classical I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The Deutsch-Jozsa algorithm is used to determine if the given function oracle, f : {0, 1}n → {0, 1} is constant (0 or 1 for all input strings) or balanced (0 for half the input strings and 1 for the rest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' From [25, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='51], the quantum state before measurement under the absence of noise is ψ1 = � z,x∈{0,1}n (−1)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='z+f(x) 2n |z⟩ |y⟩ , where |y⟩ = |0⟩−|1⟩ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Measuring the first n qubits yields either |0⟩⊗n if f(·) is constant, or an n-qubit state from {|0⟩ , |1⟩}⊗n \\ {|0⟩⊗n} if f(·) is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose the quantum states are corrupted by erasure right before measurement as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' then the state of the circuit becomes ψ2 = � z,x∈{0,1}n (−1)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='z+f(x) 2n |z⟩(e) |y⟩(e) , where |z⟩(e) and |y⟩(e) are the corrupted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' erased) versions of |z⟩ and |y⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For example, if f(·) is constant, |z⟩(e) = |e00e00 · · ·e0⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', each qubit state |0⟩ is replaced i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' with probability p by qubit state |e⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Now, consider our modified algorithm: 1) Run Deutsch-Jozsa algorithm T times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2) If no |1⟩ state was measured in any run, declare function oracle f(·) a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' When f(·) is balanced, the measurement in the no-erasure case must have one or more |1⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Such an oracle can be incorrectly declared as constant when all of these |1⟩ states are erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' So the probability of error is: Pe = P{f(·) is declared constant |f(·) is balanced}, = P \uf8f1 \uf8f2 \uf8f3 � j,t |zj⟩(e) t ̸= |1⟩ ���f(·) is balanced \uf8fc \uf8fd \uf8fe , ≤ P �� t |zj⟩(e) t = |e⟩ ��� |zj⟩ = |1⟩ � , for some j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since erasures are independent, Pe ≤ T � t=1 P � |zj⟩(e) t = |e⟩ ��� |zj⟩ = |1⟩ � = pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Choosing T ≥ ��� ln ǫ ln p ���, one can achieve Pe ≤ ǫ for any p ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This counterexample proves that the bound for redundancy N ≥ n Q(N) proposed in [14] does not hold for quantum computation with classical input/output, since for an erasure channel, the quantum capacity Q(N) = max{0, 1 − 2p} = 0 as p > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This motivates the need for a different bound which holds for classical I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that this does not imply the prior bounds are incorrect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' the apparent contradiction is due differences in the definition of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Prior work [14] uses a notion of distance (or similarity) between the output quantum states of noiseless and noisy circuits to quantify accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This requirement is too stringent when the output bits are classical and error probability is a more suitable performance criterion [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As such, we obtain a lower bound on the redundancy under the error probability criterion and then study the effect of resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Model of Computation Consider the computational model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2, which is a quantum circuit with classical inputs and classical outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This is denoted by CQC : {0, 1}n → {0, 1}n or equivalently CQC(x) for x ∈ {0, 1}n, where n is the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The goal of the circuit is to realize a function f : {0, 1}n → {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The circuit consists of l layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The first layer takes n clas- sical inputs (x) as orthogonal quantum states |0⟩ and |1⟩ along with N − n ancillas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' It maps the input to a density operator of dimension 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Any subsequent layer i, for 2 ≤ i ≤ l − 1, takes the output of the previous layer, layer i − 1 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The output of any layer i, 1 ≤ i ≤ l − 1, is a density operator of dimension 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The final layer, layer l, performs a POVM measurement and obtains classical output CQC(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Each layer i, with i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', l − 1} is a noisy quantum operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This is modeled as a noiseless quantum operation Li on density operators of dimensions 2N followed by N i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' quantum channels N (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Finally, the last layer, layer l, performs a measurement (POVM), which yields a classical output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus the quantum circuit can be represented as a composition of quantum operations as CQC(x) = Ll ◦ N ⊗N ◦ Ll−1 ◦ · · · ◦ L2 ◦ N ⊗N ◦ L1(x), where ◦ has the usual meaning of function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We use the notation QCl−1 to denote the combined opera- tions of layers 2 to l, given by Ll◦N ⊗N ◦Ll−1◦· · ·◦N ⊗N ◦L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Noise models Here, we consider only Holevo-additive channels charac- terized by a single parameter p ∈ [0, 1] and whose Holevo capacity is monotonically decreasing in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We use the generic notation Np for such a channel with parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Examples include erasure, depolarizing, and symmetric GAD channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' a) Erasure Channel: In a quantum erasure channel (QEC), each qubit flips to |e⟩⟨e|, which is orthogonal to every ρ ∈ L(Cd), with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, whenever a qubit gets corrupted, the location of corruption is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Np(ρ) = (1 − p)ρ + ρTr[ρ] |e⟩⟨e| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The classical capacity is [28]: χ(Np) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (1) b) Depolarizing Channel: When a qubit undergoes de- polarizing noise, it is replaced by a maximally mixed state I/2 with probability p [28]: Np(ρ) = (1 − p)ρ + p 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In contrast to the erasure channel, the receiver (or the decoder) is not aware of the location of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The Holevo informa- tion of the depolarizing channel is: χ(Np) = 1 − h2 � p 2 � , (2) where h2(·) is the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that the Holevo information is similar to the capacity of a binary symmetric channel with crossover probability p/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' c) Generalized Amplitude Damping Channel (GADC): Amplitude damping channels model the transformation of an excited atom to ground state by spontaneous emission of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The changes are expressed using |0⟩ for the ground (no photon) state and |1⟩ for the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' If the initial state of the environment |0⟩⟨0|, is replaced by the state θµ ≜ (1 − µ) |0⟩⟨0| + µ |1⟩⟨1| , µ ∈ [0, 1] where, µ is thermal noise, we get the generalized ADC described using the following four Kraus operators [29]: A1 = � 1 − µ �1 0 0 √1 − p � , A2 = � 1 − µ �0 √p 0 0 � , A3 = √µ �√1 − p 0 0 1 � , A4 = √µ � 0 0 √p 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' GADC is not additive in general (for arbitrary µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, in the special case of symmetric generalized amplitude damping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', generalized amplitude damping with µ = 1/2, it is a Holevo additive channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The classical capacity of symmetric GADC (µ = 1/2) is [29]: χ(Np) = 1 − h2 � 1−√1−p 2 � , (3) where p is the probability an atom decays from excited to ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that we have used p to describe different impairments in different channels, so p must be interpreted appropriately based on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Resource Constraints and Scale-Dependent Noise In [24], it was shown that resource constraints can lead to an increase in noise with increase in redundancy, scale- dependent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' A few models of scale-dependent noise have been studied in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Let k ≜ N/n ≥ 1 be the redundancy and p(k) be the noise strength when the redundancy is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (Recall that we consider Holevo-additive noise models that can be characterized by a single parameter 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=') In the polynomial model, p(k) = min(p0(1 + α(k − 1))γ, 1) and in the exponential model, p(k) = min(p0 exp(α(k − 1)γ), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, p0 ∈ [0, 1] is the noise strength in the absence of any redundancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', k = 1, and α and γ are positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Motivated by practically useful noise models like erasure, depolarization, and models for scale dependence in [24], we consider the following generic scale-dependent noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Noise Np is parameterized by a single parameter p ∈ [0, 1] and the Holevo information χ(Np) is non-increasing in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The parameter p is a function of redundancy k, given by min(p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ), 1), where θ is a tuple of non-negative parameters from the set K, and 1) p0 = p(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) for all θ, 2) for any k ≥ 1, p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is non-decreasing in any component of θ and in p0, given the other parameters are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, p0 represents the noise without redundancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', the initial noise without any resource constraint arising due to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' CQC model of computation: classical input, quantum computation, and classical output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Clearly, the polynomial and exponential models are special cases with θ = (α, γ) ∈ K = R2 ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The threshold for p0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', the minimum p0 beyond which reliable quantum computation is not possible, was studied in [24] assuming concatenated codes for error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, we obtain a universal threshold for all fault tolerance schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' LOWER BOUND ON REQUIRED REDUNDANCY We first define the accuracy criterion for computation using quantum circuits with classical input/output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then we show how to convert the noisy computation problem to a communi- cation problem over i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This finally leads to the redundancy bound in Theorem 1, which we use to obtain thresholds for p0 under resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose f(·) is a classical function realized by a quantum circuit CQC(·) as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then the ǫ-accuracy is: P{CQC(x) ̸= f(x)} < ǫ, for all x ∈ Zn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (4) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (4) holds for all x ∈ Zn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, the ǫ-accuracy condition holds for any subset of Zn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The following lemma states a necessary condition for ǫ-accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Consider x(1), x(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' , x(R) ∈ Zn 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' |{f(x(i)) : 1 ≤ i ≤ R}| = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then a necessary condition for ǫ-accuracy condition (4) to hold is P{CQC(x(i)) ̸= f(x(i))} < ǫ, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that the domain is restricted to R inputs, such that the restricted mapping is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Using this bijectivity, we obtain the following simpler lemma, which connects ǫ-accurate computation with finite blocklength communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose there exists a CQC(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' P{CQC(x(i)) ̸= f(x(i))} < ǫ for all 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then there exists a classical circuit C(·) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' P{C(CQC(x(i))) ̸= x(i)} < ǫ, for all 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (5) Proof: Suppose ˆf(·) is a restriction of f(·) such that the mapping ˆf : {x(i), 1 ≤ i ≤ R} → {f(x(i)), 1 ≤ i ≤ R}, is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then the inverse map ˆf −1(·) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Choosing a C(·) that implements ˆf −1(·), the probability of error can be equivalently expressed as P{C(CQC(x(i))) ̸= x(i)} < ǫ, for all 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We define ˆx(i) to be the output of C(CQC(x(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then the condition in (5) is equivalent to max x(i) P{x(i) ̸= ˆx(i)} < ǫ, 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This implies that a necessary condition to satisfy accuracy condition (4) is inf L1,C,QCl−1 max x(i) P{x(i) ̸= ˆx(i)} ≤ max x(i) P{x(i) ̸= ˆx(i)} < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that L1 is an encoding of classical bits into a quantum state, and QCl−1 followed by C(·) can be interpreted as the decoding of the noisy version of the same quantum state (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Hence, infL1,C,QCl−1 maxx(i) P{x(i) ̸= ˆx(i)} is equivalent to the maximum probability of error for transmitting message x(i), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' , R} over the channel N ⊗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Using this reduction, we lower-bound the redundancy for any classical computation using a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Let f : {0, 1}n → {0, 1}n be a classical function and Rf = |{f(x) : x ∈ {0, 1}n}| be the cardinality of the range of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then, for computing a classical function f with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Reduction of noisy computation model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2 to noisy communi- cation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' ǫ-accuracy using a quantum circuit corrupted by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Holevo- additive noise, the required number of physical qubits N is bounded as N > (1 − ǫ) log2(Rf) − h2(ǫ) χ(N) for all ǫ ∈ [0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: For any additive quantum channel N, an upper bound for classical communication over a quantum channel using an (M, N, ǫ) code is [28]: log2(|M|) ≤ χ(N ⊗N) + h2(ǫ) 1 − ǫ , where M is the message alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Assigning |M| = Rf yields ǫ > Pe ≥ 1 − χ(N ⊗N) + h2(Pe) log2 Rf , (6) ≥ 1 − χ(N ⊗N) + h2(ǫ) log2 Rf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The last inequality holds, since h2(·) is increasing in [0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Rearranging, we obtain χ(N ⊗N) > (1 − ǫ) log2 Rf − h2(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Noting that Holevo information is sub-additive, Nχ(N) > (1 − ǫ) log2 Rf − h2(ǫ), N > (1 − ǫ) log2 Rf − h2(ǫ) χ(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (7) The bound in Theorem 1 states that the number of quantum buffers, N, needed for accurate computation of an n-bit function f is lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' III, k ≜ N n is the redundancy of the quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus, Theorem 1 can be seen as a lower bound on the required redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The quantity log2 Rf is the number of bits needed to encode the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We define η ≜ log2 Rf n as the compression factor of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' To understand the impact of scale-dependent noise, we will use the following corollary of Theorem 1 that gives a lower bound on the redundancy k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The condition for ǫ-accuracy in Theorem 1 is alternatively k > c(ǫ, η, n) χ(N) , (8) where c(ǫ, η, n) ≜ (1 − ǫ)η − h2(ǫ) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: Substituting log2 Rf = ηn and k = N/n in (7), and rearranging we obtain N > (1 − ǫ)ηn − h2(ǫ) χ(N) , k > c(ǫ, η, n) χ(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' SCALE-DEPENDENT NOISE: CONVERSE REGIONS Let Np denote the channel parameterized by a noise in- tensity term p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Examples include the probability of erasure, p, for erasure channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' the probability of a quantum state being replaced by a maximally mixed state, p, for depolarizing channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' and the amplitude decay parameter, p, for symmetric GAD channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' If the error per physical qubit p is a constant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' k, then χ(Np) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, one can ensure that the necessary condition for ǫ-accuracy in (8) is satisfied by sufficiently increasing redundancy (choosing large k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' On the other hand, if p scales (increases) with redundancy, then χ(N) decreases with k (we denote the dependency on p(k) as χ(Np(k))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, satisfying ǫ-accuracy condition k > c(ǫ, η, n)/χ(Np(k)) is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In fact, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 4 plots the error probability lower bound (6) with both scale- independent and scale-dependent erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Notice that when the physical noise is independent of k, the Pe lower bound rapidly decreases with an increase in redundancy, whereas when the noise is scale dependent, the probability of error initially decreases with increasing redundancy k, but then grows beyond a certain optimum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' With this motivation, we explore the limitations of ǫ-accurate computation under scale- dependent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We specifically aim to characterize the set of (p0, θ) for which ǫ-accurate computation is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This is equiv- alent to the noise threshold in traditional models, with scale- independent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The following corollary to Theorem 1 provides a converse in terms of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose we have, ¯Θ ≜ � (p0, θ) ∈ K ���min k≥1 g(k, p0, θ, ǫ) ≥ 0 � , where g(k, p0, θ, ǫ) ≜ c(ǫ, η, n) k − χ(Np(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then ǫ-accurate computation is not possible for θ ∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Also, if (p0, θ) ∈ ¯Θ then (p′ 0, θ′) ∈ ¯Θ if (p′ 0, θ′) ≥ (p0, θ) in a component-wise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='75 Redundancy (k) 10−2 10−1 100 Pe (lower bound) p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='15 p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='20 p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Comparison between Pe lower bound with (solid lines) and without (dashed-lines) scale-dependent physical noise for erasure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: From Definition 2, we must prove that if Pe < ǫ, then (p0, θ) /∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Considering the scale-dependent noise in Corollary 1, we have if Pe < ǫ, then k > c(ǫ, η, n) χ(Np((k)), g(k, p0, θ, ǫ) = c(ǫ, η, n) k − χ(Np(k)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (9) For any θ ∈ K, (9) is satisfied only if min k≥1 g(k, p0, θ, ǫ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In other words, (p0, θ) /∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As χ(Np) is non-increasing in p and p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is non-decreasing in each component, (p′ 0, θ′) ≥ (p0, θ) in a component-wise sense implies (p′ 0, θ′) ∈ ¯Θ whenever (p0, θ) ∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We refer to ¯Θ as the converse region since ǫ-accurate classical computation on quantum circuits is not possible if the parameters of the scale-dependent noise are in ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As any fault- tolerant implementation has to avoid this region, characterizing ¯Θ is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' By Corollary 2, for characterizing ¯Θ, it is enough to find the minimum p0 for each θ such that (p0, θ) ∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' More precisely, for a fixed θ, the threshold pth(θ) can be defined as: pth(θ) := inf{p0 | (p0, θ) ∈ ¯Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (10) The threshold pth(θ) (or pth for brevity) can be obtained by solving the following optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' minimize p0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' min k≥1, 0≤p(k)≤1 gθ(k, p0) ≥ 0, (11) where, gθ(k, p0) := g(k, p0, θ, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Consider the following optimization problem PL : min k≥1, 0≤p(k)≤1 gθ(k, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Clearly, (11) has the optimization problem PL, which we refer to as the lower-level optimization problem, as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus, (11) is a bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For a given set of θ the solution to PL is a function of p0, which we denote as g∗ θ(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus, the bi-level optimization problem in (11) can also be written as min p0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' g∗ θ(p0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (12) In general, to compute the threshold pth one needs to solve (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, for erasure noise and some special classes of p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ), closed-form expressions for pth can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For erasure noise, thresholds are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1) If p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0) = p0 (constant), then pth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2) If p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, α) = p0(1 + α(k − 1)), then pth = \uf8f1 \uf8f2 \uf8f3 1 − c, if α ≥ c 1−c, and ( √cα−√cα−α+1) 2 (α−1)2 , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3) If p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, γ) = p0kγ, then pth = \uf8f1 \uf8f2 \uf8f3 1 − c, if γ ≥ c 1−c, and ( γ c ) γ (γ+1)γ+1 , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here c = c(ǫ, η, n), defined in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that θ = ∅, α and γ in cases 1), 2) and 3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: Consider a procedure to find a closed-form expres- sion for pth as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1) Minimize gθ(k, p0) over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is non- decreasing in k, it is enough to minimize gθ(k, p0) over [1, kmax], where kmax = max{k | p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) ≤ 1} (see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The minimum occurs at either k = 1, k = kmax or a stationary point of gθ(k, p0) in [1, kmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2) Substitute the minimizer k into gθ(k, p0) ≥ 0, which yields an equation in p0, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3) Solving the equation for p0 yields a closed-form expres- sion for pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The derivation of pth for corresponding p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For a general p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ), however, a closed-form expression for pth in terms of θ cannot be obtained, and therefore, pth must be computed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We develop Algorithm 1 to obtain pth by solving bi-level optimization problem (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In Algorithm 1, we solve the alternate formulation (12) using the bisection method, while assuming access to an oracle that computes g∗ θ(p0) for any p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Later, we also develop efficient algorithms that solve PL and obtain g∗ θ(p0) for any p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Algorithm 1 computes the threshold pth (up to an error of δp0), for a pre-determined set of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' First, a channel-specific Lipschitz constant L is computed using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (15), (17), or (19) for a given (p0, θ), which determines how quickly PL is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Lines 7–17 describe the bisection method to compute pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Depending on whether PL is convex or non-convex, Algorithm 2 or Algorithm 3 is used to compute g∗ θ(p0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The following theorem provides a proof of global conver- gence of Algorithm 1, with only a monotonicity assumption in θ (note that continuity in θ is not needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose a quantum circuit is corrupted by a scale- dependent noise-per-physical qubit, p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' θ) that is monotonic in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then for any given θ ∈ K, the sequence {p0i} generated using Algorithm 1 converges to the threshold pth in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: Algorithm 1 generates a non-increasing sequence {p+ 0i} and a non-decreasing sequence {p− 0i}, which at every iteration yields g∗(p+ 0i) ≥ 0 and g∗(p− 0i) < 0, with p0i = p+ 0i +p− 0i 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since the bisection method halves the difference between p+ 0i and p− 0i at every iteration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', p+ 0i+1 − p− 0i+1 = p+ 0i −p− 0i 2 ), we have that for all ǫ > 0, there exists an i0 such that for all i ≥ i0, we get p+ 0i −p− 0i < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Also, since both {p+ 0i} and {p− 0i} are bounded, they converge, and since for all i ≥ i0, p+ 0i −p− 0i < ǫ, they converge to a common limit point (say p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since, K is closed, (p∗ 0, α, γ) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Due to the monotonicity of g∗ θ(p0) (non-decreasing with p0), the following inequality holds: g∗ θ(p− 0i) ≤ g∗ θ(p∗ 0) ≤ g∗ θ(p+ 0i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, g∗ θ(p0) < 0, for all p0 < p∗, and g∗ θ(p0) ≥ 0, for all p0 > p∗, which is by definition p∗ = pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Obtaining g∗ θ(·) requires solving PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Next, we present efficient algorithms for solving PL for erasure, depolarizing, and symmetric GAD channels, and numerically obtain the converse surface for those noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Algorithm 1 Algorithm to obtain pth/¯Θs numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1: Initialize the set K′ ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2: Initialize max iters, δp0, δ, ¯Θs = {}, k ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3: for each θ ∈ K′ do 4: Initialize i ← 0, ∆p0 ← 1, p0 ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='5, 5: p− 0 ← 0, p+ 0 ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 6: % BISECTION METHOD 7: while ∆p0 > δp0 and i ˜κ, |f⟩ and |b⟩ +exhibits cyclic exchange of optical power (as a function +of z) which is a primitive outcome for a PT -symmetric +dynamics. On the other hand, a strong coupling regime + +3 +FIG. 1. a) Shows the reflection spectrum of a conventional +(periodic) 1D-PC. b) Shows the variation in Re(e1) (dotted +black curve), Im(e1) (dotted maroon curve), Re(e2) (solid +black curve) and Im(e2) (solid maroon curve) as a function +of frequency (ν). c) and d) Shows the mode-field intensity for +frequencies within the PBG 240 THz and that outside the +PBG 310 THz respectively. The solid red arrow represents +the direction of incidence of light. +where ∆k < ˜κ manifests through a monotonic growth +of backscattered mode (|b⟩) that is a signature of PT - +symmetry broken phase. It is worthwhile to reiterate the +point that the two regimes depicted by the inequality +of ∆k and ˜κ (in the parameter space) could be mapped +onto the PBG and pass or transmission band (s) in the +reflected spectrum. Subsequently, each PBG is necessar- +ily bounded by two EPs in this framework. Additionally, +these two EPs are fixed and could not be tailored for +a given 1D-PC with a fixed duty cycle and fixed period. +Also, the conventional 1D-PC geometry excludes the pos- +sibility of realizing higher-order exceptional points [31]. +Taking a cue from this critical viewpoint, we note that +a small apodization or gradual change in dielectric fill- +ing fraction (ζ) of each unit cell of the 1D-PC would +allow us to realize discretely spaced (multiple) EPs at +different optical frequencies (or wavelengths). In order +to elucidate this point, we recall that ∆k as well as ˜κ is +a function of ζ. An optimum spatial variation in ζ could +essentially give rise to the possibility of EPs at different +physical locations (along z) in a 1D-PC. As an example, +we show below that an optimally apodized 1D-PC (1D- +APC) which satisfies the adiabatic constraints enables us +to observe EPs at discreetly separated points along z. +A. +Design of an 1D apodized PC and intermodal +coupling +We consider a 1D-PC configuration that exhibits +varying dielectric filling fraction (ζ) in each unit cell. +This variation is essentially dictated through the relation +d1M = d1 −Mδ and d2M = Λ−d1M. Here, d1M and d2M +are the thickness of TiO2 and SiO2 layers respectively +in M th unit cell (M = 0, 1, 2, 3, ..., (N − 1) for N number +of unit cells). The unit cell period, however remains un- +changed i.e. Λ = d1M +d2M = d1 +d2. This apodization +FIG. 2. a) Shows the reflection spectrum for designed 1D- +APC. (b) and (c) Shows the mode-field intensities for two +different frequencies νa = 250 THz and νb = 300 THz which +are within the PBG of 1D-APC. (d) and (e) Shows the vari- +ation in Re(e1) (dotted black curve), Im(e1) (dotted maroon +curve), Re(e2) (solid black curve) and Re(e2) (solid maroon +curve) as a function of TiO2 layer thickness for each unit cell +(i.e. d1M) at frequencies νa = 250 THz and νb = 300 THz +respectively. +in 1D-PC could be visualized through a longitudinal +variation in ∆k as well as ˜κ by virtue of a monotonic +change in average refractive index (¯n) for an unit cell. +This variation in ∆k and ˜κ in a 1D-APC geometry +leads to an adiabatic evolution of the Stokes vector +along the propagation direction and manifests through +a broader PBG (≈ 140 THz) in comparison with a +conventional (periodic) 1D-PC [30]. +This is presented +in Fig. 2(a) which shows a broader reflection spectrum +for the 1D-APC in comparison with the conventional +1D-PC (Fig.1(a)). In addition, a flat transmission band +and the absence of sharp transmission resonances is +a distinct feature of 1D-APC. The mode-propagation +characteristics for the frequencies within the PBG (of +1D-APC) is explored by drawing a comparison with the +mode-field distributions for the equivalent modes within +the PBG of a conventional 1D-PC. Figures 2(b) and (c) +shows the mode-field distribution for two frequencies +νa = 250 THz and νb = 300 THz which are within the +PBG of 1D-APC. In comparison with the mode-field +distribution shown in Fig. 1(c), it could be observed that +different modes are reflected from spatially separated +z values. +The smaller frequency (νa = 250 THz) is +reflected from the regions which are closer to z = 0 edge +of the 1D-APC in comparison to that for νb = 300 THz. +This variation is indicative of the fact that the field +is localized and exhibits instantaneous localization in +different 1D-APC sections. From a different perspective, +it is apparent that the variation in dielectric filling +fraction (ζ) would result in different eigenvalues (and +corresponding eigenvectors) for each unit cell. Accord- +ingly, we plot the eigenvalues e1 and e2 as a function of +d1M for two frequencies νa = 250 THz (Fig. 2(d)) and +νb = 300 THz (Fig. 2(e)) which are within the PBG of +1D-APC. Each one of the figures shows that the eigen- + +1 +0.15 +a) +b) +0.1 +0.8 +0.05 +0.05 +0.6 +R +1,2′ +0 +0 +0.4 +e +-0.05 +Re( +-0.05 +0.2 +-0.1 +-0.15 +-0.1 +0 +175 +200 +225 +250 +175 +200 +225 +250 +275 +300 +325 +275 +300 +325 +v (THz) +v (THz) +C +310THz +240THza) +0.8 +250THz +0.6 +R +0.4 +c) +300THz +0.2 +0 +160 +200 +250 +300 +350 +380 +V (THz) +0.15 +0.3 +Re(e,)... Ree,)Im(e,.m(e,) +0.04 +d) +e) +0.1 +0.2 +(V/) ( +0.05 +(V /z) +(V /Z) +0.02 +(V /) +0.05 +0.1 +0 +0 +0 +0 +? +1 +Re( +-0.05 +-0.1 +-0.1 +-0.2 +250THz +300THz +-0.15 +-0.1 +-0.3 +-0.04 +0 +50 +100 +150 +200 +250 +300 +0 +50 +100 +150 +200 +250 +300 +d +(nm) +d +(nm) +1M +1M4 +FIG. 3. +(a) Shows the variation of ⃗B in parameter space +(spanned by κ and ∆k) at different operating frequencies +(ν1 += +400 THz, ν2 += +250 THz, ν3 += +160 THz) for +the designed 1D-APC. The blue and green solid lines repre- +sent the ∆k = κ and ∆k = −κ curves. (b) Shows the location +of EPs in different unit cells (with different filling fraction ζ) +as a function of frequency (ν). +values (e1 and e2) vanish at two different values of d1M +i.e. at the location of two different unit cells. Therefore, +the 1D-APC geometry hosts two EPs for every d1M. +Consequently, for a multitude of ζ, there would be +multiple EPs in the 1D-APC for a forward-propagating +mode to a backscattered mode-coupling process. +As +discussed before, the regions where ℜe1 and ℜe2 are +non-zero in Figs. 2(d) and 2(e) exhibit a PT -symmetric +coupling dynamics between the forward-propagating and +backscattered modes. On the other hand, in the regions +where e1 and e2 are purely imaginary, the mode-coupling +process exhibits PT -symmetry broken manifolds. +The +illustrations presented in Figs. 2(d) and 2(e) show that +for each frequency within the PBG, the 1D-APC hosts +two EPs at two different d1M. This essentially implies +that there exists one or more than one EPs hosted by +each unit cell of the 1D-APC. Therefore, an 1D-APC +is expected to host multiple EPs which are spectrally +as well as spatially separated from each other. In order +ascertain the spectral location of EPs in the 1D-APC, we +plot the evolution of ⃗B in the parameter space for three +different frequencies ν1 = 400 THz, ν2 = 250 THz, +and ν3 += 160 THz as shown in Fig.3(a). It could be +noted at ν1 and ν3 are situated outside PBG of 1D-APC +(see Fig. +2(a)). +Since, the EPs are depicted by the +condition ∆k = |κ|, Fig.3(a) also contains the curve +∆k = ±κ (solid blue and green curves). It is apparent +that ∆k = ±κ curve intersects ⃗Bν2 at two points and +it does not intersect the ⃗Bν1 curve as well as the ⃗Bν3 +curve in the parameter space. For frequencies close to +the band-edge of 1D-APC (say 200 THz or 350 THz), +it could be ascertained that there exists only one EP in +the eigenvalue spectrum. +This is primarily due to the +adiabatic constraints followed by the 1D-APC design. In +other words, for the band-edge frequencies, the forward +and backward propagating modes are decoupled (˜κ) +at entry (z = 0) and exit (z = L) face of the crystal. +Additionally, d1M = Λ for m = 0 (or d2M = Λ for +m = N) in case of band-edge frequencies that leads to +∆k = 0 for ζ = (or ζ = 1). Therefore, ˜κ = ∆k = 0 +depicts the only EP for the band-edge frequencies. +In order to elucidate the aforementioned point, we +present the spectral location of EPs as a function of di- +electric filling fraction (ζ) or propagation direction (z) +in Fig. 3(b). It could be observed that there exists two +(2) EPs (at different ζ or z) for all the frequencies well +within the PBG of 1D-APC. However, for the band-edge +frequencies (νl = 200 THz and νh = 330 THz), the 1D- +APC hosts one EP only. Nevertheless, the area enclosed +by the EPs in Fig. +3(b) represents the region of PT - +symmetry broken phase for the 1D-APC. It is interesting +to note that the separation between the two EPs for fre- +quencies closer to the band-edges (say ν ≤ 210 THz or +ν ≥ 310 THz) very less and they tend to overlap at the +same filling fraction. It is important to note that these +EPs are physically positioned close to the entry (z = 0) +and exit (z = L) face of the 1D-APC where ˜κ is very +small. By virtue of this, the PBG corresponding to that +unit cell of 1D-APC is relatively smaller in comparison +with the PBG for a unit cell close to the center (z ≈ L +2 ) +of 1D-APC. Due to the fact that the EPs exist at the +band-edges of PBG for each unit cell of APC, a smaller +PBG would essentially imply closely spaced EPs near the +band-edges (see Fig. 3(b)). +B. +Geometric phase estimation of reflection band +It is well known that the geometric phase of a pass- +band (or transmission band) for a one-dimensional con- +ventional photonic crystal is quantized (0 or π) and it +is known as the ‘Zak’ phase. +However, the geomet- +ric interpretation of backscattered (or reflection) phase +from a 1D-PC remains irrelevant. However, in case of +1D-APC, the reflection of different spectral components +(within the PBG) takes place from different unit cells +(or z) along the propagation direction [30]. For exam- +ple, the adiabatic following constraint leads to conver- +sion of optical power from the forward-propagating to +the backscattered mode predominantly towards the exit +face of 1D-APC for frequency ν = 250 THz which could +be seen in Fig. 4(a). Through a similar route, it could +be shown that different spectral components within the +PBG are reflected strongly from different unit cells of +1D-APC [30]. The primary underlying reason could be +traced to the variation in ˜κ and ∆k for each spectral +component in the PBG which are non-identical. Conse- +quently, the estimation of geometric phase acquired by +different backscattered modes is expected to be differ- +ent and must play a crucial role in establishing the bulk- +boundary correspondence in case of 1D-APC. In order to +obtain the geometric phase γ, we consider a triad defining +the state vector ⃗S (≡ [u, v, w]) where u = ˜Ai ˜A∗ +r + ˜Ar ˜A∗ +i , +v = −i[ ˜Ai ˜A∗ +r − ˜Ar ˜A∗ +i ] and w = | ˜Ar| +2 − | ˜Ai| +2 [30]. +The z-component of the state-vector (w) represents the +conversion efficiency of optical power from a forward- +propagating to a backscattered mode [30]. It is also worth + +10 +1 +B(v/) +a) +b) +0.8 +5 +△k (μm" +0.6 +米米 +△k= +S +0.4 +米 +B(v) +0 +0.2 +B(v) +Ak= - k +0 +-5 +0.5 +2 +2.5 +190 +210 +230 +250 +270 +290 +310 +330 +0 +1 +350 +-11.5 +k(um +v (THz)5 +FIG. 4. a) Shows the variation in conversion efficiency ( w+1 +2 ) +for optical power transfer between a forward-propagating +mode to a backscattered mode as a function of 1D-APC length +(z) for a frequency ν2 = 250 THz which is within the PBG. +(b) Presents the state-vector (⃗S = [u, v, w]) trajectory on +the Bloch sphere for ν2 = 250 THz. +noting that the trajectory of state-vector (⃗S) correspond- +ing to the frequencies within the PBG is non-closed. Al- +ternatively, the geometric phase is not a conserved quan- +tity during the dynamical evolution of states owing to the +PT -symmetry broken phase. In general, the solid angle +subtended by the state-vector trajectory at the center +of the Bloch sphere is used for computing the geomet- +ric phase. +However, in case of an adiabatic evolution, +the state-vector trajectory could be very complicated. In +Fig. +4(b), we have plotted such a state-vector trajec- +tory (on the Bloch sphere) corresponding to a frequency +ν = 250 THz (which is within the PBG of 1D-APC). It +is important to note that ⃗S = [0, 0, −1] and ⃗S = [0, 0, 1] +represent states in which all the optical power (∝ | ˜Af,b|2) +is present in the forward-propagating and backscattered +mode respectively. +Although, the adiabatic evolution +of state-vector results in complete optical power trans- +fer from the forward to backward-propagating mode i.e. +w = −1 to w = 1, the estimation of acquired geometric +phase is quite complicated owing to the spiralling trajec- +tory of ⃗S on the Bloch-sphere. However, it is interest- +ing to note that ⃗S goes from [0, 0, −1] to [0, 0, 1] for all +the frequencies within the PBG of 1D-APC by virtue of +satisfying the adiabatic following constraints. The most +important point is to note that the conversion efficiency +(or reflectivity) is ‘unity’ for all the frequencies within +the PBG of 1D-APC [30]. In other words, ⃗B goes from +[0, 0, −∆k] to [0, 0, ∆k] in the parameter space for all the +PBG frequencies (through any trajectory) when the adi- +abatic following constraints are satisfied [30]. +By virtue of the fact that the state-vector ⃗S adiabat- +ically follows ⃗B (as per the Bloch equation), the initial +and the final value of ⃗B could also yield the geometric +phase (γ). It is known that γ is estimated from angle +φ (subtended by ⃗B at the origin ∆k = ˜κ = 0) through +the relation γ = +φ +2 . In that case, the geometric phase +for each spectral component within the PBG is +π +2 . In +order to elucidate this point, we plot ⃗B at different z +of 1D-APC in the parameter space for ν = 250 THz as +FIG. 5. +Represents the evolution of ⃗B as a function of +length (L) of 1D-APC in parameter (∆k − κ) space for a) +ν2 = 250 THz and b) ν4 = 180 THz. φ represents the angle +subtended by curve ⃗B at the origin. +shown in Fig. 5(a). At the entry face of 1D-APC (z = 0), +⃗B(z = 0) = [0, 0, −2.7 µm−1] (black arrow) and gradu- +ally goes to ⃗B(z = L) = [0, 0, +2.7 µm−1] (red arrow) +at z = L. At z = L +2 , ∆k = 0 and ˜κ is maximum (green +arrow in Fig. 5(a)) The evolution of ⃗B in Fig. 5(a) yields +φ = π and consequently, γ = π +2 . In a similar manner, γ +for all the frequencies within the PBG would be π +2 by +virtue of adhering to the constraints imposed by adia- +batic following. Hence, it could be asserted that a geo- +metric phase of π +2 is acquired by a reflected beam in a 1D- +APC for the values of parameters which results in PT - +symmetry broken phase. On the contrary, the variation +in ⃗B is plotted as a function of z for ν = 180 THz which +is outside the PBG of 1D-APC (see Fig. 5(b)). ⃗B(z = 0) +(black arrow) and ⃗B(z = L) (red dashed arrow) are both +negative as well as co-parallel in this case. Consequently, +the geometric phase γ = φ +2 = 0 for ν = 180 THz. In +addition, it is apparent that ∆k ̸= 0 at any point (or any +z) in the 1D-APC. +C. +Tamm-plasmon excitations in 1D-APC and +topological connection +The presence of a plasmon-active layer adjacent to +the all-dielectric 1D-APC results in excitation of mul- +tiple Tamm-plasmon modes which are non-degenerate. +As an example, we consider a thin (dAu = 30 nm) layer +of gold placed in contact with high index layer (TiO2) +of 1D-APC (see Fig.6(a)). The simulated reflection spec- +trum (using transfer matrix method) exhibits a sharp res- +onance within the PBG as shown in Fig.6(b). These res- +onances are essentially due to Tamm-plasmon mode exci- +tations which are highly localized electromagnetic states. +Figure 6(b) depicts the existence of 10 Tamm-plasmon +modes within the PBG of 1D-APC. Although there are a +few sharp resonances outside the PBG, their mode-field +signatures do not resemble that for a Tamm-plasmon +mode [32]. In general, the existence of Tamm-plasmon +modes is governed by the condition φAP C + φAu = 2sπ +where s += +0, 1, 2, 3.... is an integer [33–35]. Here, +φAP C is the total phase acquired by the reflected beam +from the 1D-APC (light incident from Au side), and φAu + +a) +b) +(0,0,1) +0.8 +0.6 +0.4 +0.2 +(0,0,-1) +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9.3 +z (μm)3 +B(z = L) +0 +B (z = 0) +a) +b) +2 +1 +B (z = L/2) +) +.2 +0 +B(z = L/2) +-3 +.4 +-2 +250 THz +B (z = L) +180 THz +B(z = 0) +-3 +-5 +0 +0.5 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +k (μm"1) +k (um=1)6 +FIG. 6. +a) Shows the schematic of the Au-1D-APC het- +erostructure. +The Au-layer is placed adjacent to the high- +index TiO2 layer. The thick brown arrow depicts the direction +of light incidence on the Au-1D-APC b) Shows the simulated +reflection spectrum of 1D-APC without Au (black solid curve) +and that of Au-1D-APC (maroon solid curve). +is the phase acquired by reflected beam at the Au−TiO2 +interface. It is worthwhile to reiterate that the dielec- +tric layer (of 1D-APC) adjacent to the Au-film is TiO2 +which is the high index layer. +In the present context +φAP C = γ + α, where α is the dynamic phase acquired +by the reflected beam [30]. This could be estimated by +noting the fact that the EPs (for a given frequency) are +situated in different unit cells (or ζ) of the 1D-APC. For +a frequency ν, if the nearest EP (with respect to z = 0) +is present in the pth-unit cell of 1D-APC, then α could +be determined using +α = 2πν +c +p +� +M=0 +[n1d1M + n2d2M] +(5) +The knowledge of location for EPs in the 1D-APC (ob- +tained from the eigenvalue spectrum of ˆH) would accu- +rately yield the dynamic phase (α) for any frequency of +operation (ν). +In conjunction with the estimate of γ, +this information would allow us to determine the Tamm- +plasmon mode resonance frequencies (νr). +This recipe +provides a flexibility in terms of designing an 1D-APC +which would facilitate excitation of Tamm-plasmon mode +at a target (desirable) frequency (or wavelength) of op- +eration. One such application could be the generation +of higher harmonics or frequency downconversion using +optical surface states [36]. +In this case, the 1D-APC +could be designed such that the Tamm-plasmon modes +(localized modes) have resonance frequencies that are +governed by the energy conservation and phase-matching +constraints imposed by the frequency conversion process. +III. +CONCLUSIONS +In conclusion, we presented an all-dielectric 1D-APC +design which hosts multiple exceptional points in its +eigenvalue spectrum by virtue of exhibiting a non- +Hermitian dynamics for a mode-coupling process between +a forward-propagating mode to its backscattered coun- +terpart. +Although, the 1D-APC does not include any +dissipative component, the intermodal coupling mecha- +nism could be classified in terms of PT -symmetric and +PT -broken phases which are connected through a quan- +tum phase-transition. We also showed that the reflected +beam (within the PBG) acquires a geometric phase of π +2 +in the PT -symmetry broken phase. As a consequence of +this outcome, the 1D-APC could be designed for excit- +ing the optical Tamm-plasmon modes at any desirable +frequency within the PBG. This design flexibility allows +us to employ such architectures for quite a few appli- +cations such as efficiently carrying out optical frequency +conversion using surface states [36]. +IV. +DISCLOSURES +The authors declare that there are no conflicts of in- +terest related to this article. +[1] M. +Berry, +Physics +of +nonhermitian +degeneracies, +Czechoslovak Journal of Physics 54, 1039 (2004). +[2] W. D. Heiss, The physics of exceptional points, Journal +of Physics A: Mathematical and Theoretical 45, 444016 +(2012). +[3] X.-F. Zhu, Y.-G. Peng, and D.-G. Zhao, Anisotropic re- +flection oscillation in periodic multilayer structures of +parity-time symmetry, Opt. Express 22, 18401 (2014). +[4] Z. Lin, H. 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B 97, 115438 (2018). + diff --git a/6NE2T4oBgHgl3EQfkgd1/content/tmp_files/load_file.txt b/6NE2T4oBgHgl3EQfkgd1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38a1a14a7364ae7709b3219feac5d4786905c8f8 --- /dev/null +++ b/6NE2T4oBgHgl3EQfkgd1/content/tmp_files/load_file.txt @@ -0,0 +1,606 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf,len=605 +page_content='Apodized photonic crystals: A non-dissipative system hosting multiple exceptional points Abhishek Mondal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Shailja Sharma and Ritwick Das∗ School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' National Institute of Science Education and Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' An OCC of Homi Bhabha National Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Jatni - 752050,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Odisha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' India (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2023) Optical systems obeying non-Hermitian dynamics have been the subject of intense and concerted investigation over the last two decades owing to their broad implications in photonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' acoustics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' electronics as well as atomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A vast majority of such investigations rely on a dissipative, balanced loss-gain system which introduces unavoidable noise and consequently, this limits the coherent control of propagation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, we show that an all-dielectric, non-dissipative photonic crystal (PC) could host, at least two exceptional points in its eigenvalue spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By introducing optimum apodization in the PC architecture, namely 1D-APC, we show that such a configuration supports a spectrum of exceptional points which distinctly demarcates the PT - symmetric region from the region where PT -symmetry is broken in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The analytical framework allows us to estimate the geometric phase of the reflected beam and derive the constraint that governs the excitation of topologically-protected optical Tamm-plasmon modes in 1D-APCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' INTRODUCTION Optical systems which are governed by non-Hermitian Hamiltonian dynamics through an engineered gain and dissipation mechanism, provide a route to overcome the limitations imposed by closed optical systems that obey the Hermitian-Hamiltonian led dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such non- Hermitian systems give rise to a real eigenvalue spec- trum when the Hamiltonian commutes with the parity- time (PT ) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A continuous change in the pa- rameter governing the Hermiticity (of the Hamiltonian) breaks the PT symmetry which manifests in the form of complex eigenvalues for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the phase space, such points where the real and complex eigenval- ues coalesce are termed as exceptional points (EPs) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This spontaneous PT -symmetry breaking has catalyzed a plethora of non-intuitive outcomes such as directional invisibility [3, 4], coherent perfect lasing and absorption [5–9], negative refraction [10], single-particle based sens- ing [11–13], distortion-free wireless optical power trans- fer [14] and a few more [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is, however, worth noting that the incommensurate gain and loss distribu- tion in non-Hermitian systems impose the primary limi- tation on the practical applications due to unpredictable signal-to-noise ratio near EP [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to cir- cumvent such bottlenecks, a few possibilities have been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' One such promising route is to create an asym- metric loss in the system (without gain) whose dynamics could be explored using a non-Hermitian Hamiltonian with a uniform background loss [20, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such a con- figuration would exhibit PT -symmetry which could be broken through scaling up the loss asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In a dif- ferent scheme, a pseudo-Hermitian system was explored which allowed strong coupling between a large number ∗ ritwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='das@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='in of modes via manipulation of the parameters governing the Hamiltonian [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This led to the existence of EPs of multiple order and the interaction of eigenvalues around each EP provides a robust control on the propagation dynamics [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In spite of the aforementioned devel- opments, a useful and practical proposition would be to devise a configuration hosting a multitude of EPs with the constraint that the electromagnetic (EM) energy lost due to the non-Hermitian dynamics is stored in a reser- voir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This essentially implies that the dissipative channel associated with a non-Hermitian system drives a separate Hermitian system which could allow reverse flow of EM energy by virtue of cyclical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such systems have been explored in the area of parametric frequency con- version processes where the EM energy lost in one of the parametric processes (obeying non-Hermitian dynamics) is coherently added to the other parametric process that follows a Hermitian dynamics [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A plausible transla- tion of such an idea in the non-absorptive linear systems would be to introduce a virtual loss in an intermodal interaction process thereby generating multiple EPs in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' One of the simplest configurations imitating such a process is a multimodal interaction in an all-dielectric one-dimensional (1D) photonic-crystal (PC) with a gradually varying duty cycle (for each unit cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In such an apodized 1D-PC, the forward (source) to backward (sink) mode-coupling dynamics is essentially governed by a pseudo-Hermitian Hamiltonian whose Her- miticity is determined by the apodization along the prop- agation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the present work, we show the exis- tence of multiple EPs in an apodized 1D-PC and develop an analytical framework for ascertaining the possibility of exciting topologically-protected optical edge modes in such aperiodically stratified configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='03979v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='optics] 10 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' THEORETICAL FRAMEWORK AND COUPLED-MODE FORMALISM We consider a 1D-PC comprised of periodic bilayers with refractive indices n1 and n2 with thicknesses d1 and d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such conventional 1D-PCs or alternatively, dis- tributed Bragg reflectors (DBRs) are usually character- ized by photonic bandgaps (PBGs) which are separated from each other by high transmission (or pass) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to appreciate the EM wave propagation dynamics, we consider the coupling between pth-mode (|p⟩) with qth-mode (|q⟩) which could be represented employing the coupled-amplitude equations given by [29] dAq dz = −i βq |βq| � p � m ˜κ(m) qp Ape−i(βq−βp−m 2π Λ )z (1) where βp and βq are the longitudinal (z) components of wavevector kp and kq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' ˜κ(m) qp defines the strength of coupling (or coupling coefficient) between the pth and qth mode that is coupled through the mth Fourier component of the periodic dielectric distribution ( Λ = d1 + d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The factor ∆β = βq − βp − m 2π Λ (known as the phase-mismatch) is one of the dynamical variables (along with κqp) which dictate the measure of optical power transferred from one mode to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For the present work, we consider a contra-directional coupling set-up where a forward (along +z) propagating mode (|p⟩ ≡ |f⟩) is coupled to a backward (along −z) prop- agating mode (|q⟩ ≡ |b⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Accordingly, it could asserted that βb = −βf or alternatively ∆β = 2βf − 2π Λ and there- fore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (1) could be simplified to [29] dAb dz = i˜κAfe−i∆βz (2) dAf dz = −i˜κ∗Abei∆βz (3) where ˜κ = i(1−cos 2πζ) 2λ (n1 2−n2 2) ¯n = iκ and ζ is the di- electric filling fraction of layer with refractive index n1 in the unit cell i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' d1 Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The mean refractive index for an unit cell of thickness Λ is ¯n = � d1n12+d2n22 Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By using a Gauge transformation given by [Af, Ab] → [ ˜Af, ˜Ab]ei/2[∆β0z− � 0 zq(z′)dz′], we obtain [30] i d dz � ˜Ab ˜Af � = � −∆k −˜κ ˜κ∗ ∆k � � ˜Ab ˜Af � (4) Equation (4) is analogous to time-dependent Schr¨odinger’s equation with t-coordinate being replaced by z-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, ∆k (= ∆β 2 ) and q(z) = 0 remains constant (for a given frequency) across the 1D-PC which has a fixed duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The autonomous Hamiltonian ˆH = −⃗σ · ⃗B with ⃗σ ≡ [σx, σy, σz] are the Pauli’s spin matrices and ⃗B ≡ [0, κ, ∆k] (magnetic field analog) rep- resents a pseudo-Hermitian evolution dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to appreciate this point, we note that the eigenvalues of ˆH which are given by e1,2 = ± � ∆k2 − κ2 whereas the eigenfunctions are |ψ1⟩ = � −i (∆k+√ ∆k2−κ2) κ 1 � and |ψ2⟩ = � +i (−∆k+√ ∆k2−κ2) κ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, ˜κ = iκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A closer look into the eigenvectors reveals that the equality κ = ± ∆k manifests as coalescing of eigenvectors accompanied by vanishing eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such points in parameter space where κ equals ±∆k are termed as exceptional points (EPs) and they distinctly demarcate the regions exhibiting Hermitian (PT -symmetric phase) and non-Hermitian (PT -broken phase) dynamical evolution of states (or modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to appreciate the aforementioned idea, we consider a practical 1D-PC with n1 ≡ TiO2 layer and n2 ≡ SiO2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The layer thicknesses are d1 = d2 = 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The reflection spectrum for N = 20 unit cells is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(a) which exhibits a high reflection band (or PBG) spreading over a 75 THz band- width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to obtain the reflection spectrum, finite element method (FEM) based simulations were carried out using the commercially available computational tool (COMSOL Multiphysics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the simulations, the peri- odic boundary condition is imposed along the transverse direction and a mesh size of 5 nm is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' We ignore the material dispersion for the simulations and assume n1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 (≡ TiO2) and n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 (≡ SiO2) across the entire spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For this 1D-PC, we also plotted the eigenvalues e1 and e2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(b)) as a function of the frequency of the incident electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is ap- parent that the eigenvalues vanish at ν1 ≈ 210 THz and ν2 ≈ 285 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' These two frequencies (ν1 and ν2) define the EPs (κ = + ∆k and κ = − ∆k) for the periodic 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A closer look would also reveal that the eigenval- ues are purely imaginary within the PBG and the band edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1 (a)) coincide with ν1 and ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The mode fields for frequencies lying inside the PBG (240 THz) and outside the PBG (310 THz) are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(c) and (d) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is worth noting that the investigations on systems exhibiting PT -symmetry (or PT -broken symmetry) led dynamics in photonics essen- tially involve optimally balanced gain-loss architectures such as segmented waveguides and photonic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In such systems, a complex relative permittivity in different sections depicting actual gain or loss for the propagat- ing light beam gives rise to the PT -symmetry (or PT - broken symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The present configuration involving 1D-PC does not include an actual dissipative component for achieving the PT -symmetric to PT -symmetry bro- ken phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Alternatively, the coupling of opti- cal power to the backscattered mode |b⟩ is analogous to a virtual loss for a forward propagating |f⟩ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' When this coupling is relatively weak i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' ∆k > ˜κ, |f⟩ and |b⟩ exhibits cyclic exchange of optical power (as a function of z) which is a primitive outcome for a PT -symmetric dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' On the other hand, a strong coupling regime 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the reflection spectrum of a conventional (periodic) 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' b) Shows the variation in Re(e1) (dotted black curve), Im(e1) (dotted maroon curve), Re(e2) (solid black curve) and Im(e2) (solid maroon curve) as a function of frequency (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' c) and d) Shows the mode-field intensity for frequencies within the PBG 240 THz and that outside the PBG 310 THz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The solid red arrow represents the direction of incidence of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' where ∆k < ˜κ manifests through a monotonic growth of backscattered mode (|b⟩) that is a signature of PT - symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is worthwhile to reiterate the point that the two regimes depicted by the inequality of ∆k and ˜κ (in the parameter space) could be mapped onto the PBG and pass or transmission band (s) in the reflected spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Subsequently, each PBG is necessar- ily bounded by two EPs in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Additionally, these two EPs are fixed and could not be tailored for a given 1D-PC with a fixed duty cycle and fixed period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Also, the conventional 1D-PC geometry excludes the pos- sibility of realizing higher-order exceptional points [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Taking a cue from this critical viewpoint, we note that a small apodization or gradual change in dielectric fill- ing fraction (ζ) of each unit cell of the 1D-PC would allow us to realize discretely spaced (multiple) EPs at different optical frequencies (or wavelengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to elucidate this point, we recall that ∆k as well as ˜κ is a function of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' An optimum spatial variation in ζ could essentially give rise to the possibility of EPs at different physical locations (along z) in a 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As an example, we show below that an optimally apodized 1D-PC (1D- APC) which satisfies the adiabatic constraints enables us to observe EPs at discreetly separated points along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Design of an 1D apodized PC and intermodal coupling We consider a 1D-PC configuration that exhibits varying dielectric filling fraction (ζ) in each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This variation is essentially dictated through the relation d1M = d1 −Mδ and d2M = Λ−d1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, d1M and d2M are the thickness of TiO2 and SiO2 layers respectively in M th unit cell (M = 0, 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=', (N − 1) for N number of unit cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The unit cell period, however remains un- changed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Λ = d1M +d2M = d1 +d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This apodization FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the reflection spectrum for designed 1D- APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (b) and (c) Shows the mode-field intensities for two different frequencies νa = 250 THz and νb = 300 THz which are within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (d) and (e) Shows the vari- ation in Re(e1) (dotted black curve), Im(e1) (dotted maroon curve), Re(e2) (solid black curve) and Re(e2) (solid maroon curve) as a function of TiO2 layer thickness for each unit cell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' d1M) at frequencies νa = 250 THz and νb = 300 THz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' in 1D-PC could be visualized through a longitudinal variation in ∆k as well as ˜κ by virtue of a monotonic change in average refractive index (¯n) for an unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This variation in ∆k and ˜κ in a 1D-APC geometry leads to an adiabatic evolution of the Stokes vector along the propagation direction and manifests through a broader PBG (≈ 140 THz) in comparison with a conventional (periodic) 1D-PC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(a) which shows a broader reflection spectrum for the 1D-APC in comparison with the conventional 1D-PC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In addition, a flat transmission band and the absence of sharp transmission resonances is a distinct feature of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The mode-propagation characteristics for the frequencies within the PBG (of 1D-APC) is explored by drawing a comparison with the mode-field distributions for the equivalent modes within the PBG of a conventional 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Figures 2(b) and (c) shows the mode-field distribution for two frequencies νa = 250 THz and νb = 300 THz which are within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In comparison with the mode-field distribution shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(c), it could be observed that different modes are reflected from spatially separated z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The smaller frequency (νa = 250 THz) is reflected from the regions which are closer to z = 0 edge of the 1D-APC in comparison to that for νb = 300 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This variation is indicative of the fact that the field is localized and exhibits instantaneous localization in different 1D-APC sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' From a different perspective, it is apparent that the variation in dielectric filling fraction (ζ) would result in different eigenvalues (and corresponding eigenvectors) for each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Accord- ingly, we plot the eigenvalues e1 and e2 as a function of d1M for two frequencies νa = 250 THz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(d)) and νb = 300 THz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(e)) which are within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Each one of the figures shows that the eigen- 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 a) b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 R 1,2′ 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 Re( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0 175 200 225 250 175 200 225 250 275 300 325 275 300 325 v (THz) v (THz) C 310THz 240THza) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 250THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 c) 300THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0 160 200 250 300 350 380 V (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3 Re(e,).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Ree,)Im(e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='m(e,) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='04 d) e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 (V/) ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 (V /z) (V /Z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='02 (V /) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0 0 0 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1 Re( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 250THz 300THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='04 0 50 100 150 200 250 300 0 50 100 150 200 250 300 d (nm) d (nm) 1M 1M4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (a) Shows the variation of ⃗B in parameter space (spanned by κ and ∆k) at different operating frequencies (ν1 = 400 THz, ν2 = 250 THz, ν3 = 160 THz) for the designed 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The blue and green solid lines repre- sent the ∆k = κ and ∆k = −κ curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (b) Shows the location of EPs in different unit cells (with different filling fraction ζ) as a function of frequency (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' values (e1 and e2) vanish at two different values of d1M i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' at the location of two different unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Therefore, the 1D-APC geometry hosts two EPs for every d1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Consequently, for a multitude of ζ, there would be multiple EPs in the 1D-APC for a forward-propagating mode to a backscattered mode-coupling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As discussed before, the regions where ℜe1 and ℜe2 are non-zero in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(d) and 2(e) exhibit a PT -symmetric coupling dynamics between the forward-propagating and backscattered modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' On the other hand, in the regions where e1 and e2 are purely imaginary, the mode-coupling process exhibits PT -symmetry broken manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The illustrations presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(d) and 2(e) show that for each frequency within the PBG, the 1D-APC hosts two EPs at two different d1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This essentially implies that there exists one or more than one EPs hosted by each unit cell of the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Therefore, an 1D-APC is expected to host multiple EPs which are spectrally as well as spatially separated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order ascertain the spectral location of EPs in the 1D-APC, we plot the evolution of ⃗B in the parameter space for three different frequencies ν1 = 400 THz, ν2 = 250 THz, and ν3 = 160 THz as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It could be noted at ν1 and ν3 are situated outside PBG of 1D-APC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Since, the EPs are depicted by the condition ∆k = |κ|, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3(a) also contains the curve ∆k = ±κ (solid blue and green curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is apparent that ∆k = ±κ curve intersects ⃗Bν2 at two points and it does not intersect the ⃗Bν1 curve as well as the ⃗Bν3 curve in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For frequencies close to the band-edge of 1D-APC (say 200 THz or 350 THz), it could be ascertained that there exists only one EP in the eigenvalue spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This is primarily due to the adiabatic constraints followed by the 1D-APC design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In other words, for the band-edge frequencies, the forward and backward propagating modes are decoupled (˜κ) at entry (z = 0) and exit (z = L) face of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Additionally, d1M = Λ for m = 0 (or d2M = Λ for m = N) in case of band-edge frequencies that leads to ∆k = 0 for ζ = (or ζ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Therefore, ˜κ = ∆k = 0 depicts the only EP for the band-edge frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to elucidate the aforementioned point, we present the spectral location of EPs as a function of di- electric filling fraction (ζ) or propagation direction (z) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It could be observed that there exists two (2) EPs (at different ζ or z) for all the frequencies well within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, for the band-edge frequencies (νl = 200 THz and νh = 330 THz), the 1D- APC hosts one EP only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Nevertheless, the area enclosed by the EPs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3(b) represents the region of PT - symmetry broken phase for the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is interesting to note that the separation between the two EPs for fre- quencies closer to the band-edges (say ν ≤ 210 THz or ν ≥ 310 THz) very less and they tend to overlap at the same filling fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is important to note that these EPs are physically positioned close to the entry (z = 0) and exit (z = L) face of the 1D-APC where ˜κ is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By virtue of this, the PBG corresponding to that unit cell of 1D-APC is relatively smaller in comparison with the PBG for a unit cell close to the center (z ≈ L 2 ) of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Due to the fact that the EPs exist at the band-edges of PBG for each unit cell of APC, a smaller PBG would essentially imply closely spaced EPs near the band-edges (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Geometric phase estimation of reflection band It is well known that the geometric phase of a pass- band (or transmission band) for a one-dimensional con- ventional photonic crystal is quantized (0 or π) and it is known as the ‘Zak’ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, the geomet- ric interpretation of backscattered (or reflection) phase from a 1D-PC remains irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, in case of 1D-APC, the reflection of different spectral components (within the PBG) takes place from different unit cells (or z) along the propagation direction [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For exam- ple, the adiabatic following constraint leads to conver- sion of optical power from the forward-propagating to the backscattered mode predominantly towards the exit face of 1D-APC for frequency ν = 250 THz which could be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Through a similar route, it could be shown that different spectral components within the PBG are reflected strongly from different unit cells of 1D-APC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The primary underlying reason could be traced to the variation in ˜κ and ∆k for each spectral component in the PBG which are non-identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Conse- quently, the estimation of geometric phase acquired by different backscattered modes is expected to be differ- ent and must play a crucial role in establishing the bulk- boundary correspondence in case of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to obtain the geometric phase γ, we consider a triad defining the state vector ⃗S (≡ [u, v, w]) where u = ˜Ai ˜A∗ r + ˜Ar ˜A∗ i , v = −i[ ˜Ai ˜A∗ r − ˜Ar ˜A∗ i ] and w = | ˜Ar| 2 − | ˜Ai| 2 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The z-component of the state-vector (w) represents the conversion efficiency of optical power from a forward- propagating to a backscattered mode [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is also worth 10 1 B(v/) a) b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 5 △k (μm" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 米米 △k= S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 米 B(v) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 B(v) Ak= - k 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 190 210 230 250 270 290 310 330 0 1 350 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 k(um v (THz)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the variation in conversion efficiency ( w+1 2 ) for optical power transfer between a forward-propagating mode to a backscattered mode as a function of 1D-APC length (z) for a frequency ν2 = 250 THz which is within the PBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (b) Presents the state-vector (⃗S = [u, v, w]) trajectory on the Bloch sphere for ν2 = 250 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' noting that the trajectory of state-vector (⃗S) correspond- ing to the frequencies within the PBG is non-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Al- ternatively, the geometric phase is not a conserved quan- tity during the dynamical evolution of states owing to the PT -symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In general, the solid angle subtended by the state-vector trajectory at the center of the Bloch sphere is used for computing the geomet- ric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, in case of an adiabatic evolution, the state-vector trajectory could be very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 4(b), we have plotted such a state-vector trajec- tory (on the Bloch sphere) corresponding to a frequency ν = 250 THz (which is within the PBG of 1D-APC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is important to note that ⃗S = [0, 0, −1] and ⃗S = [0, 0, 1] represent states in which all the optical power (∝ | ˜Af,b|2) is present in the forward-propagating and backscattered mode respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Although, the adiabatic evolution of state-vector results in complete optical power trans- fer from the forward to backward-propagating mode i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' w = −1 to w = 1, the estimation of acquired geometric phase is quite complicated owing to the spiralling trajec- tory of ⃗S on the Bloch-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, it is interest- ing to note that ⃗S goes from [0, 0, −1] to [0, 0, 1] for all the frequencies within the PBG of 1D-APC by virtue of satisfying the adiabatic following constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The most important point is to note that the conversion efficiency (or reflectivity) is ‘unity’ for all the frequencies within the PBG of 1D-APC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In other words, ⃗B goes from [0, 0, −∆k] to [0, 0, ∆k] in the parameter space for all the PBG frequencies (through any trajectory) when the adi- abatic following constraints are satisfied [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By virtue of the fact that the state-vector ⃗S adiabat- ically follows ⃗B (as per the Bloch equation), the initial and the final value of ⃗B could also yield the geometric phase (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is known that γ is estimated from angle φ (subtended by ⃗B at the origin ∆k = ˜κ = 0) through the relation γ = φ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In that case, the geometric phase for each spectral component within the PBG is π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to elucidate this point, we plot ⃗B at different z of 1D-APC in the parameter space for ν = 250 THz as FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Represents the evolution of ⃗B as a function of length (L) of 1D-APC in parameter (∆k − κ) space for a) ν2 = 250 THz and b) ν4 = 180 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' φ represents the angle subtended by curve ⃗B at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' At the entry face of 1D-APC (z = 0), ⃗B(z = 0) = [0, 0, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='7 µm−1] (black arrow) and gradu- ally goes to ⃗B(z = L) = [0, 0, +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='7 µm−1] (red arrow) at z = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' At z = L 2 , ∆k = 0 and ˜κ is maximum (green arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(a)) The evolution of ⃗B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(a) yields φ = π and consequently, γ = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In a similar manner, γ for all the frequencies within the PBG would be π 2 by virtue of adhering to the constraints imposed by adia- batic following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Hence, it could be asserted that a geo- metric phase of π 2 is acquired by a reflected beam in a 1D- APC for the values of parameters which results in PT - symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' On the contrary, the variation in ⃗B is plotted as a function of z for ν = 180 THz which is outside the PBG of 1D-APC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' ⃗B(z = 0) (black arrow) and ⃗B(z = L) (red dashed arrow) are both negative as well as co-parallel in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Consequently, the geometric phase γ = φ 2 = 0 for ν = 180 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In addition, it is apparent that ∆k ̸= 0 at any point (or any z) in the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Tamm-plasmon excitations in 1D-APC and topological connection The presence of a plasmon-active layer adjacent to the all-dielectric 1D-APC results in excitation of mul- tiple Tamm-plasmon modes which are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As an example, we consider a thin (dAu = 30 nm) layer of gold placed in contact with high index layer (TiO2) of 1D-APC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The simulated reflection spec- trum (using transfer matrix method) exhibits a sharp res- onance within the PBG as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' These res- onances are essentially due to Tamm-plasmon mode exci- tations which are highly localized electromagnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Figure 6(b) depicts the existence of 10 Tamm-plasmon modes within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Although there are a few sharp resonances outside the PBG, their mode-field signatures do not resemble that for a Tamm-plasmon mode [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In general, the existence of Tamm-plasmon modes is governed by the condition φAP C + φAu = 2sπ where s = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='. is an integer [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, φAP C is the total phase acquired by the reflected beam from the 1D-APC (light incident from Au side), and φAu a) b) (0,0,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 (0,0,-1) 0 0 1 2 3 4 5 6 7 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3 z (μm)3 B(z = L) 0 B (z = 0) a) b) 2 1 B (z = L/2) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0 B(z = L/2) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 2 250 THz B (z = L) 180 THz B(z = 0) 3 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 k (μm"1) k (um=1)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the schematic of the Au-1D-APC het- erostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The Au-layer is placed adjacent to the high- index TiO2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The thick brown arrow depicts the direction of light incidence on the Au-1D-APC b) Shows the simulated reflection spectrum of 1D-APC without Au (black solid curve) and that of Au-1D-APC (maroon solid curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' is the phase acquired by reflected beam at the Au−TiO2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is worthwhile to reiterate that the dielec- tric layer (of 1D-APC) adjacent to the Au-film is TiO2 which is the high index layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the present context φAP C = γ + α, where α is the dynamic phase acquired by the reflected beam [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This could be estimated by noting the fact that the EPs (for a given frequency) are situated in different unit cells (or ζ) of the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For a frequency ν, if the nearest EP (with respect to z = 0) is present in the pth-unit cell of 1D-APC, then α could be determined using α = 2πν c p � M=0 [n1d1M + n2d2M] (5) The knowledge of location for EPs in the 1D-APC (ob- tained from the eigenvalue spectrum of ˆH) would accu- rately yield the dynamic phase (α) for any frequency of operation (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In conjunction with the estimate of γ, this information would allow us to determine the Tamm- plasmon mode resonance frequencies (νr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This recipe provides a flexibility in terms of designing an 1D-APC which would facilitate excitation of Tamm-plasmon mode at a target (desirable) frequency (or wavelength) of op- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' One such application could be the generation of higher harmonics or frequency downconversion using optical surface states [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In this case, the 1D-APC could be designed such that the Tamm-plasmon modes (localized modes) have resonance frequencies that are governed by the energy conservation and phase-matching constraints imposed by the frequency conversion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we presented an all-dielectric 1D-APC design which hosts multiple exceptional points in its eigenvalue spectrum by virtue of exhibiting a non- Hermitian dynamics for a mode-coupling process between a forward-propagating mode to its backscattered coun- terpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Although, the 1D-APC does not include any dissipative component, the intermodal coupling mecha- nism could be classified in terms of PT -symmetric and PT -broken phases which are connected through a quan- tum phase-transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' We also showed that the reflected beam (within the PBG) acquires a geometric phase of π 2 in the PT -symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As a consequence of this outcome, the 1D-APC could be designed for excit- ing the optical Tamm-plasmon modes at any desirable frequency within the PBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This design flexibility allows us to employ such architectures for quite a few appli- cations such as efficiently carrying out optical frequency conversion using surface states [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' DISCLOSURES The authors declare that there are no conflicts of in- terest related to this article.' metadata={'source': 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GARGb, Chethan KRISHNANc, Arnab KUNDUa +aTheory Division, Saha Institute of Nuclear Physics, +A CI of Homi Bhabha National Institute, +1/AF, Bidhannagar, Kolkata 700064, India +Email: suman.das@saha.ac.in, arnab.kundu@saha.ac.in +bManipal Centre for Natural Sciences, +Manipal Academy of Higher Education, +Dr. +T.M.A. Pai Planetarium Building, +Manipal-576104, Karnataka, India +Email: +sumit.kumar@manipal.edu +cCenter for High Energy Physics, Indian Institute of Science, +C.V. Raman Road, Bangalore 560012, India. +Email: +chethan.krishnan@gmail.com +Black holes are believed to have the fast scrambling properties of random matrices. If the fuzzball +proposal is to be a viable model for quantum black holes, it should reproduce this expectation. +This is considered challenging, because it is natural for the modes on a fuzzball microstate to +follow Poisson statistics. In a previous paper, we noted a potential loophole here, thanks to the +modes depending not just on the n-quantum number, but also on the J-quantum numbers of the +compact dimensions. For a free scalar field φ, by imposing a Dirichlet boundary condition φ = 0 +at the stretched horizon, we showed that this J-dependence leads to a linear ramp in the Spectral +Form Factor (SFF). Despite this, the status of level repulsion remained mysterious. In this letter, +motivated by the profile functions of BPS fuzzballs, we consider a generic profile φ = φ0(θ) instead +of φ = 0 at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients +of φ0(θ) are suitably Gaussian distributed), we find that the J-dependence of the spectrum exhibits +striking evidence of level repulsion, along with the linear ramp. We also find that varying the profile +leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra. The linear +ramp in our previous work can be understood as arising via an extreme version of level repulsion in +such a limiting spectrum. We also explain how the stretched horizon/fuzzball is different in these +aspects from simply putting a cut-off in flat space or AdS (ie., without a horizon). +Introduction: The quest for an understanding of quan- +tum black holes has been one of the engines driving re- +search in quantum gravity in the last half century. In +particular, the recent revival of the black hole informa- +tion paradox [1, 2] due to the works of Mathur [3] and +AMPS [4] has raised questions about the smoothness of +the horizon which are still not fully settled. +In the context of holography/string theory, there are +two broad lines along which work on quantum black holes +has progressed. The first approach, which we will call +the semi-classical approach following [5], is built on in- +sights from bulk (often Euclidean) effective field theory, +toy models of 2D gravity, and holographic entanglement +entropy. Considerable intuition has been gleaned about +the quantum nature of black holes from this approach +(eg. [6–10]) with the crowning achievement being a semi- +classical reproduction of the Page curve [11, 12]. Despite +this, the precise status of detailed unitarity and smooth- +ness are still unclear from this perspective, because the +calculation is fundamentally Euclidean. The second line +of approach is the fuzzball program of Mathur and others +which argues that black hole microstates cap off smoothly +before the horizon. In our opinion, the operational mean- +ing of this bulk statement in the full quantum setting is +not yet completely clear. But the mere existence of large +classes of such solutions [13–20] in the supergravity limit +of stringy BPS black holes is surprising. In conventional +general relativity, they would not exist thanks to the no +hair theorems. See [5] for a more detailed discussion of +the pros and cons of the two approaches. +It was suggested in [5] that one way to make progress +may be to try and reproduce general lessons of the semi- +classical approach, from fuzzball-motivated considera- +tions. The hope is that since many of these expectations +are generic, this may teach us something about how to +think about quantum fuzzballs at finite temperature even +arXiv:2301.11780v1 [hep-th] 27 Jan 2023 + +2 +though constructing explicit solutions is possible only +in the supergravity BPS limit. Conversely, if realizing +these lesson from fuzzball-motivated ideas is impossible +or highly contrived, that could be viewed as an argument +against the fuzzball program. +A particularly sharp setting in which one could explore +this tension is in the expectation that black holes are fast +scramblers [6], and that they exhibit dynamical features +of random matrices [21]. A linear ramp [22] in the spec- +tral form factor (SFF) and repulsion in the level spacing +distribution (LSD), are viewed as indicators of chaos in +random matrix theory (RMT) [23]. However, these RMT +signatures are generally thought to be challenging to re- +alize from the fuzzball paradigm, see eg. [25] – we expect +capped geometries to have roughly evenly spaced levels, +in loose analogy with the standing waves of a cylindrical +column. This makes conventional level repulsion and the +linear ramp, difficult to understand from the fuzzball per- +spective. Note also that simply declaring that the black +hole is an ensemble of such spectra, does not solve the +problem [26] – While this will certainly allow a richer set +of level spacings in the collective spectrum, there is still +no mechanism to ensure level repulsion [27]. Instead, an +ensemble of fuzzballs will give rise to Poisson statistics, +just as an ensemble of simple harmonic oscillators (SHO) +would [28]. +These expectations are reasonable, but they are also +difficult to test. This is because solving wave equations +in generic fuzzball microstate geometries is both difficult +(because the metric is complicated) and not immediately +useful (because explicit metrics in BPS cases are at zero +temperature). Exploiting the fact that the questions we +wish to tackle are generic, in [5] it was suggested that +one may be able to make progress by studying a black +hole at finite temperature with a stretched horizon. In +particular, the normal modes of a scalar field were stud- +ied in [5], by computing the spectrum of modes that re- +sult from a φ = 0 boundary condition at the stretched +horizon. The results of [5] showed that the expectations +listed in the previous paragraph have a major caveat, +they are true only if one ignored the dependence of the +spectrum on the angular quantum numbers of the com- +pact dimensions. Unlike the dependence on the principal +n-quantum number, the dependence on the J-quantum +numbers was found not to be (approximately) linear. In- +stead there was a quasi-degeneracy of levels as a function +of J for moderately large J. Most strikingly, it was found +that the SFF computed from the spectrum showed very +clear evidence of a linear ramp, even though conventional +level repulsion was not present in the J-direction [29]. It +should be emphasized here that this is the only case in +the literature that we are aware of, where a linear ramp +in the SFF exists without an underlying RMT spectrum +with Wigner-Dyson (WD) level spacing [31]. +While the results of [5] were a tantalizing hint of RMT +behavior in fuzzballs, a coherent understanding of them +could not be found. In particular, the presence of a lin- +ear ramp together with the absence of conventional level +repulsion, made a compelling interpretation impossible. +The purpose of this letter, is to shed some light on this +mysterious state of affairs. We will place the results of +[5] in context by finding a more general calculation that +can interpolate between Poisson and RMT-like spectra. +The idea (at least in hindsight) is extremely simple, and +motivated by the fact that the known BPS fuzzball solu- +tions [13, 15, 17] are described by profile functions that +are supposed to capture the fluctuations of the cap. This +suggests that a natural generalization of our simple φ = 0 +boundary conditions of [5] is to consider a generic pro- +file φ = φ0(θ) at the stretched horizon, where θ is a +mnemonic for the angular directions of the metric. In +this paper, we will consider profiles of this type, where +“genericity” will be implemented via choosing Fourier +coefficients of φ0(θ) from suitable random distributions. +This is a natural implementation of the intuitive notion +of “fluctuation at the horizon”. Remarkably, in this very +natural set up, we see both level repulsion as well as the +linear ramp. By tuning the variance of the distribution +from which φ0(θ) is chosen, we show that the LSD can +interpolate from Poisson to WD-like spectra. In partic- +ular, as the variance collapses to zero and the boundary +condition reduces to φ = 0, we find that the LSD col- +lapses to a very sharp (almost delta-function-like) peak, +as found in [5]. It was speculated in [5] that this should +be viewed as an “extreme” version of level-repulsion, and +our present paper clarifies the precise sense in which +this is true. Conversely, as the variance is steadily in- +creased, the LSD transitions from “extreme” to conven- +tional Wigner-Dyson spectra and eventually to Poisson +[32]. +Our results demonstrate that fuzzball/stretched hori- +zon modes can exhibit the spectral features of RMT and +late time chaos. We emphasize that this is a bulk cal- +culation of RMT behavior. +The expectation of RMT +behavior and eigenstate thermalization in black hole mi- +crostates is natural in the dual holographic theory, be- +cause it is strongly coupled. +This has been explicitly +demonstrated in the setting of toy dual theories like SYK +and tensor models [34]. From the bulk however, while +early time chaos is captured by out-of-time-ordered cor- +relators [7, 8], late-time chaos as captured by level repul- +sion and discreteness of the spectrum are very difficult + +3 +to understand. Fuzzballs can exhibit discreteness in the +spectrum trivially, by virtue of the fact that they do not +have a horizon. On the other hand as we noted earlier, +the origin of RMT behavior from fuzzballs is supposedly +non-trivial to arrange. Our results show on the contrary, +that there are generic bulk mechanisms that can enable +fuzzballs to capture RMT features. +In the following section, we will present our main re- +sults while relegating the technical details to various Sup- +plementary Material. +To give further confidence that +these results really do have to do with the magic of black +holes and horizons, we will also discuss some examples +where there are no horizons. Putting a cut-off in such +geometries leads to major qualitative differences from +stretched horizons, which we elaborate. In the Conclu- +sions section we review and emphasize the salient points +of our results and extract some lessons. Some related fur- +ther observations and comments about future directions +[30], as well as various technical details, are presented in +various Supplementary Material. +Main Results: We will solve the massless scalar field +equation in a black hole geometry with a stretched hori- +zon, while demanding the boundary condition φ = φ0(θ) +at the stretched horizon. We will do this for the BTZ +black hole as well as for the Rindler wedge (times a com- +pact space); these were the two cases studied in detail in +[5]. The primary virtue of these choices is that the wave +equation is solvable in terms of well-known special func- +tions. We will see that the resulting physics is identical in +both cases, and we do not expect qualitative changes in +our conclusions for other black holes, in 2+1 dimensions +and higher. +The details of the wave equations and how we obtain +the normal modes for a general stretched horizon pro- +file are presented in the Supplementary Material. The +scalar field boundary condition profile can be described +in terms of its Fourier coefficients. We will choose each of +these Fourier coefficients randomly from a suitable Gaus- +sian distribution (see the discussion in the Supplementary +Material, for details on how this is done). This means +that there are two choices we need to make in order to +fully define the problem – the mean and the variance of +this Gaussian distribution [35]. To make sure that the +Fourier series sum converges and leads to a well-defined +profile, we will also cut-off the sum at some J. +This +should be compared to the cut-off in J that is required +to define the SFF [5]. It turns out that the mean and +the variance have a heuristic (but suggestive) interpre- +tation in terms of the location and the fluctuations of +the stretched horizon, see again the Supplementary Ma- +terial. To have a natural interpretation as the stretched +horizon at a Planck length, we will take the mean to be +very large in tortoise coordinates (and therefore close to +the horizon). Note that since we are working with a fixed +background geometry, the Planck length is an arbitrary +choice. +Our conclusions are entirely analogous for both BTZ +and Rindler, so we will discuss BTZ here for concreteness; +see Figures. +More plots and discussions are provided +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +s +p(s) +FIG. 1: LSD for BTZ black hole normal modes +ω(n = 1, J), with ⟨λ⟩ = −103, Jmax = 800 and +σλJ = σ0/J with σ0 = 0.3 . Supplementary Material +contains definitions and explanations of the notation. +The blue curve is the GUE level spacing curve. +β=0 +105 +106 +107 +108 +109 +1010 +1011 +10-7 +10-5 +0.001 +0.100 +t +g(t) +FIG. 2: SFF for BTZ black hole normal modes; same +parameters as above. The slope of the line is unity. +Together these two figures (and the many others in the +Supplementary Material) show that we can get both the +linear ramp as well as level repulsion from “synthetic” +fuzzball normal modes. +in the Supplementary Material. To summarize – Our re- +sults for the SFF and the LSD reduce to those of [5] when +the variance is zero; the SFF has a linear ramp, but the +LSD is of the “extreme” delta function-like form. But +remarkably, for small but non-zero choices of the vari- + +4 +ance, one finds LSDs that fit Wigner-Dyson [36], while +the linear ramp remains intact. Finally, as the variance +becomes large, the LSD reduces to the Poisson form and +the ramp goes away. +These results are qualitatively different from corre- +sponding results in a geometry where a cut-off is intro- +duced without a horizon. To demonstrate this, we also +study flat space and AdS with a cut-off. +Once again +the details of the computation and plots are presented +in the Supplementary Material. +In flat space, we find +that there is never a ramp of slope ∼ 1, but for moderate +variances, there is a clear non-linear ramp of slope ∼ 1.7. +The level-spacing distribution when there is no variance +is again a strongly peaked delta-function-like form. But +the origin of this fact has a simple (and less interesting) +understanding, as opposed to when there was a horizon. +In flat space the levels are roughly evenly spaced and +therefore the spectrum is analogous to that of an SHO +(which also has a delta function LSD, even though it is +the farthest thing from RMT). Indeed, we have directly +checked that the SFF of an SHO with a small amount of +noise added to its energy levels, reproduces precisely the +non-linear ramp of slope ∼ 1.7 we noted above. This, and +some interesting related results are presented in some of +the Supplementary Material and a follow-up paper [30]. +The bottom line is that the linearity of the ramp is lost +when we simply put a cut-off in flat space as opposed to +at a stretched horizon. Loosely similar statements hold +in AdS as well. We will suppress the details, except to +mention that one has to take care of two separate cases. +One where the cut-off size is much larger than the AdS +scale, and another where it is much smaller. The latter +turns out to yield a discussion identical to the flat space +case above (as expected). In the former case, there is +no well-defined constant slope ramp at all in the log-log +plot, so it will not be of interest to us here. +A second distinction between the modes of a horizon- +less cut-off and a stretched horizon is that the variance +one introduces in the former case can heuristically be in- +terpreted as due to macroscopic fluctuations at the cut- +off. In the stretched horizon case, the fluctuations are +in the tortoise coordinate, and therefore have a natu- +ral interpretation as Planckian suppressed. This is again +very natural from the membrane paradigm/fuzzball per- +spective. +These matters are discussed in detail in the +Supplementary Material. +Conclusions: Our goal in [5] and this paper has been +to see whether the fuzzball/stretched horizon paradigm +can be useful for reproducing some of the successes of +the semi-classical approach to quantum black holes. As +pointed out in [5], both approaches have produced inter- +esting results, yet major open problems remain. While +the stretched horizon/fuzzball will trivially get rid of +some aspects of the information paradox, finding hints +of RMT behavior is considered challenging. +We demonstrated that we can find both the linear +ramp and conventional level repulsion (as well as RMT +level spacing ratios) from a stretched horizon. The linear +ramp is a direct consequence of a cut-off near the hori- +zon. In a cut-off geometry without a horizon, the linear +ramp never exists, and a non-linear ramp when it exists, +can be understood as related to an SHO spectrum with +noise. We also found that conventional level repulsion +is easy to realize, by simply incorporating angular de- +pendence in the boundary condition. This is interesting, +because such angle-dependence is generic in BPS fuzzball +microstates. +The existence of the linear ramp is usually taken as an +indicator of rigidity in the spectrum. It is a signature of +strong chaos. Finding the linear ramp in our previous +paper [5] was encouraging, but the absence of conven- +tional level repulsion made the result puzzling. But given +the ramp, it is natural to suspect that some small per- +turbation may be able to produce the nearest-neighbor +correlations [37] as well. The challenge was to identify +the right kind of perturbation. The fluctuations at the +stretched horizon that we have included in this paper +can be viewed as a natural candidate for such a small +perturbation. The variance in the Fourier modes of the +fluctuation profile leads to a small noise in the spectrum, +which leads to the requisite spread in the LSD. +Our results also strengthen the case that level repulsion +is a weaker hint of chaos than the linear ramp. This is +because it is only sensitive to nearest neighbor physics. +We explicitly demonstrate this using the example of the +SHO in the Supplementary Material, where it is shown +that adding a small amount of noise to the SHO energy +levels is sufficient to produce conventional WD-like LSD +plots. +But this is not sufficient to produce the linear +ramp, which is sensitive to long range correlations within +the spectrum. This again ties nicely with the observation +that the linear ramp is present only when the cut-off is +near the black hole horizon, while level repulsion can be +realized in a cut-off geometry with or without a horizon +as long as we are working with a fluctuating profile [38]. +The SFFs of horizonless cases with moderate variance +have a power law ramp of slope ∼ 1.7 – This is the same +as that of a moderately noisy SHO. +A natural proposal that ties together our observations +then, is as follows – Signatures of robust chaos (in the +sense of spectral rigidity) emerge when we consider a +stretched horizon close to the black hole. +Such signa- + +5 +tures are not present when the cut off is in empty space +or far from the horizon. These statements are indepen- +dent of the profile choices at the cut-off. But the profiles +do play a role, when we are discussing nearest neighbor +physics and level repulsion in the system. A profile with +non-vanishing variance can lead to nearest-neighbor level +repulsion both with or without a horizon, but the natural +length scale associated to the variance has to be macro- +scopic for this to happen in a horizonless geometry. In +other words, even if we allow macroscopic fluctuations, +we can at best see nearest neighbor effects in a horizon- +less geometry with a cut-off. On the contrary, stretched +horizon/fuzzball modes automatically carry signatures of +robust chaos and a linear ramp, with or without a non- +trivial profile. If the profile is generic in the sense of hav- +ing a small non-zero variance, they reproduce the correct +nearest neighbor effects as well. +Semi-classical +bulk +calculations +involving +replica +wormholes (and implicitly, ensemble averages) are known +to produce a smooth linear ramp without fluctuations. +The challenge for quantum gravity is to reproduce a lin- +ear ramp without any ensemble average from a single +microstate, and with fluctuations. Our calculation, de- +spite its simplicity has reproduced both these features. +This may seem surprising because our set up is super- +ficially (semi-)classical. +But this is misleading – The +boundary conditions we are imposing at the stretched +horizon, while technically simple, are conceptually highly +non-trivial from the dual CFT. 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Chen, “Spectral form +factor for free large N gauge theory and strings,” +JHEP 06, +137 (2022) doi:10.1007/JHEP06(2022)137 +[arXiv:2202.04741 [hep-th]]. +[35] The precise distribution does not seem too important for +our results. This is natural because (as noted in our mo- +tivations), we are looking for results like linear ramp and +level repulsion, which are semi-qualitative and robust. We +have checked that similar statements hold also for uni- +formly distributed Fourier modes, but we will not elabo- +rate on it here. +[36] By choosing the variance suitably, we can get good fits +with GSE, GUE or GOE. We will mostly present GUE +fits in this paper. A very interesting feature of these re- +sults is that since they arise by tuning certain continu- +ous boundary conditions and not the (discrete choice of) +ensemble from which the Hamiltonian matrix is chosen, +they seem to allow a continuum of LSDs that naturally +generalize WD. +[37] We thank M. Hanada for some encouraging comments on +this point. +[38] Let us also re-iterate that the fluctuations should nat- +urally be viewed as macroscopic (and not Planck sup- +pressed) if they are to give rise to level repulsion in a +cut-off geometry without a horizon. +[39] M. B. Marcus and G. Pisier, “Random Fourier Series with +Applications to Harmonic Analysis”, (AM-101), Volume +101 (Annals of Mathematics Studies, 101), Princeton +University Press. +[40] B. Bhattacharjee and C. Krishnan, “A General Prescrip- +tion for Semi-Classical Holography,” [arXiv:1908.04786 +[hep-th]]. +[41] C. Krishnan and V. Mohan, “Hints of gravitational +ergodicity: +Berry’s ensemble and the universality of +the semi-classical Page curve,” JHEP 05, 126 (2021) +doi:10.1007/JHEP05(2021)126 [arXiv:2102.07703 [hep- +th]]. + +8 +Supplementary material +CASE STUDY: BTZ +As in [5], we will start by considering a scalar field Φ of mass m in the BTZ background, +ds2 = −(r2 − r2 +h) +L2 +dt2 + +L2 +(r2 − r2 +h)dr2 + r2dψ2 +(1) +with −∞ < t < ∞, 0 < r < ∞ and 0 ≤ ψ < 2π. In [5] we fixed units by setting L = 1 and worked with the numerical +choice rh = 1 from the outset. Here, we will present the more general expressions because it is useful in comparisons +with cut-off empty space. The new boundary conditions and the corresponding results/plots start only after (14). +So a reader who is familiar with the results of [5] and is willing to believe the slightly more general expressions we +present here, can skip directly to the discussion after (14). +The wave equation +□Φ ≡ +1 +� +|g| +∂µ +�� +|g|∂µΦ +� += m2Φ +(2) +can be solved by writing +Φ = +1 +√r +� +ω,J +e−iωteiJψφω,J(r) +(3) +with integer J. The radial part of (2) satisfies, +(r2 − r2 +h)2φ +′′ +ω,J(r) + 2r(r2 − r2 +h)φ +′ +ω,J(r) + ω2L4φω,J(r) − VJ(r)φω,J(r) = 0 +(4) +where +V (r) = (r2 − r2 +h) +� 1 +r2 +� +J2L2 + r2 +h +4 +� ++ ν2 − 1 +4 +� +, +ν2 = 1 + m2. +(5) +We will generally work with the massless case, ν = 1. The solution1 of this is given in terms of hypergeometric +functions as +φ(r) = (r) +1 +2 − iJL +rh � +r2 − r2 +h +�− iωL2 +2rh +� +e− πJL +rh +� r +rh +� 2iJL +rh C2H (r) + C1G (r) +� +, +(6) +where we are suppressing the subscripts ω, J on the LHS as well as on C1 and C2. Here, +G (r) = 2F1 +�1 +2 +� +1 − iωL2 +rh +− iJL +rh +− ν +� +, 1 +2 +� +1 − iωL2 +rh +− iJL +rh ++ ν +� +; 1 − iJL +rh +, r2 +r2 +h +� +(7) +H (r) = 2F1 +�1 +2 +� +1 − iωL2 +rh ++ iJL +rh +− ν +� +, 1 +2 +� +1 − iωL2 +rh ++ iJL +rh ++ ν +� +; 1 + iJL +rh +, r2 +r2 +h +� +. +(8) +1We will work with the massless scalar and the J = 0 mode needs special treatment. See footnote 13 of [5]. + +9 +Near the AdS boundary (r → ∞), the radial solution (6) becomes +φbdry(r) ≈ −ir +iωL2 +rh +−ν− 1 +2 r +1− iωL2 +rh +− iJL +rh +ν +h +(r2 − r2 +h)− iωL2 +2rh e− πL(J+ωL) +2rh +× +× +� +e−i π +2 ν� +γ (J, −ν) C1 + γ (−J, −ν) C2 +� ++ O +� +1/r3/2� ++ r2νei π +2 ν +r2ν +h +� +γ (J, ν) C1 + γ (−J, ν) C2 + O +� +1/r3/2��� +, (9) +where +γ (J, ν) ≡ +Γ(1 − iJL +rh )Γ(ν) +Γ +� +1 +2(1 − iωL2 +rh +− iJL +rh + ν) +� +Γ +� +1 +2(1 + iωL2 +rh +− iJL +rh + ν) +�, +(10) +Normalizability at r → ∞ sets the 2nd term of equation (9) to zero, which leads to +C2 = − γ (J, ν) +γ (−J, ν)C1, +(11) +fixing the constant of integration C2 in terms of C1 or vice versa. +We will eventually place our boundary condition at a stretched horizon, to be thought of as a Planck length or so +outside the horizon. Near the horizon, the radial solution can be approximated as +φhor(r) ≈ C1 +� +P1 (r/rh − 1)− iωL2 +2rh + Q1 (r/rh − 1) +iωL2 +2rh +� +, +(12) +where +P1 = − +2− iωL2 +2rh e− πJL +rh (JπL) +� +e +2πJL +rh +− 1 +� +r +− 1 +2 − iωL2 +rh +− iJL +rh +h +csch( πωL2 +rh )Γ(− iJL +rh ) +� +e +πJL +rh + eπ(iν+ ωL2 +rh ) +� +Γ(1 − iωL2 +rh )Γ( 1 +2(1 + iωL2 +rh +− iJL +rh − ν))Γ( 1 +2(1 + iωL2 +rh +− iJL +rh + ν)) +(13) +Q1 = +(−1) +iωL2 +rh 21+ iωL2 +2rh e +2πωL2 +rh +π2r +1 +2 − iωL2 +rh +− iJL +rh +h +(coth( πωL2 +rh ) − 1) +� +eiπν + e +πL(J+ωL) +rh +� +Γ( iJL +rh )Γ(1 + iωL2 +rh )Γ( 1 +2(1 − iωL2 +rh +− iJL +rh − ν))Γ( 1 +2(1 − iωL2 +rh +− iJL +rh + ν)) +. +(14) +In [5] we demanded a vanishing condition for the scalar at the stretched horizon r = r0. Motivated by the angle- +dependent profiles that are found in BPS fuzzballs, we will generalize this in the present paper. We will demand that +at r = r0 the scalar field takes the form of a given profile φ0(ψ). In terms of the notation introduced in (3), we will +write +1 +√r0 +� +J,ω +eiJψe−iωtφω,J(r0) = φ0(ψ, t) +(15) +Expanding the RHS in terms of the Fourier modes eiJψ and e−iωt and absorbing some constants suitably, we get an +equation of the form φhor(r = r0) = C0 where on both LHS and RHS we have suppressed the ω and J subscripts. +Note that ultimately we will get a quantization condition for our ω’s, and this means that an implicit assumption in +the above approach is that the φ0(ψ, t) can be expanded in terms of these modes. Our explicit boundary conditions +below and their solution can be viewed as a self-consistent way to do this. +Concretely, this leads to +C1 +� +P1 (r0/rh − 1)− iωL2 +2rh + Q1 (r0/rh − 1) +iωL2 +2rh +� += C0, +(16) +=⇒ P1 +Q1 += +C0 +C1Q1 +(r0/rh − 1) +iωL2 +2rh − (r0/rh − 1) +iωL2 +rh . +(17) + +10 +As in [5], it is possible to show that |P1| = |Q1|. So by writing P1 = |P1|eiα and Q1 = |Q1|eiβ, (17) can be written as +ei(α−β) = µJe +i +� +λJ ωL2 +rh ++ θ +2 +� +− eiθ +(18) +where +θ = Arg +� +(r0/rh − 1) +iωL2 +rh +� +, +µJ = +��� C0 +C1Q1 +���, +and λJωL2 +rh += Arg +� C0 +C1Q1 +� +(19) +We have emphasized the J-dependence of µ and λ in the notation, but it should be noted that with these definitions, +they have an n-dependence as well. The real and imaginary parts of (18) lead to the definition +µJ = 2 cos +�λJωL2 +rh +− θ +2 +� +(20) +as well as the quantization condition on ω, +cos(α − β) = cos +�2λJωL2 +rh +� +(21) +This last equation is a key equation for our purposes. Since this is a phase equation, the modes depend on a free +integer n. It is possible to check that these two equations together reduce to the quantization condition we had in [5] +when we set µJ = 0. More generally, one can solve the quantization condition by choosing λJ from a distribution, +which we will usually take to be Gaussian. +We will take λ for each value of J from the same distribution. Note that heuristically, λJ is comparable to the +stretched horizon location. One way to see this is to note that (20) implies (if there are no fluctuations, and λ and µ +are taken to be J-independent constants) that fixing +λJ = 1 +2 ln +� r0 +rh +− 1 +� +(22) +fixes µJ. More generally, the fact that the difference between λJ and 1 +2 ln +� +r0 +rh − 1 +� +is what shows up in (20) suggests +that the natural scale of λJ is the stretched horizon radius in (essentially) tortoise coordinates. Eqn (20) also makes +it tempting to view the fluctuations in µJ as due not to the fluctuations in λJ but due to the fluctuations of the +stretched horizon. This last interpretation is of course simply a heuristic, because it is not meaningful to have a +J-dependent notion of stretched horizon radius. Nonetheless, we view this as highly suggestive, in light of the usual +claim that the profile functions in fuzzball geometries are supposed to capture the fluctuations of the cap. Indeed, +our initial motivation for considering the scalar field profile, was as a proxy for this. +It is worth emphasizing in the above discussion (and elsewhere), that there is some leftover freedom in fixing C1 +in terms of C0 and the rest of the quantities. An analogous freedom existed in [5] as well – our demands do not +completely fix the boundary conditions, but they fix them enough to determine the normal modes. We can fix this +extra freedom by setting C1Q1 = 1 so that µJ and λJ have the nice interpretation as (essentially) the magnitude and +phase of C0. Remember that C0 has J-dependence which we often suppress to avoid notational congestion, it is the +Fourier coefficient of the scalar profile. +There is one choice we have made in the above definitions, which may be worth further study. In defining λJ via the +last equation in (19), we have extracted an ω on the LHS. It may also be natural to define the λ variable without this, +in which case our quantization conditions should be solved after the replacement λJ → λJ/ω and choosing the new λ’s +from some suitable distribution. Since the target results we are aiming for are believed to be robust semi-qualitative +statements like level repulsion and the linear ramp, these choices should not affect them. We have checked that indeed +this is the case. Ultimately these choices all correspond to how we parametrize the Fourier modes C0 of the profile +φ0(ψ, t) in (15). Explicitly, the profile should be written as +φ0(ψ, t) = +� +n,J +C0(n,J)eiJψe−iω(n,J)t +(23) + +11 +and our choice corresponds to the parametrization +C0(n,J) = µJ,nei λJ ω(n,J)L2 +rh +(24) +where we have kept the n and J dependencies, fully explicit. If we absorb the ω into the definition of λ as discussed +above, then the µ (and therefore the C0) have only J-dependence. (Superficially, this may seem illegal because ω’s +have an n-dependence. But remember that the ω’s are determined after the definition of λ, so one can check that +this is perfectly well-defined.) This leads to some nice features in some expressions, but also some compensating +complications/ugliness in others. So we will stick to the form defined by (18) and (19), or (24). It may be interesting +to investigate the naturalness of the choices involved here from the perspective of Haar typicality in the phase space +of the scalar field, but we will not undertake it here. +With these caveats, one way to get some intuition for the profile is to consider the quantity +˜φ(ψ) ≡ +Jcut +� +J=0 +C0(n=0,J)eiJψ = +Jcut +� +J=0 +µJ,n=0ei λJ ω(n=0,J)L2 +rh +eiJψ. +(25) +This is what we will often call the profile function. It should be emphasized that our quantization condition arises +essentially as a condition on the phase of the Fourier coefficient. The various arbitrary choices we discussed above +can be understood as arising from the fact that it does not unambiguously fix C0. In writing the second equality of +(25) we have fixed C1Q1 = 1 as mentioned above, but this is an ad-hoc choice. Similar statements were true in the +discussion in [5] as well, where the magnitude information was again not needed to determine the normal modes. One +way to understand this in the present setting is to note that the last two equations in (19) basically determine the +phase and the magnitude of the profile C0 via +µJei λJ ωL2 +rh += +C0 +C1Q1 +. +(26) +Once we make a choice of λ (which is a single real variable that captures the phase information) the quantization +condition is obtained via (21). Then µJ is completely fixed via (20). All of this only fixes the ratio on the RHS of +(26), while the profile itself is controlled by C0. Fourier series where the phase is suitably random have been studied +extensively by mathematicians, see eg. the book [39]. It seems significant that this structure naturally arises in our +discussions; this is clearly worthy of further study. +In the plots in this section, we have set L = rh = 1, and ⟨λ⟩ = 1 +2 ln +� +r0 +rh − 1 +� +, as we change the variance of the +Gaussian distribution from which λ is chosen. This choice of ⟨λ⟩ ensures that µJ = 2 in the zero-variance limit. This is +slightly different from the µJ = 0 condition in [5] but it is natural (and straightforward to check) that the qualitative +results on LSD and SFF remain identical. One can also in principle treat µJ (instead of λJ) as the quantity chosen from +a distribution. This is slightly more convenient to connect to the language of [5]. This changes some of our formulas +in minor ways, but the essential point that there is one real parameter worth of freedom that we are fixing, remains +intact. We have experimented with various choices of λ-variance as a function of J, eg. σλJ ≡ σ0, σ0/J, σ0/ +√ +J. In +the plots in this section, we present the σ0/ +√ +J case and we quote the value of σ0. We will sometimes refer to σ0 +loosely as the variance. A suppression of the variance with J is useful because the normal mode level-spacing gets +smaller as J increases, and therefore too large a variance can completely destabilize the structure of the spectrum +(and along with it, the linear ramp and level repulsion). Let us also mention that when we juxtapose the plots of an +SFF and an LSD for some choice of variance, we show it for the same realization that we choose from the Gaussian +distribution. This statement applies to the Rindler plots of the next section as well. +For zero variance, we reproduce the “extreme” Wigner-Dyson plots for the level spacing that we found in [5] as well +as the linear ramp. If we increase the variance slightly, the ramp remains intact, but the level-spacing takes the more +conventional WD form. We can find fits with GSE, GUE or GOE with minor increments in variance, we present GUE +in the plots. Eventually, as we increase the variance to very large values, the level spacing degenerates to a Poisson +form and the ramp is lost. + +12 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0 +1 +2 +3 +4 +5 +s +p(s) +β=0 +107 +108 +109 +1010 +1011 +1012 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +t +g(t) +FIG. 3: LSD (left) and SFF (right) for BTZ with ⟨λ⟩ = −104 and Jmax = 400 with σ0 = 0.0. We are working with +ω(n = 2, J). These results are a version of the results in [5]. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +s +p(s) +β=0 +107 +108 +109 +1010 +1011 +1012 +1013 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +t +g(t) +FIG. 4: Same as before, but with σ0 = 0.025. The blue curve on the left is GUE. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +s +p(s) +β=0 +107 +108 +109 +1010 +1011 +1012 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +t +g(t) +FIG. 5: Same as in the previous figures, but with σ0 = 2.0. The red curve on the left is Poisson. + +13 +0 +100 +200 +300 +400 +0.00015711 +0.00015712 +0.00015713 +0.00015714 +0.00015715 +0.00015716 +0.00015717 +0.00015718 +J +ω(2,J) +0 +100 +200 +300 +400 +0.00015712 +0.00015714 +0.00015716 +0.00015718 +J +ω(2,J) +FIG. 6: Spectrum of BTZ with σ0 = 0 (left) vs σ0 = 2.0 (right). ⟨λ⟩ = −104 and Jmax = 400. We show ω(n = 2, J). +CASE STUDY: RINDLER × COMPACT SPACE +We will follow the motivations and discussion in section 4.2 of [5] when developing the Rindler case, which the +reader should consult for notations. We solve the wave equation in the metric +ds2 = e2aξ(−dη2 + dξ2) + R2dφ2 +(27) +and introduce A ≡ ω/a and y ≡ eaξ(J/aR) as in [5]. In terms of y variable the position of boundary and horizon are +given by y → ∞ and y → 0 respectively. In the notations of [5], we require that the field φ(y) vanish at boundary. +We also demand that it has a profile at some small y0 (or ξ = ξ0). When y → ∞, the relevant equation is [5] +φ(y) → (C1 + C2) +ey +√2πy + (C1eπA + C2e−πA) e−y +√2πy . +(28) +The boundary condition at infinity leads to C1 = −C2, and at y0 implies (in notation that is parallel to the BTZ case +before): +C1(I[−iA, y0] − I[iA, y0]) = C0, +=⇒ I[−iA, y0] − I[iA, y0] = C0 +C1 +(29) +Near horizon i.e. in the limit y0 → 0 the above expressions can be approximated by +C1 +� +y−iA +2iA +Γ(1 − iA) − yiA +2−iA +Γ(1 + iA) +� += C0 +(30) +C0 +C1 +� J +aR +�−iA �eaξ +2 +�iA +− +�eaξ +2 +�2iA += +� J +aR +�−2iA Γ(iA) +Γ(−iA) +(31) +Now Abs +�� J +aR +�−2iA +Γ(iA) +Γ(−iA) +� += 1, so (31) can be written, again in notation that simulates the BTZ case as +µJeiωλJeiθ/2 − eiθ = eiα +(32) +with +µJ = Abs +� +C0 +C1 +� J +aR +�−iA� +, +ωλJ = Arg +� +C0 +C1 +�eaξ +2 +�iA� +, +α = Arg +�� J +aR +�−2iA Γ(iA) +Γ(−iA) +� +, +θ = Arg +��eaξ +2 +�2iA� +(33) + +14 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +s +p(s) +β=0 +105 +106 +107 +108 +109 +1010 +1011 +10-7 +10-5 +0.001 +0.100 +t +g(t) +FIG. 7: LSD (left) and SFF (right) for Rindler with parameters described in the text. σ0 = 0.0. We are working +with ω(n = 1, J). These results should be compared to the results in [5]. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +s +p(s) +β=0 +105 +106 +107 +108 +109 +1010 +1011 +10-7 +10-5 +0.001 +0.100 +t +g(t) +FIG. 8: Same as the previous figure, but with σ0 = 0.02. The blue curve on the left is GUE. +0 +1 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +s +p(s) +β=0 +105 +106 +107 +108 +109 +1010 +10-7 +10-5 +0.001 +0.100 +t +g(t) +FIG. 9: Same as the two previous figures, but with σ0 = 1. The red curve on the left is Poisson. + +15 +0 +100 +200 +300 +400 +500 +600 +700 +0.001561 +0.001562 +0.001563 +0.001564 +0.001565 +0.001566 +0.001567 +J +ω(1, J) +0 +100 +200 +300 +400 +500 +600 +700 +0.001560 +0.001561 +0.001562 +0.001563 +0.001564 +0.001565 +0.001566 +0.001567 +J +ω(1, J) +FIG. 10: Spectrum of Rindler with σ0 = 0 (left) vs σ0 = 1.0 (right). We show ω(n = 1, J). +This therefore again leads to similar structures as in BTZ. We find +µJ = 2 cos(λJω − θ/2) +(34) +as well as the quantization condition +cos(α) = cos(2λJω) +(35) +Because the structure is precisely parallel to BTZ, we will not repeat the discussion; it is clear that the normal mode +calculation proceeds in an identical manner. The mean value of λ can be related to the stretched horizon location. +Once we choose R, ξ0 and a, the normal modes ω(n, J) can be numerically solved for as a function of J (and an integer +n). We present the plots in precise parallel to the BTZ case. The qualitative results are identical, despite the fact +that the special functions that showed up in the wave equations here are different. In the plots we present, we have +chosen a = 1, R = 2, Jmax = 700, ⟨λ⟩ = −103 and σJ = σo/ +√ +J. The σ0 values are quoted in the plots. +THE HAIRY HARMONIC OSCILLATOR AND CUT-OFF IN EMPTY SPACE: +LEVEL REPULSION WITHOUT LINEAR RAMP +We noted that the linear ramp in the SFF and repulsion in the LSD can both be seen in the stretched horizon +spectrum if the boundary condition is generic. We also pointed out that the level spacing ratio discussed in [33] is also +consistent with RMT expectations. Together, these constitute very strong evidence that fuzzball/stretched horizon +spectra have strong connections to random matrices and chaos. +In this section, we will ask a slightly more resolved question: which of these is a more robust indicator of chaos? Is +it the linear ramp or is it level repulsion? Or are both these features always present in systems concomitantly? We +will present some hints in this section that the linear ramp may be a more robust diagnostic of strong chaos than +nearest-neighbor data. This is not an entirely new suggestion (the length of the ramp is often viewed as an indicator +of the “strength” of chaos), but we will give some examples which we feel are instructive. +We will start (as often in physics) with the simple harmonic oscillator (SHO). For our purposes, the SHO is +interesting because even though it is the farthest thing from a chaotic system, it exhibits a naive (or extreme) version +of level repulsion – the levels are equally spaced, and the LSD is a delta function shifted from the origin. Motivated +by the results of this paper, we can ask if there is a natural way to “perturb” the SHO spectrum so that the level +spacing becomes a more conventional Wigner-Dyson-like form. It turns out that a simple way to engineer this exists +– we simply allow a small amount of (Gaussian) noise in the levels of the SHO. We will call this set up a hairy or +noisy SHO. See Figure 11 right panel, for a typical LSD of an SHO perturbed in this way. We present a GOE fit +for concreteness. But again, by adjusting the variance, we can find fits with GSE or GUE. We are not aware of a +previous observation of this simple but striking fact in the literature, but it is easy enough to understand – Random +noise in the energy levels directly affects the nearest neighbor data, which explains why the delta function in the LSD +gets spread out. + +16 +0 +1 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +s +p(s) +0 +1 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +s +p(s) +FIG. 11: LSDs of Cut-off flat space with fluctuation profile (left) vs hairy SHO (right). Flat space data: +Jmax = 300, rcut = 1, λ-variance = 0.0174. We are working with ω(n = 1, J). SHO data: nmax = 600, ω = 1, +spectral noise variance = 0.36. Both fits are GOE. +β=0 +0.01 +0.10 +1 +10 +100 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +t +g(t) +β=0 +0.001 +0.010 +0.100 +1 +10 +100 +10-7 +10-5 +0.001 +0.100 +t +g(t) +FIG. 12: SFFs of the same systems (SHO on the right). The yellow line has slope 1.7 (both left and right). In other +words, this is a power law ramp. +0 +50 +100 +150 +200 +250 +300 +0 +50 +100 +150 +200 +250 +300 +J +ω(1,J) +0 +100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +400 +500 +600 +n +ω(n) +FIG. 13: Shapes of spectra, for the same systems as above (SHO again on the right). It is clear that both the +spectra are approximately evenly spaced. The punchline of the figures in this page is that the spectral features of +the two systems have crucial similarities. +On the other hand, strong chaos is characterized by spectral rigidity which is encoded in the linear ramp in the +SFF. And indeed, if one computes the SFF of the SHO with noise in the spectrum, one finds that the ramp is in fact + +17 +non-linear. This is illustrated in Figure 12, right panel. We emphasize that it is remarkable that a well-defined ramp +exists, even though it is not linear. In fact, we find that on a log-log plot, it has a well-defined slope of ∼ 1.7. In other +words, a hairy SHO has a power law ramp, at least within the context of our numerical results. +These SHO results shed light on the distinctions between a black hole with a stretched horizon, and a cut-off in +empty space. If we impose a simple Dirichlet condition φ = 0 at the cut-off, in the former case we find a linear ramp +[5], but in empty space there is no clear ramp, certainly nothing of slope ∼ 1. See Figure 14. But as we add variance +to the profile, we see the emergence of a power law ramp, see Figure 12 left panel. The SHO example above provides +us a clear understanding of this. A cut-off in flat space leads to eigenvalues that are connected to the zeros of Bessel +functions (as we will see). These are roughly evenly spaced – so the spectrum looks crudely like that of an SHO. +Relatedly, the level spacing in the φ = 0 case is essentially a delta function. But this can be made to look like a +more spread out (WD-like) form by demanding instead that the boundary condition is φ ∼ φ0(θ) where the profile +has some variance in its Fourier modes. The noise in the spectrum increases when we do this, and as a result (as +pointed out above for the hairy SHO) we find that the LSD takes a more conventional WD form. Of course, when +the variance is very large, the spectrum ends up becoming Poisson. Crucially, the slope of the ramp is never ∼ 1 in +these cases. For moderate values of the variance, it is consistent with the ∼ 1.7 quoted above for the noisy SHO – see +figures. (Note that when the variance is steadily increased, the ramp gets increasingly washed out. So this statement +applies only to those values of the variance for which there is a clear ramp.) +The basic message we extract from these calculations is that the spectrum on a cut-off geometry without a horizon +is essentially a hairy SHO spectrum. When we have a horizon on the other hand, the spectrum is not that of an SHO +in any sense (as we saw in previous sections). Together with the striking linearity of the ramp, we are therefore lead +to conclude that the physics in the latter case is not simply due to nearest-neighbor physics. +We conclude this section by providing some of the details of the flat space calculation. We will work with 2+1 +dimensions, the physics we aim for is unaffected by increase in dimensions: +ds2 = −dt2 + dr2 + r2dψ2 +(36) +Separating the scalar field as (say) in the BTZ case, we find the radial part +φ +′′ +ω,J(r) + 1 +r φ +′ +ω,J(r) + ω2φω,J(r) − V (r)φω,J(r) = 0 +(37) +with +V (r) = 1 +r2 +� +J2 + m2� +. +(38) +We will consider the solution of this equation (37) in the massless case, which is given in terms of Bessel functions: +φ(r) = C1JJ(ωr) + C2YJ(ωr), +(39) +where, JJ and YJ are Bessel functions of first and second kind respectively. We suppress the J and ω (or n) subscripts +of C1 and C2. +As before, we need one boundary condition to fix a relationship between C1 and C2 and another condition at a +cut-off to fix the normal modes. The former role was played by AdS-normalizability in the BTZ case. We could +likewise demand a suitably chosen bulk condition here as well that relates C1 and C2. By numerical experimentation +we have found that the qualitative features of the ramp and LSD that we are after, are insensitive to this choice. +This is unsurprising because the physics we are interested in, is the result of the quantization condition, and not the +relationship between C1 and C2. In the following, we will simply demand that C2 = 0. Note that this sets the bulk +source mode (which is singular at the origin) to zero, while retaining the homogeneous mode. It was noted in [40] +that the bulk source mode is the analogue in flat space, to the non-normalizable mode in AdS. So this choice is a +natural analogue of the normalizability demand in AdS. But we emphasize that large classes of choices are likely to +give similar results. + +18 +Using this boundary condition, equation (39) becomes +φ(r) = C1JJ(ωr). +(40) +Demanding a profile at the cut-off r = r0 leads to an equations analogous to what we found for BTZ: φ(r = r0) = C0. +C1JJ(ωr0) = C0 =⇒ JJ(ωr0) = C0 +C1 +≡ λJ. +(41) +Note that we could also define the RHS to be ωλJ, which is more analogous to some of our discussions in BTZ and +Rindler. But as we mentioned, these choices do not affect the semi-qualitative features we are after, so we will stick +with this simple choice here for concreteness. +We will take λJ to be Gaussian distributed with mean zero, and adjustable variance. The equation is easy to solve +numerically, by taking the seed for the root search to be the 1st zero of the J-the Bessel function. When the variance +is zero, we find an “extreme” delta-function like distribution in the LSD. The ramp of the SFF is not particularly +well-defined, but we can already see a crude similarity to an SHO with a very small amount of noise – See Figure 14 +below. +0.01 +1 +100 +104 +10-7 +10-5 +0.001 +0.100 +t +g(t) +β=0 +0.01 +1 +100 +104 +10-8 +10-5 +0.01 +t +g(t) +FIG. 14: Cut-off flat space with no variance vs SHO with a tiny amount of noise. The precise values are +unimportant. Our goal here is not to make a detailed comparison, but to observe the crude similarity which +becomes more striking as we increase the variance/noise, see Figure 12. The two lines are of slope ∼ 1.7 and ∼ 1. +When we steadily add variance, we find more conventional level repulsion and the emergence of a robust ramp +of slope ∼ 1.7, which we presented in Figure 12 left panel. As noted above, this is precisely what one finds from a +noisy SHO as well. Eventually we find a Poisson distributed LSD. The (power law) ramp gets washed out, when the +variance becomes very large. These features are identical to what we find in the hairy/noisy SHO case. +To summarize – flat space with a cut-off is qualitatively identical to hairy SHO. Unlike in the case of the stretched +horizon cut-off, the levels are essentially evenly spaced. We have done a similar calculation in empty AdS as well, +as discussed in the main body of the paper, and the results are again consistent. These results mean that the linear +ramp (which is often viewed as an indicator of strong chaos) does not arise from a cut-off in flat space. But for the +same reason that a hairy SHO can mimic the LSD of an RMT (which in itself is a fact not emphasized previously in +the literature, to our knowledge), the spectrum of cut-off flat space can also exhibit level repulsion – the variance in +the boundary condition simply introduces a variance in the nearest neighbor levels. But this is not sufficient to create +conventional spectral rigidity or robust chaos. +A further distinction between empty space with cut-off and the stretched horizon is discussed in the next section. +PLANCK-SCALE HIERARCHY +We observed that the fluctuations at the cut-off in empty space translate to fluctuations in the energy levels and +therefore lead to level repulsion. In other words, nearest neighbor effects of chaos can be produced simply by having + +19 +fluctuations at the cut-off. We also noted however that the linear ramp (which is a deeper signature of chaos) cannot +be realized this way, and requires the presence of a horizon. +In fact there is another interesting distinction between the stretched horizon and a cut-off in empty space. This +has to do with the fact that the fluctuations at the cut-off needed in the stretched horizon scenario are hierarchically +suppressed, allowing the interpretation that they are Planck-scale. The fluctuations in the empty space cut-off on the +other hand are naturally macroscopic. To see this, first note that in (41), the first zero of the J-th Bessel function +is linearly spaced in J with the scale controlled by r0. The natural scale controlling the fluctuations in the RHS is +therefore r0 (this dependence is approximately linear if we define the RHS of (41) to be ωλJ instead of λJ). On the +other hand in the horizon case, the situation is more interesting. To see this in detail, let us work with the concrete +case of BTZ, and observe that the conventional tortoise coordinate here is defined via +z = L2 +2 rh +ln +�r + rh +r − rh +� +(42) +This means that the usual radial coordinate of the stretched horizon x ≡ r − rh is approximately +x = 2 rhe−2rhz/L2, +(43) +from which it follows that the fluctuation in the stretched horizon location goes as +|∆x| ∼ 4 (rh/L)2 e−2rhz/L2|∆z| +(44) +where we have instated a magnitude sign because z → ∞ corresponds to the horizon. Now, from (21) it follows that +e2λ = (x/rh) and therefore +2 e2λ∆λ = ∆x +rh +=⇒ 2 x ∆λ ∼ ∆x. +(45) +Using (43) and (44) in this final relation, we get +L2 +rh +|∆λ| = |∆z|. +(46) +Since the horizon size and AdS length scale are both macroscopic, this means that the fluctuations in λ are naturally +in tortoise coordinate, implying via (44) that the stretched horizon fluctuations are suppressed by a factor of +e−2rhz0/L2 +(47) +where z0 is the mean stretched horizon in tortoise coordinate. We also see that L2/z0 is a natural candidate for the +Planck length. In units where L = 1, note that this is a small quantity because z0 is very large when the cut-off +is close to the horizon. Of course, since we are working with a fixed background, these are all somewhat heuristic +statements. +To summarize: The variance in both cases (with and without horizon) can be used as a heuristic proxy for fluctu- +ations of the cut-off surface. But a key distinction in the stretched horizon is that there, the variance captures the +tortoise coordinate and therefore the fluctuations can naturally be viewed as Planck suppressed. +OPEN QUESTIONS AND FUTURE DIRECTIONS +In this section, we discuss some questions that are worth understanding better in the wake of our results. Some of +these are more conceptual than others. +• Are there more natural choices for the profile functions? We have considered the most simple-minded notion of +a “generic” profile – choose some randomly distributed Fourier coefficients. The BPS fuzzball profiles, at least +in the 2-charge case [14, 16] are known to contain enough phase space to reproduce the entropy of the black +hole. This suggests that perhaps Haar typicality in some form is a better notion of genericity than our present +proposal. It will be interesting to incorporate this in some systematic way. + +20 +• Despite the simplicity of our calculation, we have managed to find a linear ramp with fluctuations and level +repulsion in (a heuristic candidate for) a single microstate. The price we have paid is that we have sacrificed a +(manifestly) smooth horizon. But the emergence of RMT behavior in our calculation suggests that thermality +(and therefore smoothness) may emerge via a suitable ensemble replacement of the microstate. Understanding +this operationally is clearly a problem of outstanding interest. +• In our previous paper [5], the LSD was not one of the conventional RMT distributions, but there was a clear +linear ramp. Our main point in that paper was that this is a generic feature of normal modes at stretched +horizons, when the boundary condition φ = 0 was imposed. In this paper, we have seen systems which exhibit +the opposite behavior – The ramp is non-linear, but one has level spacing that matches well with conventional +Wigner-Dyson-like statistics. In fact, we noticed that the latter can be arranged very simply via an SHO with a +noisy spectrum. Together the results of these papers are a very clear demonstration that the folk wisdom that the +linear ramp is a smoking gun of conventional Wigner-Dyson classes (or their Altland-Zernbauer generalizations) +is not always true. It will be good to understand the broader setting in which these features arise as special +cases. +• We did not have to introduce any form of ensemble average. +Our profile curve is chosen via a Gaussian +distribution in the Fourier coefficients, but it should be emphasized that once the curve is chosen, there is +absolutely nothing “averaged” about the calculation. The emergence of RMT behavior is entirely deterministic. +It has been suggested in [41] that semi-classical gravity should be viewed as a tool for capturing ergodic averaged +gravitational dynamics, for evolution that is in bulk local equilibrium. This would give an understanding of the +surprising utility of Euclidean gravity in each epoch of Hawking radiation in obtaining the Page curve [11]. It +will be very interesting to connect these two perspectives. +• In [5] we had observed that there was a kink-like structure at the top of the ramp. A tangential consequence of +the calculations in the present paper is that we have understood that this kink becomes less and less prominent, +as we bring the stretched horizon closer and closer to the horizon. This is a strong indication that one of the +worries expressed in [5] – that the ramp may be an artefact – is very unlikely to be true. +• Inspired by the results of this paper and [5], we have been able to identify a broader class of spectra which lead +to interesting ramps and level spacing structures. These results together suggest the notion of a generalized +RMT spectrum, which will be elaborated elsewhere [30]. A key message is that boundary conditions are often +a crucial ingredient in quantum chaos. This is true in our black hole problem, but note that the idea is much +more general. Eg., the Hamiltonian of the hard sphere gas is simply that of a collection of free particles – it is +the boundary conditions that breathe life (and chaos) into the system. +• One of the technical features underlying the results of this paper and [5] is the observation that the dependence +of the spectrum on the angular quantum numbers is not linear. Instead it gets pulled logarithmically along J. +The resulting quasi-degeneracy was essential for our results. It will be good to get a more mechanical/conceptual +understanding of this observation as well as to explore its consequences more broadly. +• We found a clear ramp with slope ∼ 1.7 in our SFF plots for hairy SHO and cut-off flat space. This is an +extremely simple example of a non-linear ramp, whose slope is a constant (̸= 1) in a log-log plot. It seems +surprising and interesting that it is closely related to the SHO. Can this shed light on the fact that despite being +the “ultimate” integrable system, the SHO exhibits an extreme version of level repulsion (ie., its LSD has no +support at the origin, and has a delta function form)? +• Relatedly, and more speculatively – does the fact that extreme WD spectra arise from Dirichlet boundary +conditions at stretched horizons indicate that black holes are the “ultimate” RMT systems? If this is true, black +holes can be viewed as the natural counterpoint to SHOs from our previous item. Note that the suggestion +we are making here is distinct from the chaos bound of [8], which is about early time chaos and OTOCs. The +observation about LSDs that we are making here is related to late time chaos. Black holes may not just be fast +scramblers [6], they may also be the most robust scramblers. Clearly, more work remains to be done. + diff --git a/6dFKT4oBgHgl3EQfTi2q/content/tmp_files/load_file.txt b/6dFKT4oBgHgl3EQfTi2q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72bac10fd52f2c5a35b218a383515b5f12e1f494 --- /dev/null +++ b/6dFKT4oBgHgl3EQfTi2q/content/tmp_files/load_file.txt @@ -0,0 +1,1047 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf,len=1046 +page_content='Fuzzballs and Random Matrices Suman DASa, Sumit K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' GARGb, Chethan KRISHNANc, Arnab KUNDUa aTheory Division, Saha Institute of Nuclear Physics, A CI of Homi Bhabha National Institute, 1/AF, Bidhannagar, Kolkata 700064, India Email: suman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='das@saha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='in, arnab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='kundu@saha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='in bManipal Centre for Natural Sciences, Manipal Academy of Higher Education, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Pai Planetarium Building, Manipal-576104, Karnataka, India Email: sumit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='kumar@manipal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='edu cCenter for High Energy Physics, Indian Institute of Science, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Raman Road, Bangalore 560012, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Email: chethan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='krishnan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='com Black holes are believed to have the fast scrambling properties of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' If the fuzzball proposal is to be a viable model for quantum black holes, it should reproduce this expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is considered challenging, because it is natural for the modes on a fuzzball microstate to follow Poisson statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In a previous paper, we noted a potential loophole here, thanks to the modes depending not just on the n-quantum number, but also on the J-quantum numbers of the compact dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' For a free scalar field φ, by imposing a Dirichlet boundary condition φ = 0 at the stretched horizon, we showed that this J-dependence leads to a linear ramp in the Spectral Form Factor (SFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Despite this, the status of level repulsion remained mysterious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In this letter, motivated by the profile functions of BPS fuzzballs, we consider a generic profile φ = φ0(θ) instead of φ = 0 at the stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' For various notions of genericity (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' when the Fourier coefficients of φ0(θ) are suitably Gaussian distributed), we find that the J-dependence of the spectrum exhibits striking evidence of level repulsion, along with the linear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also find that varying the profile leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The linear ramp in our previous work can be understood as arising via an extreme version of level repulsion in such a limiting spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also explain how the stretched horizon/fuzzball is different in these aspects from simply putting a cut-off in flat space or AdS (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=', without a horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Introduction: The quest for an understanding of quan- tum black holes has been one of the engines driving re- search in quantum gravity in the last half century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In particular, the recent revival of the black hole informa- tion paradox [1, 2] due to the works of Mathur [3] and AMPS [4] has raised questions about the smoothness of the horizon which are still not fully settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the context of holography/string theory, there are two broad lines along which work on quantum black holes has progressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The first approach, which we will call the semi-classical approach following [5], is built on in- sights from bulk (often Euclidean) effective field theory, toy models of 2D gravity, and holographic entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Considerable intuition has been gleaned about the quantum nature of black holes from this approach (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [6–10]) with the crowning achievement being a semi- classical reproduction of the Page curve [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Despite this, the precise status of detailed unitarity and smooth- ness are still unclear from this perspective, because the calculation is fundamentally Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The second line of approach is the fuzzball program of Mathur and others which argues that black hole microstates cap off smoothly before the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In our opinion, the operational mean- ing of this bulk statement in the full quantum setting is not yet completely clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But the mere existence of large classes of such solutions [13–20] in the supergravity limit of stringy BPS black holes is surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In conventional general relativity, they would not exist thanks to the no hair theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' See [5] for a more detailed discussion of the pros and cons of the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It was suggested in [5] that one way to make progress may be to try and reproduce general lessons of the semi- classical approach, from fuzzball-motivated considera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The hope is that since many of these expectations are generic, this may teach us something about how to think about quantum fuzzballs at finite temperature even arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='11780v1 [hep-th] 27 Jan 2023 2 though constructing explicit solutions is possible only in the supergravity BPS limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Conversely, if realizing these lesson from fuzzball-motivated ideas is impossible or highly contrived, that could be viewed as an argument against the fuzzball program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A particularly sharp setting in which one could explore this tension is in the expectation that black holes are fast scramblers [6], and that they exhibit dynamical features of random matrices [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A linear ramp [22] in the spec- tral form factor (SFF) and repulsion in the level spacing distribution (LSD), are viewed as indicators of chaos in random matrix theory (RMT) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' However, these RMT signatures are generally thought to be challenging to re- alize from the fuzzball paradigm, see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [25] – we expect capped geometries to have roughly evenly spaced levels, in loose analogy with the standing waves of a cylindrical column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This makes conventional level repulsion and the linear ramp, difficult to understand from the fuzzball per- spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note also that simply declaring that the black hole is an ensemble of such spectra, does not solve the problem [26] – While this will certainly allow a richer set of level spacings in the collective spectrum, there is still no mechanism to ensure level repulsion [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Instead, an ensemble of fuzzballs will give rise to Poisson statistics, just as an ensemble of simple harmonic oscillators (SHO) would [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These expectations are reasonable, but they are also difficult to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is because solving wave equations in generic fuzzball microstate geometries is both difficult (because the metric is complicated) and not immediately useful (because explicit metrics in BPS cases are at zero temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Exploiting the fact that the questions we wish to tackle are generic, in [5] it was suggested that one may be able to make progress by studying a black hole at finite temperature with a stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In particular, the normal modes of a scalar field were stud- ied in [5], by computing the spectrum of modes that re- sult from a φ = 0 boundary condition at the stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The results of [5] showed that the expectations listed in the previous paragraph have a major caveat, they are true only if one ignored the dependence of the spectrum on the angular quantum numbers of the com- pact dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Unlike the dependence on the principal n-quantum number, the dependence on the J-quantum numbers was found not to be (approximately) linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In- stead there was a quasi-degeneracy of levels as a function of J for moderately large J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Most strikingly, it was found that the SFF computed from the spectrum showed very clear evidence of a linear ramp, even though conventional level repulsion was not present in the J-direction [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It should be emphasized here that this is the only case in the literature that we are aware of, where a linear ramp in the SFF exists without an underlying RMT spectrum with Wigner-Dyson (WD) level spacing [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' While the results of [5] were a tantalizing hint of RMT behavior in fuzzballs, a coherent understanding of them could not be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In particular, the presence of a lin- ear ramp together with the absence of conventional level repulsion, made a compelling interpretation impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The purpose of this letter, is to shed some light on this mysterious state of affairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will place the results of [5] in context by finding a more general calculation that can interpolate between Poisson and RMT-like spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The idea (at least in hindsight) is extremely simple, and motivated by the fact that the known BPS fuzzball solu- tions [13, 15, 17] are described by profile functions that are supposed to capture the fluctuations of the cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This suggests that a natural generalization of our simple φ = 0 boundary conditions of [5] is to consider a generic pro- file φ = φ0(θ) at the stretched horizon, where θ is a mnemonic for the angular directions of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In this paper, we will consider profiles of this type, where “genericity” will be implemented via choosing Fourier coefficients of φ0(θ) from suitable random distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is a natural implementation of the intuitive notion of “fluctuation at the horizon”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Remarkably, in this very natural set up, we see both level repulsion as well as the linear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' By tuning the variance of the distribution from which φ0(θ) is chosen, we show that the LSD can interpolate from Poisson to WD-like spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In partic- ular, as the variance collapses to zero and the boundary condition reduces to φ = 0, we find that the LSD col- lapses to a very sharp (almost delta-function-like) peak, as found in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It was speculated in [5] that this should be viewed as an “extreme” version of level-repulsion, and our present paper clarifies the precise sense in which this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Conversely, as the variance is steadily in- creased, the LSD transitions from “extreme” to conven- tional Wigner-Dyson spectra and eventually to Poisson [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our results demonstrate that fuzzball/stretched hori- zon modes can exhibit the spectral features of RMT and late time chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We emphasize that this is a bulk cal- culation of RMT behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The expectation of RMT behavior and eigenstate thermalization in black hole mi- crostates is natural in the dual holographic theory, be- cause it is strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This has been explicitly demonstrated in the setting of toy dual theories like SYK and tensor models [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' From the bulk however, while early time chaos is captured by out-of-time-ordered cor- relators [7, 8], late-time chaos as captured by level repul- sion and discreteness of the spectrum are very difficult 3 to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Fuzzballs can exhibit discreteness in the spectrum trivially, by virtue of the fact that they do not have a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' On the other hand as we noted earlier, the origin of RMT behavior from fuzzballs is supposedly non-trivial to arrange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our results show on the contrary, that there are generic bulk mechanisms that can enable fuzzballs to capture RMT features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the following section, we will present our main re- sults while relegating the technical details to various Sup- plementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To give further confidence that these results really do have to do with the magic of black holes and horizons, we will also discuss some examples where there are no horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Putting a cut-off in such geometries leads to major qualitative differences from stretched horizons, which we elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the Conclu- sions section we review and emphasize the salient points of our results and extract some lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Some related fur- ther observations and comments about future directions [30], as well as various technical details, are presented in various Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Main Results: We will solve the massless scalar field equation in a black hole geometry with a stretched hori- zon, while demanding the boundary condition φ = φ0(θ) at the stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will do this for the BTZ black hole as well as for the Rindler wedge (times a com- pact space);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' these were the two cases studied in detail in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The primary virtue of these choices is that the wave equation is solvable in terms of well-known special func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will see that the resulting physics is identical in both cases, and we do not expect qualitative changes in our conclusions for other black holes, in 2+1 dimensions and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The details of the wave equations and how we obtain the normal modes for a general stretched horizon pro- file are presented in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The scalar field boundary condition profile can be described in terms of its Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will choose each of these Fourier coefficients randomly from a suitable Gaus- sian distribution (see the discussion in the Supplementary Material, for details on how this is done).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This means that there are two choices we need to make in order to fully define the problem – the mean and the variance of this Gaussian distribution [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To make sure that the Fourier series sum converges and leads to a well-defined profile, we will also cut-off the sum at some J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This should be compared to the cut-off in J that is required to define the SFF [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It turns out that the mean and the variance have a heuristic (but suggestive) interpre- tation in terms of the location and the fluctuations of the stretched horizon, see again the Supplementary Ma- terial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To have a natural interpretation as the stretched horizon at a Planck length, we will take the mean to be very large in tortoise coordinates (and therefore close to the horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note that since we are working with a fixed background geometry, the Planck length is an arbitrary choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our conclusions are entirely analogous for both BTZ and Rindler, so we will discuss BTZ here for concreteness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' see Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' More plots and discussions are provided 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 s p(s) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 1: LSD for BTZ black hole normal modes ω(n = 1, J), with ⟨λ⟩ = −103, Jmax = 800 and σλJ = σ0/J with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Supplementary Material contains definitions and explanations of the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The blue curve is the GUE level spacing curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' β=0 105 106 107 108 109 1010 1011 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 2: SFF for BTZ black hole normal modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' same parameters as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The slope of the line is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Together these two figures (and the many others in the Supplementary Material) show that we can get both the linear ramp as well as level repulsion from “synthetic” fuzzball normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To summarize – Our re- sults for the SFF and the LSD reduce to those of [5] when the variance is zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' the SFF has a linear ramp, but the LSD is of the “extreme” delta function-like form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But remarkably, for small but non-zero choices of the vari- 4 ance, one finds LSDs that fit Wigner-Dyson [36], while the linear ramp remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Finally, as the variance becomes large, the LSD reduces to the Poisson form and the ramp goes away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These results are qualitatively different from corre- sponding results in a geometry where a cut-off is intro- duced without a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To demonstrate this, we also study flat space and AdS with a cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Once again the details of the computation and plots are presented in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In flat space, we find that there is never a ramp of slope ∼ 1, but for moderate variances, there is a clear non-linear ramp of slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The level-spacing distribution when there is no variance is again a strongly peaked delta-function-like form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But the origin of this fact has a simple (and less interesting) understanding, as opposed to when there was a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In flat space the levels are roughly evenly spaced and therefore the spectrum is analogous to that of an SHO (which also has a delta function LSD, even though it is the farthest thing from RMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Indeed, we have directly checked that the SFF of an SHO with a small amount of noise added to its energy levels, reproduces precisely the non-linear ramp of slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7 we noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This, and some interesting related results are presented in some of the Supplementary Material and a follow-up paper [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The bottom line is that the linearity of the ramp is lost when we simply put a cut-off in flat space as opposed to at a stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Loosely similar statements hold in AdS as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will suppress the details, except to mention that one has to take care of two separate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' One where the cut-off size is much larger than the AdS scale, and another where it is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The latter turns out to yield a discussion identical to the flat space case above (as expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the former case, there is no well-defined constant slope ramp at all in the log-log plot, so it will not be of interest to us here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A second distinction between the modes of a horizon- less cut-off and a stretched horizon is that the variance one introduces in the former case can heuristically be in- terpreted as due to macroscopic fluctuations at the cut- off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the stretched horizon case, the fluctuations are in the tortoise coordinate, and therefore have a natu- ral interpretation as Planckian suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is again very natural from the membrane paradigm/fuzzball per- spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These matters are discussed in detail in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Conclusions: Our goal in [5] and this paper has been to see whether the fuzzball/stretched horizon paradigm can be useful for reproducing some of the successes of the semi-classical approach to quantum black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' As pointed out in [5], both approaches have produced inter- esting results, yet major open problems remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' While the stretched horizon/fuzzball will trivially get rid of some aspects of the information paradox, finding hints of RMT behavior is considered challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We demonstrated that we can find both the linear ramp and conventional level repulsion (as well as RMT level spacing ratios) from a stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The linear ramp is a direct consequence of a cut-off near the hori- zon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In a cut-off geometry without a horizon, the linear ramp never exists, and a non-linear ramp when it exists, can be understood as related to an SHO spectrum with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also found that conventional level repulsion is easy to realize, by simply incorporating angular de- pendence in the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is interesting, because such angle-dependence is generic in BPS fuzzball microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The existence of the linear ramp is usually taken as an indicator of rigidity in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It is a signature of strong chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Finding the linear ramp in our previous paper [5] was encouraging, but the absence of conven- tional level repulsion made the result puzzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But given the ramp, it is natural to suspect that some small per- turbation may be able to produce the nearest-neighbor correlations [37] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The challenge was to identify the right kind of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The fluctuations at the stretched horizon that we have included in this paper can be viewed as a natural candidate for such a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The variance in the Fourier modes of the fluctuation profile leads to a small noise in the spectrum, which leads to the requisite spread in the LSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our results also strengthen the case that level repulsion is a weaker hint of chaos than the linear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is because it is only sensitive to nearest neighbor physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We explicitly demonstrate this using the example of the SHO in the Supplementary Material, where it is shown that adding a small amount of noise to the SHO energy levels is sufficient to produce conventional WD-like LSD plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But this is not sufficient to produce the linear ramp, which is sensitive to long range correlations within the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This again ties nicely with the observation that the linear ramp is present only when the cut-off is near the black hole horizon, while level repulsion can be realized in a cut-off geometry with or without a horizon as long as we are working with a fluctuating profile [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The SFFs of horizonless cases with moderate variance have a power law ramp of slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7 – This is the same as that of a moderately noisy SHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A natural proposal that ties together our observations then, is as follows – Signatures of robust chaos (in the sense of spectral rigidity) emerge when we consider a stretched horizon close to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Such signa- 5 tures are not present when the cut off is in empty space or far from the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These statements are indepen- dent of the profile choices at the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But the profiles do play a role, when we are discussing nearest neighbor physics and level repulsion in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A profile with non-vanishing variance can lead to nearest-neighbor level repulsion both with or without a horizon, but the natural length scale associated to the variance has to be macro- scopic for this to happen in a horizonless geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In other words, even if we allow macroscopic fluctuations, we can at best see nearest neighbor effects in a horizon- less geometry with a cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' On the contrary, stretched horizon/fuzzball modes automatically carry signatures of robust chaos and a linear ramp, with or without a non- trivial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' If the profile is generic in the sense of hav- ing a small non-zero variance, they reproduce the correct nearest neighbor effects as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Semi-classical bulk calculations involving replica wormholes (and implicitly, ensemble averages) are known to produce a smooth linear ramp without fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The challenge for quantum gravity is to reproduce a lin- ear ramp without any ensemble average from a single microstate, and with fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our calculation, de- spite its simplicity has reproduced both these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This may seem surprising because our set up is super- ficially (semi-)classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But this is misleading – The boundary conditions we are imposing at the stretched horizon, while technically simple, are conceptually highly non-trivial from the dual CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It is clearly of interest to understand this boundary condition better from an in- trinsically CFT perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It may be useful to re-visit the many questions about (quantum) black holes at finite temperature, armed with the perspectives we have added in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In this section, we have only emphasized black hole physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A more detailed discussion of open questions and questions more intrinsic to RMT physics are presented in the Sup- plementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Preetham Kumar for crucial contribu- tions in our previous collaboration [5], and Masanori Hanada, Romesh Kaul, Alok Laddha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Loganayagam, Ayan Mukhopadhyay, Onkar Parrikar, Ashoke Sen, Kostas Skenderis and Amitabh Virmani for discussions and/or correspondence.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Gur-Ari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Hanada, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Polchin- ski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Saad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Shenker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Stanford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Stre- icher and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Tezuka, “Black Holes and Random Ma- trices,” JHEP 05, 118 (2017) [erratum: JHEP 09, 002 (2018)] doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='1007/JHEP05(2017)118 [arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='04650 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [22] In this paper, by linear ramp we will mean a ramp of slope ∼ 1 on the log-log plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A constant slope ramp on the log-log plot, but with a slope different from unity, is still non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [23] See [21] for the general definition of SFF and [5] for dis- cussions on it in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We follow the notations of [5] and always work at infinite temperature, β = 0, in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' LSD is defined and discussed in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [24] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Haake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Gnutzmann and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Kus, “Quantum Signa- tures of Chaos” (Springer Series in Synergetics) 4th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [25] www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='v=0BO-p58Pypc&t=3397s [26] Even with an ensemble, there are conceptual questions on when/how an ensemble should replace a microstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Ensembles arise in physics typically as effective repre- sentations of microscopic physics, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' when an ensemble average can stand in for a time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' So it is not clear in the first place that one should simply adjoin the nor- mal modes of all the separate microstates in order to get the “effective” spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [27] It is generally expected that level repulsion and linearity of the ramp go hand in hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our results in [5] and this paper demonstrate that this is very far from a theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Nonetheless the general expectation that RMT behavior is connected to level repulsion and linear ramp is broadly true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [28] See Appendix C of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [29] It was speculated in [5] that the level spacing found there may perhaps be viewed as an “extreme” version of a Wigner-Dyson-like distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The grounds for this speculation were quite scanty, but in this paper we will see that there is a systematic sense in which it is true!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note that just because a level spacing plot has no support at the origin does not guarantee that we are dealing with a random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The simplest illustration of this fact is the SHO – the LSD of the SHO is a delta function sep- arated from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will have more to say about this in the Supplementary Material and also in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Garg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Krishnan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Kundu, “Gener- alized Random Matrix Spectra”, To Appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [31] We have since been able to construct many examples of this type, this will be presented elsewhere [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [32] Even though we do no report the details here, we have also studied the level-spacing ratios γ [33] of these spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is another diagnostic of RMT behavior along with SFF and LSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' For small/zero variance, we find γ values that are consistent with RMT spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But it steadily increases with the variance and becomes (very) large, matching the expectation that γ = ∞ for Poisson systems [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' γ is a diagnostic defined via nearest neigh- bor data and is therefore somewhat redundant with the LSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is one reason why we do not consider this as truly distinct diagnostic, and do not emphasize it in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In all the examples we consider, the behavior of LSD and LSR are mutually consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The LSD and the (linear ramp of the) SFF on the other hand, do genuinely capture somewhat distinct aspects of random matrix be- havior as we will elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [33] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Atas, E.' metadata={'source': 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[hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Chen, “Spectral form factor for free large N gauge theory and strings,” JHEP 06, 137 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='1007/JHEP06(2022)137 [arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='04741 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [35] The precise distribution does not seem too important for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is natural because (as noted in our mo- tivations), we are looking for results like linear ramp and level repulsion, which are semi-qualitative and robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We have checked that similar statements hold also for uni- formly distributed Fourier modes, but we will not elabo- rate on it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [36] By choosing the variance suitably, we can get good fits with GSE, GUE or GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will mostly present GUE fits in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A very interesting feature of these re- sults is that since they arise by tuning certain continu- ous boundary conditions and not the (discrete choice of) ensemble from which the Hamiltonian matrix is chosen, they seem to allow a continuum of LSDs that naturally generalize WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [37] We thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Hanada for some encouraging comments on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [38] Let us also re-iterate that the fluctuations should nat- urally be viewed as macroscopic (and not Planck sup- pressed) if they are to give rise to level repulsion in a cut-off geometry without a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Marcus and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Pisier, “Random Fourier Series with Applications to Harmonic Analysis”, (AM-101), Volume 101 (Annals of Mathematics Studies, 101), Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [40] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Bhattacharjee and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Krishnan, “A General Prescrip- tion for Semi-Classical Holography,” [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='04786 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Krishnan and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Mohan, “Hints of gravitational ergodicity: Berry’s ensemble and the universality of the semi-classical Page curve,” JHEP 05, 126 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='1007/JHEP05(2021)126 [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='07703 [hep- th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 8 Supplementary material CASE STUDY: BTZ As in [5], we will start by considering a scalar field Φ of mass m in the BTZ background, ds2 = −(r2 − r2 h) L2 dt2 + L2 (r2 − r2 h)dr2 + r2dψ2 (1) with −∞ < t < ∞, 0 < r < ∞ and 0 ≤ ψ < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In [5] we fixed units by setting L = 1 and worked with the numerical choice rh = 1 from the outset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Here, we will present the more general expressions because it is useful in comparisons with cut-off empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The new boundary conditions and the corresponding results/plots start only after (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' So a reader who is familiar with the results of [5] and is willing to believe the slightly more general expressions we present here, can skip directly to the discussion after (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The wave equation □Φ ≡ 1 � |g| ∂µ �� |g|∂µΦ � = m2Φ (2) can be solved by writing Φ = 1 √r � ω,J e−iωteiJψφω,J(r) (3) with integer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The radial part of (2) satisfies, (r2 − r2 h)2φ ′′ ω,J(r) + 2r(r2 − r2 h)φ ′ ω,J(r) + ω2L4φω,J(r) − VJ(r)φω,J(r) = 0 (4) where V (r) = (r2 − r2 h) � 1 r2 � J2L2 + r2 h 4 � + ν2 − 1 4 � , ν2 = 1 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (5) We will generally work with the massless case, ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The solution1 of this is given in terms of hypergeometric functions as φ(r) = (r) 1 2 − iJL rh � r2 − r2 h �− iωL2 2rh � e− πJL rh � r rh � 2iJL rh C2H (r) + C1G (r) � , (6) where we are suppressing the subscripts ω, J on the LHS as well as on C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Here, G (r) = 2F1 �1 2 � 1 − iωL2 rh − iJL rh − ν � , 1 2 � 1 − iωL2 rh − iJL rh + ν � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 1 − iJL rh , r2 r2 h � (7) H (r) = 2F1 �1 2 � 1 − iωL2 rh + iJL rh − ν � , 1 2 � 1 − iωL2 rh + iJL rh + ν � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 1 + iJL rh , r2 r2 h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (8) 1We will work with the massless scalar and the J = 0 mode needs special treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' See footnote 13 of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 9 Near the AdS boundary (r → ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' the radial solution (6) becomes φbdry(r) ≈ −ir iωL2 rh −ν− 1 2 r 1− iωL2 rh − iJL rh +ν h (r2 − r2 h)− iωL2 2rh e− πL(J+ωL) 2rh × × � e−i π 2 ν� γ (J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' −ν) C1 + γ (−J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' −ν) C2 � + O � 1/r3/2� + r2νei π 2 ν r2ν h � γ (J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ν) C1 + γ (−J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ν) C2 + O � 1/r3/2��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (9) where γ (J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ν) ≡ Γ(1 − iJL rh )Γ(ν) Γ � 1 2(1 − iωL2 rh − iJL rh + ν) � Γ � 1 2(1 + iωL2 rh − iJL rh + ν) �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (10) Normalizability at r → ∞ sets the 2nd term of equation (9) to zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' which leads to C2 = − γ (J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ν) γ (−J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ν)C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (11) fixing the constant of integration C2 in terms of C1 or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will eventually place our boundary condition at a stretched horizon, to be thought of as a Planck length or so outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Near the horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' the radial solution can be approximated as φhor(r) ≈ C1 � P1 (r/rh − 1)− iωL2 2rh + Q1 (r/rh − 1) iωL2 2rh � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='P1 = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2− iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2rh e− πJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh (JπL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2πJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 − iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='csch( πωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh )Γ(− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='πJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh + eπ(iν+ ωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='Γ(1 − iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh )Γ( 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2(1 + iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh − ν))Γ( 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2(1 + iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh + ν)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='(13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='Q1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh 21+ iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2rh e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2πωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='π2r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 − iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='(coth( πωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ) − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='eiπν + e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='πL(J+ωL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='Γ( iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh )Γ(1 + iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh )Γ( 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2(1 − iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh − ν))Γ( 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2(1 − iωL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='− iJL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='rh + ν)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (14) In [5] we demanded a vanishing condition for the scalar at the stretched horizon r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Motivated by the angle- dependent profiles that are found in BPS fuzzballs, we will generalize this in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will demand that at r = r0 the scalar field takes the form of a given profile φ0(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In terms of the notation introduced in (3), we will write 1 √r0 � J,ω eiJψe−iωtφω,J(r0) = φ0(ψ, t) (15) Expanding the RHS in terms of the Fourier modes eiJψ and e−iωt and absorbing some constants suitably, we get an equation of the form φhor(r = r0) = C0 where on both LHS and RHS we have suppressed the ω and J subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note that ultimately we will get a quantization condition for our ω’s, and this means that an implicit assumption in the above approach is that the φ0(ψ, t) can be expanded in terms of these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our explicit boundary conditions below and their solution can be viewed as a self-consistent way to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Concretely, this leads to C1 � P1 (r0/rh − 1)− iωL2 2rh + Q1 (r0/rh − 1) iωL2 2rh � = C0, (16) =⇒ P1 Q1 = C0 C1Q1 (r0/rh − 1) iωL2 2rh − (r0/rh − 1) iωL2 rh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (17) 10 As in [5], it is possible to show that |P1| = |Q1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' So by writing P1 = |P1|eiα and Q1 = |Q1|eiβ, (17) can be written as ei(α−β) = µJe i � λJ ωL2 rh + θ 2 � − eiθ (18) where θ = Arg � (r0/rh − 1) iωL2 rh � , µJ = ��� C0 C1Q1 ���, and λJωL2 rh = Arg � C0 C1Q1 � (19) We have emphasized the J-dependence of µ and λ in the notation, but it should be noted that with these definitions, they have an n-dependence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The real and imaginary parts of (18) lead to the definition µJ = 2 cos �λJωL2 rh − θ 2 � (20) as well as the quantization condition on ω, cos(α − β) = cos �2λJωL2 rh � (21) This last equation is a key equation for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Since this is a phase equation, the modes depend on a free integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It is possible to check that these two equations together reduce to the quantization condition we had in [5] when we set µJ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' More generally, one can solve the quantization condition by choosing λJ from a distribution, which we will usually take to be Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will take λ for each value of J from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note that heuristically, λJ is comparable to the stretched horizon location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' One way to see this is to note that (20) implies (if there are no fluctuations, and λ and µ are taken to be J-independent constants) that fixing λJ = 1 2 ln � r0 rh − 1 � (22) fixes µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' More generally, the fact that the difference between λJ and 1 2 ln � r0 rh − 1 � is what shows up in (20) suggests that the natural scale of λJ is the stretched horizon radius in (essentially) tortoise coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Eqn (20) also makes it tempting to view the fluctuations in µJ as due not to the fluctuations in λJ but due to the fluctuations of the stretched horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This last interpretation is of course simply a heuristic, because it is not meaningful to have a J-dependent notion of stretched horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Nonetheless, we view this as highly suggestive, in light of the usual claim that the profile functions in fuzzball geometries are supposed to capture the fluctuations of the cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Indeed, our initial motivation for considering the scalar field profile, was as a proxy for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It is worth emphasizing in the above discussion (and elsewhere), that there is some leftover freedom in fixing C1 in terms of C0 and the rest of the quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' An analogous freedom existed in [5] as well – our demands do not completely fix the boundary conditions, but they fix them enough to determine the normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We can fix this extra freedom by setting C1Q1 = 1 so that µJ and λJ have the nice interpretation as (essentially) the magnitude and phase of C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Remember that C0 has J-dependence which we often suppress to avoid notational congestion, it is the Fourier coefficient of the scalar profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' There is one choice we have made in the above definitions, which may be worth further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In defining λJ via the last equation in (19), we have extracted an ω on the LHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It may also be natural to define the λ variable without this, in which case our quantization conditions should be solved after the replacement λJ → λJ/ω and choosing the new λ’s from some suitable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Since the target results we are aiming for are believed to be robust semi-qualitative statements like level repulsion and the linear ramp, these choices should not affect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We have checked that indeed this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Ultimately these choices all correspond to how we parametrize the Fourier modes C0 of the profile φ0(ψ, t) in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Explicitly, the profile should be written as φ0(ψ, t) = � n,J C0(n,J)eiJψe−iω(n,J)t (23) 11 and our choice corresponds to the parametrization C0(n,J) = µJ,nei λJ ω(n,J)L2 rh (24) where we have kept the n and J dependencies, fully explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' If we absorb the ω into the definition of λ as discussed above, then the µ (and therefore the C0) have only J-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (Superficially, this may seem illegal because ω’s have an n-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But remember that the ω’s are determined after the definition of λ, so one can check that this is perfectly well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=') This leads to some nice features in some expressions, but also some compensating complications/ugliness in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' So we will stick to the form defined by (18) and (19), or (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It may be interesting to investigate the naturalness of the choices involved here from the perspective of Haar typicality in the phase space of the scalar field, but we will not undertake it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' With these caveats, one way to get some intuition for the profile is to consider the quantity ˜φ(ψ) ≡ Jcut � J=0 C0(n=0,J)eiJψ = Jcut � J=0 µJ,n=0ei λJ ω(n=0,J)L2 rh eiJψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (25) This is what we will often call the profile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It should be emphasized that our quantization condition arises essentially as a condition on the phase of the Fourier coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The various arbitrary choices we discussed above can be understood as arising from the fact that it does not unambiguously fix C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In writing the second equality of (25) we have fixed C1Q1 = 1 as mentioned above, but this is an ad-hoc choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Similar statements were true in the discussion in [5] as well, where the magnitude information was again not needed to determine the normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' One way to understand this in the present setting is to note that the last two equations in (19) basically determine the phase and the magnitude of the profile C0 via µJei λJ ωL2 rh = C0 C1Q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (26) Once we make a choice of λ (which is a single real variable that captures the phase information) the quantization condition is obtained via (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Then µJ is completely fixed via (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' All of this only fixes the ratio on the RHS of (26), while the profile itself is controlled by C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Fourier series where the phase is suitably random have been studied extensively by mathematicians, see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' the book [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It seems significant that this structure naturally arises in our discussions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' this is clearly worthy of further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the plots in this section, we have set L = rh = 1, and ⟨λ⟩ = 1 2 ln � r0 rh − 1 � , as we change the variance of the Gaussian distribution from which λ is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This choice of ⟨λ⟩ ensures that µJ = 2 in the zero-variance limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is slightly different from the µJ = 0 condition in [5] but it is natural (and straightforward to check) that the qualitative results on LSD and SFF remain identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' One can also in principle treat µJ (instead of λJ) as the quantity chosen from a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is slightly more convenient to connect to the language of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This changes some of our formulas in minor ways, but the essential point that there is one real parameter worth of freedom that we are fixing, remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We have experimented with various choices of λ-variance as a function of J, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' σλJ ≡ σ0, σ0/J, σ0/ √ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the plots in this section, we present the σ0/ √ J case and we quote the value of σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will sometimes refer to σ0 loosely as the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A suppression of the variance with J is useful because the normal mode level-spacing gets smaller as J increases, and therefore too large a variance can completely destabilize the structure of the spectrum (and along with it, the linear ramp and level repulsion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Let us also mention that when we juxtapose the plots of an SFF and an LSD for some choice of variance, we show it for the same realization that we choose from the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This statement applies to the Rindler plots of the next section as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' For zero variance, we reproduce the “extreme” Wigner-Dyson plots for the level spacing that we found in [5] as well as the linear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' If we increase the variance slightly, the ramp remains intact, but the level-spacing takes the more conventional WD form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We can find fits with GSE, GUE or GOE with minor increments in variance, we present GUE in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Eventually, as we increase the variance to very large values, the level spacing degenerates to a Poisson form and the ramp is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 0 1 2 3 4 5 s p(s) β=0 107 108 109 1010 1011 1012 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 1 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 3: LSD (left) and SFF (right) for BTZ with ⟨λ⟩ = −104 and Jmax = 400 with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We are working with ω(n = 2, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These results are a version of the results in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 s p(s) β=0 107 108 109 1010 1011 1012 1013 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 1 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 4: Same as before, but with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The blue curve on the left is GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 s p(s) β=0 107 108 109 1010 1011 1012 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 1 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 5: Same as in the previous figures, but with σ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The red curve on the left is Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 13 0 100 200 300 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015718 J ω(2,J) 0 100 200 300 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='00015718 J ω(2,J) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 6: Spectrum of BTZ with σ0 = 0 (left) vs σ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ⟨λ⟩ = −104 and Jmax = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We show ω(n = 2, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' CASE STUDY: RINDLER × COMPACT SPACE We will follow the motivations and discussion in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 of [5] when developing the Rindler case, which the reader should consult for notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We solve the wave equation in the metric ds2 = e2aξ(−dη2 + dξ2) + R2dφ2 (27) and introduce A ≡ ω/a and y ≡ eaξ(J/aR) as in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In terms of y variable the position of boundary and horizon are given by y → ∞ and y → 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the notations of [5], we require that the field φ(y) vanish at boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also demand that it has a profile at some small y0 (or ξ = ξ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' When y → ∞, the relevant equation is [5] φ(y) → (C1 + C2) ey √2πy + (C1eπA + C2e−πA) e−y √2πy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (28) The boundary condition at infinity leads to C1 = −C2, and at y0 implies (in notation that is parallel to the BTZ case before): C1(I[−iA, y0] − I[iA, y0]) = C0, =⇒ I[−iA, y0] − I[iA, y0] = C0 C1 (29) Near horizon i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' in the limit y0 → 0 the above expressions can be approximated by C1 � y−iA 2iA Γ(1 − iA) − yiA 2−iA Γ(1 + iA) � = C0 (30) C0 C1 � J aR �−iA �eaξ 2 �iA − �eaξ 2 �2iA = � J aR �−2iA Γ(iA) Γ(−iA) (31) Now Abs �� J aR �−2iA Γ(iA) Γ(−iA) � = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' so (31) can be written,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' again in notation that simulates the BTZ case as µJeiωλJeiθ/2 − eiθ = eiα (32) with µJ = Abs � C0 C1 � J aR �−iA� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' ωλJ = Arg � C0 C1 �eaξ 2 �iA� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' α = Arg �� J aR �−2iA Γ(iA) Γ(−iA) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' θ = Arg ��eaξ 2 �2iA� (33) 14 0 1 2 3 4 0 1 2 3 4 5 s p(s) β=0 105 106 107 108 109 1010 1011 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 7: LSD (left) and SFF (right) for Rindler with parameters described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We are working with ω(n = 1, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These results should be compared to the results in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 s p(s) β=0 105 106 107 108 109 1010 1011 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 8: Same as the previous figure, but with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The blue curve on the left is GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 s p(s) β=0 105 106 107 108 109 1010 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 9: Same as the two previous figures, but with σ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The red curve on the left is Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 15 0 100 200 300 400 500 600 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001567 J ω(1, J) 0 100 200 300 400 500 600 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001560 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001567 J ω(1, J) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 10: Spectrum of Rindler with σ0 = 0 (left) vs σ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We show ω(n = 1, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This therefore again leads to similar structures as in BTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We find µJ = 2 cos(λJω − θ/2) (34) as well as the quantization condition cos(α) = cos(2λJω) (35) Because the structure is precisely parallel to BTZ, we will not repeat the discussion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' it is clear that the normal mode calculation proceeds in an identical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The mean value of λ can be related to the stretched horizon location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Once we choose R, ξ0 and a, the normal modes ω(n, J) can be numerically solved for as a function of J (and an integer n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We present the plots in precise parallel to the BTZ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The qualitative results are identical, despite the fact that the special functions that showed up in the wave equations here are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the plots we present, we have chosen a = 1, R = 2, Jmax = 700, ⟨λ⟩ = −103 and σJ = σo/ √ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The σ0 values are quoted in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' THE HAIRY HARMONIC OSCILLATOR AND CUT-OFF IN EMPTY SPACE: LEVEL REPULSION WITHOUT LINEAR RAMP We noted that the linear ramp in the SFF and repulsion in the LSD can both be seen in the stretched horizon spectrum if the boundary condition is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also pointed out that the level spacing ratio discussed in [33] is also consistent with RMT expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Together, these constitute very strong evidence that fuzzball/stretched horizon spectra have strong connections to random matrices and chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In this section, we will ask a slightly more resolved question: which of these is a more robust indicator of chaos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Is it the linear ramp or is it level repulsion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Or are both these features always present in systems concomitantly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will present some hints in this section that the linear ramp may be a more robust diagnostic of strong chaos than nearest-neighbor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is not an entirely new suggestion (the length of the ramp is often viewed as an indicator of the “strength” of chaos), but we will give some examples which we feel are instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will start (as often in physics) with the simple harmonic oscillator (SHO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' For our purposes, the SHO is interesting because even though it is the farthest thing from a chaotic system, it exhibits a naive (or extreme) version of level repulsion – the levels are equally spaced, and the LSD is a delta function shifted from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Motivated by the results of this paper, we can ask if there is a natural way to “perturb” the SHO spectrum so that the level spacing becomes a more conventional Wigner-Dyson-like form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It turns out that a simple way to engineer this exists – we simply allow a small amount of (Gaussian) noise in the levels of the SHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will call this set up a hairy or noisy SHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' See Figure 11 right panel, for a typical LSD of an SHO perturbed in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We present a GOE fit for concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But again, by adjusting the variance, we can find fits with GSE or GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We are not aware of a previous observation of this simple but striking fact in the literature, but it is easy enough to understand – Random noise in the energy levels directly affects the nearest neighbor data, which explains why the delta function in the LSD gets spread out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 16 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 s p(s) 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='4 s p(s) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 11: LSDs of Cut-off flat space with fluctuation profile (left) vs hairy SHO (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Flat space data: Jmax = 300, rcut = 1, λ-variance = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='0174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We are working with ω(n = 1, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' SHO data: nmax = 600, ω = 1, spectral noise variance = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Both fits are GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' β=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='10 1 10 100 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 1 t g(t) β=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 1 10 100 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 12: SFFs of the same systems (SHO on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The yellow line has slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7 (both left and right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In other words, this is a power law ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 0 50 100 150 200 250 300 0 50 100 150 200 250 300 J ω(1,J) 0 100 200 300 400 500 600 0 100 200 300 400 500 600 n ω(n) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 13: Shapes of spectra, for the same systems as above (SHO again on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It is clear that both the spectra are approximately evenly spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The punchline of the figures in this page is that the spectral features of the two systems have crucial similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' On the other hand, strong chaos is characterized by spectral rigidity which is encoded in the linear ramp in the SFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' And indeed, if one computes the SFF of the SHO with noise in the spectrum, one finds that the ramp is in fact 17 non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is illustrated in Figure 12, right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We emphasize that it is remarkable that a well-defined ramp exists, even though it is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In fact, we find that on a log-log plot, it has a well-defined slope of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In other words, a hairy SHO has a power law ramp, at least within the context of our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These SHO results shed light on the distinctions between a black hole with a stretched horizon, and a cut-off in empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' If we impose a simple Dirichlet condition φ = 0 at the cut-off, in the former case we find a linear ramp [5], but in empty space there is no clear ramp, certainly nothing of slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' See Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But as we add variance to the profile, we see the emergence of a power law ramp, see Figure 12 left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The SHO example above provides us a clear understanding of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A cut-off in flat space leads to eigenvalues that are connected to the zeros of Bessel functions (as we will see).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These are roughly evenly spaced – so the spectrum looks crudely like that of an SHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Relatedly, the level spacing in the φ = 0 case is essentially a delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But this can be made to look like a more spread out (WD-like) form by demanding instead that the boundary condition is φ ∼ φ0(θ) where the profile has some variance in its Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The noise in the spectrum increases when we do this, and as a result (as pointed out above for the hairy SHO) we find that the LSD takes a more conventional WD form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Of course, when the variance is very large, the spectrum ends up becoming Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Crucially, the slope of the ramp is never ∼ 1 in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' For moderate values of the variance, it is consistent with the ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7 quoted above for the noisy SHO – see figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (Note that when the variance is steadily increased, the ramp gets increasingly washed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' So this statement applies only to those values of the variance for which there is a clear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=') The basic message we extract from these calculations is that the spectrum on a cut-off geometry without a horizon is essentially a hairy SHO spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' When we have a horizon on the other hand, the spectrum is not that of an SHO in any sense (as we saw in previous sections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Together with the striking linearity of the ramp, we are therefore lead to conclude that the physics in the latter case is not simply due to nearest-neighbor physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We conclude this section by providing some of the details of the flat space calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will work with 2+1 dimensions, the physics we aim for is unaffected by increase in dimensions: ds2 = −dt2 + dr2 + r2dψ2 (36) Separating the scalar field as (say) in the BTZ case, we find the radial part φ ′′ ω,J(r) + 1 r φ ′ ω,J(r) + ω2φω,J(r) − V (r)φω,J(r) = 0 (37) with V (r) = 1 r2 � J2 + m2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (38) We will consider the solution of this equation (37) in the massless case, which is given in terms of Bessel functions: φ(r) = C1JJ(ωr) + C2YJ(ωr), (39) where, JJ and YJ are Bessel functions of first and second kind respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We suppress the J and ω (or n) subscripts of C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' As before, we need one boundary condition to fix a relationship between C1 and C2 and another condition at a cut-off to fix the normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The former role was played by AdS-normalizability in the BTZ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We could likewise demand a suitably chosen bulk condition here as well that relates C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' By numerical experimentation we have found that the qualitative features of the ramp and LSD that we are after, are insensitive to this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is unsurprising because the physics we are interested in, is the result of the quantization condition, and not the relationship between C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In the following, we will simply demand that C2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note that this sets the bulk source mode (which is singular at the origin) to zero, while retaining the homogeneous mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It was noted in [40] that the bulk source mode is the analogue in flat space, to the non-normalizable mode in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' So this choice is a natural analogue of the normalizability demand in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But we emphasize that large classes of choices are likely to give similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 18 Using this boundary condition, equation (39) becomes φ(r) = C1JJ(ωr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (40) Demanding a profile at the cut-off r = r0 leads to an equations analogous to what we found for BTZ: φ(r = r0) = C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' C1JJ(ωr0) = C0 =⇒ JJ(ωr0) = C0 C1 ≡ λJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (41) Note that we could also define the RHS to be ωλJ, which is more analogous to some of our discussions in BTZ and Rindler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But as we mentioned, these choices do not affect the semi-qualitative features we are after, so we will stick with this simple choice here for concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We will take λJ to be Gaussian distributed with mean zero, and adjustable variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The equation is easy to solve numerically, by taking the seed for the root search to be the 1st zero of the J-the Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' When the variance is zero, we find an “extreme” delta-function like distribution in the LSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The ramp of the SFF is not particularly well-defined, but we can already see a crude similarity to an SHO with a very small amount of noise – See Figure 14 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='01 1 100 104 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='100 t g(t) β=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='01 1 100 104 10-8 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='01 t g(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 14: Cut-off flat space with no variance vs SHO with a tiny amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The precise values are unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our goal here is not to make a detailed comparison, but to observe the crude similarity which becomes more striking as we increase the variance/noise, see Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The two lines are of slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7 and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' When we steadily add variance, we find more conventional level repulsion and the emergence of a robust ramp of slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7, which we presented in Figure 12 left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' As noted above, this is precisely what one finds from a noisy SHO as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Eventually we find a Poisson distributed LSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The (power law) ramp gets washed out, when the variance becomes very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These features are identical to what we find in the hairy/noisy SHO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To summarize – flat space with a cut-off is qualitatively identical to hairy SHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Unlike in the case of the stretched horizon cut-off, the levels are essentially evenly spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We have done a similar calculation in empty AdS as well, as discussed in the main body of the paper, and the results are again consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These results mean that the linear ramp (which is often viewed as an indicator of strong chaos) does not arise from a cut-off in flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But for the same reason that a hairy SHO can mimic the LSD of an RMT (which in itself is a fact not emphasized previously in the literature, to our knowledge), the spectrum of cut-off flat space can also exhibit level repulsion – the variance in the boundary condition simply introduces a variance in the nearest neighbor levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But this is not sufficient to create conventional spectral rigidity or robust chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A further distinction between empty space with cut-off and the stretched horizon is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' PLANCK-SCALE HIERARCHY We observed that the fluctuations at the cut-off in empty space translate to fluctuations in the energy levels and therefore lead to level repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In other words, nearest neighbor effects of chaos can be produced simply by having 19 fluctuations at the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also noted however that the linear ramp (which is a deeper signature of chaos) cannot be realized this way, and requires the presence of a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In fact there is another interesting distinction between the stretched horizon and a cut-off in empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This has to do with the fact that the fluctuations at the cut-off needed in the stretched horizon scenario are hierarchically suppressed, allowing the interpretation that they are Planck-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The fluctuations in the empty space cut-off on the other hand are naturally macroscopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To see this, first note that in (41), the first zero of the J-th Bessel function is linearly spaced in J with the scale controlled by r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The natural scale controlling the fluctuations in the RHS is therefore r0 (this dependence is approximately linear if we define the RHS of (41) to be ωλJ instead of λJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' On the other hand in the horizon case, the situation is more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To see this in detail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' let us work with the concrete case of BTZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' and observe that the conventional tortoise coordinate here is defined via z = L2 2 rh ln �r + rh r − rh � (42) This means that the usual radial coordinate of the stretched horizon x ≡ r − rh is approximately x = 2 rhe−2rhz/L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (43) from which it follows that the fluctuation in the stretched horizon location goes as |∆x| ∼ 4 (rh/L)2 e−2rhz/L2|∆z| (44) where we have instated a magnitude sign because z → ∞ corresponds to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Now, from (21) it follows that e2λ = (x/rh) and therefore 2 e2λ∆λ = ∆x rh =⇒ 2 x ∆λ ∼ ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (45) Using (43) and (44) in this final relation, we get L2 rh |∆λ| = |∆z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' (46) Since the horizon size and AdS length scale are both macroscopic, this means that the fluctuations in λ are naturally in tortoise coordinate, implying via (44) that the stretched horizon fluctuations are suppressed by a factor of e−2rhz0/L2 (47) where z0 is the mean stretched horizon in tortoise coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We also see that L2/z0 is a natural candidate for the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In units where L = 1, note that this is a small quantity because z0 is very large when the cut-off is close to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Of course, since we are working with a fixed background, these are all somewhat heuristic statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' To summarize: The variance in both cases (with and without horizon) can be used as a heuristic proxy for fluctu- ations of the cut-off surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But a key distinction in the stretched horizon is that there, the variance captures the tortoise coordinate and therefore the fluctuations can naturally be viewed as Planck suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' OPEN QUESTIONS AND FUTURE DIRECTIONS In this section, we discuss some questions that are worth understanding better in the wake of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Some of these are more conceptual than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Are there more natural choices for the profile functions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We have considered the most simple-minded notion of a “generic” profile – choose some randomly distributed Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The BPS fuzzball profiles, at least in the 2-charge case [14, 16] are known to contain enough phase space to reproduce the entropy of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This suggests that perhaps Haar typicality in some form is a better notion of genericity than our present proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It will be interesting to incorporate this in some systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' 20 Despite the simplicity of our calculation, we have managed to find a linear ramp with fluctuations and level repulsion in (a heuristic candidate for) a single microstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The price we have paid is that we have sacrificed a (manifestly) smooth horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' But the emergence of RMT behavior in our calculation suggests that thermality (and therefore smoothness) may emerge via a suitable ensemble replacement of the microstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Understanding this operationally is clearly a problem of outstanding interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In our previous paper [5], the LSD was not one of the conventional RMT distributions, but there was a clear linear ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our main point in that paper was that this is a generic feature of normal modes at stretched horizons, when the boundary condition φ = 0 was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In this paper, we have seen systems which exhibit the opposite behavior – The ramp is non-linear, but one has level spacing that matches well with conventional Wigner-Dyson-like statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In fact, we noticed that the latter can be arranged very simply via an SHO with a noisy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Together the results of these papers are a very clear demonstration that the folk wisdom that the linear ramp is a smoking gun of conventional Wigner-Dyson classes (or their Altland-Zernbauer generalizations) is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It will be good to understand the broader setting in which these features arise as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We did not have to introduce any form of ensemble average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Our profile curve is chosen via a Gaussian distribution in the Fourier coefficients, but it should be emphasized that once the curve is chosen, there is absolutely nothing “averaged” about the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The emergence of RMT behavior is entirely deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It has been suggested in [41] that semi-classical gravity should be viewed as a tool for capturing ergodic averaged gravitational dynamics, for evolution that is in bulk local equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This would give an understanding of the surprising utility of Euclidean gravity in each epoch of Hawking radiation in obtaining the Page curve [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It will be very interesting to connect these two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' In [5] we had observed that there was a kink-like structure at the top of the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A tangential consequence of the calculations in the present paper is that we have understood that this kink becomes less and less prominent, as we bring the stretched horizon closer and closer to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is a strong indication that one of the worries expressed in [5] – that the ramp may be an artefact – is very unlikely to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Inspired by the results of this paper and [5], we have been able to identify a broader class of spectra which lead to interesting ramps and level spacing structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' These results together suggest the notion of a generalized RMT spectrum, which will be elaborated elsewhere [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' A key message is that boundary conditions are often a crucial ingredient in quantum chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is true in our black hole problem, but note that the idea is much more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=', the Hamiltonian of the hard sphere gas is simply that of a collection of free particles – it is the boundary conditions that breathe life (and chaos) into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' One of the technical features underlying the results of this paper and [5] is the observation that the dependence of the spectrum on the angular quantum numbers is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Instead it gets pulled logarithmically along J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The resulting quasi-degeneracy was essential for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It will be good to get a more mechanical/conceptual understanding of this observation as well as to explore its consequences more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' We found a clear ramp with slope ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content='7 in our SFF plots for hairy SHO and cut-off flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' This is an extremely simple example of a non-linear ramp, whose slope is a constant (̸= 1) in a log-log plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' It seems surprising and interesting that it is closely related to the SHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Can this shed light on the fact that despite being the “ultimate” integrable system, the SHO exhibits an extreme version of level repulsion (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=', its LSD has no support at the origin, and has a delta function form)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Relatedly, and more speculatively – does the fact that extreme WD spectra arise from Dirichlet boundary conditions at stretched horizons indicate that black holes are the “ultimate” RMT systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' If this is true, black holes can be viewed as the natural counterpoint to SHOs from our previous item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Note that the suggestion we are making here is distinct from the chaos bound of [8], which is about early time chaos and OTOCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' The observation about LSDs that we are making here is related to late time chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Black holes may not just be fast scramblers [6], they may also be the most robust scramblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} +page_content=' Clearly, more work remains to be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQfTi2q/content/2301.11780v1.pdf'} diff --git a/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/2301.02628v1.pdf.txt b/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/2301.02628v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f46fdeae5e6f15587f71772238a9ab5687aca40d --- /dev/null +++ b/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/2301.02628v1.pdf.txt @@ -0,0 +1,1228 @@ +arXiv:2301.02628v1 [math.CO] 6 Jan 2023 +PINNACLE SETS OF SIGNED PERMUTATIONS +NICOLLE GONZ´ALEZ, PAMELA E. HARRIS, GORDON ROJAS KIRBY, MARIANA SMIT VEGA GARCIA, +AND BRIDGET EILEEN TENNER +Abstract. Pinnacle sets record the values of the local maxima for a given family of permutations. +They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previ- +ously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for +the type A symmetric group have been widely studied. In this article we define the pinnacle set +of signed permutations of types B and D. We give a closed formula for the number of type B/D +admissible pinnacle sets and answer several other related enumerative questions. +1. Introduction +The study of permutation statistics is an active subdiscipline of combinatorics. Given a per- +mutation w = w(1)w(2) · · · w(n), two particularly well-studied statistics are descents and peaks. +Respectively, these statistics refer to indices i such that w(i) > w(i + 1), and indices i such that +w(i − 1) < w(i) > w(i + 1). The collection of a permutation’s descent indices is its descent set, +with a permutation’s peak set being similarly defined. Two fundamental goals in the study of these +particular statistics are (1) understanding which subsets can arise as descent sets or peak sets (i.e., +which sets are admissible as descent or peak sets), and (2) enumerating the permutations that have +a given admissible descent or peak set. +For descents of permutations in the (type A) symmetric group Sn, this question was answered +by Stanley [15, Ex. 2.2.4] and is well known to give rise to the Eulerian numbers. Inspired by +Stembridge’s study of peaks in the context of poset partitions [16], Billey, Burdzy, and Sagan [1] +introduced the study of admissible peak sets for Sn with an interest in probabilistic applications, and +established that the number of permutations with peak set I is given by 2n−|I|−1p(n), where p(n) is +a polynomial of degree max(I) − 1. Shortly thereafter, their results were extended to permutations +in type B by Castro-Velez et al. [2] where it was shown that the number of permutations with a +given peak set I is 22n−|I|−1p(n), with p(n) the same as in [1] above. The second author and various +collaborators went further by extending these results to types C and D [7], using peaks to study +properties of the descent polynomial [6], and then initiating the study of peaks in the context of +graphs [4]. +A notion that is closely related to peaks is the pinnacle set of a permutation. Pinnacles are the +set of values held by the permutation at the peak indices. More precisely, given a permutation w = +w(1)w(2) · · · w(n) with peak set Peak(w), the pinnacle set of w is Pin(w) = {w(i) : i ∈ Peak(w)}. +Given a subset I ⊆ [n], if there exists a permutation w whose pinnacle set is I, we say that +I is an admissible pinnacle set. +In [3], Davis, Nelson, Petersen, and the last author pioneered +the study of pinnacles for permutations in Sn and gave a complete characterization of admissible +pinnacle sets. They provided a closed formula for the number of admissible pinnacle sets with a +given maximum value, as well as a refinement to those appearing in Sn. In particular, Davis et +Date: January 9, 2023. +P.E.H. was partially supported through a Karen Uhlenbeck EDGE Fellowship. +M.S.V.G was partially supported by the NSF grant DMS 2054282. +B.E.T. was partially supported by the NSF grant DMS-2054436. +1 + +al. gave a recursive formula for the number of permutations in Sn with a given pinnacle set p(n) +and asked whether a more efficient expression could be computed. This paper led to a sequence +of articles in recent years, many focused on improved and faster formulas for p(n), by realizing +permutations with given pinnacle sets as invariants under certain modified Sn-actions [5, 9] or via +more traditional enumerative methods [8, 10, 11]. In related work, Rusu [13] and Rusu-Tenner [14] +deepened the knowledge of pinnacles in Sn by investigating further properties of these statistics +and characterizing admissible pinnacle orderings. +In this article we look beyond type A and study pinnacles and admissible pinnacle sets for the +type B and type D signed symmetric groups, SB +n and SD +n . Our main results are the following, +where we write APSX +n to denote the admissible pinnacle sets in SX +n for X ∈ {A, B, D}: +(1) Theorem 3.12 gives a closed formula for the number of admissible pinnacle sets in SB +n , +|APSB +n | = +⌊ n−1 +2 ⌋ +� +k=0 +�n +k +��n − 1 − k +� n−1 +2 +� +− k +� +. +(2) Theorem 4.2 proves that any admissible pinnacle set in SB +2k is also admissible in SD +2k; that +is, APSD +2k = APSB +2k. +(3) In counterpoint to Theorem 4.2, Theorem 4.9 counts the admissible pinnacle sets of type +B that are not in type D when n = 2k + 1, +|APSB +2k+1 \ APSD +2k+1| = +�2k − 1 +k +� +. +(4) Theorems 4.11 and 4.12 count the all-positive admissible pinnacle sets of type B that are +not admissible in type A. Namely, defining APS+ +n := {S ∈ APSB +n : S ⊂ N}; we prove that +the sets APS+ +n \ APSn are enumerated by, +��APS+ +n \ APSn +�� = +� +4k − +�2k +k +� +if n = 2k + 1, and +22k−1 − +�2k +k +� +if n = 2k. +This article is organized as follows. In Section 2, we introduce all the necessary background and +notation, defining pinnacles and related notions in type B. In Section 3, we give a characterization +of admissible signed pinnacle sets and a formula for their enumeration. In Section 4, we provide +relations between admissible pinnacle sets of type A, B, and D. Lastly, in Section 5, we describe +some future directions and open conjectures. +Acknowledgements. The authors thank Patrek K´arason Ragnarsson for the coding and data +that facilitated the research in this project, and Freyja K´arad´ottir Ragnarsson for the key insight +to the proof of Theorem 4.9. The authors also thank the American Institute of Mathematics and +the National Science Foundation for sponsoring the Latinx Mathematicians Research Community, +which brought together a subset of the authors initially for collaboration. +2. Background +Let N = {1, 2, 3, . . .} and for n ∈ N we write [n] := {1, 2, . . . , n}. For any set X, typically of +positive values, although we make the definition more generally, we define +−X := {−x : x ∈ X}. +Finally, we define +±X = X ∪ −X. +2 + +Throughout this paper, we let Sn denote the (type A) symmetric group. That is, Sn is the group +of bijections from [n] → [n] under function composition. We often write w ∈ Sn using one-line +notation, as w = w(1)w(2) · · · w(n). +The type B symmetric group (that is, the hyperoctahedral group) is the group of signed +permutations SB +n . These are bijections ±[n] → ±[n] such that +w(−i) = −w(i) for all i ∈ [n]. +In particular, any w ∈ SB +n satisfies the property that {|w(1)|, . . . , |w(n)|} = [n]. +The type D symmetric group is the subgroup SD +n of SB +n consisting of signed permutations with +an even number of signs. Namely, these are the signed permutations w for which +|{i ∈ [n] : w(i) < 0}| is even. +As in type A, we use one-line notation to denote signed permutations w ∈ SB +n , where we may +write only w = w(1)w(2) · · · w(n) since this uniquely determines w(−i) for all positive i. Following +convention, we write −i = ¯i to ease notation. For example, w = ¯12¯3 is the signed permutation with +w(1) = −1, w(2) = 2, and w(3) = −3. +Recall that a permutation w ∈ Sn has a peak at index i ∈ {2, . . . , n − 1} if +w(i − 1) < w(i) > w(i + 1), +and the value w(i) is a pinnacle of w. We denote by Peak(w) the set of all peaks of w ∈ Sn. The +pinnacle set of w ∈ Sn is +Pin(w) = {w(i) : i ∈ Peak(w)}. +Definition 2.1. A set P ⊆ [n] is an n-admissible pinnacle set in type A if there exists a permutation +w ∈ Sn such that Pin(w) = P, and we call the permutation w a witness for the set P. +For example, the identity permutation is a witness for the admissible pinnacle set ∅ (as is any +peak-less permutation). Denote by APSn the set of all n-admissible pinnacle sets in type A. +In order to facilitate our discussions about pinnacles, we introduce terminology about their +minimal counterparts: a permutation w ∈ Sn has a valley at index i ∈ {2, . . . , n − 1} if +w(i − 1) > w(i) < w(i + 1), +and the value w(i) is a vale of w. +1 +2 +3 +4 +5 +6 +7 +8 +1 +2 +3 +4 +5 +6 +7 +8 +• +• +• +• +• +• +• +• +Figure 1. The graph of the permutation 23715648 ∈ S8 with the pinnacles/peaks +circled in red and the vales/valleys in blue. +Example 2.2. Consider the permutation w = 23715648 ∈ S8 shown in Figure 1. +We have +Peak(w) = {3, 6} and Pin(w) = {6, 7}, and valleys and vales {4, 7} and {1, 4}, respectively. +3 + +2.1. Pinnacles in types B and D. Pinnacles were defined in [3] for unsigned permutations, but +they could just as easily have been defined for signed permutations—or, in fact, for arbitrary strings +of distinct real numbers. We now expand the type A definitions to type B, and note that since +SD +n ⊂ SB +n , these definitions also hold for type D. +Definition 2.3. Let w be a signed permutation. A pinnacle of w is a value w(i) that is larger than +both w(i − 1) and w(i + 1). The pinnacle set of w is the set of its pinnacles. +In order to define admissible pinnacle sets, it is important to establish which subsets could even +appear among the one-line notation of a signed permutation. +Definition 2.4. A signed set (or signed subset, depending on context) is a set I such that x ∈ I +implies −x ̸∈ I. +Throughout this paper, we assume that all subsets of ±[n] are signed subsets. +Definition 2.5. A signed subset S ⊂ ±[n] is an admissible pinnacle set if S is the pinnacle set of +some signed permutation. That permutation is a witness for S. +Note that when we study sets that are admissible as pinnacle sets in type D, any witness +permutation will be required to be in SD +n for some n. +As before, we denote by APSB +n (resp., +APSD +n ) the set of all n-admissible pinnacle sets in type B (resp., type D). Once again, we have +∅ ∈ APSD +n ⊆ APSB +n . For example, 123 · · · n and ¯2¯134 · · · n are both witnesses for ∅. +While there can be multiple witness permutations for a given admissible pinnacle set, we will +often refer to a particular witness permutation that we call “canonical.” +Definition 2.6. For S ∈ APSB +n , write S = {s1 < s2 < · · · < sk}, and set +S′ := −[n] \ {−|s| : s ∈ S} = {s′ +1 < s′ +2 < · · · < s′ +n−k}. +Then the canonical witness permutation is +w := s′ +1 s1 s′ +2 s2 · · · s′ +k sk s′ +k+1 · · · s′ +n−k ∈ SB +n . +If S ∈ APSD +n , then its canonical (type D) witness permutation is w as defined above if w is in SD +n , +and otherwise its canonical witness is obtained from w by replacing s′ +n−k with |s′ +n−k|. +Next we establish that the “canonical witness permutations” are, in fact, witnesses and follow +this by providing canonical witness permutations in Example 2.8. +Lemma 2.7. The canonical witness permutation for an admissible set S is a witness for S. +Proof. The set S is admissible, so there is some permutation whose pinnacle set is S. The canonical +witness has been constructed to have minimal possible non-pinnacle values, and to position the +smallest non-pinnacle values beside the smallest pinnacle values. Therefore, if any permutations +were to have S as a pinnacle set (and we know that some permutation does), the permutation given +in Definition 2.6 would be among them. +□ +Although SB +n contains both Sn and SD +n as subgroups, there are interesting subtleties to the +pinnacle sets that become admissible when witness permutations can be signed. First, some sets +will be admissible with type B permutations, but not with type D permutations. And second, +some sets of all-positive values will be admissible with type B permutations, but not with type A +(unsigned) permutations. We demonstrate each of these scenarios below. +4 + +(a) +1 +2 +3 +4 +5 +6 +7 +1 +2 +3 +4 +5 +6 +7 +0 +−1 +−2 +−3 +−4 +−5 +−6 +−7 +• +• +• +• +• +• +• +(b) +(b) +1 +2 +3 +4 +5 +6 +7 +1 +2 +3 +4 +5 +6 +7 +0 +−1 +−2 +−3 +−4 +−5 +−6 +−7 +• +• +• +• +• +• +• +Figure 2. (a) The graph of the permutation ¯7¯4¯61¯52¯3 ∈ SB +7 +with the pinna- +cles/peaks circled in red and the vales/valleys in blue. (b) The graph of the permu- +tation ¯63¯54¯17¯2 ∈ SB +7 with the pinnacles/peaks circled in red and the vales/valleys +in blue. +Example 2.8. The set {¯4, 1, 2} is admissible in SB +7 , with canonical witness permutation ¯7¯4¯61¯52¯3 +as shown in Figure 2(a). +However, there is no element of SD +7 having this pinnacle set. +That +is, {¯4, 1, 2} ̸∈ APSD +7 . The set {3, 4, 7} is admissible in SB +7 , with canonical witness permutation +¯63¯54¯17¯2, as shown in Figure 2(b). However, despite its pinnacle set having all positive values, there +is no type A permutation having this pinnacle set. That is, {3, 4, 7} ̸∈ APSn for any n. +3. Admissible signed pinnacle sets in type B +In this section, we characterize and enumerate the admissible pinnacle sets among signed +permutations. This expands on the work begun in [3] for unsigned permutations, but, as we show, +the results for signed permutations are subtly different from those in type A. +3.1. Characterization of admissible pinnacle sets. For the remainder of the article, we will +often use the fact that given an admissible pinnacle set S ∈ APSB +n , we can always write +S = P(S) ⊔ N(S) +with +P(S) := S ∩ [n] and N(S) := S ∩ −[n]. +When no confusion will arise, we simply write P := P(S) and N := N(S). +To give a first inkling of how admissible pinnacle sets in type B are fundamentally different +from those in type A, we note that there are some sets of positive integers that are never in APSn +for any n. For example, any set containing 1 or 2 will never be the pinnacle set of any permutation +in Sn. On the other hand, such a statement is not true in type B. +Lemma 3.1. Every finite signed subset S is admissible in SB +n , for some n ∈ N. That is, there +exists w ∈ SB +n such that S = PinB(w). +5 + +Proof. Write S = {s1 < · · · < sk}. Let m = max{|s| : s ∈ S} (that is, m = max{|s1|, |sk|}). Define +the set S′ := −[2m + 1] \ {−|s| : s ∈ S}, which we write as S′ = {s′ +1 < · · · < s′ +2m+1−k}. Then +w = s′ +1 s1 s′ +2 s2 · · · s′ +k sk s′ +k+1 s′ +k+2 · · · s′ +2m+1−k ∈ SB +2m+1, +and PinB(w) = S. +□ +Using a similar argument as the one proving Lemma 3.1, it follows that any finite set of all +positive values is admissible in some SB +n . +Corollary 3.2. Any subset P ⊂ [n] with |P| ≤ n−1 +2 +is admissible in SB +n . +Proof. Let P = {p1 < · · · < pk}, and set P ′ := −([n] \ P) = {p′ +1 < · · · < p′ +n−k}. Then +w = p′ +1 p1 p′ +2 p2 · · · p′ +k pk p′ +k+1 p′ +k+2 · · · p′ +n−k ∈ SB +n +and PinB(w) = P. +□ +This can be particularly interesting when the set P was not admissible in Sn. +Example 3.3. Consider P = {1, 2} with n = 5. +The permutation ¯51¯42¯3 ∈ SB +5 is a witness +permutation for P, so P ∈ APSB +5 , while P ̸∈ APSn for any n. +Our goal is to establish a characterization and formula for the number of admissible pinnacle +sets in SB +n . We begin with some preliminary steps, from which those results will follow. The first +of these is a bijection between admissible pinnacle sets in Sn and those admissible pinnacle sets in +SB +n that have no positive values. +Lemma 3.4. There exists a bijection between APSn and {S ∈ APSB +n : S ⊂ −N}. +Proof. Given T ∈ APSn, define T ′ := {t − (n + 1) : t ∈ T}. The set T ′ has no positive elements. +Let w ∈ Sn be the canonical witness for T. Then w′ := (w(1) − (n + 1)) · · · (w(n) − (n + 1)) ∈ SB +n +has pinnacle set T ′, and so T ′ ∈ APSB +n . +This process can be inverted: given S ∈ APSB +n with P(S) = ∅, map this S to S′ := {s + n + 1 : +s ∈ S}. It follows that S′ ∈ APSn, as before. +□ +We illustrate Lemma 3.4 with an example. +Example 3.5. The set {3, 6, 7, 10} ∈ APS10 is in correspondence with {¯8, ¯5, ¯4, ¯1} ∈ APSB +10. The +permutations described in the proof of Lemma 3.4, which exhibit these sets as pinnacle sets, are +shown in Figure 3. +We have defined admissible pinnacle sets in types A, B, and D, referring to permutations in +Sn, SB +n , or SD +n . +However, as suggested earlier, there is a natural generalization of admissible +pinnacle sets to permutations of any totally ordered set. +Definition 3.6. For any totally ordered set X, let APS(X) be the set of admissible pinnacle sets +of X. The definitions of witness and canonical witness permutations in this general setting are +analogous to their definitions in the symmetric groups. +Because they arise so often, we have been easing notation by writing APS(Sn) as APSn, +APS(SB +n ) as APSB +n , and APS(SD +n ) as APSD +n . +Example 3.7. The set X = {−2, π, 4, 5, 100} has six admissible pinnacle sets: +∅, {4}, {5}, {100}, {4, 100}, and {5, 100}. +6 + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +• +• +• +• +• +• +• +• +• +• +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +−1 +−2 +−3 +−4 +−5 +−6 +−7 +−8 +−9 +−10 +0 +... +• +• +• +• +• +• +• +• +• +• +Figure 3. The left-hand figure shows the canonical witness for {3, 6, 7, 10} in S10. +The right-hand figure shows the corresponding witness permutation for {¯8, ¯5, ¯4, ¯1}, +as defined in the proof of Lemma 3.4. +Note that if we are only interested in how many admissible pinnacle sets X has, as opposed to +the sets themselves, then the size of X is what matters. +Lemma 3.8. For any totally ordered finite set X, |APS(X)| = |APS ([|X|]) | = |APS|X||. +This calculation will be useful in the enumeration appearing in the next subsection. +We are now are able to characterize admissible pinnacle sets for signed permutations. +Theorem 3.9. The sets in APSB +n are exactly the sets S = P(S) ⊔ N(S) for which +• |P(S)| + |N(S)| ≤ (n − 1)/2, +• P(S) ⊂ [n], +• N(S) ⊂ −([n] \ P(S)), and +• N(S) ∈ APS(−([n] \ P(S))). +Proof. First of all, it is clear that any admissible pinnacle set in SB +n must satisfy the listed require- +ments. +Now suppose that a set S satisfies the listed requirements, with P := P(S) = {p1 < · · · < pk} +and N := N(S) = {n1 < · · · < nr}. In light of the last requirement, let w be the canonical witness +permutation of the set (−([n] \ P)), having pinnacle set N. That is, +w = i1 n1 i2 n2 · · · ir nr ir+1 ir+2 ir+3 · · · in−k−r +where ij < ij+1 and {i1, . . . , in−k−r} = −([n] \ P) \ N. Then +i1 n1 i2 n2 · · · ir nr ir+1 p1 ir+2 p2 ir+3 · · · pk ir+k+1 ir+k+2 · · · in−r−k +is a canonical witness for S = P ⊔ N in SB +n . Hence S ∈ APSB +n . +□ +3.2. Enumeration of admissible pinnacle sets. The conditions listed in Theorem 3.9 inform +our enumeration of the admissible pinnacle sets in SB +n . In particular, we will construct these sets by +7 + +first fixing a collection P of positive pinnacles and then determining how many sets N of negative +pinnacles exist for which P ∪ N is admissible in SB +n . +In order not to have too many pinnacles (that is, not more than ⌊(n − 1)/2⌋), we need to +understand the following value. +Definition 3.10. Let pn(d) be the number of admissible pinnacle sets in Sn having cardinality at +most d. That is, +pn(d) := |{S ∈ APSn : |S| ≤ d}|. +This statistic has a particularly nice formula. +Proposition 3.11. For all integers d ∈ [0, ⌊(n − 1)/2⌋], +pn(d) = +�n − 1 +d +� +. +Proof. The admissible pinnacle sets in Sn having cardinality at most d can be partitioned into +two sets: those that contain n, and those that do not. We claim that the first set is counted by +pn−1(d − 1), and the second set is counted by pn−1(d). +Suppose, first, that S ∈ APSn such that n ∈ S and |S| = k ≤ d. Let w ∈ Sn be the canonical +witness for S. Deleting n from the one-line notation of w will produce a permutation v ∈ Sn−1 +with Pin(v) = S \ {n}. Conversely, given T ∈ APSn−1 with |T| = k − 1, let u ∈ Sn−1 be the +canonical witness for T. Inserting n between the non-pinnacles u(2k − 1) and u(2k) will produce a +permutation in Sn whose pinnacle set is T ∪ {n}. This establishes the first part of the claim. +For the second part of the claim, suppose that S ∈ APSn with n ̸∈ S and |S| = k ≤ d. Let +w ∈ Sn be the canonical witness for S. Because n ̸∈ S, we have w(n) = n. Thus the permutation +w(1) · · · w(n − 1) ∈ Sn−1 has pinnacle set S. Conversely, if v ∈ Sn−1 has pinnacle set S, then +appending n to the end of v will produce a permutation in Sn that also has pinnacle set S. +This gives the binomial recurrence +pn(d) = pn−1(d − 1) + pn−1(d). +To complete the argument, notice that pn(0) = 1 and pn(1) = 1 + (n − 2) = n − 1, for all positive +integers n. +□ +Combining Theorem 3.9, which characterizes admissible pinnacle sets for signed permutations, +with the enumeration in Proposition 3.11, we now count the admissible pinnacle sets for signed +permutations. +Theorem 3.12. If n ≥ 2, then +��APSB +n +�� = +⌊ n−1 +2 ⌋ +� +k=0 +�n +k +�� n − 1 − k +� n−1 +2 +� +− k +� +. +Proof. The main idea of the proof will be to construct admissible pinnacle sets in SB +n following the +requirements of Theorem 3.9. First, we will select a set P of positive pinnacles. In other words, +P ⊂ [n] and |P| ≤ (n − 1)/2. Then we add to it any set N ⊂ −([n] \ P) that is in APS(−([n] \ P)), +so long as |P| + |N| ≤ (n − 1)/2. We are interested in the number of such sets, and Lemma 3.8 +says that we only need to care about the size of P in this process. This and Lemma 3.4 mean that +such sets N can be counted in terms of admissible pinnacle sets of Sn−|P |. +8 + +Fix an integer k ∈ [0, (n − 1)/2], and choose a k-element subset P ⊂ [n]. There are +�n +k +� +ways +to do this. We can supplement P with any r-element admissible pinnacle set N ⊂ −([n] \ P), as +long as k + r ≤ ⌊(n − 1)/2⌋. The number of ways to do this is +pn−k +��n − 1 +2 +� +− k +� +. +Therefore, by Proposition 3.11, the number of admissible pinnacle sets in SB +n is +⌊ n−1 +2 ⌋ +� +k=0 +�n +k +�� n − 1 − k +� n−1 +2 +� +− k +� +, +as desired. +□ +In Table 1, we give the number of signed admissible pinnacle sets in type B for 3 ≤ n ≤ 15, +while permutations in SB +1 and SB +2 have no pinnacles. This appears in the OEIS as sequence [12, +A359066]. The even-indexed terms in the table appear in [12, A240721] and the odd-indexed terms +appear in [12, A178792]. +n +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +��APSB +n +�� +5 +7 +31 +49 +209 +351 +1471 +2561 +10625 +18943 +78079 +141569 +580865 +Table 1. The number of admissible pinnacle sets in SB +n , for 3 ≤ n ≤ 15. +In the next Section, we will be able to answer the analogous enumerative question in type D +(see Corollary 4.10). +4. Relating admissible pinnacle sets in types A, B, and D +There is a natural embedding of Sn in SD +n , and of SD +n in SB +n . Having spent Section 3 analyzing +pinnacle sets that are admissible in SB +n , it is natural to wonder how these sets are related to those +that are admissible in SD +n or, for those elements of APSB +n without negative values, to those that +are admissible in Sn. We now give complete characterization of each of these relationships. +4.1. Comparing admissible pinnacle sets in types B and D. As mentioned before, SD +n ⊂ SB +n , +thus it is natural to investigate the relationship between those sets that are admissible as pinnacle +sets in type B and those that are in type D. It is, perhaps, not surprising that this relationship +depends on the parity of n. +As a first step in this analysis, we identify a technique that will be handy in proving that a set +is admissible for type D. +Lemma 4.1. Suppose that w ∈ SB +n is a witness for a pinnacle set S. If w(n − 1) > ±w(n) or if +w(n − 1) < ±w(n), then the permutation w′, defined by +w′(i) = +� +w(i) +i < n and +−w(i) +i = n, +is also a witness for S. Moreover, S ∈ APSD +n . +9 + +Proof. First note that w′ is an element of SB +n because changing the sign of the last letter does not +alter the fact that this is a signed permutation on ±[n]. Next observe that the pinnacle set has +not changed from w to w′ because none of the inequalities between consecutive letters has been +altered. Finally, note that the numbers of negative values in w and in w′ differ by 1, meaning that +one of these permutations is in SD +n while the other is in SB +n \ SD +n . +□ +We will call on the previous result often throughout our arguments in this section, beginning +with a description of the simple relationship between APSB +n and APSD +n . +Theorem 4.2. For k ≥ 1, APSB +2k = APSD +2k. +Proof. Certainly anything admissible in type D is also admissible in type B, because signed per- +mutations include the signed permutations in type D. It remains to show that any pinnacle set +that is admissible in SB +2k is also admissible in SD +2k. Fix S := {s1 < · · · < sl} ∈ APSB +2k. Because +l ≤ ⌊(2k − 1)/2⌋, we have l ≤ k − 1. Then the canonical witness w for S satisfies the hypotheses of +Lemma 4.1, and so in fact S ∈ APSD +2k. +□ +The equality shown in Theorem 4.2 relies on the fact that there are always at least two more +non-pinnacles than there are pinnacles in signed permutations on 2k letters. This not necessarily +true for signed permutations of an odd number of letters, and hence it is not surprising that the +relationship between APSB +2k+1 and APSD +2k+1 has more nuance than the relationship presented in +Theorem 4.2. Indeed, we will show that APSD +2k+1 is a strict subset of APSB +2k+1, and we will describe +the elements of the latter that are not elements of the former. +Lemma 4.3. If S ∈ APSB +2k+1 \ APSD +2k+1, then |S| = k. +Proof. Fix S ∈ APSB +2k+1 and let w ∈ SB +2k+1 be the canonical witness for S. If |S| < k, then both +w(2k) and w(2k + 1) are non-pinnacles and w(2k) < w(2k + 1) < 0. In particular, the hypotheses +of Lemma 4.1 are satisfied by w, and so S ∈ APSD +2k+1. Hence, if S ∈ APSB +2k+1 \ APSD +2k+1, then +|S| = k. +□ +One implication of Lemma 4.3 is that if w ∈ SB +2k+1 is a witness for S ∈ APSB +2k+1\APSD +2k+1, then +w(3), w(5), . . . , w(2k−1) are all vales. With Lemma 4.3 providing a first step toward understanding +elements of APSB +2k+1 \ APSD +2k+1, we now proceed to describe these sets more clearly. +Lemma 4.4. Fix S ∈ APSB +2k+1 \ APSD +2k+1. In every witness permutation for S, the non-pinnacle +values are all negative. +Proof. Fix S ∈ APSB +2k+1 \ APSD +2k+1 and w ∈ SB +2k+1 a witness for S. Following Lemma 4.3, the non- +pinnacles of w are precisely w(1), w(3), . . . , w(2k + 1). In particular, each w(2i + 1) is less than its +immediate neighbors. Suppose, for the purpose of obtaining a contradiction, that w(2j +1) > 0 for +some j. Let w′ ∈ SB +2k+1 be the permutation obtained from w by replacing w(2j +1) by −w(2j +1). +Then w′ is still a witness for S. Either w or w′ is in SD +2k+1, meaning that S must be an element of +APSD +2k+1. This is a contradiction, so there is no such j. +□ +In fact, the negative values of S ∈ APSB +2k+1 \ APSD +2k+1 are enough to determine all of S. +Lemma 4.5. Suppose that S ∈ APSB +2k+1 \ APSD +2k+1, with P := S ∩ N and N := S ∩ −N. Then the +elements of P are the smallest k − |N| values in the set [2k + 1] \ −N. In particular, N determines +P, and hence all of S. +10 + +Proof. Fix S ∈ APSB +2k+1 \ APSD +2k+1, with P and N as defined. By Lemma 4.3, we have |S| = k, +so let S = {s1 < s2 < · · · < sk}. If |N| = k, then there is nothing to check, so assume that +|N| < k and hence sk > 0. Suppose, for the purpose of obtaining a contradiction, that there exists +q ∈ ([2k + 1] \ −N) \ P with q < sk. Let w be the canonical witness permutation for S. By +definition, w(2k) = sk and w(2k + 1) = −q. But then w′, which agrees with w everywhere except +w′(2k + 1) = q, is also a witness for S, contradicting Lemma 4.4. Therefore P consists precisely of +the smallest k − |N| values in the set [2k + 1] \ −N. +□ +Lemma 4.5 gives a necessary condition for elements of APSB +2k+1 \ APSD +2k+1. The next result +establishes that the set N ⊔ P constructed in Lemma 4.5 is, in fact, an admissible signed pinnacle +set. +Corollary 4.6. Suppose that N ⊂ −N and N ∈ APSB +2k+1. Let P be the smallest k − |N| values in +[2k + 1] \ −N. Then N ⊔ P ∈ APSB +2k+1. +Proof. This follows from Theorem 3.9. +□ +Maintaining the terminology of Corollary 4.6, note that for any set N ⊂ −N, all witness +permutations for N ⊔ P are forced by construction of P to have the same number of negative +values: k + 1 + |N|. This yields the following corollary. +Corollary 4.7. Suppose S ∈ APSB +2k+1 \ APSD +2k+1, with N := S ∩ −N. The sets |N| and |S| have +the same parity. +Proof. To have S ∈ APSB +2k+1 \ APSD +2k+1, we need |S| = k, by Lemma 4.3. Moreover, as discussed +above, the number of negative values is k + 1 + |N|, and this must be odd because S /∈ APSD +2k+1. +Thus k + |N| = |S| + |N| is even, completing the proof. +□ +The consequence of this collection of results is that if we have a set N ⊂ −N that is, itself, +admissible in SB +2k+1, and for which |N| has the same parity as k, then there is a unique ((k − |N|)- +element) set P ⊂ N for which +N ⊔ P ∈ APSB +2k+1 \ APSD +2k+1. +Therefore, to enumerate APSB +2k+1 \ APSD +2k+1, it suffices to count the elements of APSB +2k+1 that have +no positive values and that have size of the form k − 2i. +Because we want to look at the elements of APSB +2k+1 having no positive values, we can take +advantage of Lemma 3.4 to look, instead, at APS2k+1. That is, it will suffice to count +� +i≥0 +����{S ∈ APS2k+1 : |S| = k − 2i} +����. +The last step of this enumeration requires a parity result. +Lemma 4.8. For k ≥ 0, +����{S ∈ APS2k+1 : |S| is even} +���� = +����{S ∈ APS2k+1 : |S| is odd} +����. +Proof. Fix S ⊂ [2k + 1]. If 2k + 1 ∈ S, then set S′ := S \ {2k + 1}. Clearly if S ∈ APS2k+1 then +also S′ ∈ APS2k+1, and the sets |S| and |S′| have different parities. +Now consider S ∈ APS2k+1 with 2k + 1 ̸∈ S. By [3, Theorem 1.8], max(S) > 2|S|. We have +max(S) < 2k + 1, so |S| < k. Consequently, S has a witness permutation w using at most k +vales, so there are at least (2k + 1) − (k − 1 + k) = 2 non-pinnacle/non-vale values in this witness +11 + +permutation, and one of these is 2k + 1. We can create a new permutation w′ by inserting 2k + 1 +immediately to the right of the largest vale in w. Thus the pinnacle set of w′ is S ∪ {2k + 1}. +Therefore there is a bijection between even-sized elements of APS2k+1 and odd-sized ones, +obtained by adding/removing the element 2k + 1. This partitions APS2k+1 into two evenly sized +parts. +□ +We have now completed all of the steps necessary to give the desired enumeration. +Theorem 4.9. For k ≥ 1, +��APSB +2k+1 \ APSD +2k+1 +�� = +�2k − 1 +k +� +. +Proof. Following Lemmas 4.3 and 4.5 and Corollary 4.7, we can enumerate APSB +2k+1 \ APSD +2k+1 by +counting elements of APS2k+1 that have size {k − 2i : i = 0, 1, . . .}. These are either all of the +odd-sized sets in APS2k+1 or all of the even-sized ones. By Lemma 4.8, then, +��APSB +2k+1 \ APSD +2k+1 +�� = 1 +2 |APS2k+1| . +It was shown in [3, Theorem 1.8] that |APS2k+1| = +�2k +k +� +. Finally, it is straightforward to check that +1 +2 +�2k +k +� += +�2k−1 +k +� +. +□ +We can now use Theorem 3.12, which enumerated APSB +n , and Theorems 4.2 and 4.9 to enu- +merate APSD +n for all n. +Corollary 4.10. For k ≥ 1, +��APSD +2k +�� = +��APSB +2k +�� and +��APSD +2k+1 +�� = +� k +� +i=0 +�2k + 1 +i +��2k − i +k − i +�� +− +�2k − 1 +k +� +. +In Table 2, we give the number of signed admissible pinnacle sets in type D for 3 ≤ n ≤ 15, +while permutations in SD +1 and SD +2 have no pinnacles. +This appears in the OEIS as sequence +A359067. +The even-indexed terms are identical to even terms in Table 1 and the odd-indexed +terms are +�2k−1 +k +� +less than the corresponding odd-indexed terms in Table 1. +n +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +��APSD +n +�� +4 +7 +28 +49 +199 +351 +1436 +2561 +10499 +18943 +77617 +141569 +579149 +Table 2. The number of admissible pinnacle sets in SD +n , for 3 ≤ n ≤ 15. +4.2. Comparing admissible pinnacle sets in types B and A. Some elements of APSB +n have +no negative values, and so one could ask if those sets might also be admissible in Sn. In this section +we consider how those elements of APSB +n are related to the admissible pinnacle sets in APSn. To +make this discussion precise, we introduce: +APS+ +n := {S ∈ APSB +n : S ⊂ N}; +in other word, APS+ +n consists of the pinnacle sets that are admissible in SB +n and that contain no +negative values. +12 + +For example, {1, 3} ∈ APS+ +5 , with canonical witness 51432 ∈ SB +5 . In fact, by Corollary 3.2, +any subset of [n] having at most (n − 1)/2 elements is admissible in SB +n . Contrast this with APSn; +for example, +APS+ +5 \ APS5 = +� +{1}, {2}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4} +� +. +Our goal in this section is to understand APS+ +n \ APSn. As with the comparison of APSB +n and +APSD +n , this will depend on the parity of n. +Theorem 4.11. For k ≥ 0, |APS+ +2k+1 \ APS2k+1| = 4k − +�2k +k +� +. +Proof. Because APS2k+1 ⊂ APS+ +2k+1, the desired value is equal to +��APS+ +2k+1 +�� − |APS2k+1| . +Following Corollary 3.2, we can compute +��APS+ +2k+1 +�� by counting i-element subsets of [2k +1] for all +i ≤ k. The result follows by recognizing that this yields a sum that is half of a row-sum of Pascal’s +triangle, and combining this with the enumeration of APS2k+1 from [3]: +|APS+ +2k+1 \ APS2k+1| = +��APS+ +2k+1 +�� − |APS2k+1| += +�2k + 1 +0 +� ++ +�2k + 1 +1 +� ++ · · · + +�2k + 1 +k +� +− +�2k +k +� += 1 +222k+1 − +�2k +k +� += 4k − +�2k +k +� +. +□ +We now complete this analysis by considering the even-indexed case. +Theorem 4.12. For k ≥ 1, +��APS+ +2k \ APS2k +�� = 22k−1 − +�2k +k +� +. +Proof. This calculation is almost identical to that from the proof of Theorem 4.11, except that we +will also have to subtract the central term from a row of Pascal’s triangle: +|APS+ +2k \ APS2k| = +��APS+ +2k +�� − |APS2k| += +�2k +0 +� ++ +�2k +1 +� ++ · · · + +� 2k +k − 1 +� +− +�2k − 1 +k − 1 +� += 1 +2 +� +22k − +�2k +k +�� +− +�2k − 1 +k − 1 +� += 22k−1 − +�1 +2 +�2k +k +� ++ +�2k − 1 +k − 1 +�� += 22k−1 − +�2k +k +� +. +□ +We combine the enumerations of Theorems 4.11 and 4.12 in Table 3. +Specifically, we list +��APS+ +n \ APSn +�� for 3 ≤ n ≤ 15, while permutations in SB +1 and SB +2 have no pinnacles. The nth +term of this appears in the OEIS as double the (n − 1)st term of [12, A294175]. Moreover, the +odd-indexed terms, enumerated in Theorem 4.11, appear in [12, A068551] and the even-indexed +terms are double the terms of [12, A008549]. +13 + +n +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +��APS+ +n \ APSn +�� +2 +2 +10 +12 +44 +58 +186 +260 +772 +1124 +3172 +4760 +12952 +Table 3. The number of all-positive pinnacle sets that are admissible in SB +n but +not in Sn, for 3 ≤ n ≤ 15. +5. Future directions +As demonstrated by the results in this paper, admissible pinnacle sets have rich structure and +properties even beyond the symmetric group. There are many directions for further research on +this topic, including broad questions about pinnacle sets for families of permutations with certain +properties, and enumerative specializations. +As a complement to those large questions, we conclude this work by pointing out that we +uncovered a possible connection between +��APSB +n +�� and sequence [12, A119258]. In particular, we +have the following conjecture. +Conjecture 5.1. Consider the sequence [12, A119258], given by T(n, 0) = T(n, n) = 1 and +T(n, k) = 2T(n − 1, k − 1) + T(n − 1, k) for 0 < k < n. Then +��APSB +n +�� = T +� +n, +�n − 1 +2 +�� +. +Appendix A. Data +Patrek Ragnarsson’s code for computing the data in Tables 1, 2, and 3 can be found at +https://github.com/PatrekR/Signed-pinnacle-sets. +Note that the data in Table 2 is the +difference between the enumerations given in two of the files posted at this GitHub link. +References +[1] Sara Billey, Krzysztof Burdzy, and Bruce E. Sagan. Permutations with given peak set. J. Integer Seq., 6(16), +2013. +[2] F. Castro-Velez, A. Diaz-Lopez, R. Orellana, J. Pastrana, and R. Zevallos. Number of permutations with same +peak set for signed permutations. Journal of Combinatorics, 8(4):631–652, 2017. +[3] Robert Davis, Sarah A. Nelson, T. Kyle Petersen, and Bridget E. Tenner. The pinnacle set of a permutation. +Discrete Math., 341(11):3249–3270, 2018. +[4] Alexander Diaz-Lopez, Lucas Everham, Pamela E. Harris, Erik Insko, Vincent Marcantonio, and Mohamed +Omar. Counting peaks on graphs. Australas. J Comb., 75:174–189, 2019. +[5] Alexander Diaz-Lopez, Pamela E. Harris, Isabella Huang, Erik Insko, and Lars Nilsen. A formula for enumerating +permutations with a fixed pinnacle set. Discret. Math., 344:112375, 2021. +[6] Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, Mohamed Omar, and Bruce E. Sagan. Descent polynomials. +Discrete Mathematics, 342(6):1674–1686, 2019. +[7] Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin. Peak sets of classical coxeter +groups. Involve, 10(2):263–290, 2017. +[8] Rachel Domagalski, Jinting Liang, Quinn Minnich, Bruce E. Sagan, Jamie Schmidt, and Alexander Sietsema. +Pinnacle set properties. Discrete Mathematics, 345(7):112882, 2022. +[9] Justine +Falque, +Jean-Christophe +Novelli, +and +Jean-Yves +Thibon. +Pinnacle +sets +revisited. +Preprint +arXiv:2106.05248, 2021. +[10] Wenjie Fang. Efficient recurrence for the enumeration of permutations with fixed pinnacle set. Disc. Math. The- +oret. Comp. Sci., 24:#8, 2022. +[11] Quinn Minnich. Further results on pinnacle sets. Discrete Math., 346(4):Paper No. 113296, 2023. +[12] OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2022. Published electronically at +http://oeis.org. +14 + +[13] Irena Rusu. Sorting permutations with fixed pinnacle set. Electron. J. Comb., 27:P3.23, 2020. +[14] Irena Rusu and Bridget Eileen Tenner. Admissible pinnacle orderings. Graphs and Comb., 37:1205–1214, 2021. +[15] Richard P. Stanley. Enumerative combinatorics, volume 49 of Cambridge Studies in Advanced Mathematics. +Cambridge University Press, Cambridge, second edition edition, 2012. +[16] John R. Stembridge. Enriched p-partitions. Transactions of the American Mathematical Society, 349:763–788, +1997. +(N. Gonz´alez) Department of Mathematics, University of California, Berkeley, CA, 94720 +Email address: nicolle@math.berkeley.edu +(P. E. Harris) Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI 53211 +Email address: peharris@uwm.edu +(G. Rojas Kirby) Department of Mathematics and Statistics, San Diego State University, CA 92182 +Email address: gkirby@sdsu.edu +(M. Smit Vega Garcia) Department of Mathematics, Western Washington University, Bellingham, +WA 98225 +Email address: smitvem@wwu.edu +(B. E. Tenner) Department of Mathematical Sciences, DePaul University, Chicago, IL 60614 +Email address: bridget@math.depaul.edu +15 + diff --git a/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/load_file.txt b/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0eab86ec653ded98e225192a9c858c096c69158 --- /dev/null +++ b/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf,len=632 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='02628v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='CO] 6 Jan 2023 PINNACLE SETS OF SIGNED PERMUTATIONS NICOLLE GONZ´ALEZ, PAMELA E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' HARRIS, GORDON ROJAS KIRBY, MARIANA SMIT VEGA GARCIA, AND BRIDGET EILEEN TENNER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacle sets record the values of the local maxima for a given family of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previ- ously defined by Billey-Burdzy-Sagan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In recent years pinnacles and admissible pinnacles sets for the type A symmetric group have been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In this article we define the pinnacle set of signed permutations of types B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We give a closed formula for the number of type B/D admissible pinnacle sets and answer several other related enumerative questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Introduction The study of permutation statistics is an active subdiscipline of combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Given a per- mutation w = w(1)w(2) · · · w(n), two particularly well-studied statistics are descents and peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Respectively, these statistics refer to indices i such that w(i) > w(i + 1), and indices i such that w(i − 1) < w(i) > w(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The collection of a permutation’s descent indices is its descent set, with a permutation’s peak set being similarly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Two fundamental goals in the study of these particular statistics are (1) understanding which subsets can arise as descent sets or peak sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', which sets are admissible as descent or peak sets), and (2) enumerating the permutations that have a given admissible descent or peak set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For descents of permutations in the (type A) symmetric group Sn, this question was answered by Stanley [15, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4] and is well known to give rise to the Eulerian numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Inspired by Stembridge’s study of peaks in the context of poset partitions [16], Billey, Burdzy, and Sagan [1] introduced the study of admissible peak sets for Sn with an interest in probabilistic applications, and established that the number of permutations with peak set I is given by 2n−|I|−1p(n), where p(n) is a polynomial of degree max(I) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Shortly thereafter, their results were extended to permutations in type B by Castro-Velez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [2] where it was shown that the number of permutations with a given peak set I is 22n−|I|−1p(n), with p(n) the same as in [1] above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The second author and various collaborators went further by extending these results to types C and D [7], using peaks to study properties of the descent polynomial [6], and then initiating the study of peaks in the context of graphs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A notion that is closely related to peaks is the pinnacle set of a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacles are the set of values held by the permutation at the peak indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' More precisely, given a permutation w = w(1)w(2) · · · w(n) with peak set Peak(w), the pinnacle set of w is Pin(w) = {w(i) : i ∈ Peak(w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Given a subset I ⊆ [n], if there exists a permutation w whose pinnacle set is I, we say that I is an admissible pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In [3], Davis, Nelson, Petersen, and the last author pioneered the study of pinnacles for permutations in Sn and gave a complete characterization of admissible pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' They provided a closed formula for the number of admissible pinnacle sets with a given maximum value, as well as a refinement to those appearing in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, Davis et Date: January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' was partially supported through a Karen Uhlenbeck EDGE Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='G was partially supported by the NSF grant DMS 2054282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' was partially supported by the NSF grant DMS-2054436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 1 al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' gave a recursive formula for the number of permutations in Sn with a given pinnacle set p(n) and asked whether a more efficient expression could be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This paper led to a sequence of articles in recent years, many focused on improved and faster formulas for p(n), by realizing permutations with given pinnacle sets as invariants under certain modified Sn-actions [5, 9] or via more traditional enumerative methods [8, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In related work, Rusu [13] and Rusu-Tenner [14] deepened the knowledge of pinnacles in Sn by investigating further properties of these statistics and characterizing admissible pinnacle orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In this article we look beyond type A and study pinnacles and admissible pinnacle sets for the type B and type D signed symmetric groups, SB n and SD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Our main results are the following, where we write APSX n to denote the admissible pinnacle sets in SX n for X ∈ {A, B, D}: (1) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12 gives a closed formula for the number of admissible pinnacle sets in SB n , |APSB n | = ⌊ n−1 2 ⌋ � k=0 �n k ��n − 1 − k � n−1 2 � − k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (2) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2 proves that any admissible pinnacle set in SB 2k is also admissible in SD 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' that is, APSD 2k = APSB 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (3) In counterpoint to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9 counts the admissible pinnacle sets of type B that are not in type D when n = 2k + 1, |APSB 2k+1 \\ APSD 2k+1| = �2k − 1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (4) Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12 count the all-positive admissible pinnacle sets of type B that are not admissible in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Namely, defining APS+ n := {S ∈ APSB n : S ⊂ N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' we prove that the sets APS+ n \\ APSn are enumerated by, ��APS+ n \\ APSn �� = � 4k − �2k k � if n = 2k + 1, and 22k−1 − �2k k � if n = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Section 2, we introduce all the necessary background and notation, defining pinnacles and related notions in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Section 3, we give a characterization of admissible signed pinnacle sets and a formula for their enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Section 4, we provide relations between admissible pinnacle sets of type A, B, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lastly, in Section 5, we describe some future directions and open conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The authors thank Patrek K´arason Ragnarsson for the coding and data that facilitated the research in this project, and Freyja K´arad´ottir Ragnarsson for the key insight to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The authors also thank the American Institute of Mathematics and the National Science Foundation for sponsoring the Latinx Mathematicians Research Community, which brought together a subset of the authors initially for collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Background Let N = {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='} and for n ∈ N we write [n] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For any set X, typically of positive values, although we make the definition more generally, we define −X := {−x : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Finally, we define ±X = X ∪ −X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 2 Throughout this paper, we let Sn denote the (type A) symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, Sn is the group of bijections from [n] → [n] under function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We often write w ∈ Sn using one-line notation, as w = w(1)w(2) · · · w(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The type B symmetric group (that is, the hyperoctahedral group) is the group of signed permutations SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' These are bijections ±[n] → ±[n] such that w(−i) = −w(i) for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, any w ∈ SB n satisfies the property that {|w(1)|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , |w(n)|} = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The type D symmetric group is the subgroup SD n of SB n consisting of signed permutations with an even number of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Namely, these are the signed permutations w for which |{i ∈ [n] : w(i) < 0}| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As in type A, we use one-line notation to denote signed permutations w ∈ SB n , where we may write only w = w(1)w(2) · · · w(n) since this uniquely determines w(−i) for all positive i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following convention, we write −i = ¯i to ease notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, w = ¯12¯3 is the signed permutation with w(1) = −1, w(2) = 2, and w(3) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Recall that a permutation w ∈ Sn has a peak at index i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , n − 1} if w(i − 1) < w(i) > w(i + 1), and the value w(i) is a pinnacle of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We denote by Peak(w) the set of all peaks of w ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The pinnacle set of w ∈ Sn is Pin(w) = {w(i) : i ∈ Peak(w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A set P ⊆ [n] is an n-admissible pinnacle set in type A if there exists a permutation w ∈ Sn such that Pin(w) = P, and we call the permutation w a witness for the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, the identity permutation is a witness for the admissible pinnacle set ∅ (as is any peak-less permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Denote by APSn the set of all n-admissible pinnacle sets in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In order to facilitate our discussions about pinnacles, we introduce terminology about their minimal counterparts: a permutation w ∈ Sn has a valley at index i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , n − 1} if w(i − 1) > w(i) < w(i + 1), and the value w(i) is a vale of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The graph of the permutation 23715648 ∈ S8 with the pinnacles/peaks circled in red and the vales/valleys in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consider the permutation w = 23715648 ∈ S8 shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We have Peak(w) = {3, 6} and Pin(w) = {6, 7}, and valleys and vales {4, 7} and {1, 4}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacles in types B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacles were defined in [3] for unsigned permutations, but they could just as easily have been defined for signed permutations—or, in fact, for arbitrary strings of distinct real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We now expand the type A definitions to type B, and note that since SD n ⊂ SB n , these definitions also hold for type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w be a signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A pinnacle of w is a value w(i) that is larger than both w(i − 1) and w(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The pinnacle set of w is the set of its pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In order to define admissible pinnacle sets, it is important to establish which subsets could even appear among the one-line notation of a signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A signed set (or signed subset, depending on context) is a set I such that x ∈ I implies −x ̸∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Throughout this paper, we assume that all subsets of ±[n] are signed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A signed subset S ⊂ ±[n] is an admissible pinnacle set if S is the pinnacle set of some signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That permutation is a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Note that when we study sets that are admissible as pinnacle sets in type D, any witness permutation will be required to be in SD n for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As before, we denote by APSB n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', APSD n ) the set of all n-admissible pinnacle sets in type B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', type D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Once again, we have ∅ ∈ APSD n ⊆ APSB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, 123 · · · n and ¯2¯134 · · · n are both witnesses for ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' While there can be multiple witness permutations for a given admissible pinnacle set, we will often refer to a particular witness permutation that we call “canonical.” Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For S ∈ APSB n , write S = {s1 < s2 < · · · < sk}, and set S′ := −[n] \\ {−|s| : s ∈ S} = {s′ 1 < s′ 2 < · · · < s′ n−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then the canonical witness permutation is w := s′ 1 s1 s′ 2 s2 · · · s′ k sk s′ k+1 · · · s′ n−k ∈ SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If S ∈ APSD n , then its canonical (type D) witness permutation is w as defined above if w is in SD n , and otherwise its canonical witness is obtained from w by replacing s′ n−k with |s′ n−k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Next we establish that the “canonical witness permutations” are, in fact, witnesses and follow this by providing canonical witness permutations in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The canonical witness permutation for an admissible set S is a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set S is admissible, so there is some permutation whose pinnacle set is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The canonical witness has been constructed to have minimal possible non-pinnacle values, and to position the smallest non-pinnacle values beside the smallest pinnacle values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore, if any permutations were to have S as a pinnacle set (and we know that some permutation does), the permutation given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6 would be among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Although SB n contains both Sn and SD n as subgroups, there are interesting subtleties to the pinnacle sets that become admissible when witness permutations can be signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First, some sets will be admissible with type B permutations, but not with type D permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' And second, some sets of all-positive values will be admissible with type B permutations, but not with type A (unsigned) permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We demonstrate each of these scenarios below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4 (a) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 −1 −2 −3 −4 −5 −6 −7 (b) (b) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 −1 −2 −3 −4 −5 −6 −7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (a) The graph of the permutation ¯7¯4¯61¯52¯3 ∈ SB 7 with the pinna- cles/peaks circled in red and the vales/valleys in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (b) The graph of the permu- tation ¯63¯54¯17¯2 ∈ SB 7 with the pinnacles/peaks circled in red and the vales/valleys in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set {¯4, 1, 2} is admissible in SB 7 , with canonical witness permutation ¯7¯4¯61¯52¯3 as shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' However, there is no element of SD 7 having this pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, {¯4, 1, 2} ̸∈ APSD 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set {3, 4, 7} is admissible in SB 7 , with canonical witness permutation ¯63¯54¯17¯2, as shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' However, despite its pinnacle set having all positive values, there is no type A permutation having this pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, {3, 4, 7} ̸∈ APSn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Admissible signed pinnacle sets in type B In this section, we characterize and enumerate the admissible pinnacle sets among signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This expands on the work begun in [3] for unsigned permutations, but, as we show, the results for signed permutations are subtly different from those in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Characterization of admissible pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For the remainder of the article, we will often use the fact that given an admissible pinnacle set S ∈ APSB n , we can always write S = P(S) ⊔ N(S) with P(S) := S ∩ [n] and N(S) := S ∩ −[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' When no confusion will arise, we simply write P := P(S) and N := N(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To give a first inkling of how admissible pinnacle sets in type B are fundamentally different from those in type A, we note that there are some sets of positive integers that are never in APSn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, any set containing 1 or 2 will never be the pinnacle set of any permutation in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' On the other hand, such a statement is not true in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Every finite signed subset S is admissible in SB n , for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, there exists w ∈ SB n such that S = PinB(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Write S = {s1 < · · · < sk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let m = max{|s| : s ∈ S} (that is, m = max{|s1|, |sk|}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Define the set S′ := −[2m + 1] \\ {−|s| : s ∈ S}, which we write as S′ = {s′ 1 < · · · < s′ 2m+1−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w = s′ 1 s1 s′ 2 s2 · · · s′ k sk s′ k+1 s′ k+2 · · · s′ 2m+1−k ∈ SB 2m+1, and PinB(w) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Using a similar argument as the one proving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1, it follows that any finite set of all positive values is admissible in some SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Any subset P ⊂ [n] with |P| ≤ n−1 2 is admissible in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let P = {p1 < · · · < pk}, and set P ′ := −([n] \\ P) = {p′ 1 < · · · < p′ n−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w = p′ 1 p1 p′ 2 p2 · · · p′ k pk p′ k+1 p′ k+2 · · · p′ n−k ∈ SB n and PinB(w) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ This can be particularly interesting when the set P was not admissible in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consider P = {1, 2} with n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The permutation ¯51¯42¯3 ∈ SB 5 is a witness permutation for P, so P ∈ APSB 5 , while P ̸∈ APSn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Our goal is to establish a characterization and formula for the number of admissible pinnacle sets in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We begin with some preliminary steps, from which those results will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The first of these is a bijection between admissible pinnacle sets in Sn and those admissible pinnacle sets in SB n that have no positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' There exists a bijection between APSn and {S ∈ APSB n : S ⊂ −N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Given T ∈ APSn, define T ′ := {t − (n + 1) : t ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set T ′ has no positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w ∈ Sn be the canonical witness for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w′ := (w(1) − (n + 1)) · · · (w(n) − (n + 1)) ∈ SB n has pinnacle set T ′, and so T ′ ∈ APSB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This process can be inverted: given S ∈ APSB n with P(S) = ∅, map this S to S′ := {s + n + 1 : s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It follows that S′ ∈ APSn, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We illustrate Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4 with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set {3, 6, 7, 10} ∈ APS10 is in correspondence with {¯8, ¯5, ¯4, ¯1} ∈ APSB 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The permutations described in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4, which exhibit these sets as pinnacle sets, are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We have defined admissible pinnacle sets in types A, B, and D, referring to permutations in Sn, SB n , or SD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' However, as suggested earlier, there is a natural generalization of admissible pinnacle sets to permutations of any totally ordered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For any totally ordered set X, let APS(X) be the set of admissible pinnacle sets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The definitions of witness and canonical witness permutations in this general setting are analogous to their definitions in the symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because they arise so often, we have been easing notation by writing APS(Sn) as APSn, APS(SB n ) as APSB n , and APS(SD n ) as APSD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set X = {−2, π, 4, 5, 100} has six admissible pinnacle sets: ∅, {4}, {5}, {100}, {4, 100}, and {5, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 6 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The left-hand figure shows the canonical witness for {3, 6, 7, 10} in S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The right-hand figure shows the corresponding witness permutation for {¯8, ¯5, ¯4, ¯1}, as defined in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Note that if we are only interested in how many admissible pinnacle sets X has, as opposed to the sets themselves, then the size of X is what matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For any totally ordered finite set X, |APS(X)| = |APS ([|X|]) | = |APS|X||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This calculation will be useful in the enumeration appearing in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We are now are able to characterize admissible pinnacle sets for signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The sets in APSB n are exactly the sets S = P(S) ⊔ N(S) for which |P(S)| + |N(S)| ≤ (n − 1)/2, P(S) ⊂ [n], N(S) ⊂ −([n] \\ P(S)), and N(S) ∈ APS(−([n] \\ P(S))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First of all, it is clear that any admissible pinnacle set in SB n must satisfy the listed require- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Now suppose that a set S satisfies the listed requirements, with P := P(S) = {p1 < · · · < pk} and N := N(S) = {n1 < · · · < nr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In light of the last requirement, let w be the canonical witness permutation of the set (−([n] \\ P)), having pinnacle set N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, w = i1 n1 i2 n2 · · · ir nr ir+1 ir+2 ir+3 · · · in−k−r where ij < ij+1 and {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , in−k−r} = −([n] \\ P) \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then i1 n1 i2 n2 · · · ir nr ir+1 p1 ir+2 p2 ir+3 · · · pk ir+k+1 ir+k+2 · · · in−r−k is a canonical witness for S = P ⊔ N in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Hence S ∈ APSB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Enumeration of admissible pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The conditions listed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9 inform our enumeration of the admissible pinnacle sets in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, we will construct these sets by 7 first fixing a collection P of positive pinnacles and then determining how many sets N of negative pinnacles exist for which P ∪ N is admissible in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In order not to have too many pinnacles (that is, not more than ⌊(n − 1)/2⌋), we need to understand the following value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let pn(d) be the number of admissible pinnacle sets in Sn having cardinality at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, pn(d) := |{S ∈ APSn : |S| ≤ d}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This statistic has a particularly nice formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For all integers d ∈ [0, ⌊(n − 1)/2⌋], pn(d) = �n − 1 d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The admissible pinnacle sets in Sn having cardinality at most d can be partitioned into two sets: those that contain n, and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We claim that the first set is counted by pn−1(d − 1), and the second set is counted by pn−1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose, first, that S ∈ APSn such that n ∈ S and |S| = k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w ∈ Sn be the canonical witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Deleting n from the one-line notation of w will produce a permutation v ∈ Sn−1 with Pin(v) = S \\ {n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Conversely, given T ∈ APSn−1 with |T| = k − 1, let u ∈ Sn−1 be the canonical witness for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Inserting n between the non-pinnacles u(2k − 1) and u(2k) will produce a permutation in Sn whose pinnacle set is T ∪ {n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This establishes the first part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For the second part of the claim, suppose that S ∈ APSn with n ̸∈ S and |S| = k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w ∈ Sn be the canonical witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because n ̸∈ S, we have w(n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Thus the permutation w(1) · · · w(n − 1) ∈ Sn−1 has pinnacle set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Conversely, if v ∈ Sn−1 has pinnacle set S, then appending n to the end of v will produce a permutation in Sn that also has pinnacle set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This gives the binomial recurrence pn(d) = pn−1(d − 1) + pn−1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To complete the argument, notice that pn(0) = 1 and pn(1) = 1 + (n − 2) = n − 1, for all positive integers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9, which characterizes admissible pinnacle sets for signed permutations, with the enumeration in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, we now count the admissible pinnacle sets for signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If n ≥ 2, then ��APSB n �� = ⌊ n−1 2 ⌋ � k=0 �n k �� n − 1 − k � n−1 2 � − k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The main idea of the proof will be to construct admissible pinnacle sets in SB n following the requirements of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First, we will select a set P of positive pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In other words, P ⊂ [n] and |P| ≤ (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then we add to it any set N ⊂ −([n] \\ P) that is in APS(−([n] \\ P)), so long as |P| + |N| ≤ (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We are interested in the number of such sets, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8 says that we only need to care about the size of P in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4 mean that such sets N can be counted in terms of admissible pinnacle sets of Sn−|P |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 8 Fix an integer k ∈ [0, (n − 1)/2], and choose a k-element subset P ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' There are �n k � ways to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We can supplement P with any r-element admissible pinnacle set N ⊂ −([n] \\ P), as long as k + r ≤ ⌊(n − 1)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of ways to do this is pn−k ��n − 1 2 � − k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, the number of admissible pinnacle sets in SB n is ⌊ n−1 2 ⌋ � k=0 �n k �� n − 1 − k � n−1 2 � − k � , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ In Table 1, we give the number of signed admissible pinnacle sets in type B for 3 ≤ n ≤ 15, while permutations in SB 1 and SB 2 have no pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This appears in the OEIS as sequence [12, A359066].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The even-indexed terms in the table appear in [12, A240721] and the odd-indexed terms appear in [12, A178792].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' n 3 4 5 6 7 8 9 10 11 12 13 14 15 ��APSB n �� 5 7 31 49 209 351 1471 2561 10625 18943 78079 141569 580865 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of admissible pinnacle sets in SB n , for 3 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In the next Section, we will be able to answer the analogous enumerative question in type D (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Relating admissible pinnacle sets in types A, B, and D There is a natural embedding of Sn in SD n , and of SD n in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Having spent Section 3 analyzing pinnacle sets that are admissible in SB n , it is natural to wonder how these sets are related to those that are admissible in SD n or, for those elements of APSB n without negative values, to those that are admissible in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We now give complete characterization of each of these relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Comparing admissible pinnacle sets in types B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As mentioned before, SD n ⊂ SB n , thus it is natural to investigate the relationship between those sets that are admissible as pinnacle sets in type B and those that are in type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It is, perhaps, not surprising that this relationship depends on the parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As a first step in this analysis, we identify a technique that will be handy in proving that a set is admissible for type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose that w ∈ SB n is a witness for a pinnacle set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If w(n − 1) > ±w(n) or if w(n − 1) < ±w(n), then the permutation w′, defined by w′(i) = � w(i) i < n and −w(i) i = n, is also a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Moreover, S ∈ APSD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First note that w′ is an element of SB n because changing the sign of the last letter does not alter the fact that this is a signed permutation on ±[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Next observe that the pinnacle set has not changed from w to w′ because none of the inequalities between consecutive letters has been altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Finally, note that the numbers of negative values in w and in w′ differ by 1, meaning that one of these permutations is in SD n while the other is in SB n \\ SD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We will call on the previous result often throughout our arguments in this section, beginning with a description of the simple relationship between APSB n and APSD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, APSB 2k = APSD 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Certainly anything admissible in type D is also admissible in type B, because signed per- mutations include the signed permutations in type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It remains to show that any pinnacle set that is admissible in SB 2k is also admissible in SD 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S := {s1 < · · · < sl} ∈ APSB 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because l ≤ ⌊(2k − 1)/2⌋, we have l ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then the canonical witness w for S satisfies the hypotheses of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1, and so in fact S ∈ APSD 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ The equality shown in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2 relies on the fact that there are always at least two more non-pinnacles than there are pinnacles in signed permutations on 2k letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This not necessarily true for signed permutations of an odd number of letters, and hence it is not surprising that the relationship between APSB 2k+1 and APSD 2k+1 has more nuance than the relationship presented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Indeed, we will show that APSD 2k+1 is a strict subset of APSB 2k+1, and we will describe the elements of the latter that are not elements of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If S ∈ APSB 2k+1 \\ APSD 2k+1, then |S| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 and let w ∈ SB 2k+1 be the canonical witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If |S| < k, then both w(2k) and w(2k + 1) are non-pinnacles and w(2k) < w(2k + 1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, the hypotheses of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1 are satisfied by w, and so S ∈ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Hence, if S ∈ APSB 2k+1 \\ APSD 2k+1, then |S| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ One implication of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3 is that if w ∈ SB 2k+1 is a witness for S ∈ APSB 2k+1\\APSD 2k+1, then w(3), w(5), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , w(2k−1) are all vales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3 providing a first step toward understanding elements of APSB 2k+1 \\ APSD 2k+1, we now proceed to describe these sets more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 \\ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In every witness permutation for S, the non-pinnacle values are all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 \\ APSD 2k+1 and w ∈ SB 2k+1 a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3, the non- pinnacles of w are precisely w(1), w(3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , w(2k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, each w(2i + 1) is less than its immediate neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose, for the purpose of obtaining a contradiction, that w(2j +1) > 0 for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w′ ∈ SB 2k+1 be the permutation obtained from w by replacing w(2j +1) by −w(2j +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w′ is still a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Either w or w′ is in SD 2k+1, meaning that S must be an element of APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This is a contradiction, so there is no such j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ In fact, the negative values of S ∈ APSB 2k+1 \\ APSD 2k+1 are enough to determine all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose that S ∈ APSB 2k+1 \\ APSD 2k+1, with P := S ∩ N and N := S ∩ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then the elements of P are the smallest k − |N| values in the set [2k + 1] \\ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, N determines P, and hence all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 \\ APSD 2k+1, with P and N as defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3, we have |S| = k, so let S = {s1 < s2 < · · · < sk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If |N| = k, then there is nothing to check, so assume that |N| < k and hence sk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose, for the purpose of obtaining a contradiction, that there exists q ∈ ([2k + 1] \\ −N) \\ P with q < sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w be the canonical witness permutation for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By definition, w(2k) = sk and w(2k + 1) = −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' But then w′, which agrees with w everywhere except w′(2k + 1) = q, is also a witness for S, contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore P consists precisely of the smallest k − |N| values in the set [2k + 1] \\ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5 gives a necessary condition for elements of APSB 2k+1 \\ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The next result establishes that the set N ⊔ P constructed in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5 is, in fact, an admissible signed pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose that N ⊂ −N and N ∈ APSB 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let P be the smallest k − |N| values in [2k + 1] \\ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then N ⊔ P ∈ APSB 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Maintaining the terminology of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6, note that for any set N ⊂ −N, all witness permutations for N ⊔ P are forced by construction of P to have the same number of negative values: k + 1 + |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This yields the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose S ∈ APSB 2k+1 \\ APSD 2k+1, with N := S ∩ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The sets |N| and |S| have the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To have S ∈ APSB 2k+1 \\ APSD 2k+1, we need |S| = k, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Moreover, as discussed above, the number of negative values is k + 1 + |N|, and this must be odd because S /∈ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Thus k + |N| = |S| + |N| is even, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ The consequence of this collection of results is that if we have a set N ⊂ −N that is, itself, admissible in SB 2k+1, and for which |N| has the same parity as k, then there is a unique ((k − |N|)- element) set P ⊂ N for which N ⊔ P ∈ APSB 2k+1 \\ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore, to enumerate APSB 2k+1 \\ APSD 2k+1, it suffices to count the elements of APSB 2k+1 that have no positive values and that have size of the form k − 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because we want to look at the elements of APSB 2k+1 having no positive values, we can take advantage of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4 to look, instead, at APS2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, it will suffice to count � i≥0 ����{S ∈ APS2k+1 : |S| = k − 2i} ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The last step of this enumeration requires a parity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 0, ����{S ∈ APS2k+1 : |S| is even} ���� = ����{S ∈ APS2k+1 : |S| is odd} ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ⊂ [2k + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If 2k + 1 ∈ S, then set S′ := S \\ {2k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Clearly if S ∈ APS2k+1 then also S′ ∈ APS2k+1, and the sets |S| and |S′| have different parities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Now consider S ∈ APS2k+1 with 2k + 1 ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8], max(S) > 2|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We have max(S) < 2k + 1, so |S| < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consequently, S has a witness permutation w using at most k vales, so there are at least (2k + 1) − (k − 1 + k) = 2 non-pinnacle/non-vale values in this witness 11 permutation, and one of these is 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We can create a new permutation w′ by inserting 2k + 1 immediately to the right of the largest vale in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Thus the pinnacle set of w′ is S ∪ {2k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore there is a bijection between even-sized elements of APS2k+1 and odd-sized ones, obtained by adding/removing the element 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This partitions APS2k+1 into two evenly sized parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We have now completed all of the steps necessary to give the desired enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, ��APSB 2k+1 \\ APSD 2k+1 �� = �2k − 1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7, we can enumerate APSB 2k+1 \\ APSD 2k+1 by counting elements of APS2k+1 that have size {k − 2i : i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' These are either all of the odd-sized sets in APS2k+1 or all of the even-sized ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8, then, ��APSB 2k+1 \\ APSD 2k+1 �� = 1 2 |APS2k+1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It was shown in [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8] that |APS2k+1| = �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Finally, it is straightforward to check that 1 2 �2k k � = �2k−1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We can now use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12, which enumerated APSB n , and Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9 to enu- merate APSD n for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, ��APSD 2k �� = ��APSB 2k �� and ��APSD 2k+1 �� = � k � i=0 �2k + 1 i ��2k − i k − i �� − �2k − 1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Table 2, we give the number of signed admissible pinnacle sets in type D for 3 ≤ n ≤ 15, while permutations in SD 1 and SD 2 have no pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This appears in the OEIS as sequence A359067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The even-indexed terms are identical to even terms in Table 1 and the odd-indexed terms are �2k−1 k � less than the corresponding odd-indexed terms in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' n 3 4 5 6 7 8 9 10 11 12 13 14 15 ��APSD n �� 4 7 28 49 199 351 1436 2561 10499 18943 77617 141569 579149 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of admissible pinnacle sets in SD n , for 3 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Comparing admissible pinnacle sets in types B and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Some elements of APSB n have no negative values, and so one could ask if those sets might also be admissible in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In this section we consider how those elements of APSB n are related to the admissible pinnacle sets in APSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To make this discussion precise, we introduce: APS+ n := {S ∈ APSB n : S ⊂ N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' in other word, APS+ n consists of the pinnacle sets that are admissible in SB n and that contain no negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 12 For example, {1, 3} ∈ APS+ 5 , with canonical witness 51432 ∈ SB 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In fact, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2, any subset of [n] having at most (n − 1)/2 elements is admissible in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Contrast this with APSn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' for example, APS+ 5 \\ APS5 = � {1}, {2}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Our goal in this section is to understand APS+ n \\ APSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As with the comparison of APSB n and APSD n , this will depend on the parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 0, |APS+ 2k+1 \\ APS2k+1| = 4k − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because APS2k+1 ⊂ APS+ 2k+1, the desired value is equal to ��APS+ 2k+1 �� − |APS2k+1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2, we can compute ��APS+ 2k+1 �� by counting i-element subsets of [2k +1] for all i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The result follows by recognizing that this yields a sum that is half of a row-sum of Pascal’s triangle, and combining this with the enumeration of APS2k+1 from [3]: |APS+ 2k+1 \\ APS2k+1| = ��APS+ 2k+1 �� − |APS2k+1| = �2k + 1 0 � + �2k + 1 1 � + · · · + �2k + 1 k � − �2k k � = 1 222k+1 − �2k k � = 4k − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We now complete this analysis by considering the even-indexed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, ��APS+ 2k \\ APS2k �� = 22k−1 − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This calculation is almost identical to that from the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, except that we will also have to subtract the central term from a row of Pascal’s triangle: |APS+ 2k \\ APS2k| = ��APS+ 2k �� − |APS2k| = �2k 0 � + �2k 1 � + · · · + � 2k k − 1 � − �2k − 1 k − 1 � = 1 2 � 22k − �2k k �� − �2k − 1 k − 1 � = 22k−1 − �1 2 �2k k � + �2k − 1 k − 1 �� = 22k−1 − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We combine the enumerations of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12 in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Specifically, we list ��APS+ n \\ APSn �� for 3 ≤ n ≤ 15, while permutations in SB 1 and SB 2 have no pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The nth term of this appears in the OEIS as double the (n − 1)st term of [12, A294175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Moreover, the odd-indexed terms, enumerated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, appear in [12, A068551] and the even-indexed terms are double the terms of [12, A008549].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 13 n 3 4 5 6 7 8 9 10 11 12 13 14 15 ��APS+ n \\ APSn �� 2 2 10 12 44 58 186 260 772 1124 3172 4760 12952 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of all-positive pinnacle sets that are admissible in SB n but not in Sn, for 3 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Future directions As demonstrated by the results in this paper, admissible pinnacle sets have rich structure and properties even beyond the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' There are many directions for further research on this topic, including broad questions about pinnacle sets for families of permutations with certain properties, and enumerative specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As a complement to those large questions, we conclude this work by pointing out that we uncovered a possible connection between ��APSB n �� and sequence [12, A119258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, we have the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consider the sequence [12, A119258], given by T(n, 0) = T(n, n) = 1 and T(n, k) = 2T(n − 1, k − 1) + T(n − 1, k) for 0 < k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then ��APSB n �� = T � n, �n − 1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Data Patrek Ragnarsson’s code for computing the data in Tables 1, 2, and 3 can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='com/PatrekR/Signed-pinnacle-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Note that the data in Table 2 is the difference between the enumerations given in two of the files posted at this GitHub link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' References [1] Sara Billey, Krzysztof Burdzy, and Bruce E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sagan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Permutations with given peak set.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Peak sets of classical coxeter groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Involve, 10(2):263–290, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [8] Rachel Domagalski, Jinting Liang, Quinn Minnich, Bruce E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sagan, Jamie Schmidt, and Alexander Sietsema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacle set properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discrete Mathematics, 345(7):112882, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [9] Justine Falque, Jean-Christophe Novelli, and Jean-Yves Thibon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacle sets revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='05248, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [10] Wenjie Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Efficient recurrence for the enumeration of permutations with fixed pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The- oret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 24:#8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [11] Quinn Minnich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Further results on pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 346(4):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 113296, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [12] OEIS Foundation Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The On-Line Encyclopedia of Integer Sequences, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Published electronically at http://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 14 [13] Irena Rusu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sorting permutations with fixed pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 27:P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='23, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [14] Irena Rusu and Bridget Eileen Tenner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Admissible pinnacle orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Graphs and Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 37:1205–1214, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [15] Richard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Enumerative combinatorics, volume 49 of Cambridge Studies in Advanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Cambridge University Press, Cambridge, second edition edition, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [16] John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Stembridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Enriched p-partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Transactions of the American Mathematical Society, 349:763–788, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Gonz´alez) Department of Mathematics, University of California, Berkeley, CA, 94720 Email address: nicolle@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Harris) Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI 53211 Email address: peharris@uwm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Rojas Kirby) Department of Mathematics and Statistics, San Diego State University, CA 92182 Email address: gkirby@sdsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Smit Vega Garcia) Department of Mathematics, Western Washington University, Bellingham, WA 98225 Email address: smitvem@wwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Tenner) Department of Mathematical Sciences, DePaul University, Chicago, IL 60614 Email address: bridget@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='depaul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} diff --git a/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf b/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e63007d7de80dc7a16326517289d6262397f14e8 --- /dev/null +++ b/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:27cf40e87d295163cdd7eced9d9049c0cac673bfc6eaaf44be3e6113d23a4905 +size 6495181 diff --git a/8dE2T4oBgHgl3EQf8Ait/vector_store/index.pkl b/8dE2T4oBgHgl3EQf8Ait/vector_store/index.pkl new file mode 100644 index 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Baldwin, and Loren N. Pfeiffer +Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA +The Wigner crystal, an ordered array of electrons, is one of the very first proposed many-body +phases stabilized by the electron-electron interaction. This electron solid phase has been reported +in ultra-clean two-dimensional electron systems at extremely low temperatures, where the Coulomb +interaction dominants over the kinetic energy, disorder potential and thermal fluctuation. We closely +examine this quantum phase with capacitance measurements where the device length-scale is com- +parable with the crystal’s correlation length. The extraordinarily high performance of our technique +makes it possible to quantitatively study the dynamic response of the Wigner crystal within the +single crystal regime. Our result will greatly boost the study of this inscrutable electron solid. +Interacting two-dimensional electron system (2DES) +subjected to high perpendicular magnetic fields (B) and +cooled to low temperatures exhibits a plethora of exotic +states [1]. The Wigner crystal (WC) [2] terminates the +sequence of fractional quantum Hall states at very small +landau level filling factor [3–24]. This electron solid is +pinned by the ubiquitous residual disorder, manifests as +an insulating phase in DC transport [3–11], and the elec- +trons’ collective motion is evidenced by a resonance in +AC transport [12–19]. A series of experiments have been +applied to investigate this correlated solid, such as the +nonlinear I − V response [4, 16], the noise spectrum [5], +the huge dielectric constant [20], the weak screening effi- +ciency [21], the melting process [21–23], the nuclear mag- +netic resonance [24] and the optics [25, 26]. +Capacitance measurements have revealed a series of +quantum phenomena [21, 27–38]. In this work, we ex- +amine the WC formed in an ultra-high mobility 2DES +at ν <∼ 1/5 using high-precision capacitance measure- +ment [39, 40]. We find an exceedingly large capacitance +at low measurement frequency f while the conductance +is almost zero. +This phenomenon is inconsistent with +transporting electrons, but rather an evidence that the +synchronous vibration of electrons induces a polarization +current. When we increase f, our high-precision mea- +surement captures the fine structure of the resonance re- +sponse with a puzzling ”half-dome” structure. Our sys- +tematic, quantitative results provide an in-depth insight +of this murky quantum phase. +Our sample consists an ultra-clean low-density 2DES +confined in a 70-nm-wide GaAs quantum well with elec- +tron density n ≃ 4.4 × 1010 cm−2 and mobility µ ≃ 17 +×106 cm2/(V·s). Each device has a pair of front concen- +tric gates G1 and G2, whose outer and inner radius are +r1 and r2, respectively; see the inset of Fig. 1(a) [41]. We +study four devices with r1 =60 µm and r2 = 60, 80, 100 +and 140 µm, respectively. We measure the capacitance C +and conductance G between the two gates using a cryo- +genic bridge and analyze its output with a custom-made +radio-frequency lock-in amplifier [39–41]. +Fig. +1(a) shows the C and G measured from the +r1 = r2 = 60 µm sample. Both C and G decrease as +we increase the magnetic field B, owing to the mag- +netic localization where the 2DES conductance σ ∝ +(ne2τ)/m⋆(1+ω2 +cτ 2), m⋆, ωc and τ are the effective mass, +cyclotron frequency and transport scattering time of the +electrons, respectively [40]. The C and G are finite at +ν = 1/2 and 1/4 where the 2DES forms compressible +composite Fermion Fermi sea. When ν is an integer or +a certain fraction such as 1/3 and 1/5, the 2DES forms +incompressible quantum Hall liquids so that both C and +G vanish [42]. +In all the above cases, the current is carried by trans- +porting electrons, so that C has a positive dependence +on G, i.e. C ∝ G3/2, as shown in Fig. 1(b) [40]. Such +a correlation discontinues when the WC forms at very +low filling factors ν <∼ 1/5, see the blue shaded regions +of Fig. 1(a). The vanishing conductance G suggests that +the electrons are immovable, however, the surprisingly +large capacitance C evidences that the WC hosts a cur- +rent even surpassing the conducting Fermi sea at ν = 1/2 +and 1/4 at much lower magnetic field! The phase tran- +sition between the WC and the liquid states are clearly +evidenced by spikes in G (marked by solid circles in Fig. +1(a)) and sharp raises in C. A developing minimum is +seen in G at 1/5 < ν < 2/9 (marked by the up-arrow) +when C has a peak. This G minimum develops towards +zero and the C peak saturates when the solid phase is +stronger (see black traces in Fig. 3(a)). This is consistent +with the reentrant insulating phase [3–5, 16, 19, 43, 44]. +It is important to mention that the 2DES in our de- +vices is effectively “isolated” and we are merely transfer- +ring charges between different regions within one quan- +tum phase. Similar to the dielectric materials which also +have no transporting electrons, the collective motion of +all electrons, i.e. the k → 0 phonon mode of WC, can +generate polarization charges and corresponding polar- +ization current in response to the in-plane component +of applied electric field. +An infinitesimally small but +ubiquitous disorder pins the WC so that electrons can +arXiv:2301.01475v1 [cond-mat.mes-hall] 4 Jan 2023 + +2 +0 +2 +4 +6 +8 +10 +12 +14 +B (T) +0 +0.4 +C (pF) +1 +0 +G (µS) +ν=1 +1 +3 +1 +4 +1 +2 +1 +5 +2 +9 +0.0 +0.2 +C (pF) +0 +1 +G3/2 (µS3/2) +WC +Liquid +(b) +0.0 +0.2 +0.2 +0.1 +0.0 +ν(∝1/B) +C (pF) +(d) +f=7 MHz +T=30 mK +r2=100 µm +0.0 +0.2 +0.4 +ln(r2/r1) +8 +4 +0 +1/CWC (1/pF) +13.5 T +(e) + + +12.0 T +lB +(c) +a0 +x +CWC +2DES +E +G2 +G1 +<< +<< +Q ∝ e-d/ζ +d +0 +h +r1=r2=60 µm +f=7 MHz +T=30 mK +(a) +G2 +G1 +Al2O3 +2DES +r1 +r2 +FIG. 1. (color online) (a) C and G measured from the r1 = r2 = 60 µm sample with 7 MHz excitation at 30 mK. The horizontal +dashed lines represent the zeros of C or G. The blue shaded regions mark the presence of WC. Inset is the cartoon of our +device. (b) The correlation between C and G in panel (a) data. Transporting current dominates at B < 8 T where C ∝ G3/2, +indicated the red solid line. When the WC polarization current dominates, C ≃ 0.2 pF and G is about zero (the blue box). +(c) The schematic model describing the collective motion of electrons in the pinned WC. h is the depth of 2DES. The equally +spaced (by the lattice constant a0) vertical bars represent the equilibrium position of electrons. The gray-scaled solid circles +represent the electron position at finite external electric field E. The darker gray corresponds to larger electron displacement +x. The radius of individual electron is about the magnetic length lB. The accumulated charge Q is proportional to ∇ · x, +and decays exponentially as a function of the distance d from the gate boundary. ζ is the decay length. CWC is the effective +capacitance of WC in the un-gated region between the two gates. (d) C v.s. ν of the r2=100 µm sample. The black dashed +line is the zero of C. The red dashed line is the linear extension of data, showing that C = 0 at the extreme quantum limit +ν = 0. (e) 1/CWC v.s. ln(r2/r1) at two different magnetic field. +only be driven out of their equilibrium lattice site by +a small displacement x, as shown in Fig. +1(c). +Dur- +ing the experiments, we use excitation Vin ≃ 0.1 mVrms +and the measured WC capacitance is ∼ 0.15 pF at 13.5 +T. The polarization charge accumulated under the inner +gate is Q = CVin ∼ 100 e. +The corresponding elec- +tron displacement at the boundary of the inner gate, +|x(r1)| ≃ Q/(2πr1ne) ∼ 0.6 nm, is much smaller than +the magnetic length lB = +� +¯h/eB ∼ 8 nm, substanti- +ating our assumption that the electrons vibrate diminu- +tively around their equilibrium lattice sites. +An ideal, disorder-free WC is effectively a perfect di- +electric with infinite permittivity, so that the device ca- +pacitance should be close to its zero-field value C0 ∼ 1 pF +when 2DES is an excellent conductor. We note that C0 is +consistent with the device geometry, ϵ0ϵGaAsπr2 +1/h ≃ 1.3 +pF, where ϵGaAs = 12.8 is the relative dielectric constant +of GaAs and h ≃ 960 nm is the depth of 2DES. How- +ever, the measured C ∼ 0.15 pF in the WC regime is +much smaller than C0. This discrepancy is likely caused +by the friction-like disorder which poses a pinning force +≃ −βx on the electrons. When the crystal’s inversion +symmetry is broken, i.e. x is non-uniform and J (x) is +finite, the electron-electron interaction generates a restor- +ing force ≃ −a0µijJ (x), where µij, a0 and J (x) are the +elastic tensor, WC lattice constant and the Jacobi ma- +trix of x, respectively. At the low frequency limit, the +WC is always at equilibrium and all forces are balanced, +eE − a0µijJ (x) − βx = 0, E is the total parallel electric +field on the WC. +E is approximately zero under the metal gates, since +the gate-to-2DES distance h is small. Therefore, x de- +creases exponentially when the distance from the gate +boundary d increases, x ∝ exp(−d/ζ), where ζ = µa0/β +is the decay length. Deeply inside the gates, electrons +feel neither parallel electric field nor net pressure from +nearby electrons, so that their displacement x remains +approximately zero. This region does not contribute to +the capacitive response, and the effective gate area re- +duces to about 2πr1ζ and 2πr2ζ at the inner and outer +gate, respectively. Because r1 = r2 = 60 µm in Fig. 1(a), +the experimentally measured C ≈ ϵ0ϵGaAs/h · 2πr1ζ/2 ≃ +0.15 pF at 13.5 T corresponds to a decay length ζ ≃ 6.7 +µm. Interestingly, our result shows a linear dependence +C ∝ 1/B in Fig. 1(d), suggesting that β ∝ l−2 +B +if we +assume µij is independent on B. Especially, the pinning +becomes infinitely strong, i.e. β → ∞, at the extreme +quantum limit lB → 0. + +3 +The permittivity of a disorder-pinned WC is no longer +infinitely large, since a non-zero electric field E is neces- +sary to sustain a finite x. If we assume x is a constant +in the ring area between the two gates, so that eE = βx. +The residual E can be modeled as a serial capacitance +CWC ≈ 2πne2/β · [ln(r2/r1)]−1 in our device. We then +measure different devices with r1= 60 µm and r2 = 60, +80, 100 and 140 µm, and calculate the corresponding +CWC through C−1 +WC = C−1 −(r1 +r2)/r2 ·C−1 +r1=r2, see Fig. +1(e). By fitting the linear dependence C−1 +WC ∝ ln(r2/r1), +we estimate the pinning strength β to be about 1.3 ×10−9 +and 1.1 ×10−9 N/m at B = 13.5 and 12 T, respectively +[45]. +Finally, assuming µij ≈ µ · δij, we can estimate +the WC elastic modulus µ ≈ β · ζ/a0. For example, µ is +about 1.6 × 10−7 N/m at 13.5 T. +0.14 +0.16 +0.18 +0.20 +0.22 +0 +0.2 +C (pF) +2 +0 +G (μS) +ν +30 mK +95 mK +110 mK +125 mK +145 mK +200 mK +1/5 +2/11 +1/7 +(a) +0 +100 +200 +T (mK) +0 +0.2 +C (pF) +ν=0.18 +ν=0.14 +2 +0 +G (μS) +0.14 +0.20 +ν +0 +200 +TC (mK) +FQH liquid +WC +Compressible +liquid +(b) +TC +TC +r2=80 μm f=17 MHz +FIG. 2. (color online) (a) C and G vs. ν measured at vari- +ous temperatures from the r2 = 80 µm sample with 17 MHz +excitation. (b) Summarized C and G vs. T at ν = 0.14 and +0.18 from the panel (a) data. A critical temperature Tc at +certain ν is defined either as the temperature when G has +a peak at ν in panel (a) or as the temperature when G vs. +T trace reaches maximum in panel (b); marked by the black +and red arrows. The panel (b) inset summarizes the Tc us- +ing the two equivalent definitions using black and red circles, +respectively. The diagram can be separated into three differ- +ent regions corresponding to the WC, the fractional quantum +Hall (FQH) liquid and the compressible liquid. +Fig. 2 reveals an intriguing temperature-induced solid- +liquid phase transition when the WC melts. Fig. 2(a) +shows C and G taken from the r2 = 80 µm sample at +various temperatures. At a certain temperature, e.g. at +T ≈ 110 mK, C ∼ 0.2 pF when the 2DES forms WC +at ν <∼ 0.16 and vanishes when it is a liquid phase at +ν >∼ 0.18. G has a peak at ν ≃ 0.175 when C vs. ν +has the maximal negative slope, and it is small when the +2DES is either a WC at ν < 0.17 or a liquid at ν > 0.19 +[46]. At very high temperature T >∼ 200 mK, both C and +G are close to zero. In Fig. 2(b), we summarized C and +G as a function of T at two different filling factors to bet- +ter illustrate this solid-liquid transition. At ν ≃ 0.14, for +example, C is large and G is small at T <∼ 100 mK when +the WC is stable [47], while both of them become small +at T >∼ 200 mK when the 2DES is a liquid. The G has +a peak at a critical temperature TC, marked by the red +arrows, around which the precipitous decrease of C hap- +pens. Alternatively, TC at a certain filling factor ν can be +defined as the temperature when the G has a peak (black +arrow in Fig. 2(a)) at ν. We summarize TC obtained us- +ing these two equivalent procedures in the Fig. 2(b) inset +with corresponding red and black symbols. TC has a lin- +ear dependence on ν whose two intercepts are TC ≃ 340 +mK at the extreme quantum limit ν = 0, and ν ≃ 1/4 at +TC = 0 mK. +The Fig. +2(b) evolution can be qualitatively under- +stood by the coexistence of transport and polarization +currents at the solid-liquid transition. The large C re- +duces to almost zero when the transport current domi- +nates over the polarization current. G is a measure of +the 2DES’s capacity to absorb and dissipate power. It is +negligible if either of these two currents dominates, since +the polarization current is dissipation-less and the dissi- +pating transport current is difficult to excite. G becomes +large when these two currents coexist nip and tuck at +intermediate T when the excited polarization charge can +be just dissipated by the transport current. +The WC exhibits a resonance when we increase the +excitation frequency. In Fig. 3(a), the C and G measured +from the r2 = 100 µm sample using different excitation +frequencies change enormously when the WC presents +(blue shaded region). G is almost zero and C is large +at f ≃ 7 MHz, and G becomes finite and C becomes +even larger at f ≃ 23 MHz. At slightly higher frequency +27 MHz, G reaches its maximum and C drops to about +zero. Further increasing f, G gradually declines while +C first becomes negative at 35 MHz and then gradually +approaches zero. +The summarized C and G vs. +f at +two certain fillings in Fig. 3(b), resembles qualitatively +a resonant behavior with resonance frequency fr ≃ 26 +MHz (when C = 0). Fig. 3(c) studies this resonance +at different temperatures. The data is taken from the +r2 ≃ 80 µm sample whose resonance frequency is about +35 MHz [48]. The abrupt change of C near fr becomes +gradual and the G peak flattens at higher temperatures. + +4 +10 +0 +G (μS) +0.14 +0.16 +0.18 +0.20 +0.22 +0.24 +ν +0.4 +0 +C (pF) + 7 +23 +27 +35 +77 +r2=100 μm +0 +0.2 +-0.2 +ν=0.14 +30 +60 +140 +280 +0 +0.2 +10 +0 +10 +100 +f (MHz) +r2=80 μm +10 +0 +10 +100 +f (MHz) +ν=0.14 +ν=0.213 +r2=100 μm +T=30 mK +T=30 mK +(b) +(a) +(c) +fr=35 MHz +fr=26 MHz +ν=0.213 + f (MHz) + T (mK) +FIG. 3. (color online) (a) C and G vs. ν taken from the r2=100 µm sample using different excitation frequencies f. We see +a violent change of C and G at different f in the blue region where the WC appears. (b) The C and G vs. f extracted from +the panel (a) trace at ν = 0.14 and 0.213. The resonance frequency fr, defined as the frequency when C changes its sign, is +about 26 MHz. (c) The C and G vs. f at ν = 0.14 and different temperatures, data taken from the r2=80 µm sample. The +resonance disappears at T ≃ 280 mK when C and G remain nearly zero. +Both C and G become flat zero at T >∼ 280 mK. It is +noteworthy that, as long as a resonance is seen, fr is +nearly independent on the filling factor (Fig. 3(b)) and +temperatures (Fig. 3(c)). This is consistent with another +experimental study using surface acoustic wave [23]. +The resonance of WC is usually explained by the +pinning mode [18, 49]. +The resonance frequency is +related to the mean free path LT of the transverse +phonon through LT = (2πµt,cl/neBfr)1/2, where µt,cl = +0.245e2n3/2/4πϵ0ϵGaAs is the classical shear modulus of +WC. fr = 26 MHz corresponds to LT ≃ 3.2 µm, very +similar to ζ ≃ 6.7 µm in our Fig. +1(c) discussion. +This is justifiable because both LT and ζ describe the +length-scale within which the collective motion of WC is +damped/scattered by the random pinning potential. +Before ending the discussion, we would like to highlight +the puzzling ”half-dome” structure of the resonance. G +has a regular-shaped resonance peak, i.e. +G decreases +gradually on both sides of fr, when either the WC is weak +( ν ≃ 0.213 in Fig. 3(b)) or the temperature is high (T ≃ +140 mK in Fig. 3(c)). Surprisingly, the resonance peak +becomes quite peculiar when the WC is strong at ν ≃ +0.14 and T ≃ 30 mK. G gradually decreases from its peak +at fr on the high frequency side f > fr, while it vanishes +instantly when the frequency is lower than fr, resulting in +a ”half-dome” G vs. f trace. Meanwhile, the C increases +by ∼ 2 times and then abruptly changes to negative at +fr. This anomalous ”half-dome” feature is seen in all of +our devices as long as the WC is strong and temperature +is sufficiently low, suggesting a threshold frequency for +the power dissipation. +In conclusion, using the extraordinarily high-precision +capacitance measurement technique, we investigate the +dynamic response of WC systematically. From the quan- +titative results and using a simple model, we can study +several physical properties of the WC such as elastic mod- +ulus, dielectric constant, pinning strength, etc., and dis- +cover a puzzling ”half-dome” feature in the resonance +peak. Our results certainly shine light on the study of +WC and provides new insight on its dynamics. +We acknowledge support by the National Nature Sci- +ence Foundation of China (Grant No. +92065104 and +12074010) and the National Basic Research Program of +China (Grant No. 2019YFA0308403) for sample fabrica- +tion and measurement. This research is funded in part by +the Gordon and Betty Moore Foundation’s EPiQS Initia- +tive, Grant GBMF9615 to L. N. Pfeiffer, and by the Na- +tional Science Foundation MRSEC grant DMR 2011750 +to Princeton University. We thank L. W. Engel, Bo Yang +and Xin Lin for valuable discussion. +∗ liuyang02@pku.edu.cn +[1] J. K. Jain, Composite Fermions (Cambridge University +Press, Cambridge, UK, 2007). +[2] E. 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Liu, Chinese Physics Letters 39, 097301 +(2022). +[41] See Supplemental Material for detailed description of our +sample information and measurement techniques. +[42] The zero of C and G can be defined either by extrapolat- +ing their field dependence to B = ∞, or by their values +at strong quantum hall states such as ν = 1. These two +approaches are consistent with each other and the dash +lines in Fig. 1(a) represent the deduced zero. +[43] M. Shayegan, in High Magnetic Fields: Science and Tech- +nology, Vol. 3, edited by F. Herlach and N. Miura (World +Scientific, Singapore, 2006) pp. 31–60. +[44] M. Shayegan, in Perspectives in Quantum Hall Effects, +edited by S. D. Sarma and A. Pinczuk (Wiley, New York, +1998) pp. 343–383. +[45] Alternatively, CWC can be modeled as a cylinder ca- +pacitor whose height equals the effective thickness of +the 2DES, Z0 ≈ 45 nm. The WC dielectric constant is +ϵWC = (2πϵ0Z0∂(C−1 +WC)/∂ ln(r2/r1))−1 ≈ 2 × 104 at 13.5 +T, consistent with previous reported value in ref. [20]. +[46] We observe developing minimum at ν = 1/7, 2/11 dur- +ing the solid-liquid phase transition, signaling that the +fractional quantum Hall state emerges [8, 11]. +[47] C vs. T has a slightly positive slope in the WC region, +possibly due to the softening of disorder pinning. +[48] fr has no obvious dependence with sample geometry, +which is about 35, 35, 26 and 29 MHz for samples with +r2 = 60, 80, 100, 140 µm, respectively. +[49] M. M. Fogler and D. A. Huse, Phys. Rev. B 62, 7553 +(2000). + +6 +SUPPLEMENTARY MATERIALS +Samples +The sample we studied is made from a GaAs/AlGaAs +heterostructure wafer grown by molecular beam epitaxy. +A 70 nm-wide GaAs quantum well is bound by AlGaAs +spacer-layers and δ-doped layers on each side, and locates +h ≃ 960 nm below the sample surface. The as-grown den- +sity of the 2DES is n ≃ 4.4×1010 cm−2, and its mobility +at 300 mK is µ ≃ 17 ×106 cm2/(V·s). Our sample is a +2 mm × 2 mm square piece with four In/Sn contacts at +each corner. The contacts are grounded through a re- +sistor to avoid signal leaking. We evaporate concentric, +Au/Ti front gate pair G1 and G2 using standard lift- +off process, whose outer and inner radius is r1 and r2, +respectively. We deposit a 20 nm thick Al2O3 layer be- +tween the two gates to prevent them from shorting with +each other. +The four outer-gates are merged into one +piece so that the area of the outer gate G2 is much larger +than the inner gate G1. +Capacitance Measurement Setup +The capacitance and conductance response is mea- +sured with a cryogenic bridge similar to refs. [39, 40]. +The kernel of the bridge consists four devices, Rh, Rr, +Cr and C, as shown in Fig. S1(a). C is the capacitance of +sample. We change the value of Rh to reach the balance +condition +C +Cr += Rh +Rr +. +(1) +The bridge output Vout is minimum at the balance con- +dition, from which we calculate the C. This is the so-call +“V-curve” procedure, see refs. [39, 40] for more informa- +tion. +In order to expand the allowed bandwidth of the ex- +citation frequency, we add an impedance match network +to the input of the bridge, shown as the Fig. S1(a). Vext +is the signal source with 50 Ω output impedance. Vext +drives a signal splitter box (the red dashed box) located +at the top of the dilution refrigerator through a ∼2 m- +long semi-rigid coaxial cable. +The box input is a 1:5 +transformer in series with a 50 Ω resistor. The trans- +former output drives two serial connected 50 Ω resistors +differentially. The differential signals are transmitted to +the cryogenic sample holder (the blue dotted box) by +two rigid coaxial cables of ∼2 m length. Another pair +of impedance matching 50 Ω resistors are added at the +input of the cryogenic bridge, and the 360 Ω resistors are +chosen by balancing the competition between the perfor- +mance and heating. The characteristic impedance of all +coaxial cables in the work is 50 Ω. +The low-frequency signals Vquasi-DC1 and Vquasi-DC2 +used to measure the value of Rh and Rr, respectively. The +0.1 µF capacitors are used to separate the high-frequency +excitation signals and the quasi-DC signal. +The output Vout is approximately +Vout ∝ S · ( +Rh +360 + Rh +− C +Cr +· +Rr +360 + Rr +) · Vext. +(2) +S can be obtain from the “V-curve” procedure by linear +fitting the VX vs. Rh/(360+Rh), as shown in Fig. S1(b). +VX and VY are the orthogonal component of Vout, +� VX = |Vout| · cos(θ), +(3) +VY = |Vout| · sin(θ), +(4) +where θ is the phase of Vout. We can derive the value of +C using Eq. (2) and (3). The new balance condition of +the revised bridge is +C +Cr += Rh +Rr +· 360 + Rr +360 + Rh +, +(5) +where the VX = 0. +Note that the capacitance C and the conductance G +of sample lead to the orthogonal component VX and VY, +respectively. Therefore, the G can be obtained from Eq. +(2) and (4) by replacing C/Cr with G/2πfCr, where f is +the excitation frequency. +Fig. S1(c) shows our calibration measurement using +different excitation frequencies. The data is almost flat +from 7 to ∼100 MHz. The measured capacitance begins +to decline slowly above ∼100 MHz, possibly due to the +parasitic inductance of bonding wires. + +7 +Rh +Rr +Cr +C +Vin ++ +Vin +- +Vout +360 Ω +360 Ω +50 Ω +50 Ω +1:5 +50 Ω +50 Ω +50 Ω +0.1 μF +0.1 μF +0.1 μF +0.1 μF +Vext +Vquasi-DC1 +Vquasi-DC2 +COAX +COAX +(a) + + +40 +-40 +0 +V (μV) +0.0 +0.2 +0.4 +0.6 +Rh/(Rh+360) +Vx +Vy +(b) +Cr= 0.1 pF +f= 7 MHz +Rr= 50 Ω +0.6 +0.0 +0.4 +0.2 +C (pF) +10 +100 +f (MHz) +0.5 pF +0.3 pF +0.1 pF +(c) +COAX +FIG. S1. (color online) (a) Circuit diagram of measurement bridge with 50 Ω impedance match networks. (b) The VX and VY +from a typical “V-curve” procedure. C is about 0.25 pF from the balance condition Eq. (5). (c) The calibration results, by +measuring commercial capacitors with different frequencies. + diff --git a/9dAzT4oBgHgl3EQfgvzi/content/tmp_files/load_file.txt b/9dAzT4oBgHgl3EQfgvzi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fad9f65ad693feced68360f9991eaf28507d81f7 --- /dev/null +++ b/9dAzT4oBgHgl3EQfgvzi/content/tmp_files/load_file.txt @@ -0,0 +1,853 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf,len=852 +page_content='Dynamic Response of Wigner Crystals Lili Zhao, Wenlu Lin, and Yang Liu∗ International Center for Quantum Materials, Peking University, Haidian, Beijing 100871, China Yoon Jang Chung, Adbhut Gupta, Kirk W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Baldwin, and Loren N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Pfeiffer Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA The Wigner crystal, an ordered array of electrons, is one of the very first proposed many-body phases stabilized by the electron-electron interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This electron solid phase has been reported in ultra-clean two-dimensional electron systems at extremely low temperatures, where the Coulomb interaction dominants over the kinetic energy, disorder potential and thermal fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We closely examine this quantum phase with capacitance measurements where the device length-scale is com- parable with the crystal’s correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The extraordinarily high performance of our technique makes it possible to quantitatively study the dynamic response of the Wigner crystal within the single crystal regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Our result will greatly boost the study of this inscrutable electron solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Interacting two-dimensional electron system (2DES) subjected to high perpendicular magnetic fields (B) and cooled to low temperatures exhibits a plethora of exotic states [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The Wigner crystal (WC) [2] terminates the sequence of fractional quantum Hall states at very small landau level filling factor [3–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This electron solid is pinned by the ubiquitous residual disorder, manifests as an insulating phase in DC transport [3–11], and the elec- trons’ collective motion is evidenced by a resonance in AC transport [12–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' A series of experiments have been applied to investigate this correlated solid, such as the nonlinear I − V response [4, 16], the noise spectrum [5], the huge dielectric constant [20], the weak screening effi- ciency [21], the melting process [21–23], the nuclear mag- netic resonance [24] and the optics [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Capacitance measurements have revealed a series of quantum phenomena [21, 27–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' In this work, we ex- amine the WC formed in an ultra-high mobility 2DES at ν <∼ 1/5 using high-precision capacitance measure- ment [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We find an exceedingly large capacitance at low measurement frequency f while the conductance is almost zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This phenomenon is inconsistent with transporting electrons, but rather an evidence that the synchronous vibration of electrons induces a polarization current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' When we increase f, our high-precision mea- surement captures the fine structure of the resonance re- sponse with a puzzling ”half-dome” structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Our sys- tematic, quantitative results provide an in-depth insight of this murky quantum phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Our sample consists an ultra-clean low-density 2DES confined in a 70-nm-wide GaAs quantum well with elec- tron density n ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4 × 1010 cm−2 and mobility µ ≃ 17 ×106 cm2/(V·s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Each device has a pair of front concen- tric gates G1 and G2, whose outer and inner radius are r1 and r2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' see the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(a) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We study four devices with r1 =60 µm and r2 = 60, 80, 100 and 140 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We measure the capacitance C and conductance G between the two gates using a cryo- genic bridge and analyze its output with a custom-made radio-frequency lock-in amplifier [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(a) shows the C and G measured from the r1 = r2 = 60 µm sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Both C and G decrease as we increase the magnetic field B, owing to the mag- netic localization where the 2DES conductance σ ∝ (ne2τ)/m⋆(1+ω2 cτ 2), m⋆, ωc and τ are the effective mass, cyclotron frequency and transport scattering time of the electrons, respectively [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The C and G are finite at ν = 1/2 and 1/4 where the 2DES forms compressible composite Fermion Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' When ν is an integer or a certain fraction such as 1/3 and 1/5, the 2DES forms incompressible quantum Hall liquids so that both C and G vanish [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' In all the above cases, the current is carried by trans- porting electrons, so that C has a positive dependence on G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' C ∝ G3/2, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(b) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Such a correlation discontinues when the WC forms at very low filling factors ν <∼ 1/5, see the blue shaded regions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The vanishing conductance G suggests that the electrons are immovable, however, the surprisingly large capacitance C evidences that the WC hosts a cur- rent even surpassing the conducting Fermi sea at ν = 1/2 and 1/4 at much lower magnetic field!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The phase tran- sition between the WC and the liquid states are clearly evidenced by spikes in G (marked by solid circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(a)) and sharp raises in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' A developing minimum is seen in G at 1/5 < ν < 2/9 (marked by the up-arrow) when C has a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This G minimum develops towards zero and the C peak saturates when the solid phase is stronger (see black traces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This is consistent with the reentrant insulating phase [3–5, 16, 19, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' It is important to mention that the 2DES in our de- vices is effectively “isolated” and we are merely transfer- ring charges between different regions within one quan- tum phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Similar to the dielectric materials which also have no transporting electrons, the collective motion of all electrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' the k → 0 phonon mode of WC, can generate polarization charges and corresponding polar- ization current in response to the in-plane component of applied electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' An infinitesimally small but ubiquitous disorder pins the WC so that electrons can arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='01475v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='mes-hall] 4 Jan 2023 2 0 2 4 6 8 10 12 14 B (T) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4 C (pF) 1 0 G (µS) ν=1 1 3 1 4 1 2 1 5 2 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 C (pF) 0 1 G3/2 (µS3/2) WC Liquid (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 ν(∝1/B) C (pF) (d) f=7 MHz T=30 mK r2=100 µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4 ln(r2/r1) 8 4 0 1/CWC (1/pF) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 T (e) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 T lB (c) a0 x CWC 2DES E G2 G1 << << Q ∝ e-d/ζ d 0 h r1=r2=60 µm f=7 MHz T=30 mK (a) G2 G1 Al2O3 2DES r1 r2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (color online) (a) C and G measured from the r1 = r2 = 60 µm sample with 7 MHz excitation at 30 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The horizontal dashed lines represent the zeros of C or G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The blue shaded regions mark the presence of WC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Inset is the cartoon of our device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (b) The correlation between C and G in panel (a) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Transporting current dominates at B < 8 T where C ∝ G3/2, indicated the red solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' When the WC polarization current dominates, C ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 pF and G is about zero (the blue box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (c) The schematic model describing the collective motion of electrons in the pinned WC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' h is the depth of 2DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The equally spaced (by the lattice constant a0) vertical bars represent the equilibrium position of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The gray-scaled solid circles represent the electron position at finite external electric field E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The darker gray corresponds to larger electron displacement x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The radius of individual electron is about the magnetic length lB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The accumulated charge Q is proportional to ∇ · x, and decays exponentially as a function of the distance d from the gate boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ζ is the decay length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' CWC is the effective capacitance of WC in the un-gated region between the two gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (d) C v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ν of the r2=100 µm sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The black dashed line is the zero of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The red dashed line is the linear extension of data, showing that C = 0 at the extreme quantum limit ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (e) 1/CWC v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ln(r2/r1) at two different magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' only be driven out of their equilibrium lattice site by a small displacement x, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Dur- ing the experiments, we use excitation Vin ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 mVrms and the measured WC capacitance is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='15 pF at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The polarization charge accumulated under the inner gate is Q = CVin ∼ 100 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The corresponding elec- tron displacement at the boundary of the inner gate, |x(r1)| ≃ Q/(2πr1ne) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='6 nm, is much smaller than the magnetic length lB = � ¯h/eB ∼ 8 nm, substanti- ating our assumption that the electrons vibrate diminu- tively around their equilibrium lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' An ideal, disorder-free WC is effectively a perfect di- electric with infinite permittivity, so that the device ca- pacitance should be close to its zero-field value C0 ∼ 1 pF when 2DES is an excellent conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We note that C0 is consistent with the device geometry, ϵ0ϵGaAsπr2 1/h ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='3 pF, where ϵGaAs = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='8 is the relative dielectric constant of GaAs and h ≃ 960 nm is the depth of 2DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' How- ever, the measured C ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='15 pF in the WC regime is much smaller than C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This discrepancy is likely caused by the friction-like disorder which poses a pinning force ≃ −βx on the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' When the crystal’s inversion symmetry is broken, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' x is non-uniform and J (x) is finite, the electron-electron interaction generates a restor- ing force ≃ −a0µijJ (x), where µij, a0 and J (x) are the elastic tensor, WC lattice constant and the Jacobi ma- trix of x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' At the low frequency limit, the WC is always at equilibrium and all forces are balanced, eE − a0µijJ (x) − βx = 0, E is the total parallel electric field on the WC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' E is approximately zero under the metal gates, since the gate-to-2DES distance h is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Therefore, x de- creases exponentially when the distance from the gate boundary d increases, x ∝ exp(−d/ζ), where ζ = µa0/β is the decay length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Deeply inside the gates, electrons feel neither parallel electric field nor net pressure from nearby electrons, so that their displacement x remains approximately zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This region does not contribute to the capacitive response, and the effective gate area re- duces to about 2πr1ζ and 2πr2ζ at the inner and outer gate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Because r1 = r2 = 60 µm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(a), the experimentally measured C ≈ ϵ0ϵGaAs/h · 2πr1ζ/2 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='15 pF at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 T corresponds to a decay length ζ ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Interestingly, our result shows a linear dependence C ∝ 1/B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(d), suggesting that β ∝ l−2 B if we assume µij is independent on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Especially, the pinning becomes infinitely strong, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' β → ∞, at the extreme quantum limit lB → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3 The permittivity of a disorder-pinned WC is no longer infinitely large, since a non-zero electric field E is neces- sary to sustain a finite x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' If we assume x is a constant in the ring area between the two gates, so that eE = βx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The residual E can be modeled as a serial capacitance CWC ≈ 2πne2/β · [ln(r2/r1)]−1 in our device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We then measure different devices with r1= 60 µm and r2 = 60, 80, 100 and 140 µm, and calculate the corresponding CWC through C−1 WC = C−1 −(r1 +r2)/r2 ·C−1 r1=r2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' By fitting the linear dependence C−1 WC ∝ ln(r2/r1), we estimate the pinning strength β to be about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='3 ×10−9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 ×10−9 N/m at B = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 and 12 T, respectively [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Finally, assuming µij ≈ µ · δij, we can estimate the WC elastic modulus µ ≈ β · ζ/a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' For example, µ is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='6 × 10−7 N/m at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='22 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 C (pF) 2 0 G (μS) ν 30 mK 95 mK 110 mK 125 mK 145 mK 200 mK 1/5 2/11 1/7 (a) 0 100 200 T (mK) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 C (pF) ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='18 ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 2 0 G (μS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='20 ν 0 200 TC (mK) FQH liquid WC Compressible liquid (b) TC TC r2=80 μm f=17 MHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (color online) (a) C and G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ν measured at vari- ous temperatures from the r2 = 80 µm sample with 17 MHz excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (b) Summarized C and G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' T at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='18 from the panel (a) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' A critical temperature Tc at certain ν is defined either as the temperature when G has a peak at ν in panel (a) or as the temperature when G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' T trace reaches maximum in panel (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' marked by the black and red arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The panel (b) inset summarizes the Tc us- ing the two equivalent definitions using black and red circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The diagram can be separated into three differ- ent regions corresponding to the WC, the fractional quantum Hall (FQH) liquid and the compressible liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2 reveals an intriguing temperature-induced solid- liquid phase transition when the WC melts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2(a) shows C and G taken from the r2 = 80 µm sample at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' At a certain temperature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' at T ≈ 110 mK, C ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 pF when the 2DES forms WC at ν <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='16 and vanishes when it is a liquid phase at ν >∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G has a peak at ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='175 when C vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ν has the maximal negative slope, and it is small when the 2DES is either a WC at ν < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='17 or a liquid at ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='19 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' At very high temperature T >∼ 200 mK, both C and G are close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2(b), we summarized C and G as a function of T at two different filling factors to bet- ter illustrate this solid-liquid transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' At ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14, for example, C is large and G is small at T <∼ 100 mK when the WC is stable [47], while both of them become small at T >∼ 200 mK when the 2DES is a liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The G has a peak at a critical temperature TC, marked by the red arrows, around which the precipitous decrease of C hap- pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Alternatively, TC at a certain filling factor ν can be defined as the temperature when the G has a peak (black arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2(a)) at ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We summarize TC obtained us- ing these two equivalent procedures in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2(b) inset with corresponding red and black symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' TC has a lin- ear dependence on ν whose two intercepts are TC ≃ 340 mK at the extreme quantum limit ν = 0, and ν ≃ 1/4 at TC = 0 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2(b) evolution can be qualitatively under- stood by the coexistence of transport and polarization currents at the solid-liquid transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The large C re- duces to almost zero when the transport current domi- nates over the polarization current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G is a measure of the 2DES’s capacity to absorb and dissipate power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' It is negligible if either of these two currents dominates, since the polarization current is dissipation-less and the dissi- pating transport current is difficult to excite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G becomes large when these two currents coexist nip and tuck at intermediate T when the excited polarization charge can be just dissipated by the transport current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The WC exhibits a resonance when we increase the excitation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(a), the C and G measured from the r2 = 100 µm sample using different excitation frequencies change enormously when the WC presents (blue shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G is almost zero and C is large at f ≃ 7 MHz, and G becomes finite and C becomes even larger at f ≃ 23 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' At slightly higher frequency 27 MHz, G reaches its maximum and C drops to about zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Further increasing f, G gradually declines while C first becomes negative at 35 MHz and then gradually approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The summarized C and G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' f at two certain fillings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(b), resembles qualitatively a resonant behavior with resonance frequency fr ≃ 26 MHz (when C = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(c) studies this resonance at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The data is taken from the r2 ≃ 80 µm sample whose resonance frequency is about 35 MHz [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The abrupt change of C near fr becomes gradual and the G peak flattens at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 4 10 0 G (μS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='24 ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4 0 C (pF) 7 23 27 35 77 r2=100 μm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 30 60 140 280 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 10 0 10 100 f (MHz) r2=80 μm 10 0 10 100 f (MHz) ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='213 r2=100 μm T=30 mK T=30 mK (b) (a) (c) fr=35 MHz fr=26 MHz ν=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='213 f (MHz) T (mK) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (color online) (a) C and G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ν taken from the r2=100 µm sample using different excitation frequencies f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We see a violent change of C and G at different f in the blue region where the WC appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (b) The C and G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' f extracted from the panel (a) trace at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The resonance frequency fr, defined as the frequency when C changes its sign, is about 26 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (c) The C and G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' f at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 and different temperatures, data taken from the r2=80 µm sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The resonance disappears at T ≃ 280 mK when C and G remain nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Both C and G become flat zero at T >∼ 280 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' It is noteworthy that, as long as a resonance is seen, fr is nearly independent on the filling factor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(b)) and temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This is consistent with another experimental study using surface acoustic wave [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The resonance of WC is usually explained by the pinning mode [18, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The resonance frequency is related to the mean free path LT of the transverse phonon through LT = (2πµt,cl/neBfr)1/2, where µt,cl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='245e2n3/2/4πϵ0ϵGaAs is the classical shear modulus of WC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' fr = 26 MHz corresponds to LT ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 µm, very similar to ζ ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='7 µm in our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(c) discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This is justifiable because both LT and ζ describe the length-scale within which the collective motion of WC is damped/scattered by the random pinning potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Before ending the discussion, we would like to highlight the puzzling ”half-dome” structure of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G has a regular-shaped resonance peak, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G decreases gradually on both sides of fr, when either the WC is weak ( ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='213 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(b)) or the temperature is high (T ≃ 140 mK in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Surprisingly, the resonance peak becomes quite peculiar when the WC is strong at ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='14 and T ≃ 30 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' G gradually decreases from its peak at fr on the high frequency side f > fr, while it vanishes instantly when the frequency is lower than fr, resulting in a ”half-dome” G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' f trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Meanwhile, the C increases by ∼ 2 times and then abruptly changes to negative at fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This anomalous ”half-dome” feature is seen in all of our devices as long as the WC is strong and temperature is sufficiently low, suggesting a threshold frequency for the power dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' In conclusion, using the extraordinarily high-precision capacitance measurement technique, we investigate the dynamic response of WC systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' From the quan- titative results and using a simple model, we can study several physical properties of the WC such as elastic mod- ulus, dielectric constant, pinning strength, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=', and dis- cover a puzzling ”half-dome” feature in the resonance peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Our results certainly shine light on the study of WC and provides new insight on its dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We acknowledge support by the National Nature Sci- ence Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 92065104 and 12074010) and the National Basic Research Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 2019YFA0308403) for sample fabrica- tion and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This research is funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initia- tive, Grant GBMF9615 to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Pfeiffer, and by the Na- tional Science Foundation MRSEC grant DMR 2011750 to Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Engel, Bo Yang and Xin Lin for valuable discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' ∗ liuyang02@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='cn [1] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 67, 1630 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Santos, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Suen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Shayegan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Engel, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Tsui, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 68, 1188 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Sajoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Engel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Tsui, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Shayegan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Rev.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Tsui, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Baldwin, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' West, Phys.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 128, 026802 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lozovik and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Yudson, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 22, 11 (1975).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Williams, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Paris, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Etienne, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lett.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Liu, Review of Scientific Instruments 93, 053910 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [40] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Chung, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Baldwin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Pfeiffer, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Liu, Chinese Physics Letters 39, 097301 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [41] See Supplemental Material for detailed description of our sample information and measurement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [42] The zero of C and G can be defined either by extrapolat- ing their field dependence to B = ∞, or by their values at strong quantum hall states such as ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' These two approaches are consistent with each other and the dash lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 1(a) represent the deduced zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Shayegan, in High Magnetic Fields: Science and Tech- nology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 3, edited by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Herlach and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Miura (World Scientific, Singapore, 2006) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 31–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Shayegan, in Perspectives in Quantum Hall Effects, edited by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Sarma and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Pinczuk (Wiley, New York, 1998) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 343–383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [45] Alternatively, CWC can be modeled as a cylinder ca- pacitor whose height equals the effective thickness of the 2DES, Z0 ≈ 45 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The WC dielectric constant is ϵWC = (2πϵ0Z0∂(C−1 WC)/∂ ln(r2/r1))−1 ≈ 2 × 104 at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 T, consistent with previous reported value in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [46] We observe developing minimum at ν = 1/7, 2/11 dur- ing the solid-liquid phase transition, signaling that the fractional quantum Hall state emerges [8, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [47] C vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' T has a slightly positive slope in the WC region, possibly due to the softening of disorder pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [48] fr has no obvious dependence with sample geometry, which is about 35, 35, 26 and 29 MHz for samples with r2 = 60, 80, 100, 140 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fogler and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Huse, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' B 62, 7553 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 6 SUPPLEMENTARY MATERIALS Samples The sample we studied is made from a GaAs/AlGaAs heterostructure wafer grown by molecular beam epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' A 70 nm-wide GaAs quantum well is bound by AlGaAs spacer-layers and δ-doped layers on each side, and locates h ≃ 960 nm below the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The as-grown den- sity of the 2DES is n ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4×1010 cm−2, and its mobility at 300 mK is µ ≃ 17 ×106 cm2/(V·s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Our sample is a 2 mm × 2 mm square piece with four In/Sn contacts at each corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The contacts are grounded through a re- sistor to avoid signal leaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We evaporate concentric, Au/Ti front gate pair G1 and G2 using standard lift- off process, whose outer and inner radius is r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We deposit a 20 nm thick Al2O3 layer be- tween the two gates to prevent them from shorting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The four outer-gates are merged into one piece so that the area of the outer gate G2 is much larger than the inner gate G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Capacitance Measurement Setup The capacitance and conductance response is mea- sured with a cryogenic bridge similar to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The kernel of the bridge consists four devices, Rh, Rr, Cr and C, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' C is the capacitance of sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We change the value of Rh to reach the balance condition C Cr = Rh Rr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (1) The bridge output Vout is minimum at the balance con- dition, from which we calculate the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' This is the so-call “V-curve” procedure, see refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' [39, 40] for more informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' In order to expand the allowed bandwidth of the ex- citation frequency, we add an impedance match network to the input of the bridge, shown as the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Vext is the signal source with 50 Ω output impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Vext drives a signal splitter box (the red dashed box) located at the top of the dilution refrigerator through a ∼2 m- long semi-rigid coaxial cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The box input is a 1:5 transformer in series with a 50 Ω resistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The trans- former output drives two serial connected 50 Ω resistors differentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The differential signals are transmitted to the cryogenic sample holder (the blue dotted box) by two rigid coaxial cables of ∼2 m length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Another pair of impedance matching 50 Ω resistors are added at the input of the cryogenic bridge, and the 360 Ω resistors are chosen by balancing the competition between the perfor- mance and heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The characteristic impedance of all coaxial cables in the work is 50 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The low-frequency signals Vquasi-DC1 and Vquasi-DC2 used to measure the value of Rh and Rr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 µF capacitors are used to separate the high-frequency excitation signals and the quasi-DC signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The output Vout is approximately Vout ∝ S · ( Rh 360 + Rh − C Cr Rr 360 + Rr ) · Vext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (2) S can be obtain from the “V-curve” procedure by linear fitting the VX vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Rh/(360+Rh), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' VX and VY are the orthogonal component of Vout, � VX = |Vout| · cos(θ), (3) VY = |Vout| · sin(θ), (4) where θ is the phase of Vout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' We can derive the value of C using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The new balance condition of the revised bridge is C Cr = Rh Rr 360 + Rr 360 + Rh , (5) where the VX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Note that the capacitance C and the conductance G of sample lead to the orthogonal component VX and VY, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Therefore, the G can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (2) and (4) by replacing C/Cr with G/2πfCr, where f is the excitation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' S1(c) shows our calibration measurement using different excitation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The data is almost flat from 7 to ∼100 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' The measured capacitance begins to decline slowly above ∼100 MHz, possibly due to the parasitic inductance of bonding wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' 7 Rh Rr Cr C Vin + Vin Vout 360 Ω 360 Ω 50 Ω 50 Ω 1:5 50 Ω 50 Ω 50 Ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 μF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 μF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 μF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 μF Vext Vquasi-DC1 Vquasi-DC2 COAX COAX (a) 40 40 0 V (μV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='6 Rh/(Rh+360) Vx Vy (b) Cr= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 pF f= 7 MHz Rr= 50 Ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='2 C (pF) 10 100 f (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='5 pF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='3 pF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='1 pF (c) COAX FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (color online) (a) Circuit diagram of measurement bridge with 50 Ω impedance match networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (b) The VX and VY from a typical “V-curve” procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' C is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content='25 pF from the balance condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} +page_content=' (c) The calibration results, by measuring commercial capacitors with different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf'} diff --git a/9dE0T4oBgHgl3EQffwCf/vector_store/index.faiss b/9dE0T4oBgHgl3EQffwCf/vector_store/index.faiss new file mode 100644 index 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Aldo Faisal1,2 +Abstract— Reaching disabilities affect the quality of life. +Functional Electrical Stimulation (FES) can restore lost motor +functions. Yet, there remain challenges in controlling FES +to induce desired movements. Neuromechanical models are +valuable tools for developing FES control methods. However, +focusing on the upper extremity areas, several existing models +are either overly simplified or too computationally demanding +for control purposes. Besides the model-related issues, finding +a general method for governing the control rules for different +tasks and subjects remains an engineering challenge. +Here, we present our approach toward FES-based restoration +of arm movements to address those fundamental issues in +controlling FES. Firstly, we present our surface-FES-oriented +neuromechanical models of human arms built using well- +accepted, open-source software. The models are designed to +capture significant dynamics in FES controls with minimal +computational cost. Our models are customisable and can be +used for testing different control methods. Secondly, we present +the application of reinforcement learning (RL) as a general +method for governing the control rules. In combination, our +customisable models and RL-based control method open the +possibility of delivering customised FES controls for different +subjects and settings with minimal engineering intervention. +We demonstrate our approach in planar and 3D settings. +Functional Electrical Stimulation, FES, Neuromechanical +Model, Reinforcement Learning, Arm Movements +I. INTRODUCTION +Every year, stroke and spinal cord injury cause the loss of +motor functions in individuals worldwide through paralysis. +In these cases the limbs are technically functional but fail to +receive motor commands from the brain. Arm movement are +one of the commonly lost motor functions and cause severe +limitations in performing daily tasks. Functional Electrical +Stimulation (FES) or neuromuscular stimulation uses low- +energy electrical signals to stimulate the muscle and induce +its contraction and eventually movement of the limb. FES can +be used to animate paralysed muscles and restore lost motor +functions and may help to restore nature motor functions +in incomplete paralysis [1]. Early successes included work +on low limb paralysis, where the ability to cycling through +rhythmic pedalling motions of the legs was restored and has +now become an internally recognised bionic sports discipline. +These success beg the question how to extend this work to +upper body restoration of movement. While FES induced +cycling entails the periodic stimulation of muscles without +1Brain & Behaviour Lab, Imperial College London, London SW7 +2AZ, United Kingdom. (email:nat.wannawas18@imperial.ac.uk). 2 Chair +in Digital Health & Data Science, University of Bayreuth, Bayreuth, +(aldo.faisal@imperial.ac.uk). NW acknowledges his support by the Royal +Thai Government Scholarship. AAF acknowledges his support by UKRI +Turing AI Fellowship (EP/V025449/1). +the need for gravity compensation or end-point precision +in their control (the feet are strapped to the pedals so all +motions are on sagittal plane), so movements are relatively +constrained and requirements on control strategies can be +relatively, low precision (often involving simple periodic +stimulation). This is very different in arm movements, where +reaches towards an end-point require non-linear muscle co- +ordination in the plane, and gravity compensating activity in +3D movements in a volume. Early work in this nascent field +therefore focus on single-joint control, e.g., of the elbow joint +[2], [3]. The literature on multiple-joint control of arm cases, +however, is quite limited. This is partly because controlling +FES in single-joint cases can be achieved through simple, +model-free, error-based control such as PID controllers , +while the multiple-joint cases require significantly more +complex controls that have to include dynamical models in +the systems. +Contrary to the fact that dynamic models play important +roles in the controls and that the neuromechanics of the +human arm is complex, many models used in FES control +studies are relatively simple. In planar arm motion, for +example, two-joint linkages with six muscles represented by +straight lines models (Fig.1a) are one of the most commonly +used [4]–[7]. These models offer fast computation, but may +not well capture the effects of muscle routes and their +deformation during the movements. In addition, the muscles’ +properties themselves vary across the studies, thereby lacking +standardization and making the results difficult to reproduce +or compare. On the other end, there exist commercial neu- +romechanical simulation software such as LifeModeler with +highly detailed models. These models, however, are suitable +for detailed analyses of particular situations such as er- +gonomic designs. Besides, the closed-source and commercial +nature of the software could limit its usage among research +communities, thereby not addressing the standardization and +reproducibility issues. +Besides the model-related issues, governing the control +rules itself is a major challenge in inducing movements +using FES. Regards specifically to inducing multiple-joint +arm movements, successes in real-world settings are limited +and, oftentimes, require assistive devices [2], [8], [9]. For +example, the PID controller with inverse dynamics [2] can +induce a narrow range of movements, while iterative learning +control [8] can induce a longer range but is limited to +repetitive trajectories. These limited successes are partly +attributed to the difficulties of conventional methods in +dealing with complexities and variations of human arms’ +neuromechanics. To our knowledge, an FES control method +arXiv:2301.04004v1 [eess.SY] 10 Jan 2023 + +Fig. 1. +(a) An example of simple planar arm models. (b) Our neuromechanics Arm-Planar model. (c) An illustration of muscle wrapping at the elbow. +(d) Our neuromechanics Arm-3D model and (e) its shoulder muscles. +that can induce arbitrary movements across different subjects +without intensive parameter tuning has not yet been reported. +Keeping both model-related and control governing issues +in focus, we here present our approach toward FES-based +restoration of arm movements that comprises two elements +which, as separate entities, can address those issues. The first +element is the neuromechanics models of the human arm +built using OpenSim [10], a freely-available, open-source +neuromechanical simulation software that is well-accepted in +the communities. This allows us to build the models using +established biomechanical components, e.g., muscle and joint +models, thereby addressing the standardisation issue and +providing state-of-the-art performances [11]. Additionally, +the open source nature of OpenSim facilitate its uses for +designing and testing different control methods which help +promote reproducibility. In this work, we present our two +arm models designed for surface FES control usages, i.e., +they are designed for fast computation while maintaining +important details. These models could be used as standards +for comparing different control methods. +The second element of our approach is to govern the +control policy using Reinforcement Learning (RL), a ma- +chine learning algorithm with a learning agent (RL agent) +that learns to control an environment by interacting with it. +RL can learn to control complex environments, for which +hand-crafted control policies are difficult to govern. In FES +control applications, RL is a promising method for governing +control policies for any FES control settings. In addition, +the fact that RL can provide customised stimulation for +different subjects without intensive manual configuration can +be an important factor that drives FES-based restoration of +movements outside the laboratory and toward at-home usage. +In this work, we present a generic RL setup to learn control +policies for arbitrary arm-reaching tasks. We demonstrate the +usage in planar and 3D arbitrary reaching tasks using our +OpenSim models. +II. RELATED WORKS +It is worth mentioning some related works to highlight +their limitations and the gaps that this work fulfils. Regarding +neuromechanical arm models, there exist several OpenSim +models built by the communities. Closely related models are +the OpenSim core Arm26 and MoBL-ARMS Dynamic Upper +Limb models [12]. These models have a few critical and +minor issues as follows. The first critical issue is that they +produce singularity computation in OpenSim4.4, the latest +version, at some postures, causing crashes. Secondly, there +is no mechanism such as joint limits to prevent unnatural +postures. The minor issues are that there is no joint damping +that prevents unnatural joint speed and, in some postures, +the muscle paths are in the wrong positions, e.g., they wrap +around the wrong side of the joint or go through the bone. +Regarding the applications of RL in FES control, the early +studies were based on old RL algorithms, simple planar arm +models (Fig.1a), and a single, fixed target [4]–[6]. A recent +study has extended these settings to multiple targets [7]. Our +previous works investigate cycling motions in simulation [13] +and single-joint arm movements [3] in the real world. A +simulation study on 3D arm motions was conducted in [14] +using a model that does not have muscle, i.e., the RL agent +directly controls joint torque rather than muscle stimulation. +III. METHODS +a) Neuromechanical models: We use two human arm +models; one is for planar motions (hereafter referred to as +Arm-Planar) which can be viewed as the detailed version +of Fig.1a-like model, and the other one is for 3D motions +(hereafter referred to as Arm-3D). Both models are designed +at a suitable detail level for surface FES control applications, +e.g., the muscles that are impossible to be stimulated sepa- +rately via surface FES are bundled together to minimise the +computation. The common properties and designs of both +models are as follows. Both models have the right arms +connected to the upper bodies located at fixed points in 3D +space. The elbow and shoulder joints are modelled as pin and +ball joints, respectively. Both joints have damping and joint +limit mechanisms that prevent unnatural joint speed and pos- +tures. The muscles are built using a variant of Hill-type mus- +cle model DeGrooteFregly2016Muscle. Both models have 4 +muscles crossing elbows: Triceps Medial, Triceps long head +(biarticular), Brachialis, and Biceps short head (biarticular). +Note that Triceps lateral head is bundled with Triceps long + +Pectoralis +Arm-Planar +Arm-3D +Deltoid Anterior +major C +Deltoid Lateral +Deltoid Posterior +Deltoid +Posterior + Pectoralis major C +N +Biceps short head +Triceps long head +Brachialis +Triceps +Table +Medial +Arm Support +a +b +dhead, and Biceps long head is bundled with Biceps short +head. These muscles wrap around a cylindrical object at the +elbows (Fig.1c). The muscles’ excitation-activation delay is +changed from the default setting of 40 ms to 100 ms to +capture a longer delay of FES-induced muscle activation. +The other muscle parameters such as maximum isometric +force follow those in the Arm26 and MoBL-ARMS Dynamic +Upper Limb models. The tendon slack length parameters are +optimised using a genetic algorithm called CMAES [15] to +equilibrate the passive forces. The other parts of both models +have slightly different designs described as follows. +The Arm-Planar model has 6 muscles in total. The +other muscles besides the aforementioned 4 muscles are the +Pectoralis Major Clavicular head (Pectoralis Major C) and +Deltoid posterior. These muscles wrap around a cylindrical +object at the shoulder. The shoulder joint is only allowed +to rotate around the vertical axis. The arm is supported at +the wrist by an arm supporter (Fig.1b) that moves on a +table with low friction and provides gravity compensation +to the arm. The Arm-3D model has 8 muscles in total which +are those 6 muscles of the Arm-Planar model plus Deltoid +lateral and Deltoid anterior. At the shoulder, there are three +half ellipsoids functioning as muscle wrapper objects. These +ellipsoids are carefully placed to support the full range of +movements and prevent the wrong muscle path issues of the +existing OpenSim models. The shoulder joint can rotate in all +directions except the direction that causes the arm to twist. +b) Reinforcement Learning controllers: The overview +of RL algorithms is briefly described as follows. RL learns +a task through reward signals collected from the interaction +with an environment. The interactions occur in a discrete- +time fashion, starting with the agent observing the envi- +ronment’s state st and selecting an action at based on its +policy π. The action causes the environment to be in a new +state st+1. The agent then receives an immediate reward +rt and observes the new state. This interaction experience +is collected as a tuple (st, at, rt, st+1) which is stored in +a replay buffer D. This tuple is used to learn an optimal +policy π∗ that maximises a return R–the sum of discounted +immediate rewards. +The RL task here is to apply the muscle stimulation to +move the arm to the desired pose which is specified by target +joint angles–shoulder and elbow (θtar,t). The state vector st +is [θt, ˙θt, θtar,t]T , where θt and ˙θt are the joint angles and +angular velocities measured at time t, respectively. Note that +appending the targets into the state vector allows the agents +to learn goal-directed policies that can perform arbitrary +reaching tasks. The action vector at comprises normalised +stimulation intensities (i ∈ [0, 1]). The immediate reward rt +is simply computed using the square error and action penalty +as rt = −(θt+1 − θtar,t)2 − Σn +i=0ai +n +, where n is the number +of stimulated muscles. With this setting, the optimal policy +π∗ is simply the policy that causes the angles to be close to +the targets with minimal stimulation. +The mechanism of finding the optimal policy varies across +different RL algorithms. In this work, we choose the soft +actor-critic (SAC) algorithm [16] because of its state-of-the- +art performance in terms of both sample efficiency and stabil- +ity across different environments. SAC has two components: +an actor and a critic. In simple terms, the critic learns to +estimate the expected return of a state-action pair, known as +the Q value. The Q value is used to adjust the actor’s policy +π by increasing the probability of choosing an action with +a high Q value. Both actor and critic are parameterised by +neural networks; we, based on empirical experiments and our +previous works [3], [13], use fully-connected neural networks +that have two hidden layers. The output layer of the actor +has a sigmoid activation function to squash the outputs. +The setups for the planar and 3D cases are slightly +different. In the planar case, the involved angles are the elbow +and shoulder angles which rotate about the vertical axes. The +state vector is therefore s ∈ R6. The action vector a has 4 +elements (ai ∈ [0, 1]) which are the normalised stimulation +intensities of the Brachialis and Biceps short head, Triceps +Medial and Triceps long head, Pectoralis Major C, and +Deltoid posterior. Note that we set the Biceps stimulation +to affect two muscles because, normally in a real situation, +only a single pair of electrodes are placed above Biceps (and +similarly for Triceps). In the 3D case, the shoulder joint can +rotate in 2 directions, and the Deltoid lateral and Deltoid +anterior are stimulated via the same pair of electrodes. Hence, +the state vector becomes s ∈ R9, and the action vector has +5 elements. +The RL training is episodic. Each episode starts with a +random arm pose and target. Each episode has 100 time steps +with 100 ms time-step size. The target changes to a new +random value at the 50th time step. Every 5 training episodes, +the agents’ performances are evaluated on 50 test episodes. +IV. RESULTS +RL agents are trained for 250 and 500 episodes on the +Arm-Planar and Arm-3D models, respectively. The training +is repeated 10 times to evaluate the robustness. The perfor- +mance evaluations along the training are shown in Fig.2a. +In both cases, the best RL’s performances in rmse measure +are approximately 10◦. The performance development in the +planar case is significantly quicker than in the 3D case. The +standard deviations in both cases are in low, confined ranges +which suggests the robustness. +Fig.2b and d show examples of control performances +in planar and 3D cases, respectively. In both cases, the +RL agents can track arbitrary trajectories that have never +been assigned during the training. The performance in the +planar case is slightly better than that in the 3D case as +the planar movements are less complex. Fig.2c and e show +the stimulation applied during the tracking tasks. In both +cases, brief bursts of stimulation appear when the targets +change, followed by steady stimulation that co-contraction +the muscles to stabilise the arms. The bursts do not appear +when the targets change in a ramping manner. +V. DISCUSSION & CONCLUSION +We present our approach toward FES-based restoration +of arm movements. Our approach has two elements. The + +Fig. 2. +(a) Performance evaluation along the training in (red) Arm-3D and (blue) Arm-Planar cases. The solid lines and the shades show the mean and +standard deviation of 10 runs. The examples of trajectory tracking in (b) Arm-Planar and (d) Arm-3D cases. The dash and solid lines are the targets and +actual angles that the RL agents achieve, respectively. (c) and (e) show the stimulation along the tracking. +first element is to use OpenSim to build neuromechanical +models of the arm. This strategy can help facilitate the build- +ing process and standardise the models on which different +control methods are tested and compared. Furthermore, we +present our two OpenSim models: Arm-Planar and Arm- +3D. The second element is to govern the control rules by +using reinforcement learning which can provide customised +stimulation for different subjects and settings with minimal +technical intervention. We present a generic RL training +setup, demonstrate its applications on our OpenSim models +and show our RL’s performances in performing arbitrary +reaching tasks. +Although this approach has promising simulation results, +several further steps have to be taken to translate it into +real-world usages. One step is to optimise the models to +accurately represent the dynamics of a certain subject’s arm. +This is yet a process that can be done using OpenSimMoCo +[17]. The customised model can be used for pre-training the +RL before transferring it to the real subject. Another step is +to take muscle fatigue into account. The fatigue behaviour +can be included in OpenSim models without touching the +source code by using the method presented in our previ- +ous work [13]. The fatigue will cause the environment’s +state to become partially observable. Based on [5], [7] and +our empirical investigation, the fatigue does not cause RL +to completely fail, but the control performance decreases. +Lastly, in the early period of the training, the RL-controlled +stimulation is unpredictable and random. This raises an issue +about safety. This issue can be mitigated by using offline RL +in the early period. +To summarise, the combination of neuromechanical mod- +els and RL can address existing challenges in FES control. +Although the translation into real-world usages involves +several further steps, its potential is emerging. +REFERENCES +[1] N. Donaldson, T. Perkins, R. Fitzwater, D. Wood, and F. Middleton, +“Fes cycling may promote recovery of leg function after incomplete +spinal cord injury,” Spinal Cord, vol. 38, no. 11, pp. 680–682, 2000. +[2] D. N. Wolf, Z. A. Hall, and E. M. Schearer, “Model learning for con- +trol of a paralyzed human arm with functional electrical stimulation,” +in IEEE Intl. Conf. on Robotics and Automation (ICRA), 2020. +[3] N. Wannawas, A. Shafti, and A. A. Faisal, “Neuromuscular reinforce- +ment learning to actuate human limbs through fes,” in IFESS22, 2022. +[4] J. Izawa et al., “Biological arm motion through reinforcement learn- +ing,” Biological Cybernetics, vol. 91, pp. 10–22, 2004. +[5] P. Thomas et al., “Creating a reinforcement learning controller for +functional electrical stimulation of a human arm,” in 14th Yale Work- +shop on Adaptive and Learning Systems, 2008. +[6] K. M. Jagodnik et al., “Human-like rewards to train a reinforcement +learning controller for planar arm movement,” IEEE Trans on Human- +Machine Systems, vol. 46, pp. 723–733, 10 2016. +[7] J. Abreu et al., “Deep reinforcement learning for control of time- +varying musculoskeletal systems with high fatigability: a feasibility +study,” in IEEE Trans. Neural Sys. and Rehab. Eng., 2022. +[8] C. T. Freeman, “Upper limb electrical stimulation using input-output +linearization and iterative learning control,” IEEE Trans. on Control +Systems Technology, vol. 23, pp. 1546–1554, 2015. +[9] E. Ambrosini et al., “The combined action of a passive exoskeleton +and an emg-controlled neuroprosthesis for upper limb stroke rehabil- +itation: First results of the retrainer project,” in IEEE Intl. Conf. on +Rehab. Robotics. +IEEE, 2017, pp. 56–61. +[10] S. L. Delp et al., “Opensim: Open-source software to create and an- +alyze dynamic simulations of movement,” IEEE Trans on Biomedical +Engineering, vol. 54, pp. 1940–1950, 2007. +[11] K. R. Saul et al., “Benchmarking of dynamic simulation predictions in +two software platforms using an upper limb musculoskeletal model,” +Computer Methods in Biomechanics and Biomedical Engineering, +vol. 18, pp. 1445–1458, 10 2015. +[12] [Online]. +Available: +https://simtk-confluence.stanford.edu:8443/ +display/OpenSim/Musculoskeletal+Models +[13] N. Wannawas, M. Subramanian, and A. A. Faisal, “Neuromechanics- +based deep reinforcement learning of neurostimulation control in fes +cycling,” in Intl. Conf. on Neural Engineering (NER), 2021. +[14] F. Fischer et al., “Reinforcement learning control of a biomechanical +model of the upper extremity,” Scientific Reports, vol. 11, 12 2021. +[15] N. +Hansen, +“The +cma +evolution +strategy: +A +tutorial,” +arXiv:1604.007722v1 [cs.LG], 2016. +[16] T. Haarnoja et al., “Soft actor-critic algorithms and applications,” +arXiv:1812.05905v2 [cs.LG], 2019. +[17] C. L. Dembia et al., “Opensim moco: Musculoskeletal optimal con- +trol,” PLoS Computational Biology, vol. 16, pp. 1–21, 2020. + +Case: Arm-Planar +RMSE: 5.21 ° +Case: Arm-3D +RMSE: 6.82 +100 +100 +a +b +-- +Elbow +Arm-3D +q +Shoulder-x +60 +Arm-Planar +80 +80 - + Shoulder-Z +60 +60 - +Angle +50 +40 +40 +20 +20 + Shoulder +Elbow +0 +100 +100- +Biceps +Delt. Post; +Biceps +Pect.Maj. +c +e +30 +Triceps +Delt. Lat. +Triceps +Deltoid Post. +Stimulation (%) +80 +Pect.Maj. +80 +60 +60 - +20 + +40 +40 +W +000000 +20 +20 - +10: +0 : +0 +50 +100 +150 +200 +250 +300 +0 +1015 +2025 +30 +35 +40 +45 +50 +55 +60 +0 +5 +10 +15 +20 +25 +30 +Episode +time [s] +time [s] \ No newline at end of file diff --git a/AdE2T4oBgHgl3EQfnAiB/content/tmp_files/load_file.txt b/AdE2T4oBgHgl3EQfnAiB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33020bf27b964a21f10e96d3b4d92b29ff6a5a08 --- /dev/null +++ b/AdE2T4oBgHgl3EQfnAiB/content/tmp_files/load_file.txt @@ -0,0 +1,314 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf,len=313 +page_content='Towards AI-controlled FES-restoration of arm movements: neuromechanics-based reinforcement learning for 3-D reaching Nat Wannawas1 & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Aldo Faisal1,2 Abstract— Reaching disabilities affect the quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Functional Electrical Stimulation (FES) can restore lost motor functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Yet, there remain challenges in controlling FES to induce desired movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Neuromechanical models are valuable tools for developing FES control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' However, focusing on the upper extremity areas, several existing models are either overly simplified or too computationally demanding for control purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Besides the model-related issues, finding a general method for governing the control rules for different tasks and subjects remains an engineering challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Here, we present our approach toward FES-based restoration of arm movements to address those fundamental issues in controlling FES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Firstly, we present our surface-FES-oriented neuromechanical models of human arms built using well- accepted, open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The models are designed to capture significant dynamics in FES controls with minimal computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Our models are customisable and can be used for testing different control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Secondly, we present the application of reinforcement learning (RL) as a general method for governing the control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In combination, our customisable models and RL-based control method open the possibility of delivering customised FES controls for different subjects and settings with minimal engineering intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' We demonstrate our approach in planar and 3D settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Functional Electrical Stimulation, FES, Neuromechanical Model, Reinforcement Learning, Arm Movements I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' INTRODUCTION Every year, stroke and spinal cord injury cause the loss of motor functions in individuals worldwide through paralysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In these cases the limbs are technically functional but fail to receive motor commands from the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Arm movement are one of the commonly lost motor functions and cause severe limitations in performing daily tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Functional Electrical Stimulation (FES) or neuromuscular stimulation uses low- energy electrical signals to stimulate the muscle and induce its contraction and eventually movement of the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' FES can be used to animate paralysed muscles and restore lost motor functions and may help to restore nature motor functions in incomplete paralysis [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Early successes included work on low limb paralysis, where the ability to cycling through rhythmic pedalling motions of the legs was restored and has now become an internally recognised bionic sports discipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These success beg the question how to extend this work to upper body restoration of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' While FES induced cycling entails the periodic stimulation of muscles without 1Brain & Behaviour Lab, Imperial College London, London SW7 2AZ, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (email:nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='wannawas18@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' 2 Chair in Digital Health & Data Science, University of Bayreuth, Bayreuth, (aldo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='faisal@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' NW acknowledges his support by the Royal Thai Government Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' AAF acknowledges his support by UKRI Turing AI Fellowship (EP/V025449/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' the need for gravity compensation or end-point precision in their control (the feet are strapped to the pedals so all motions are on sagittal plane), so movements are relatively constrained and requirements on control strategies can be relatively, low precision (often involving simple periodic stimulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This is very different in arm movements, where reaches towards an end-point require non-linear muscle co- ordination in the plane, and gravity compensating activity in 3D movements in a volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Early work in this nascent field therefore focus on single-joint control, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=', of the elbow joint [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The literature on multiple-joint control of arm cases, however, is quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This is partly because controlling FES in single-joint cases can be achieved through simple, model-free, error-based control such as PID controllers , while the multiple-joint cases require significantly more complex controls that have to include dynamical models in the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Contrary to the fact that dynamic models play important roles in the controls and that the neuromechanics of the human arm is complex, many models used in FES control studies are relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In planar arm motion, for example, two-joint linkages with six muscles represented by straight lines models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='1a) are one of the most commonly used [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These models offer fast computation, but may not well capture the effects of muscle routes and their deformation during the movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In addition, the muscles’ properties themselves vary across the studies, thereby lacking standardization and making the results difficult to reproduce or compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' On the other end, there exist commercial neu- romechanical simulation software such as LifeModeler with highly detailed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These models, however, are suitable for detailed analyses of particular situations such as er- gonomic designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Besides, the closed-source and commercial nature of the software could limit its usage among research communities, thereby not addressing the standardization and reproducibility issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Besides the model-related issues, governing the control rules itself is a major challenge in inducing movements using FES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Regards specifically to inducing multiple-joint arm movements, successes in real-world settings are limited and, oftentimes, require assistive devices [2], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' For example, the PID controller with inverse dynamics [2] can induce a narrow range of movements, while iterative learning control [8] can induce a longer range but is limited to repetitive trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These limited successes are partly attributed to the difficulties of conventional methods in dealing with complexities and variations of human arms’ neuromechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' To our knowledge, an FES control method arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='04004v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='SY] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (a) An example of simple planar arm models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (b) Our neuromechanics Arm-Planar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (c) An illustration of muscle wrapping at the elbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (d) Our neuromechanics Arm-3D model and (e) its shoulder muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' that can induce arbitrary movements across different subjects without intensive parameter tuning has not yet been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Keeping both model-related and control governing issues in focus, we here present our approach toward FES-based restoration of arm movements that comprises two elements which, as separate entities, can address those issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The first element is the neuromechanics models of the human arm built using OpenSim [10], a freely-available, open-source neuromechanical simulation software that is well-accepted in the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This allows us to build the models using established biomechanical components, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=', muscle and joint models, thereby addressing the standardisation issue and providing state-of-the-art performances [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Additionally, the open source nature of OpenSim facilitate its uses for designing and testing different control methods which help promote reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In this work, we present our two arm models designed for surface FES control usages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=', they are designed for fast computation while maintaining important details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These models could be used as standards for comparing different control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The second element of our approach is to govern the control policy using Reinforcement Learning (RL), a ma- chine learning algorithm with a learning agent (RL agent) that learns to control an environment by interacting with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' RL can learn to control complex environments, for which hand-crafted control policies are difficult to govern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In FES control applications, RL is a promising method for governing control policies for any FES control settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In addition, the fact that RL can provide customised stimulation for different subjects without intensive manual configuration can be an important factor that drives FES-based restoration of movements outside the laboratory and toward at-home usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In this work, we present a generic RL setup to learn control policies for arbitrary arm-reaching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' We demonstrate the usage in planar and 3D arbitrary reaching tasks using our OpenSim models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' RELATED WORKS It is worth mentioning some related works to highlight their limitations and the gaps that this work fulfils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Regarding neuromechanical arm models, there exist several OpenSim models built by the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Closely related models are the OpenSim core Arm26 and MoBL-ARMS Dynamic Upper Limb models [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These models have a few critical and minor issues as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The first critical issue is that they produce singularity computation in OpenSim4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='4, the latest version, at some postures, causing crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Secondly, there is no mechanism such as joint limits to prevent unnatural postures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The minor issues are that there is no joint damping that prevents unnatural joint speed and, in some postures, the muscle paths are in the wrong positions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=', they wrap around the wrong side of the joint or go through the bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Regarding the applications of RL in FES control, the early studies were based on old RL algorithms, simple planar arm models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='1a), and a single, fixed target [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' A recent study has extended these settings to multiple targets [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Our previous works investigate cycling motions in simulation [13] and single-joint arm movements [3] in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' A simulation study on 3D arm motions was conducted in [14] using a model that does not have muscle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=', the RL agent directly controls joint torque rather than muscle stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' METHODS a) Neuromechanical models: We use two human arm models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' one is for planar motions (hereafter referred to as Arm-Planar) which can be viewed as the detailed version of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='1a-like model, and the other one is for 3D motions (hereafter referred to as Arm-3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Both models are designed at a suitable detail level for surface FES control applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=', the muscles that are impossible to be stimulated sepa- rately via surface FES are bundled together to minimise the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The common properties and designs of both models are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Both models have the right arms connected to the upper bodies located at fixed points in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The elbow and shoulder joints are modelled as pin and ball joints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Both joints have damping and joint limit mechanisms that prevent unnatural joint speed and pos- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The muscles are built using a variant of Hill-type mus- cle model DeGrooteFregly2016Muscle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Both models have 4 muscles crossing elbows: Triceps Medial, Triceps long head (biarticular), Brachialis, and Biceps short head (biarticular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Note that Triceps lateral head is bundled with Triceps long Pectoralis Arm-Planar Arm-3D Deltoid Anterior major C Deltoid Lateral Deltoid Posterior Deltoid Posterior Pectoralis major C N Biceps short head Triceps long head Brachialis Triceps Table Medial Arm Support a b dhead, and Biceps long head is bundled with Biceps short head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These muscles wrap around a cylindrical object at the elbows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The muscles’ excitation-activation delay is changed from the default setting of 40 ms to 100 ms to capture a longer delay of FES-induced muscle activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The other muscle parameters such as maximum isometric force follow those in the Arm26 and MoBL-ARMS Dynamic Upper Limb models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The tendon slack length parameters are optimised using a genetic algorithm called CMAES [15] to equilibrate the passive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The other parts of both models have slightly different designs described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The Arm-Planar model has 6 muscles in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The other muscles besides the aforementioned 4 muscles are the Pectoralis Major Clavicular head (Pectoralis Major C) and Deltoid posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These muscles wrap around a cylindrical object at the shoulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The shoulder joint is only allowed to rotate around the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The arm is supported at the wrist by an arm supporter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='1b) that moves on a table with low friction and provides gravity compensation to the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The Arm-3D model has 8 muscles in total which are those 6 muscles of the Arm-Planar model plus Deltoid lateral and Deltoid anterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' At the shoulder, there are three half ellipsoids functioning as muscle wrapper objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' These ellipsoids are carefully placed to support the full range of movements and prevent the wrong muscle path issues of the existing OpenSim models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The shoulder joint can rotate in all directions except the direction that causes the arm to twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' b) Reinforcement Learning controllers: The overview of RL algorithms is briefly described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' RL learns a task through reward signals collected from the interaction with an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The interactions occur in a discrete- time fashion, starting with the agent observing the envi- ronment’s state st and selecting an action at based on its policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The action causes the environment to be in a new state st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The agent then receives an immediate reward rt and observes the new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This interaction experience is collected as a tuple (st, at, rt, st+1) which is stored in a replay buffer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This tuple is used to learn an optimal policy π∗ that maximises a return R–the sum of discounted immediate rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The RL task here is to apply the muscle stimulation to move the arm to the desired pose which is specified by target joint angles–shoulder and elbow (θtar,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The state vector st is [θt, ˙θt, θtar,t]T , where θt and ˙θt are the joint angles and angular velocities measured at time t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Note that appending the targets into the state vector allows the agents to learn goal-directed policies that can perform arbitrary reaching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The action vector at comprises normalised stimulation intensities (i ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The immediate reward rt is simply computed using the square error and action penalty as rt = −(θt+1 − θtar,t)2 − Σn i=0ai n , where n is the number of stimulated muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' With this setting, the optimal policy π∗ is simply the policy that causes the angles to be close to the targets with minimal stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The mechanism of finding the optimal policy varies across different RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In this work, we choose the soft actor-critic (SAC) algorithm [16] because of its state-of-the- art performance in terms of both sample efficiency and stabil- ity across different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' SAC has two components: an actor and a critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In simple terms, the critic learns to estimate the expected return of a state-action pair, known as the Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The Q value is used to adjust the actor’s policy π by increasing the probability of choosing an action with a high Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Both actor and critic are parameterised by neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' we, based on empirical experiments and our previous works [3], [13], use fully-connected neural networks that have two hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The output layer of the actor has a sigmoid activation function to squash the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The setups for the planar and 3D cases are slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In the planar case, the involved angles are the elbow and shoulder angles which rotate about the vertical axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The state vector is therefore s ∈ R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The action vector a has 4 elements (ai ∈ [0, 1]) which are the normalised stimulation intensities of the Brachialis and Biceps short head, Triceps Medial and Triceps long head, Pectoralis Major C, and Deltoid posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Note that we set the Biceps stimulation to affect two muscles because, normally in a real situation, only a single pair of electrodes are placed above Biceps (and similarly for Triceps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In the 3D case, the shoulder joint can rotate in 2 directions, and the Deltoid lateral and Deltoid anterior are stimulated via the same pair of electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Hence, the state vector becomes s ∈ R9, and the action vector has 5 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The RL training is episodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Each episode starts with a random arm pose and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Each episode has 100 time steps with 100 ms time-step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The target changes to a new random value at the 50th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Every 5 training episodes, the agents’ performances are evaluated on 50 test episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' RESULTS RL agents are trained for 250 and 500 episodes on the Arm-Planar and Arm-3D models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The training is repeated 10 times to evaluate the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The perfor- mance evaluations along the training are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In both cases, the best RL’s performances in rmse measure are approximately 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The performance development in the planar case is significantly quicker than in the 3D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The standard deviations in both cases are in low, confined ranges which suggests the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='2b and d show examples of control performances in planar and 3D cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In both cases, the RL agents can track arbitrary trajectories that have never been assigned during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The performance in the planar case is slightly better than that in the 3D case as the planar movements are less complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='2c and e show the stimulation applied during the tracking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' In both cases, brief bursts of stimulation appear when the targets change, followed by steady stimulation that co-contraction the muscles to stabilise the arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The bursts do not appear when the targets change in a ramping manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' DISCUSSION & CONCLUSION We present our approach toward FES-based restoration of arm movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Our approach has two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (a) Performance evaluation along the training in (red) Arm-3D and (blue) Arm-Planar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The solid lines and the shades show the mean and standard deviation of 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The examples of trajectory tracking in (b) Arm-Planar and (d) Arm-3D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The dash and solid lines are the targets and actual angles that the RL agents achieve, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' (c) and (e) show the stimulation along the tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' first element is to use OpenSim to build neuromechanical models of the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This strategy can help facilitate the build- ing process and standardise the models on which different control methods are tested and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Furthermore, we present our two OpenSim models: Arm-Planar and Arm- 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The second element is to govern the control rules by using reinforcement learning which can provide customised stimulation for different subjects and settings with minimal technical intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' We present a generic RL training setup, demonstrate its applications on our OpenSim models and show our RL’s performances in performing arbitrary reaching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Although this approach has promising simulation results, several further steps have to be taken to translate it into real-world usages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' One step is to optimise the models to accurately represent the dynamics of a certain subject’s arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This is yet a process that can be done using OpenSimMoCo [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The customised model can be used for pre-training the RL before transferring it to the real subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Another step is to take muscle fatigue into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The fatigue behaviour can be included in OpenSim models without touching the source code by using the method presented in our previ- ous work [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' The fatigue will cause the environment’s state to become partially observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Based on [5], [7] and our empirical investigation, the fatigue does not cause RL to completely fail, but the control performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Lastly, in the early period of the training, the RL-controlled stimulation is unpredictable and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This raises an issue about safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' This issue can be mitigated by using offline RL in the early period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' To summarise, the combination of neuromechanical mod- els and RL can address existing challenges in FES control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Although the translation into 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='82 100 100 a b -- Elbow Arm-3D q Shoulder-x 60 Arm-Planar 80 80 - Shoulder-Z 60 60 - Angle 50 40 40 20 20 Shoulder Elbow 0 100 100- Biceps Delt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Post;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Biceps Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' c e 30 Triceps Delt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Triceps Deltoid Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' Stimulation (%) 80 Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content='Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} +page_content=' 80 60 60 - 20 + 40 40 W 000000 20 20 - 10: 0 : 0 50 100 150 200 250 300 0 1015 2025 30 35 40 45 50 55 60 0 5 10 15 20 25 30 Episode time [s] time [s]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'} diff --git a/C9AzT4oBgHgl3EQfiP2U/content/tmp_files/2301.01497v1.pdf.txt b/C9AzT4oBgHgl3EQfiP2U/content/tmp_files/2301.01497v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0fa20bb10a9557bda8345b4daf9c7d5a252ba41 --- /dev/null +++ b/C9AzT4oBgHgl3EQfiP2U/content/tmp_files/2301.01497v1.pdf.txt @@ -0,0 +1,1930 @@ +Complex dynamics of knowledgeable monopoly models with gradient +mechanisms +Xiaoliang Lia, Jiacheng Fub, and Wei Niu∗b,c +aSchool of Digital Economics, Dongguan City University, Dongguan, China +bSino-French Engineer School, Beihang University, Beijing, China +cBeihang Hangzhou Innovation Institute Yuhang, Hangzhou, China +Abstract +In this paper, we explore the dynamics of two monopoly models with knowledgeable players. +The first model was initially introduced by Naimzada and Ricchiuti, while the second one is sim- +plified from a famous monopoly introduced by Puu. We employ several tools based on symbolic +computations to analyze the local stability and bifurcations of the two models. To the best of our +knowledge, the complete stability conditions of the second model are obtained for the first time. We +also investigate periodic solutions as well as their stability. Most importantly, we discover that the +topological structure of the parameter space of the second model is much more complex than that +of the first one. Specifically, in the first model, the parameter region for the stability of any periodic +orbit with a fixed order constitutes a connected set. In the second model, however, the stability +regions for the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint +portions. Furthermore, we find that the basins of the two stable equilibria in the second model are +disconnected and also have complicated topological structures. In addition, the existence of chaos +in the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits in the +two models, respectively. +Keywords: monopoly; gradient mechanism; stability; periodic orbit; chaos +1 +Introduction +Unlike a competitive market with a large number of relatively small companies producing homogeneous +products and competing with each other, an oligopoly is a market supplied only by a few firms. It +is well known that Cournot developed the first formal theory of oligopoly in [7], where players are +supposed to have the naive expectations that their rivals produce the same quantity of output as in +the immediately previous period. Cournot introduced a gradient mechanism of adjusting the quantity +of output and proved that his model has one unique equilibrium, which is globally stable provided +that only two firms exist in the market. +A monopoly is the simplest oligopoly, which is a market served by one unique firm. In the existing +literature, a market supplied by two, three, or even four companies is called a duopoly [17], a triopoly +[19], or a quadropoly [21], respectively. +However, a monopoly may also exhibit complex dynamic +behaviors such as periodic orbits and chaos if the involved firm is supposed to be boundedly rational. +As distinguished by Matsumoto and Szidarovszky [22], a boundedly rational monopolist is said to be +knowledgeable if it has full information regarding the inverse demand function, and limited if it does +not know the form of the inverse demand function but possesses the values of output and price only +in the past two periods. Knowledgeable and limited players have been considered in several monopoly +models. +For example, Puu [26] introduced a monopoly where the inverse demand function is a cubic func- +tion with an inflection point, and the marginal cost is quadratic. In this model, the monopolist is +∗Corresponding author: wei.niu@buaa.edu.cn +1 +arXiv:2301.01497v1 [econ.TH] 4 Jan 2023 + +supposed to be a limited player. Puu indicated that there exist multiple (at most three) equilibria, and +complex dynamics such as chaos may appear if the reactivity of the monopolist becomes sufficiently +large. Moreover, Puu’s model was reconsidered by Al-Hdaibat and others in [1], where a numerical +continuation method is used to compute solutions with different periods and determine their stability +regions. In particular, they analytically investigated general formulae for solutions with period four. +It should be mentioned that the equilibrium multiplicity and complex dynamics of Puu’s model +might depend strictly on the inverse demand function that has an inflection point. In this regard, +Naimzada and Ricchiuti [25] introduced a simpler monopoly with a knowledgeable player, where the +inverse demand function is still cubic but has no inflection points. It was discovered that complex +dynamics can also arise, especially when the reaction coefficient to variation in profits is high. Askar [2] +and Sarafopoulos [27] generalized the inverse demand function of Naimzada and Ricchiuti to a function +of a similar form, but the degree of their function could be any positive integer. The difference is that +the cost function in Askar’s model is linear but quadratic in Sarafopoulos’s. +Cavalli and Naimzada [4] studied a monopoly model characterized by a constant elasticity demand +function, in which the firm is also assumed to be knowledgeable with a linear cost. They focused on +the equilibrium stability as the variation of the price elasticity of demand and proved that there are +two possible different cases, where elasticity has either a stabilizing or a mixed stabilizing/destabilizing +effect. Moreover, Elsadany and Awad [8] explored a monopoly game with delays where the inverse +demand is a log-concave function. Caravaggio and Sodini [3] considered a nonlinear model, where +a knowledgeable monopolist provides a fixed amount of an intermediate good and then uses this +good to produce two vertically differentiated final commodities. They found that there are chaotic +and multiple attractors. Furthermore, continuous dynamical systems have also been applied in the +study of monopolistic markets. In [23], Matsumoto and Szidarovszky proposed a monopoly model +formulated in continuous time and investigated the effect of delays in obtaining and implementing the +output information. Motivated by the aforementioned work, other remarkable contributions including +[9, 10] were done in this strand of research. +In our study, we consider two monopoly models formulated with discrete dynamical systems, where +the players are supposed to be knowledgeable. The two models are distinct mainly in their inverse +demand functions. The first model uses the inverse demand of Naimzada and Ricchiuti [25], while the +second one employs that of Puu [26]. For both models, we analyze the existence and local stability of +equilibria and periodic solutions by using tools based on symbolic computations such as the method +of triangular decomposition and the method of partial cylindrical algebraic decomposition. It should +be mentioned that different from numerical computations, symbolic computations are exact, thus the +results can be used to rigorously prove economic theorems in some sense. +The main contributions of this paper are as follows. To the best of our knowledge, the complete +stability conditions of the second model are obtained for the first time. We also investigate the periodic +solutions in the two models as well as their stability. Most importantly, we find different topological +structures of the parameter spaces of the two considered models. Specifically, in the first model, the +parameter region for the stability of any periodic solution with a fixed order constitutes a connected +set. In the second model, however, the stability regions for the 3-cycle, 4-cycle, and 5-cycle orbits are +disconnected sets formed by many disjoint portions. In other words, the topological structures of the +regions for stable periodic orbits in Model 2 are much more complex than those in Model 1. This may +be because the inverse demand function of Model 2 has an inflection point. Furthermore, according +to our numerical simulations of Model 2, it is discovered that the basins of the two stable equilibria +are disconnected and also have complex topological structures. In addition, the existence of chaos in +the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits in the two +models, respectively. +The rest of this paper is organized as follows. In Section 2, we revisit the construction of the two +models. In Section 3, the local stability of the equilibrium is thoroughly studied, and bifurcations +through which the equilibrium loses its stability are also investigated. In Section 4, the existence and +stability of periodic orbits with relatively lower orders are explored for the two models. In Section 5, +we rigorously derive the existence of chaotic dynamics in the sense of Li-Yorke. The paper is concluded +with some remarks in Section 6. +2 + +2 +Basic Models +Suppose a monopolist exists in the market, and the quantity of its output is denoted as x. We use P(x) +to denote the price function (also called inverse demand function), which is assumed to be downward +sloping, i.e., +dP(x) +dx +< 0, +for any x > 0. +(1) +It follows that P(x) is invertible. +The demand function (the inverse of P(x)) exists and is also +downward sloping. Furthermore, the cost function is denoted as C(x). Then the profit is +Π(x) = P(x)x − C(x). +The monopolist is assumed to adopt a gradient mechanism of adjusting its output to achieve +increased profits. +Suppose that the firm is a knowledgeable player, which means that it has full +information regarding the inverse demand function P(x) and has the capability of computing the +marginal profit dΠ/dx. The firm adjusts its output by focusing on how the variation of x affects the +variation of Π(x). Specifically, the adjustment process is formulated as +x(t + 1) = x(t) + K dΠ(x(t)) +dx(t) +, +K > 0. +Since K > 0, a positive marginal profit induces the monopolist to adjust the quantity of its output in +a positive direction and vice versa. +The first model considered in this paper was initially proposed by Naimzada and Ricchiuti [25], +where a cubic price function without the inflection point is employed. We restate the formulation of +this model in the sequel. +Model 1. The price function is cubic and the cost function is linear as follows. +P(x) = a − bx3, +C(x) = cx, +where a, b, c are parameters. The downward sloping condition (1) is guaranteed if dP/dx = −3bx2 < 0, +that is if b > 0. Moreover, assume that the marginal cost dC/dx = c > 0. We adopt the general +principle of setting price above marginal cost, i.e., P(x) − c > 0 for any x ≥ 0. Therefore, we must +have that a > c. One knows the profit function is +Π(x) = P(x)x − C(x) = (a − bx3)x − cx = (a − c)x − bx4. +Thus, the gradient adjustment mechanism can be described as +x(t + 1) = x(t) + K(a − c − 4bx3(t)), +K > 0. +Without loss of generality, we denote f = 4bK and e = (a − c)/4b. Then, the model is simplified into +a map with only two parameters: +x(t + 1) = x(t) + f(e − x3(t)), +e, f > 0. +(2) +The second model considered in this paper is simplified from a famous monopoly model introduced +by Puu [26]. We retain the same inverse demand function and cost function. The only difference is +that the monopolist in our model is knowledgeable, whereas the monopolist in Puu’s original model +is limited. +Model 2. The price function is cubic of a more general form +P(x) = a1 − b1x + c1x2 − d1x3, +where a1, b1, c1, d1 > 0 are parameters. The cost function is also cubic and has no fixed costs, i.e., +C(x) = a2x − b2x2 + c2x3, +3 + +where a2, b2, c2 > 0. Hence, the profit function becomes +Π(x) = P(x)x − C(x) = (a1 − a2)x − (b1 − b2)x2 + (c1 − c2)x3 − d1x4, +which can be denoted as +Π(x) = ax − bx2 + cx3 − dx4 +with +a = a1 − a2, b = b1 − b2, c = c1 − c2, and d = d1. +For the sake of simplicity, we assume that a, b, c, d > 0. The marginal profit dΠ/dx is directly obtained +and the gradient adjustment mechanism can be formulated as +x(t + 1) = x(t) + K(a − 2bx(t) + 3cx2(t) − 4dx3(t)), +a, b, c, d > 0. +(3) +3 +Local Stability and Bifurcations +Firstly, we explain the main idea of the symbolic approach used in this paper by analyzing stepwise +the local stability of Model 1. Then the theoretical results of Model 2 are reported without giving all +the calculation details. +3.1 +Model 1 +Proposition 1. Model 1 always has a unique equilibrium, which is stable if +4b(a − c)2K3 < 8 +27 +Moreover, there is a period-doubling bifurcation if +4b(a − c)2K3 = 8 +27. +The above proposition is a known result, which was first derived by Naimzada and Ricchiuti +[25]. +Indeed, this proposition can be easily proved since the analytical expression of the unique +equilibrium can be obtained, i.e., x∗ = ( a−c +4b )1/3. However, we would like to provide another proof in +a computational style to demonstrate in detail how our symbolic approach works. +In what follows, the model formulation (2) is taken. By setting x(t + 1) = x(t) = x, we acquire +the equilibrium equation x = x + f(e − x3). An equilibrium x of the one-dimensional iteration map is +locally stable if +����� +dx(t + 1) +dx(t) +���� +x(t)=x +����� = +��1 − 3fx2�� < 1. +Moreover, we say the equilibrium x to be feasible if x > 0. Thus, a stable and feasible equilibrium can +be characterized as a real solution of +� +� +� +� +� +x = x + f(e − x3), +��1 − 3fx2�� < 1, +x > 0, e > 0, f > 0. +(4) +Although system (4) is so simple that one can solve the closed-form expression of x from the +equality part, the problem is how we handle a general polynomial that may have no closed-form +solutions. +Furthermore, it is also a nontrivial task to identify the conditions on the parameters +whether a system with inequalities has real solutions. In [18], the first author of this paper and his +coworker proposed an algebraic approach to systematically tackle these problems. The main idea of +this approach is as follows. +The parametric system (4) is univariate in x. For a univariate system, we introduce a key concept +called border polynomial in the sequel. One useful property of a border polynomial is that its real +zeros divide the parameter space into separated regions and the solution number of the original system +is invariant for all parameter points in each region. +4 + +Definition 1 (Border Polynomial). Consider a univariate system +� +P(u, x) = �m +i=0 ai(u) xi = 0, +Q1(u, x) > 0, . . . , Qs(u, x) > 0, +(5) +where P and Q1, . . . , Qs are univariate polynomials in x, and u stands for all parameters. The product +am(u) · discr(P) · +s +� +i=1 +res(P, Qi) +is called the border polynomial of system (5). Here, res(F, G) stands for the resultant of two polyno- +mials F and G, while discr(F) denotes the discriminant of F. +More specifically, the formal definitions of the resultant and the discriminant in the above definition +are given as follows. Let +F = +m +� +i=0 +ai xi, +G = +l +� +j=0 +bj xj +be two univariate polynomials in x with coefficients ai, bj in the field of complex numbers, and am, bl ̸= +0. The determinant +��������������� +am +am−1 +· · · +a0 +... +... +... +... +am +am−1 +· · · +a0 +bl +bl−1 +· · · +b0 +... +... +... +... +bl +bl−1 +· · · +b0 +��������������� +� +� +� l +� +� +� m +is called the Sylvester resultant (or simply resultant) of F and G, and denoted by res(F, G). The +resultant of F and its derivative dF/dx, i.e., res(F, dF/dx), is called the discriminant of F and +denoted by discr(F). The following lemma is one of the well-known properties of resultants, which +could be found in [24]. +Lemma 1. Two univariate polynomials F and G have common zeros in the field of complex numbers +if and only if res(F, G) = 0. Moreover, a univariate polynomial F has a multiple zero in the field of +complex numbers if and only if discr(F) = 0. +It is worth noticing that the number of real zeros of P may change when the leading coefficient +am(u) or the discriminant discr(P) goes from non-zero to zero and vice versa. In addition, if res(P, Qi) +goes across zero, then the zeros of P will pass through the boundaries of Qi > 0, which means that +the number of real roots of (5) may change. Therefore, the following lemma is derived. +Lemma 2. Consider a univariate system as (5). Let A and B be two points in the space of parameters +u. Suppose that any of A, B does not annihilate the border polynomial of system (5). If there exists +a real path C from A to B such that any point on C is not a root of the border polynomial, then the +number of real solutions of system (5) evaluated at A is the same as that at B. +Since 1 − 3fx2 < 1, we know that system (4) is equivalent to +� +� +� +� +� +x3 − e = 0, +2 − 3fx2 > 0, +x > 0, e > 0, f > 0. +(6) +We have am = 1 and discr(x3 − e) = 27e2. +Moreover, res(x3 − e, 2 − 3fx2) = −27e2f3 + 8 and +res(x3−e, x) = e. According to Definition 1, the border polynomial of system (6) is 27e3(−27e2f3+8), +the zeros of which are marked in blue as shown in Figure 1. This blue curve divides the parameter +set {(e, f) | e > 0, f > 0} into two (the northeast and the southwest) regions. +5 + +S2 +S1 +A +Real path C +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +e +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +f +Figure 1: Partitions of the parameter space of Model 1 and sample points +Notice the two points S2 and A in Figure 1. One can find a real path C from A to S2 such that +it does not pass through the blue curve. According to Lemma 2, system (6) has the same number of +real roots with the parameters evaluated at S2 and A. This means that the number of real solutions +of system (6) is invariant in the northeast region. Therefore, we can choose a sample point from each +region to determine the root number. For this simple system, sample points might be selected directly +by eyes, e.g., S1 = (1, 1/2), S2 = (1, 1). However, the choosing process might be extremely complex in +general, which could be done automatically by using, e.g., the method of partial cylindrical algebraic +decomposition or called the PCAD method [5]. +For each region, one can determine the root number by counting roots of the non-parametric system +of (6) evaluated at the corresponding sample point. Take S1 as an example, where (6) becomes +� +x3 − 1 = 0, 2 − 3 +2x2 > 0, x > 0 +� +. +(7) +In order to count the number of its real roots, an obvious way is directly solving x3 −1 = 0, i.e., x = 1, +and then checking whether 2 − 3 +2x2 > 0 and x > 0 are satisfied. The result is true, which means that +there exists one unique real solution of (7). However, it is difficult to precisely obtain all real zeros +of a general univariate system since root formulae do not exist for polynomials with degrees greater +than 4. Therefore, a more systematic method called real root counting [31] is generally needed here, +and we demonstrate how this method works by using (7) as an example. +It is noted that x3 −1, 2− 3 +2x2 and x have no common zeros, i.e., they have no factors in common. +Otherwise, one needs to reduce the common factors from the inequalities first. After that, we isolate +all real zeros of 2 − 3 +2x2 and x by rational intervals, e.g., +� +−12 +10, −11 +10 +� +, +� +− 1 +10, 1 +10 +� +, +�11 +10, 12 +10 +� +. +(8) +Although it is trivial for this simple example, the isolation process could be particularly tough for +general polynomials, which may be handled by using, e.g., the modified Uspensky algorithm [6]. +Moreover, the intervals can be made as small as possible to guarantee no zeros of x3 − 1 lie in these +intervals, which could be checked by using, e.g., Sturm’s theorem [28]. Thus, the real zeros of x3 − 1 +must be in the complement of (8): +� +−∞, −12 +10 +� +, +� +−11 +10, − 1 +10 +� +, +� 1 +10, 11 +10 +� +, +�12 +10, +∞ +� +. +(9) +In each of these open intervals, the signs of 2 − 3 +2x2 and x are invariant and can be determined +by checking them at selected sample points. +For instance, to determine the sign of 2 − 3 +2x2 on +6 + +(12/10, +∞), we check the sign at a sample point, e.g., x = 2. We have that 2 − 3 +2x2|x=2 = −4 < 0, +thus 2 − 3 +2x2 < 0 on (12/10, +∞). Similarly, it is obtained that the signs of 2 − 3 +2x2 and x at (9) are +−, +, +, − and −, −, +, +, respectively. Hence, (1/10, 11/10) is the only interval such that the two +inequalities 2 − 3 +2x2 > 0 and x > 0 of system (7) are simultaneously satisfied. +We focus on (1/10, 11/10). Using Sturm’s theorem, we can count the number of the real zeros +of x3 − 1 at (1/10, 11/10), which is one. Therefore, system (6) has one real root at S1 = (1, 1/2). +The above approach works well for a system formulated with univariate polynomial equations and +inequalities although some steps seem silly and not necessary for this simple example. Similarly, we +know that system (6) has no real roots at S2 = (1, 1). +In conclusion, system (6) has one real root if the parameters take values from the southwest region +where S1 lies, and has no real roots if the parameters take values from the northeast region where S2 +lies. Furthermore, the inequalities of some factors of the border polynomial may be used to explicitly +describe a given region. It is evident that 27e2f3 −8 < 0 describes the region where S1 lies. Therefore, +Model 1 has one unique stable equilibrium provided that +e2f3 = +�a − c +4b +�2 +(4bK)3 = 4b(a − c)2K3 < 8 +27, +which is consistent with Proposition 1. +According to the classical bifurcation theory, for a one-dimensional iteration map x(t+1) = F(x(t)), +we know that bifurcations may occur if +����� +dx(t + 1) +dx(t) +���� +x(t)=x +����� = +���� +dF +dx +���� = 1. +More specifically, if dF/dx = −1, then the system may undergo a period-doubling bifurcation (also +called flip bifurcation), where the dynamics switch to a new behavior with twice the period of the +original system. On the other hand, if dF/dx = 1, then the system may undergo a saddle-node (fold), +transcritical, or pitchfork bifurcation. One might determine the type of bifurcation from the change +in the number of the (stable) equilibria. In the case of saddle-node bifurcation, one stable equilibrium +(a node) annihilates with another unstable one (a saddle). Before and after a transcritical bifurcation, +there is one unstable and one stable equilibrium, and the unstable equilibrium becomes stable and +vice versa. In the case of pitchfork bifurcation, the number of equilibria changes from one to three or +from three to one, while the number of stable equilibria changes from one to two or from one to zero. +Accordingly, it is concluded that Model 1 may undergo a period-doubling bifurcation if +e2f3 = 4b(a − c)2K3 = 8 +27, +and there are no other bifurcations. +3.2 +Model 2 +According to (3), by setting x(t + 1) = x(t) = x, we know that Model 2 has at most three equilibria. +The analytical expressions of the equilibria exist, but are complex, i.e., +x1 = +3√ +M +12d − 8bd − 3c2 +4d +3√ +M ++ c +4d, +x2,3 = − +3√ +M +24d + 8bd − 3c2 +8d +3√ +M ++ c +4d ± i +√ +3 +2 +� +3√ +M +12d + 8bd − 3c2 +4d +3√ +M +� +, +(10) +where +M = 12d +√ +3 +� +108a2d2 − 108abcd + 27 ac3 + 32b3d − 9b2c2 + 216ad2 − 108bcd + 27c3. +7 + +Furthermore, an equilibrium x is locally stable provided that +����� +dx(t + 1) +dx(t) +���� +x(t)=x +����� = +��1 + K(−2b + 6cx − 12dx2) +�� < 1. +Hence, a stable equilibrium of map (3) is a real solution of +� +� +� +� +� +� +� +� +� +� +� +x = x + K(a − 2bx + 3cx2 − 4dx3), +K(−2b + 6cx − 12dx2) < 0, +2 + K(−2b + 6cx − 12dx2) > 0, +x > 0, a > 0, b > 0, c > 0, d > 0. +(11) +Obviously, analyzing the stable equilibrium by substituting the closed-form solutions (10) into (11) +is complicated and impractical. In comparison, the approach applied in the analysis of Model 1 does +not require explicitly solving any closed-form equilibrium. If the analytical solution has a complicated +expression or even if there are no closed-form solutions, our approach still works in theory. +Concerning the border polynomial of system (11), we compute +discr(K(a − 2bx + 3cx2 − 4dx3)) = −16K5dR1, +res(K(a − 2bx + 3cx2 − 4dx3), K(−2b + 6cx − 12dx2)) = −16K5dR1, +res(K(a − 2bx + 3cx2 − 4dx3), 2 + K(−2b + 6cx − 12dx2)) = −16K2dR2, +res(K(a − 2bx + 3cx2 − 4dx3), x) = −Ka, +where +R1 = 108a2d2 − 108abcd + 27ac3 + 32b3d − 9b2c2, +R2 = 108K3a2d2 − 108K3abcd + 27K3ac3 + 32K3b3d − 9K3b2c2 − 24Kbd + 9Kc2 − 8d. +Therefore, the border polynomial is −16384 d4K14aR2 +1R2, the zeros of which divide the parameter set +{(a, b, c, d, K) | a, b, c, d, K > 0} into separated regions. The PCAD method [5] permits us to select at +least one sample point from each region. In Table 1, we list the 30 selected sample points and the +corresponding numbers of distinct real solutions of system (11). +Table 1: Selected Sample Points in {(a, b, c, d, K) | a, b, c, d, K > 0} +(a, b, c, d, K) +num +R1 +R2 +(a, b, c, d, K) +num +R1 +R2 +(1, 1, 1/4, 1/64, 1/2) +2 +− +− +(1, 1, 1/4, 1/64, 1) +1 +− ++ +(1, 1, 1/4, 1/64, 2) +0 +− +− +(1, 1, 1/4, 19/1024, 1) +2 +− +− +(1, 1, 1/4, 19/1024, 2) +1 +− ++ +(1, 1, 1/4, 19/1024, 3) +0 +− +− +(1, 1, 1/4, 1/16, 1) +1 ++ +− +(1, 1, 1/4, 1/16, 2) +0 ++ ++ +(1, 1, 1/4, 1, 1/2) +1 ++ +− +(1, 1, 1/4, 1, 1) +0 ++ ++ +(1, 1, 3/8, 1/64, 1/8) +1 ++ +− +(1, 1, 3/8, 1/64, 1) +0 ++ ++ +(1, 1, 3/8, 1/32, 1/4) +2 +− +− +(1, 1, 3/8, 1/32, 1) +1 +− ++ +(1, 1, 3/8, 1/32, 17) +0 +− +− +(1, 1, 3/8, 49/1024, 1) +2 +− +− +(1, 1, 3/8, 49/1024, 4) +1 +− ++ +(1, 1, 3/8, 49/1024, 8) +0 +− +− +(1, 1, 3/8, 1/16, 1) +1 +− ++ +(1, 1, 3/8, 1/16, 3) +0 +− +− +(1, 1, 3/8, 1, 1/2) +1 ++ +− +(1, 1, 3/8, 1, 1) +0 ++ ++ +(1, 1, 15/32, 1/16, 1/2) +1 ++ +− +(1, 1, 15/32, 1/16, 1) +0 ++ ++ +(1, 1, 15/32, 3/32, 1) +1 ++ +− +(1, 1, 15/32, 3/32, 8) +0 ++ ++ +(1, 1, 15/32, 1, 1/2) +1 ++ +− +(1, 1, 15/32, 1, 1) +0 ++ ++ +(1, 1, 1, 1, 1/2) +1 ++ +− +(1, 1, 1, 1, 1) +0 ++ ++ +According to Table 1, one can see that system (11) has one real solution if and only if R1 < 0, R2 > 0 +or R1 > 0, R2 < 0. Moreover, a necessary condition that system (11) has two real solutions is that +8 + +R1 < 0 and R2 < 0, which is not a sufficient condition, however. For example, at (a, b, c, d, K) = +(1, 1, 1/4, 1/64, 2), system (11) has no real solutions but R1 < 0 and R2 < 0 are fulfilled. To acquire +the necessary and sufficient condition, additional polynomials (R3 and R4) are needed, which can be +found in the so-called generalized discriminant list and can be picked out by repeated trials. Regarding +the generalized discriminant list, readers may refer to [32] for more details. Due to space limitations, +we directly report below the necessary and sufficient condition that system (11) has two real solutions +without giving the calculation details: +R1 < 0, R2 < 0, R3 > 0, R4 < 0, +where +R3 = 8Kbd − 3Kc2 + 8d, +R4 = 432K2a2d3 − 432K2abcd2 + 108K2ac3d + 128K2b3dt2 − 36K2b2c2d + 192Kb2d2 +− 144Kbc2d + 27Kc4 + 64bd2 − 24c2d. +We continue to analyze the bifurcations of this model. An equilibrium x of map (3) may undergo +a period-doubling bifurcation if +dx(t + 1) +dx(t) +���� +x(t)=x += 1 + K(−2b + 6cx − 12dx2) = −1. +Hence, a period-doubling bifurcation may occur if the following system has at least one real solution. +� +� +� +� +� +x = x + K(a − 2bx + 3cx2 − 4dx3), +K(−2b + 6cx − 12dx2) + 2 = 0, +x > 0, a > 0, b > 0, c > 0, d > 0. +(12) +By using the method of triangular decomposition1, we transform the solutions of the first two equations +of system (12) into zeros of the triangular set +T = [(8Kbd − 3Kc2 + 4d)x − 6adK + bcK − c, R2]. +Obviously, the system {T = 0, x > 0, a > 0, b > 0, c > 0, d > 0} has at least one real positive +solution if R2 = 0 and x = (6adK − bcK + c)/(8Kbd − 3Kc2 + 4d) > 0, i.e., +R2 = 0, R5 > 0, +where +R5 = (6adK − bcK + c)(8Kbd − 3Kc2 + 4d) += 48K2abd2 − 18K2ac2d − 8K2b2cd + 3K2bc3 + 24Kad2 + 4Kbcd − 3Kc3 + 4cd. +Similarly, concerning the occurrence of a pitchfork bifurcation, we consider +� +� +� +� +� +x = x + K(a − 2bx + 3cx2 − 4dx3), +K(−2b + 6cx − 12dx2) = 0, +x > 0, a > 0, b > 0, c > 0, d > 0, +(13) +and count the number of stable equilibria. More details are not reported here due to space limitations. +We summarize all the obtained results in the following theorem. +1The method of triangular decomposition can be viewed as an extension of the method of Gaussian elimination. The +main idea of both methods is to transform a system into a triangular form. However, the triangular decomposition +method is available for polynomial systems, while the Gaussian elimination method is just for linear systems. Refer to +[30, 16, 12, 29] for more details. +9 + +Theorem 1. Model 2 has at most two stable equilibria. +Specifically, there exists just one stable +equilibrium if +R1 < 0, R2 > 0 or R1 > 0, R2 < 0, +and there exist two stable equilibria if +R1 < 0, R2 < 0, R3 > 0, R4 < 0. +Moreover, there is a period-doubling bifurcation if +R2 = 0, R5 > 0, +and there is a pitchfork bifurcation if +R1 = 0, R2 > 0, R6 > 0 or R1 = 0, R2 > 0, R4 < 0, R6 > 0, +where +R6 = 48abd2 − 18ac2d − 8b2cd + 3bc3. +Remark 1. To the best of our knowledge, the stability results regarding the parameters a, b, c, d, K +reported in Theorem 1 are new although the special case of a = 3.6, b = 2.4, c = 0.6, d = 0.05 has +been discussed in [22]. The two parameters K, a play more ambitious roles than others in practice for +K controls the speed of adjusting the monopolist’s output and a is the difference between the initial +product price of the market without any supply and the initial marginal cost of the firm without any +production. By fixing b = 2.4, c = 0.6 and d = 0.05, we depict the (a, K) parameter plane in Figure +2, where the region for the existence of one stable equilibrium is colored in yellow, while the region for +the existence of two stable equilibria is colored in blue-gray. Model 2 behaves differently from typical +oligopolies with gradient mechanisms. As shown by Figure 2, for instance, even if the adjustment +speed K is quite large, there always exist some values of a such that Model 2 is stable. Moreover, for +a fixed value of K greater than around 1.7, Model 2 undergoes from instability to stability and then +back to instability twice as the parameter a changes from low to high. +Figure 2: The two-dimensional (a, K) parameter plane of Model 2 with the other parameters fixed: +b = 2.4, c = 0.6, and d = 0.05. The region for the existence of one stable equilibrium is colored in +yellow, while that of two stable equilibria is colored in blue-gray. +10 + +6- +pitchfork bifurcation +5 +curves (R1=0) +4- +K 3 +period-doubling bifurcation +curves (R2=0) +2 +1 +0 +0 +1 +2 +3 +4 +5 +6 +a +R=0 +R=04 +Periodic Solutions +From an economic point of view, it is realistic to assume that a boundedly rational firm can not learn +the pattern behind output and profits if periodic dynamics take place. In this regard, we investigate +the existence and stability of periodic solutions with relatively lower orders in this section. +Let I be an interval of real numbers, and let F : I → R be a function. If x ∈ I, suppose that +F 0(x) represents x and F n+1(x) denotes F(F n(x)) for n ∈ {0, 1, . . .}. A point p ∈ I is said to be a +periodic point with period n or order n if p = F n(p), and p ̸= F k(p) for any 1 ≤ k < n. If p is a point +with period n, we call p �→ F 1(p) �→ · · · �→ F n(p) = p a n-cycle orbit. Furthermore, a point y ∈ I +with period k is said to be asymptotically stable if there exists δ such that |F k(x) − y| < |x − y| for all +x ∈ (y − δ, y + δ). +The following lemma can be found in [15], which provides an algebraic criterion to verify the +stability of a periodic point. +Lemma 3. Assume that y ∈ I is a periodic point of F with period k. If F is differentiable at the points +y, F(y), . . . , F k−1(y), then y is asymptotically stable if +����� +k−1 +� +i=0 +d +dxF(yi) +����� < 1, +where yi = F i(y). +4.1 +Model 1 +We start by considering the existence of periodic orbits with order two. Assume that there is a 2-cycle +orbit x �→ y �→ x, where �→ stands for the iteration map (2). Thus, we have +y = x + f(e − x3), +x = y + f(e − y3). +(14) +Obviously, x ̸= y should be guaranteed. Otherwise, x �→ y �→ x will degenerate into an equilib- +rium. Then, the problem of determining the existence of 2-cycles is transformed into determining the +existence of real solutions of +� +� +� +� +� +� +� +� +� +� +� +y = x + f(e − x3), +x = y + f(e − y3), +x ̸= y, +x > 0, y > 0, e > 0, f > 0. +(15) +Since the above system involves two variables x and y, the approach used in Section 3 (feasible +only for univariate systems) might not be directly employed herein. +Remark 2. However, we can transform system (15) equivalently into univariate systems based on its +triangular decomposition. Specifically, the triangular decomposition method permits us to decompose +the equation part (14) into the following two triangular sets. +T11 = [y − x, x3 − e], +T12 = [y + fx3 − x − ef, f3x6 − 3f2x4 − 2ef3x3 + 3fx2 + 3ef2x + e2f3 − 2]. +Since the first polynomial in T11 is y − x, which implies that x = y. Thus, the zeros of T11 are not +of our concern. We only focus on T12, where the first polynomial y + fx3 − x − ef has degree one +with respect to y. Therefore, one can directly solve y = −fx3 + x + ef and substitute it into relative +inequalities of system (15). In short, system (15) can be equivalently transformed into the following +univariate system. +� +� +� +� +� +f3x6 − 3f2x4 − 2ef3x3 + 3fx2 + 3ef2x + e2f3 − 2 = 0, +− fx3 + x + ef > 0, +x > 0, e > 0, f > 0. +After that, the approach in Section 3 can be applied. The results show that the above system has two +real solutions if and only if 8/27 < e2f3 < 2. It is evident that these two real solutions belong to the +11 + +same 2-cycle orbit because x, y are symmetric and can be replaced with each other. Therefore, there +exists at most one 2-cycle orbit in Model 1. +According to Lemma 3, to determine whether the discovered 2-cycle is stable, we consider (15) +together with the condition +���� +d(x + f(e − x3)) +dx +× d(y + f(e − y3)) +dy +���� < 1, +i.e., +��(1 − 3fx2)(1 − 3fy2) +�� < 1. +The technique introduced in Remark 2 is needed to transform the system into a univariate one. +According to our calculations, the unique 2-cycle orbit is stable if and only if 729e4f6 − 3294e2f3 + +1664 > 0 or equivalently 8/27 < e2f3 < (61 − 11 +√ +17)/27. We collect the aforementioned results in +the following theorem. +Theorem 2. Model 1 has at most one 2-cycle orbit, which exists if +8/27 < e2f3 < 2. +Furthermore, this unique 2-cycle is stable if +8/27 < e2f3 < 61 − 11 +√ +17 +27 +, +or approximately +0.2962962963 < e2f3 < 0.5794754859. +The measurement of the magnitude of periodic orbits is economically interesting for it characterizes +the size of fluctuations in dynamic economies. For a n-cycle orbit p1 �→ p2 �→ · · · pn �→ p1, a direct +definition of the magnitude measure is +d = |p1 − p2| + |p2 − p3| + · · · + |pn−1 − pn| + |pn − p1|. +However, to obtain better mathematical properties, we square each item and define the magnitude +measure to be +d = (p1 − p2)2 + (p2 − p3)2 + · · · + (pn−1 − pn)2 + (pn − p1)2. +For a 2-cycle orbit x �→ y �→ x in Model 1, the magnitude measure becomes d = (x − y)2 + (y − x)2. +Thus, we have +� +� +� +� +� +d − (x − y)2 − (y − x)2 = 0, +− y + x + f(e − x3) = 0, +− x + y + f(e − y3) = 0. +Using the method of triangular decomposition, we decompose the solutions of the above system into +zeros of the following two triangular sets. +T21 = [ y − x, x3 − e, d ], +T22 = [ y + x3f − ef − x, +(d2f3 + 4df2 + 4f)x2 + (−6def3 − 12ef2)x + 36e2f3 − 2f2d2 − 8fd − 8, +f3d3 − 12f2d2 − 60fd + 216e2f3 − 64 ]. +The first polynomial y − x in T21 implies that x = y. Thus, T21 is not of concern since it corresponds +to equilibria rather than 2-cycle orbits. We focus on the last polynomial f3d3 − 12f2d2 − 60fd + +216e2f3 − 64 in T22. By solving d from this polynomial, we obtain three solutions: +d1 = 2 +f +�3H +2 + 6 +H + 2 +� +, +d2, d3 = 2 +f +� +−3H +4 − 3 +H + 2 ± i +√ +3 +2 +�3H +2 − 6 +H +�� +, +12 + +where +H = +3� +8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2. +Here, only the real solution d1 is meaningful. Therefore, the magnitude measure of the unique 2-cycle +orbit in Model 1 can be expressed as +d = 2 +f +� +3 3� +8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 +2 ++ +6 +3� +8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 + 2 +� +. +In the rest of this section, similar calculations as above are repeated. We omit these computation +details due to space limitations. Concerning 3-cycle orbits in Model 1, we need to count real solutions +of +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +y = x + f(e − x3), +z = y + f(e − y3), +x = z + f(e − z3), +x ̸= y, x ̸= z, +x > 0, y > 0, z > 0, e > 0, f > 0. +Based on a series of computations, we derive the following theorem. +Theorem 3. Model 1 has no 3-cycle orbits for all possible parameter values such that e, f > 0. +For a 4-cycle orbit x �→ y �→ z �→ w �→ x, we have the system +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +y = x + f(e − x3), +z = y + f(e − y3), +w = z + f(e − z3), +x = w + f(e − w3), +x ̸= y, x ̸= z, x ̸= w, +x > 0, y > 0, z > 0, e > 0, f > 0. +Furthermore, the following condition is required to guarantee that the considered 4-cycle is stable. +���� +d(x + f(e − x3)) +dx +× d(y + f(e − y3)) +dy +× d(z + f(e − z3)) +dz +× d(w + f(e − w3)) +dw +���� < 1, +i.e., +��(1 − 3fx2)(1 − 3fy2)(1 − 3fz2)(1 − 3fw2) +�� < 1. +As the polynomials involved in the conditions of the existence and stability of 4-cycle orbits are +extremely complicated, we report below the obtained results in an approximate style. +Theorem 4. Model 1 has at most one 4-cycle orbit, which exists if +0.5794754859 < e2f3 < 1.237575627. +Furthermore, this unique 4-cycle is stable if +0.5794754859 < e2f3 < 0.6673871142. +Figure 3 (a) depicts the phase diagram of the unique 4-cycle in Model 1 with e = 0.6 and f = 1.2. +Since e2f3 = 0.62208 ∈ (0.5794754859, 0.6673871142), this unique 4-cycle in Model 1 is asymptotically +stable according to Theorem 4. Actually, the horizontal coordinates of A, B, C, D, i.e., x, y, z, w, are +the four points in the 4-cycle orbit. For the sake of simplicity, we connect A, B, C, D with lines and +use the simplified phase diagram as Figure 3 (b) to demonstrate periodic solutions in the rest of this +paper. +13 + +(a) phase diagram. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +x(t-1) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +x(t) +A +B +D +C +(b) simplified phase diagram. +Figure 3: The unique stable 4-cycle in Model 1 with e = 0.6 and f = 1.2. +Furthermore, by using the same approach as we computed the magnitude of the 2-cycle orbit, we +conclude that if a 4-cycle x �→ y �→ z �→ w �→ x exists in Model 1, its magnitude measure equals to +d = 4 +f +� +3 3� +8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 +2 ++ +6 +3� +8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 + 2 +� +, +which is twice as large as that of the 2-cycle orbit. +The parameter plane of Model 1 is shown in Figure 4. One can see that the parameter region for +the stability of the unique equilibrium (2-cycle or 4-cycle orbit) constitutes a connected set. Moreover, +the three regions for the stability of the equilibrium, 2-cycle, and 4-cycle adjoin without any gap. In +the next subsection, one will find that the topological structure of the parameter space of Model 2 is +much more complex than that of Model 1. +Figure 4: The parameter plane of Model 1. +The light blue, yellow, and light orange regions are +the parameter regions for the stability of the 4-cycle orbit, the 2-cycle orbit, and the equilibrium, +respectively. +Figure 5 depicts the two-dimensional bifurcation diagram of Model 1 for (e, f) ∈ [0.6, 1.6]×[0.6, 1.6]. +For additional information regarding two-dimensional bifurcation diagrams, readers can refer to [13]. +14 + +c +A +1 +0.8 +0.6 +B +0.4 +D +0.2 +- +- +x +Z +y +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +1. +x(t-1)21.2In the numerical simulations of Figure 5, we set the initial state to be x(0) = 1.0. Parameter points +corresponding to periodic orbits with different orders are marked in different colors. For example, +parameter points are colored in dark red if the order is just one (equilibria) and are marked in black +if the order is greater than or equal to 24 (complex trajectories). In the case that the order is greater +than 24, the black points may be viewed as the parameter values where complex dynamics such as +chaos take place. Moreover, we also use black to mark those parameter points where the trajectories +diverge to ∞. One can see that Figure 5 confirms the theoretical results presented in Figure 4. +In Figure 5, the transitions between different types of periodic orbits can also be observed. One can +see that the equilibrium loses its stability through a series of period-doubling bifurcations as the value +of e or f increases. For example, along the line of e = 1.0, the unique stable equilibrium bifurcates +into a stable 2-cycle orbit at f = 0.6665, which further bifurcates into a 4-cycle orbit at f = 0.8339. +There is a stable 8-cycle orbit when f ∈ (0.8744, 0.8826). Finally, chaotic dynamics take place if the +value of f is large enough. Additional details can be found in the one-dimensional bifurcation diagram +presented in Figure 6, where we fix e = 1.0 and choose x(0) = 1.1 to be the initial state of iterations. +Figure 5: The two-dimensional bifurcation diagram of Model 1 for (e, f) ∈ [0.6, 1.6] × [0.6, 1.6]. We +choose x(0) = 1.0 to be the initial state of the iterations. +4.2 +Model 2 +The formulation (3) of Model 2 involves five parameters, which might be particularly complex for +symbolic computations of searching periodic solutions. In what follows, we keep K as the only pa- +rameter and assume that a = 3.6, b = 2.4, c = 0.6, and d = 0.05. This setting is meaningful and +has been discussed by several economists, e.g., Puu [26], Al-Hdaibat and others [1], Matsumoto and +Szidarovszky [22]. +15 + +1.6 +1.5 +20 +1.4 +1.3 +15 +1.2 +f 1.1 +1.0 +10 +0.9 +0.8 +5 +0.7 +0.6 - +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +eFigure 6: The one-dimensional bifurcation diagram of Model 1 with respect to f by fixing e = 1.0. +We choose x(0) = 1.1 to be the initial state of the iterations. +Let x �→ y �→ x be a 2-cycle orbit. Hence, we have +� +� +� +� +� +� +� +� +� +� +� +y = x + K(3.6 − 4.8x + 1.8x2 − 0.2x3), +x = y + K(3.6 − 4.8y + 1.8y2 − 0.2y3), +x ̸= y, +x > 0, y > 0, K > 0. +(16) +Furthermore, the following condition is required if the stability of the 2-cycle is considered. +|S(x) · S(y)| < 1, +where +S(x) = d(x + K(3.6 − 4.8x + 1.8x2 − 0.2x3) +dx += 1 − K(4.8 − 3.6x + 0.6x2). +(17) +According to our computations, the following theorem is obtained. +Theorem 5. In Model 2, the possible number of 2-cycle orbits is zero (no real solutions in system (16)) +or three (six real solutions in system (16)). There exist three 2-cycle orbits if K > 5/3. Moreover, +two of them are stable if +5/3 < K < (5 +√ +5 − 5)/3, +or approximately +1.666666667 < K < 2.060113296. +To measure the magnitude of a 2-cycle orbit x �→ y �→ x, we also use d = (x − y)2 + (y − x)2. The +method of triangular decomposition permits us to decompose the solutions of +� +� +� +� +� +d = (x − y)2 + (y − x)2, +y = x + K(3.6 − 4.8x + 1.8x2 − 0.2x3), +x = y + K(3.6 − 4.8y + 1.8y2 − 0.2y3) +16 + +1.5 +1.0 +0.5 +X +0.0 +-0.5 +-1.0 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +finto zeros of the following triangular systems. +T31 = [ y − 3, x − 3, d ], +T32 = [ y − x, x2 − 6x + 6, d ], +T33 = [ y + x − 6, Kx2 − 6Kx + 6K − 10, Kd − 24K − 80 ], +T34 = [ 5y + x3K − 9Kx2 + (24K − 5)x − 18K, +K2x4 − 12K2x3 + (51K2 − 5K)x2 + (−90K2 + 30K)x + 54K2 − 45K + 25, +Kd − 6K + 10 ], +where the last two polynomials Kd − 24K − 80 and Kd − 6K + 10 in T33 and T34 are of our concern. +We conclude that d = (24K + 80)/K or d = (6K − 10)/K. One can see that two of the three 2-cycle +orbits possess the same magnitude. +For a 3-cycle orbit x �→ y �→ z �→ x, we consider the system +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +y = x + K(3.6 − 4.8x + 1.8x2 − 0.2x3), +z = y + K(3.6 − 4.8y + 1.8y2 − 0.2y3), +x = z + K(3.6 − 4.8z + 1.8z2 − 0.2z3), +x ̸= y, x ̸= z, +x > 0, y > 0, z > 0, K > 0, +(18) +as well as the stability condition +|S(x) · S(y) · S(z)| < 1, +(19) +where S(x) is given in (17). Based on a series of calculations, we have the following theorem. +Theorem 6. In Model 2, all possible cases for the number of (stable) 3-cycle orbits are listed in Table +2, where +m1 ≈ 2.417401607, m2 ≈ 2.434714456, m3 ≈ 3.302953127, m4 ≈ 3.303122765. +Readers can refer to Remark 3 to understand how these mi are obtained. +Table 2: Numbers of (stable) 3-cycle orbits in Model 2 +K ∈ +(0, m1) +(m1, m2) +(m2, m3) +(m3, m4) +(m4, +∞) +3-cycles +0 +4 +4 +8 +8 +Stable 3-cycles +0 +2 +0 +2 +0 +Remark 3. As aforementioned, the border polynomial plays an important role. However, one can derive +that the properties of the border polynomial reported in Lemma 2 will retain if we use the squarefree +part of the border polynomial. The squarefree part SP of the border polynomial of (18)+(19) is +simpler, which is given in Appendix. In Theorem 6, m1, . . . , m4 are the real roots of SP. A rigorous +style of writing Theorem 6 is to express the conditions using factors in SP. However, this would be +quite tedious. Since only one parameter, i.e., K, is involved in SP, the regions divided by zeros of +SP are indeed intervals and can be approximately described as in Theorem 6. However, it should be +noticed that the values of m1, . . . , m4 can be made arbitrarily accurate if we want because the exact +expression of SP has already been obtained. +Figure 7 depicts all the 3-cycle orbits in Model 2 with K = 3.303 ∈ (m3, m4), where the 6 unstable +cycles are marked in red and the 2 stable cycles are marked in blue. It is worth noting that two of +the unstable 3-cycle orbits in red almost coincide with the stable ones in blue, but they are different. +We should underline that this dynamic phenomenon, derived by symbolic computations, may be too +subtle to observe through numerical simulations. +17 + +0 +1 +2 +3 +4 +5 +6 +x(t-1) +0 +1 +2 +3 +4 +5 +6 +x(t) +Figure 7: The 3-cycle orbits in Model 2 with K = 3.303. The 6 unstable cycles are marked in red, +while the 2 stable ones are marked in blue. +Moreover, if measuring the magnitude of the 3-cycle orbit x �→ y �→ z �→ x with d = (x − y)2 + +(y − z)2 + (z − x)2, then we have +K4d4 + (−54K4 − 90K3)d3 + (972K4 + 2700K3 + 1800K2)d2 ++ (−6696K4 − 19440K3 − 5400K2 + 27000K)d ++ 15552K4 + 38880K3 − 32400K2 − 162000K + 270000 = 0. +The above condition on K and d is plotted in Figure 8. +Figure 8: The magnitude d of the possible 3-cycle orbits in Model 2 as the variation of K. +Similarly, we analyze the 4-cycle and 5-cycle orbits in Model 2, and report the obtained results in +the sequel. +Theorem 7. In Model 2, all possible cases for the number of (stable) 4-cycle orbits are given in Table +3, where +m1 ≈ 2.060113296, m2 ≈ 2.146719591, m3 ≈ 2.579725065, m4 ≈ 2.581385365, m5 ≈ 3.062775154, +m6 ≈ 3.070194019, m7 ≈ 3.279225134, m8 ≈ 3.279260335, m9 ≈ 3.319881360, m10 ≈ 3.319889702. +18 + +Readers can refer to Remark 3 to understand how these mi are obtained. +Table 3: Numbers of (stable) 4-cycle orbits in Model 2 +K ∈ +(0, m1) +(m1, m2) +(m2, m3) +(m3, m4) +(m4, m5) +(m5, m6) +4-cycles +0 +2 +2 +6 +6 +10 +Stable 4-cycles +0 +2 +0 +2 +0 +2 +K ∈ +(m6, m7) +(m7, m8) +(m8, m9) +(m9, m10) +(m10, +∞) +4-cycles +10 +14 +14 +18 +18 +Stable 4-cycles +0 +2 +0 +2 +0 +In Figure 9, we show all the 4-cycle orbits in Model 2 with K = 3.319885 ∈ (m9, m10), where the +16 unstable cycles are marked in red and the 2 stable ones are marked in blue. If we measure the +magnitude of the 4-cycle orbit x �→ y �→ z �→ w �→ x with d = (x−y)2 +(y −z)2 +(z −w)2 +(w −x)2, +then d must satisfy one of the following equations. +Kd − 12K + 20 = 0, +Kd − 48K − 160 = 0, +K2d2 + (−36K2 − 60K)d + 288K2 + 960K + 1600 = 0, +C4(K, d) = 0, +where C4(K, d) is a complex polynomial given in Appendix. +Figure 9: The 4-cycle orbits in Model 2 with K = 3.319885. The 16 unstable cycles are marked in +red, while the 2 stable ones are marked in blue. +Theorem 8. In Model 2, all possible cases for the number of (stable) 5-cycle orbits are listed in Table +4, where +m1 ≈ 2.323208379, m2 ≈ 2.326320457, m3 ≈ 2.509741151, m4 ≈ 2.510528490, +m5 ≈ 2.632885028, m6 ≈ 2.633089005, m7 ≈ 2.997641294, m8 ≈ 2.997736262, +m9 ≈ 3.113029799, m10 ≈ 3.113069634, m11 ≈ 3.197332995, m12 ≈ 3.197354147, +m13 ≈ 3.219425160, m14 ≈ 3.219440784, m15 ≈ 3.269613400, m16 ≈ 3.269618202, +m17 ≈ 3.288059620, m18 ≈ 3.288062995, m19 ≈ 3.314977518, m20 ≈ 3.314978815, +m21 ≈ 3.324008184, m22 ≈ 3.324008826, m23 ≈ 3.332961824, m24 ≈ 3.332961850. +19 + +5 +4 +x(t) +2 +1 +2 +3 +4 +5 +x(t-1)9Readers can refer to Remark 3 to understand how these mi are obtained. +Table 4: Numbers of (stable) 5-cycle orbits in Model 2 +K ∈ +(0, m1) +(m1, m2) +(m2, m3) +(m3, m4) +(m4, m5) +5-cycles +0 +4 +4 +8 +8 +Stable 5-cycles +0 +2 +0 +2 +0 +K ∈ +(m5, m6) +(m6, m7) +(m7, m8) +(m8, m9) +(m9, m10) +5-cycles +12 +12 +16 +16 +20 +Stable 5-cycles +2 +0 +2 +0 +2 +K ∈ +(m10, m11) +(m11, m12) +(m12, m13) +(m13, m14) +(m14, m15) +5-cycles +20 +24 +24 +28 +28 +Stable 5-cycles +0 +2 +0 +2 +0 +K ∈ +(m15, m16) +(m16, m17) +(m17, m18) +(m18, m19) +(m19, m20) +5-cycles +32 +32 +36 +36 +40 +Stable 5-cycles +2 +0 +2 +0 +2 +K ∈ +(m20, m21) +(m21, m22) +(m22, m23) +(m23, m24) +(m24, +∞) +5-cycles +40 +44 +44 +48 +48 +Stable 5-cycles +0 +2 +0 +2 +0 +In Figure 10, we plot all possible 5-cycle orbits in Model 2 with K = 3.33296183 ∈ (m23, m24), +where the 46 unstable cycles are marked in red and the 2 stable ones are marked in blue. +Figure 10: The 5-cycle orbits in Model 2 with K = 3.33296183. The 46 unstable cycles are marked in +red, while the 2 stable ones are marked in blue. +By Theorems 6, 7 and 8, one can see the parameter space of Model 2 is quite different from +that of Model 1 in the sense that the stability regions for the 3-cycle, 4-cycle and 5-cycle orbits +are disconnected sets formed by many disjoint portions. Therefore, the topological structures of the +regions for stable periodic orbits in Model 2 are much more complex than those in Model 1. This may +be because the inverse demand function of Model 2 has an inflection point. However, the following +observations of Model 2 are similar to Model 1. Theorem 5 shows that the stability region for the +2-cycles is a connected interval. In Model 2, the right boundary of the stability region for the 2-cycles +is the same as the left boundary of the stability region for the 4-cycles. When a = 3.6, b = 2.4, c = 0.6, +and d = 0.05, by Theorem 1 we know that Model 2 has stable equilibria if K ∈ (0, 5/3), which adjoins +the stability region for the 2-cycles. Moreover, in Model 2, the stability regions for cycles with distinct +periods may not intersect with each other, which means that multistability might only arise among +20 + +5 +4 +x(t) 3 +2 +0 +1 +2 +3 +4 +5 +x(t-1)96periodic orbits with the same period. +Figure 11 depicts the two-dimensional bifurcation diagram of Model 2 for (a, K) ∈ [2.5, 5.0] × +[0.0, 3.0]. We fix the parameters b = 2.4, c = 0.6, d = 0.05, and set the initial state to be x(0) = 1.0. +Similarly, we use different colors to mark parameter points corresponding to trajectories with different +periods. Parameter points are marked in black if the corresponding orbits have orders greater than +24. +Furthermore, we also use black to mark the parameter points where the trajectories diverge. +One can see that Figure 11 confirms the theoretical results reported in Theorem 1. However, Fig- +ure 11 generated by numerical simulations is not accurate compared to Figure 2 based on symbolic +computations. +Figure 11: The two-dimensional bifurcation diagram of Model 2 for (a, K) ∈ [2.5, 5.0] × [0.0, 3.0]. We +fix the parameters b = 2.4, c = 0.6, d = 0.05, and choose x(0) = 1.0 to be the initial state of the +iterations. +Figure 12 depicts the one-dimensional bifurcation diagrams of Model 2 with respect to K by fixing +a = 3.3, b = 2.4, c = 0.6, and d = 0.05. The bifurcation diagrams are different if the selected initial +states of the iterations are distinct. +For example, in Figure 12 (a) and (b), the initial states are +selected to be x(0) = 1.0 and x(0) = 4.0, respectively. The difference may be because two stable +equilibria exist when K is relatively small and distinct initial states approach distinct equilibria. As +shown by Figure 12 (a), the trajectory converges to 1.058 when K < 1.1996 and converges to 4.384 +when K > 1.9874. In Figure 12, the occurrence of period-doubling bifurcations can also be observed. +Figure 13 depicts the one-dimensional bifurcation diagrams of Model 2 with respect to a by fixing +K = 2.2, b = 2.4, c = 0.6, and d = 0.05. In Figure 13 (a) and (b), the initial states of the iterations +are selected to be x(0) = 1.0 and x(0) = 4.0, respectively. Similarly, the two bifurcation diagrams are +different because of the selection of distinct initial states. Furthermore, pitchfork bifurcations can be +observed in Figure 13, where the number of stable equilibria changes from one to zero. +In Model 2, two stable equilibria may coexist (see the blue-gray region in Figure 2). The equilibrium +selection problem is interesting. The final outcome of the iterations depends not only on the values of +21 + +3.0 +2.7 +20 +2.4 +2.1 - +15 +1.8 +K 1.5 +1.2 +10 +0.9 +0.6 +5 +0.3 +0.0 - +2.5 +2.75 +3.0 +3.25 +3.5 +3.75 +4.0 +4.25 +4.5 +4.75 +5.0 +a(a) x(0) = 1.0. +(b) x(0) = 4.0. +Figure 12: The one-dimensional bifurcation diagrams of Model 2 with respect to K by fixing a = 3.3, +b = 2.4, c = 0.6, and d = 0.05. +(a) x(0) = 1.0. +(b) x(0) = 4.0. +Figure 13: The one-dimensional bifurcation diagrams of Model 2 with respect to a by fixing K = 2.2, +b = 2.4, c = 0.6, and d = 0.05. +22 + +4.5 +4.0 +3.5 +3.0 +x2.5 +2.0 +1.5- +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +K4.6 +4.5 +4.4 +4.3 +X +4.2 +4.1 +4.0 +3.9 +0 +2 +3 +1 +4 +5 +K6 +5 +4 +X +3 - +2 +1 +3.2 +3.3 +3.4 +3.5 +3.6 +3.7 +3.8 +3.9 +4.0 +a5.5 +5.0 +4.5 +4.0 +X3.5 +3.0 +2.5 +2.0 +1.5 +3.2 +3.3 +3.4 +3.5 +3.6 +3.7 +3.8 +3.9 +4.0 +athe parameters but also on the starting conditions of the game. According to our numerical simulations +of Model 2, the basins of attraction of coexisting equilibria have complicated structures. For example, +by fixing K = 0.5, a = 3.5, b = 2.4, c = 0.6, d = 0.05, we have two stable equilibria E1 = 1.19 and +E2 = 4.64. The basin of E1 is +B(E1) = (0, 3.168) ∪ (6.518, 7.577) ∪ (7.745, 7.781) ∪ (7.786, 7.789), +while that of E2 is +B(E2) = (3.168, 6.518) ∪ (7.577, 7.745) ∪ (7.781, 7.786). +Furthermore, when the initial state x(0) > 7.786, the trajectory will not converge to any of the two +stable equilibria but diverge to +∞. Take K = 1 and a = 4 as the other example. If the other +parameters keep unchanged, i.e., b = 2.4, c = 0.6, and d = 0.05, there are two stable equilibria +E1 = 4.99 and E2 = 1.99. Our simulations show that the basins of these two equilibria are +B(E1) = (0, 0.807) ∪ (2.0, 6.192) ∪ (6.431, 6.647) ∪ (6.653, 6.659), +and +B(E2) = (0.807, 2.0) ∪ (6.192, 6.431) ∪ (6.647, 6.653), +respectively. The escape set is (6.659, +∞), where the trajectory diverges. In short, in Model 2, the +basins of the two stable equilibria are disconnected sets and have complex topological structures. +5 +Chaotic Dynamics +In the bifurcation diagrams (Figures 6 and 12), one can observe that the dynamics of the two considered +models transition to chaos through period-doubling bifurcations as the adjustment speed increases. +From an economic point of view, if chaos appears, the pattern behind output and profits is nearly +impossible to learn even for completely rational players. Therefore, it is extremely hard for a firm to +handle a chaotic economy, where no market rules could be discovered and followed. +In this section, we rigorously prove the existence of chaos for the two models. +The following +famous lemma was first derived by Li and Yorke [15], which is mathematically deep and facilitates the +exploration of complicated dynamics arising in one-dimensional discrete dynamical systems. +Lemma 4. Let I be an interval of real numbers, and let F : I → R be a continuous function. Assume +that there exists a point x ∈ I such that +F 3(x) ≤ x < F(x) < F 2(x) +or +F 3(x) ≥ x > F(x) > F 2(x), +(20) +then the following two statements are true. +1. For each k ∈ {1, 2, . . .}, there is a point pk ∈ I with period k, i.e., F k(pk) = pk, and F i(pk) ̸= pk +for 1 ≤ i < k. +2. There is an uncountable set S ⊂ I (containing no periodic points), which satisfies the following +conditions: +(a) for any p, q ∈ S with p ̸= q, +lim sup +n→∞ |F n(p) − F n(q)| > 0, +(21) +and +lim inf +n→∞ |F n(p) − F n(q)| = 0; +(22) +(b) for every point p ∈ S and every periodic point q ∈ I, +lim sup +n→∞ |F n(p) − F n(q)| > 0. +(23) +23 + +Remark 4. Eq. (22) means that every trajectory in S can wander arbitrarily close to every other. +However, by (21) we know that no matter how close two distinct trajectories in S may come to each +other, they must eventually wander away. Furthermore, by (23) it is clear that every trajectory in S +goes away from any periodic orbit in I. If the two statements in the above lemma are both satisfied, +we say that there exist chaotic dynamics or chaos in the sense of Li-Yorke. +Therefore, we can conclude that “period three implies chaos” for one-dimensional discrete dynam- +ical systems. In Section 4, we have rigorously derived the existence of 3-cycle orbits in Model 2 if +K > 2.417401607, which proves that chaos would arise for an uncountable set of initial states in the +sense of Li-Yorke. +But in Model 1, we have proved that there are no solutions with period three. However, it can +not be concluded that there exist no chaotic trajectories since the existence of period three is not +a necessary but only a sufficient condition of chaos. In [20], Marotto indicated that the existence +of snapback repellers also implies chaos for general n-dimensional systems. However, Li and Chen +[14] pointed out that Marotto’s original definition of snapback repeller may result in an insufficiency, +and proposed the Marotto-Li-Chen Theorem. Thus, we give the following lemma for one-dimensional +systems by simplifying the Marotto-Li-Chen Theorem. Readers can refer to [11] for additional details. +Lemma 5. Let I be an interval of real numbers, and let F : I → R be a differentiable function. Assume +that +1. x ∈ I is an equilibrium, i.e., F(x) = x; +2. there exists a close interval S ⊂ I such that x is an inner point of S, and the derivative of F +has the absolute value greater than 1 at every point p ∈ S, i.e., |F ′(p)| > 1; +3. for some integer m > 1, there exists a point y ∈ S such that y ̸= x, F m(y) = x, and F ′(F k(y)) ̸= +0 for all 1 ≤ k ≤ m. +Then the system x(t + 1) = F(x(t)) is chaotic in the sense of Li-Yorke. +For Model 1, we have F(x) = x + f(e − x3) and F ′(x) = 1 − 3fx2. Then |F ′(x)| > 1 and x > 0 +imply that x > +� +2 +3f . Thus, if we can find x, y with x ̸= y such that both |F ′(x)| > 1 and |F ′(y)| > 1 +are satisfied, then there must exist one closed interval S containing x, y as inner points. In such a +case, it is obvious that |F ′(p)| > 1 for every point p ∈ S. Naturally, we start from m = 2 to verify the +conditions of Lemma 5 by counting real solutions of the following system. +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +x = x + f(e − x3), +x = F 2(y) = y + f(e − y3) + f(e − (y + f(e − y3))3), +|1 − 3fx2| > 1, +|1 − 3fy2| > 1, +|1 − 3f(y + f(e − y3))2| ̸= 0, +x ̸= y, +x > 0, y > 0, e > 0, f > 0. +The technique introduced in Remark 2 should be conducted first to transform the above system into +a univariate one. According to our calculations, the above system has at least one real solution if and +only if 8/27 < e2f3 < 64/27. Therefore, we conclude that Model 1 is chaotic in the sense of Li-Yorke +provided that 8/27 < e2f3 < 64/27. +6 +Concluding Remarks +It is known that a monopoly may exhibit complex dynamics such as periodic orbits and chaos al- +though it is the simplest oligopoly. In this study, we investigated two monopoly models with gradient +mechanisms, where the monopolists are knowledgeable firms. The two models are distinct mainly in +24 + +their inverse demand functions. Model 1 uses the inverse demand function of Naimzada and Ricchiuti +[25], while Model 2 employs that of Puu [26]. Different from widely applied numerical methods such +as numerical simulations and bifurcation continuation approaches, symbolic methods were applied in +this paper to analyze the local stability, periodic solutions, and even chaotic dynamics. Numerical +methods have some shortcomings, e.g., the computations may encounter the problem of instability, +which makes the results completely useless. In comparison, symbolic computations are exact, thus the +obtained results can be used to rigorously prove economic theorems in some sense. +By reproving the already-known results (Proposition 1) of the local stability and bifurcations +of Model 1, we explained in detail how our symbolic approach works. Afterward, the analysis of the +stability and bifurcations of Model 2 was conducted based on this approach. We acquired the complete +conditions of the local stability and bifurcations of Model 2 for the first time (see Theorem 1). In +Figure 2, it was observed that Model 2 behaves quite differently from typical oligopoly models with +gradient mechanisms. For example, even if the adjustment speed K is quite large, there always exist +some values of a (the difference between the initial commodity price and the initial marginal cost) +such that Model 2 has a stable equilibrium. Moreover, Model 2 may go from instability to stability +and then back to instability twice as the value of a increases. +From an economic point of view, the study of periodic solutions is of practical importance. Under +the assumption of bounded rationality, firms can not learn the pattern behind output and profits if +periodic dynamics take place. For the two models, we explored the periodic solutions with lower orders +as well as their local stability. Differences between the two models were found, e.g., 3-cycle orbits exist +in Model 2 but not in Model 1. In Model 1, the parameter region for the stability of the periodic +solution with a fixed order constitutes a connected set. In Model 2, however, the stability regions for +the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint portions. In other +words, the topological structures of the regions for stable periodic orbits in Model 2 are much more +complex than those in Model 1. The above differences may be because the inverse demand function of +Model 2 has an inflection point. According to the numerical simulations of Model 2, we found that the +basins of the two stable equilibria are disconnected sets and also have complex topological structures. +For a n-cycle orbit p1 �→ p2 �→ · · · pn �→ p1, we defined the magnitude measure to be +d = (p1 − p2)2 + (p2 − p3)2 + · · · + (pn−1 − pn)2 + (pn − p1)2. +For the two considered models, we analytically investigated the formulae for the magnitude of periodic +orbits with lower orders. +Furthermore, it is extremely hard for a firm to handle an economy when chaos appears. In such +a case, no market rules can be discovered and followed, and the pattern behind output and profits +is nearly impossible to learn even for completely rational players. In the bifurcation diagrams of the +two models, it seems that chaos occurs when the adjustment speed is large enough. We clarified this +observation analytically. By virtue of the fact “period three implies chaos”, we derived that Model 2 +is chaotic in the sense of Li-Yorke by proving the existence of 3-cycle orbits. However, there are no +3-cycles in Model 1, but the Marotto-Li-Chen Theorem permitted us to prove the existence of chaos +by finding snapback repellers. +In this paper, we take the assumption of knowledgeable players, which means the enterprise has +full information regarding the inverse demand function and can compute its marginal profit at any +time. In the real world, however, it is more reasonable to assume players to be limited rather than +knowledgeable. In this case, the enterprise does not know the form of the inverse demand function, +but possesses the values of output and price only in the past periods and estimates its marginal profit +with a simple difference formula. The investigation of the dynamics of limited firms might be an +important direction for our future study. +Acknowledgments +The authors wish to thank Dr. Bo Huang for the beneficial discussions and are grateful to the anony- +mous referees for their helpful comments. +25 + +This work has been supported by Philosophy and Social Science Foundation of Guangdong under +Grant No. GD21CLJ01, Major Research and Cultivation Project of Dongguan City University under +Grant Nos. 2021YZDYB04Z and 2022YZD05R, National Natural Science Foundation of China under +Grant No. 11601023, and Beijing Natural Science Foundation under Grant No. 1212005. +Declaration of competing interest +The authors declare no conflict of interest. +Appendix +SP = (972K8 + 19440K7 + 127575K6 + 162000K5 − 1552500K4 − 6412500K3 − 5062500K2 ++ 23437500K + 67187500)(8503056K12 + 191318760K11 + 1523464200K10 ++ 3754532250K9 − 14134854375K8 − 101982543750K7 − 146939062500K6 ++ 399469218750K5 + 1522072265625K4 + 261457031250K3 − 4576816406250K2 +− 1938867187500K + 13981445312500), +C4(K, d) = K8d8 + (−126K8 − 210K7)d7 + (6660K8 + 21300K7 + 17800K6)d6 + (−192024K8 +− 874800K7 − 1382400K6 − 731000K5)d5 + (3285360K8 + 18688320K7 + 41115600K6 ++ 39438000K5 + 13350000K4)d4 + (−33957792K8 − 221940000K7 − 588016800K6 +− 728172000K5 − 379740000K4 − 45500000K3)d3 + (206172864K8 + 1453101120K7 ++ 4191652800K6 + 5433912000K5 + 2183760000K4 − 1105200000K3 − 478000000K2)d2 ++ (−672686208K8 − 4870886400K7 − 14246409600K6 − 16185744000K5 ++ 2054160000K4 + 13262400000K3 − 7632000000K2 − 11520000000K)d + 906992640K8 ++ 6500113920K7 + 18223833600K6 + 13351392000K5 − 25284960000K4 +− 27302400000K3 + 65376000000K2 + 30720000000K − 102400000000. +References +[1] B. 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A complete algorithm for automated discovering of a class of +inequality-type theorems. Science in China Series F, 44:33–49, 2001. +28 + diff --git a/C9AzT4oBgHgl3EQfiP2U/content/tmp_files/load_file.txt b/C9AzT4oBgHgl3EQfiP2U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ecc3732bffd4a6d9323dbc81392115ccd59ca832 --- /dev/null +++ b/C9AzT4oBgHgl3EQfiP2U/content/tmp_files/load_file.txt @@ -0,0 +1,1310 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf,len=1309 +page_content='Complex dynamics of knowledgeable monopoly models with gradient mechanisms Xiaoliang Lia, Jiacheng Fub, and Wei Niu∗b,c aSchool of Digital Economics, Dongguan City University, Dongguan, China bSino-French Engineer School, Beihang University, Beijing, China cBeihang Hangzhou Innovation Institute Yuhang, Hangzhou, China Abstract In this paper, we explore the dynamics of two monopoly models with knowledgeable players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The first model was initially introduced by Naimzada and Ricchiuti, while the second one is sim- plified from a famous monopoly introduced by Puu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We employ several tools based on symbolic computations to analyze the local stability and bifurcations of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' To the best of our knowledge, the complete stability conditions of the second model are obtained for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We also investigate periodic solutions as well as their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Most importantly, we discover that the topological structure of the parameter space of the second model is much more complex than that of the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Specifically, in the first model, the parameter region for the stability of any periodic orbit with a fixed order constitutes a connected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the second model, however, the stability regions for the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, we find that the basins of the two stable equilibria in the second model are disconnected and also have complicated topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In addition, the existence of chaos in the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits in the two models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Keywords: monopoly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' gradient mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' periodic orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' chaos 1 Introduction Unlike a competitive market with a large number of relatively small companies producing homogeneous products and competing with each other, an oligopoly is a market supplied only by a few firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It is well known that Cournot developed the first formal theory of oligopoly in [7], where players are supposed to have the naive expectations that their rivals produce the same quantity of output as in the immediately previous period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Cournot introduced a gradient mechanism of adjusting the quantity of output and proved that his model has one unique equilibrium, which is globally stable provided that only two firms exist in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' A monopoly is the simplest oligopoly, which is a market served by one unique firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the existing literature, a market supplied by two, three, or even four companies is called a duopoly [17], a triopoly [19], or a quadropoly [21], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, a monopoly may also exhibit complex dynamic behaviors such as periodic orbits and chaos if the involved firm is supposed to be boundedly rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' As distinguished by Matsumoto and Szidarovszky [22], a boundedly rational monopolist is said to be knowledgeable if it has full information regarding the inverse demand function, and limited if it does not know the form of the inverse demand function but possesses the values of output and price only in the past two periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Knowledgeable and limited players have been considered in several monopoly models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, Puu [26] introduced a monopoly where the inverse demand function is a cubic func- tion with an inflection point, and the marginal cost is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this model, the monopolist is ∗Corresponding author: wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='niu@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='01497v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='TH] 4 Jan 2023 supposed to be a limited player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Puu indicated that there exist multiple (at most three) equilibria, and complex dynamics such as chaos may appear if the reactivity of the monopolist becomes sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, Puu’s model was reconsidered by Al-Hdaibat and others in [1], where a numerical continuation method is used to compute solutions with different periods and determine their stability regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In particular, they analytically investigated general formulae for solutions with period four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It should be mentioned that the equilibrium multiplicity and complex dynamics of Puu’s model might depend strictly on the inverse demand function that has an inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this regard, Naimzada and Ricchiuti [25] introduced a simpler monopoly with a knowledgeable player, where the inverse demand function is still cubic but has no inflection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It was discovered that complex dynamics can also arise, especially when the reaction coefficient to variation in profits is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Askar [2] and Sarafopoulos [27] generalized the inverse demand function of Naimzada and Ricchiuti to a function of a similar form, but the degree of their function could be any positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The difference is that the cost function in Askar’s model is linear but quadratic in Sarafopoulos’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Cavalli and Naimzada [4] studied a monopoly model characterized by a constant elasticity demand function, in which the firm is also assumed to be knowledgeable with a linear cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' They focused on the equilibrium stability as the variation of the price elasticity of demand and proved that there are two possible different cases, where elasticity has either a stabilizing or a mixed stabilizing/destabilizing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, Elsadany and Awad [8] explored a monopoly game with delays where the inverse demand is a log-concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Caravaggio and Sodini [3] considered a nonlinear model, where a knowledgeable monopolist provides a fixed amount of an intermediate good and then uses this good to produce two vertically differentiated final commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' They found that there are chaotic and multiple attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, continuous dynamical systems have also been applied in the study of monopolistic markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In [23], Matsumoto and Szidarovszky proposed a monopoly model formulated in continuous time and investigated the effect of delays in obtaining and implementing the output information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Motivated by the aforementioned work, other remarkable contributions including [9, 10] were done in this strand of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In our study, we consider two monopoly models formulated with discrete dynamical systems, where the players are supposed to be knowledgeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The two models are distinct mainly in their inverse demand functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The first model uses the inverse demand of Naimzada and Ricchiuti [25], while the second one employs that of Puu [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For both models, we analyze the existence and local stability of equilibria and periodic solutions by using tools based on symbolic computations such as the method of triangular decomposition and the method of partial cylindrical algebraic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It should be mentioned that different from numerical computations, symbolic computations are exact, thus the results can be used to rigorously prove economic theorems in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The main contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' To the best of our knowledge, the complete stability conditions of the second model are obtained for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We also investigate the periodic solutions in the two models as well as their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Most importantly, we find different topological structures of the parameter spaces of the two considered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Specifically, in the first model, the parameter region for the stability of any periodic solution with a fixed order constitutes a connected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the second model, however, the stability regions for the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In other words, the topological structures of the regions for stable periodic orbits in Model 2 are much more complex than those in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' This may be because the inverse demand function of Model 2 has an inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, according to our numerical simulations of Model 2, it is discovered that the basins of the two stable equilibria are disconnected and also have complex topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In addition, the existence of chaos in the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits in the two models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Section 2, we revisit the construction of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Section 3, the local stability of the equilibrium is thoroughly studied, and bifurcations through which the equilibrium loses its stability are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Section 4, the existence and stability of periodic orbits with relatively lower orders are explored for the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Section 5, we rigorously derive the existence of chaotic dynamics in the sense of Li-Yorke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The paper is concluded with some remarks in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 2 2 Basic Models Suppose a monopolist exists in the market, and the quantity of its output is denoted as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We use P(x) to denote the price function (also called inverse demand function), which is assumed to be downward sloping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', dP(x) dx < 0, for any x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (1) It follows that P(x) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The demand function (the inverse of P(x)) exists and is also downward sloping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, the cost function is denoted as C(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Then the profit is Π(x) = P(x)x − C(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The monopolist is assumed to adopt a gradient mechanism of adjusting its output to achieve increased profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Suppose that the firm is a knowledgeable player, which means that it has full information regarding the inverse demand function P(x) and has the capability of computing the marginal profit dΠ/dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The firm adjusts its output by focusing on how the variation of x affects the variation of Π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Specifically, the adjustment process is formulated as x(t + 1) = x(t) + K dΠ(x(t)) dx(t) , K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Since K > 0, a positive marginal profit induces the monopolist to adjust the quantity of its output in a positive direction and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The first model considered in this paper was initially proposed by Naimzada and Ricchiuti [25], where a cubic price function without the inflection point is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We restate the formulation of this model in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The price function is cubic and the cost function is linear as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' P(x) = a − bx3, C(x) = cx, where a, b, c are parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The downward sloping condition (1) is guaranteed if dP/dx = −3bx2 < 0, that is if b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, assume that the marginal cost dC/dx = c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We adopt the general principle of setting price above marginal cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', P(x) − c > 0 for any x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, we must have that a > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One knows the profit function is Π(x) = P(x)x − C(x) = (a − bx3)x − cx = (a − c)x − bx4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, the gradient adjustment mechanism can be described as x(t + 1) = x(t) + K(a − c − 4bx3(t)), K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Without loss of generality, we denote f = 4bK and e = (a − c)/4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Then, the model is simplified into a map with only two parameters: x(t + 1) = x(t) + f(e − x3(t)), e, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (2) The second model considered in this paper is simplified from a famous monopoly model introduced by Puu [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We retain the same inverse demand function and cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The only difference is that the monopolist in our model is knowledgeable, whereas the monopolist in Puu’s original model is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The price function is cubic of a more general form P(x) = a1 − b1x + c1x2 − d1x3, where a1, b1, c1, d1 > 0 are parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The cost function is also cubic and has no fixed costs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', C(x) = a2x − b2x2 + c2x3, 3 where a2, b2, c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Hence, the profit function becomes Π(x) = P(x)x − C(x) = (a1 − a2)x − (b1 − b2)x2 + (c1 − c2)x3 − d1x4, which can be denoted as Π(x) = ax − bx2 + cx3 − dx4 with a = a1 − a2, b = b1 − b2, c = c1 − c2, and d = d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For the sake of simplicity, we assume that a, b, c, d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The marginal profit dΠ/dx is directly obtained and the gradient adjustment mechanism can be formulated as x(t + 1) = x(t) + K(a − 2bx(t) + 3cx2(t) − 4dx3(t)), a, b, c, d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (3) 3 Local Stability and Bifurcations Firstly, we explain the main idea of the symbolic approach used in this paper by analyzing stepwise the local stability of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Then the theoretical results of Model 2 are reported without giving all the calculation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 Model 1 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 1 always has a unique equilibrium, which is stable if 4b(a − c)2K3 < 8 27 Moreover, there is a period-doubling bifurcation if 4b(a − c)2K3 = 8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The above proposition is a known result, which was first derived by Naimzada and Ricchiuti [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Indeed, this proposition can be easily proved since the analytical expression of the unique equilibrium can be obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', x∗ = ( a−c 4b )1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, we would like to provide another proof in a computational style to demonstrate in detail how our symbolic approach works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In what follows, the model formulation (2) is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' By setting x(t + 1) = x(t) = x, we acquire the equilibrium equation x = x + f(e − x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' An equilibrium x of the one-dimensional iteration map is locally stable if ����� dx(t + 1) dx(t) ���� x(t)=x ����� = ��1 − 3fx2�� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, we say the equilibrium x to be feasible if x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, a stable and feasible equilibrium can be characterized as a real solution of � � � � � x = x + f(e − x3), ��1 − 3fx2�� < 1, x > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (4) Although system (4) is so simple that one can solve the closed-form expression of x from the equality part, the problem is how we handle a general polynomial that may have no closed-form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, it is also a nontrivial task to identify the conditions on the parameters whether a system with inequalities has real solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In [18], the first author of this paper and his coworker proposed an algebraic approach to systematically tackle these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The main idea of this approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The parametric system (4) is univariate in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For a univariate system, we introduce a key concept called border polynomial in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One useful property of a border polynomial is that its real zeros divide the parameter space into separated regions and the solution number of the original system is invariant for all parameter points in each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 4 Definition 1 (Border Polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Consider a univariate system � P(u, x) = �m i=0 ai(u) xi = 0, Q1(u, x) > 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' , Qs(u, x) > 0, (5) where P and Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' , Qs are univariate polynomials in x, and u stands for all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The product am(u) · discr(P) · s � i=1 res(P, Qi) is called the border polynomial of system (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Here, res(F, G) stands for the resultant of two polyno- mials F and G, while discr(F) denotes the discriminant of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' More specifically, the formal definitions of the resultant and the discriminant in the above definition are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Let F = m � i=0 ai xi, G = l � j=0 bj xj be two univariate polynomials in x with coefficients ai, bj in the field of complex numbers, and am, bl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The determinant ��������������� am am−1 · · a0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' am am−1 · · a0 bl bl−1 · · b0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' bl bl−1 · · b0 ��������������� � � � l � � � m is called the Sylvester resultant (or simply resultant) of F and G, and denoted by res(F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The resultant of F and its derivative dF/dx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', res(F, dF/dx), is called the discriminant of F and denoted by discr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The following lemma is one of the well-known properties of resultants, which could be found in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Two univariate polynomials F and G have common zeros in the field of complex numbers if and only if res(F, G) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, a univariate polynomial F has a multiple zero in the field of complex numbers if and only if discr(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It is worth noticing that the number of real zeros of P may change when the leading coefficient am(u) or the discriminant discr(P) goes from non-zero to zero and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In addition, if res(P, Qi) goes across zero, then the zeros of P will pass through the boundaries of Qi > 0, which means that the number of real roots of (5) may change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, the following lemma is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Consider a univariate system as (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Let A and B be two points in the space of parameters u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Suppose that any of A, B does not annihilate the border polynomial of system (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If there exists a real path C from A to B such that any point on C is not a root of the border polynomial, then the number of real solutions of system (5) evaluated at A is the same as that at B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Since 1 − 3fx2 < 1, we know that system (4) is equivalent to � � � � � x3 − e = 0, 2 − 3fx2 > 0, x > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (6) We have am = 1 and discr(x3 − e) = 27e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, res(x3 − e, 2 − 3fx2) = −27e2f3 + 8 and res(x3−e, x) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to Definition 1, the border polynomial of system (6) is 27e3(−27e2f3+8), the zeros of which are marked in blue as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' This blue curve divides the parameter set {(e, f) | e > 0, f > 0} into two (the northeast and the southwest) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 5 S2 S1 A Real path C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 4 e 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 4 f Figure 1: Partitions of the parameter space of Model 1 and sample points Notice the two points S2 and A in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One can find a real path C from A to S2 such that it does not pass through the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to Lemma 2, system (6) has the same number of real roots with the parameters evaluated at S2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' This means that the number of real solutions of system (6) is invariant in the northeast region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, we can choose a sample point from each region to determine the root number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For this simple system, sample points might be selected directly by eyes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', S1 = (1, 1/2), S2 = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, the choosing process might be extremely complex in general, which could be done automatically by using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', the method of partial cylindrical algebraic decomposition or called the PCAD method [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For each region, one can determine the root number by counting roots of the non-parametric system of (6) evaluated at the corresponding sample point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Take S1 as an example, where (6) becomes � x3 − 1 = 0, 2 − 3 2x2 > 0, x > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (7) In order to count the number of its real roots, an obvious way is directly solving x3 −1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', x = 1, and then checking whether 2 − 3 2x2 > 0 and x > 0 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The result is true, which means that there exists one unique real solution of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, it is difficult to precisely obtain all real zeros of a general univariate system since root formulae do not exist for polynomials with degrees greater than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, a more systematic method called real root counting [31] is generally needed here, and we demonstrate how this method works by using (7) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It is noted that x3 −1, 2− 3 2x2 and x have no common zeros, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', they have no factors in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Otherwise, one needs to reduce the common factors from the inequalities first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' After that, we isolate all real zeros of 2 − 3 2x2 and x by rational intervals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', � −12 10, −11 10 � , � − 1 10, 1 10 � , �11 10, 12 10 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (8) Although it is trivial for this simple example, the isolation process could be particularly tough for general polynomials, which may be handled by using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', the modified Uspensky algorithm [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, the intervals can be made as small as possible to guarantee no zeros of x3 − 1 lie in these intervals, which could be checked by using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', Sturm’s theorem [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, the real zeros of x3 − 1 must be in the complement of (8): � −∞, −12 10 � , � −11 10, − 1 10 � , � 1 10, 11 10 � , �12 10, +∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (9) In each of these open intervals, the signs of 2 − 3 2x2 and x are invariant and can be determined by checking them at selected sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For instance, to determine the sign of 2 − 3 2x2 on 6 (12/10, +∞), we check the sign at a sample point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We have that 2 − 3 2x2|x=2 = −4 < 0, thus 2 − 3 2x2 < 0 on (12/10, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Similarly, it is obtained that the signs of 2 − 3 2x2 and x at (9) are −, +, +, − and −, −, +, +, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Hence, (1/10, 11/10) is the only interval such that the two inequalities 2 − 3 2x2 > 0 and x > 0 of system (7) are simultaneously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We focus on (1/10, 11/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Using Sturm’s theorem, we can count the number of the real zeros of x3 − 1 at (1/10, 11/10), which is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, system (6) has one real root at S1 = (1, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The above approach works well for a system formulated with univariate polynomial equations and inequalities although some steps seem silly and not necessary for this simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Similarly, we know that system (6) has no real roots at S2 = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In conclusion, system (6) has one real root if the parameters take values from the southwest region where S1 lies, and has no real roots if the parameters take values from the northeast region where S2 lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, the inequalities of some factors of the border polynomial may be used to explicitly describe a given region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It is evident that 27e2f3 −8 < 0 describes the region where S1 lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, Model 1 has one unique stable equilibrium provided that e2f3 = �a − c 4b �2 (4bK)3 = 4b(a − c)2K3 < 8 27, which is consistent with Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to the classical bifurcation theory, for a one-dimensional iteration map x(t+1) = F(x(t)), we know that bifurcations may occur if ����� dx(t + 1) dx(t) ���� x(t)=x ����� = ���� dF dx ���� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' More specifically, if dF/dx = −1, then the system may undergo a period-doubling bifurcation (also called flip bifurcation), where the dynamics switch to a new behavior with twice the period of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' On the other hand, if dF/dx = 1, then the system may undergo a saddle-node (fold), transcritical, or pitchfork bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One might determine the type of bifurcation from the change in the number of the (stable) equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the case of saddle-node bifurcation, one stable equilibrium (a node) annihilates with another unstable one (a saddle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Before and after a transcritical bifurcation, there is one unstable and one stable equilibrium, and the unstable equilibrium becomes stable and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the case of pitchfork bifurcation, the number of equilibria changes from one to three or from three to one, while the number of stable equilibria changes from one to two or from one to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Accordingly, it is concluded that Model 1 may undergo a period-doubling bifurcation if e2f3 = 4b(a − c)2K3 = 8 27, and there are no other bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 Model 2 According to (3), by setting x(t + 1) = x(t) = x, we know that Model 2 has at most three equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The analytical expressions of the equilibria exist, but are complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', x1 = 3√ M 12d − 8bd − 3c2 4d 3√ M + c 4d, x2,3 = − 3√ M 24d + 8bd − 3c2 8d 3√ M + c 4d ± i √ 3 2 � 3√ M 12d + 8bd − 3c2 4d 3√ M � , (10) where M = 12d √ 3 � 108a2d2 − 108abcd + 27 ac3 + 32b3d − 9b2c2 + 216ad2 − 108bcd + 27c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 7 Furthermore, an equilibrium x is locally stable provided that ����� dx(t + 1) dx(t) ���� x(t)=x ����� = ��1 + K(−2b + 6cx − 12dx2) �� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Hence, a stable equilibrium of map (3) is a real solution of � � � � � � � � � � � x = x + K(a − 2bx + 3cx2 − 4dx3), K(−2b + 6cx − 12dx2) < 0, 2 + K(−2b + 6cx − 12dx2) > 0, x > 0, a > 0, b > 0, c > 0, d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (11) Obviously, analyzing the stable equilibrium by substituting the closed-form solutions (10) into (11) is complicated and impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In comparison, the approach applied in the analysis of Model 1 does not require explicitly solving any closed-form equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If the analytical solution has a complicated expression or even if there are no closed-form solutions, our approach still works in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Concerning the border polynomial of system (11), we compute discr(K(a − 2bx + 3cx2 − 4dx3)) = −16K5dR1, res(K(a − 2bx + 3cx2 − 4dx3), K(−2b + 6cx − 12dx2)) = −16K5dR1, res(K(a − 2bx + 3cx2 − 4dx3), 2 + K(−2b + 6cx − 12dx2)) = −16K2dR2, res(K(a − 2bx + 3cx2 − 4dx3), x) = −Ka, where R1 = 108a2d2 − 108abcd + 27ac3 + 32b3d − 9b2c2, R2 = 108K3a2d2 − 108K3abcd + 27K3ac3 + 32K3b3d − 9K3b2c2 − 24Kbd + 9Kc2 − 8d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, the border polynomial is −16384 d4K14aR2 1R2, the zeros of which divide the parameter set {(a, b, c, d, K) | a, b, c, d, K > 0} into separated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The PCAD method [5] permits us to select at least one sample point from each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Table 1, we list the 30 selected sample points and the corresponding numbers of distinct real solutions of system (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Table 1: Selected Sample Points in {(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' K) | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' K > 0} (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' K) num R1 R2 (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' K) num R1 R2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1/64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1/2) 2 − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1/64,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1) 0 + + According to Table 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' one can see that system (11) has one real solution if and only if R1 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' R2 > 0 or R1 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' R2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, a necessary condition that system (11) has two real solutions is that 8 R1 < 0 and R2 < 0, which is not a sufficient condition, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, at (a, b, c, d, K) = (1, 1, 1/4, 1/64, 2), system (11) has no real solutions but R1 < 0 and R2 < 0 are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' To acquire the necessary and sufficient condition, additional polynomials (R3 and R4) are needed, which can be found in the so-called generalized discriminant list and can be picked out by repeated trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Regarding the generalized discriminant list, readers may refer to [32] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Due to space limitations, we directly report below the necessary and sufficient condition that system (11) has two real solutions without giving the calculation details: R1 < 0, R2 < 0, R3 > 0, R4 < 0, where R3 = 8Kbd − 3Kc2 + 8d, R4 = 432K2a2d3 − 432K2abcd2 + 108K2ac3d + 128K2b3dt2 − 36K2b2c2d + 192Kb2d2 − 144Kbc2d + 27Kc4 + 64bd2 − 24c2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We continue to analyze the bifurcations of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' An equilibrium x of map (3) may undergo a period-doubling bifurcation if dx(t + 1) dx(t) ���� x(t)=x = 1 + K(−2b + 6cx − 12dx2) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Hence, a period-doubling bifurcation may occur if the following system has at least one real solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' � � � � � x = x + K(a − 2bx + 3cx2 − 4dx3), K(−2b + 6cx − 12dx2) + 2 = 0, x > 0, a > 0, b > 0, c > 0, d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (12) By using the method of triangular decomposition1, we transform the solutions of the first two equations of system (12) into zeros of the triangular set T = [(8Kbd − 3Kc2 + 4d)x − 6adK + bcK − c, R2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Obviously, the system {T = 0, x > 0, a > 0, b > 0, c > 0, d > 0} has at least one real positive solution if R2 = 0 and x = (6adK − bcK + c)/(8Kbd − 3Kc2 + 4d) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', R2 = 0, R5 > 0, where R5 = (6adK − bcK + c)(8Kbd − 3Kc2 + 4d) = 48K2abd2 − 18K2ac2d − 8K2b2cd + 3K2bc3 + 24Kad2 + 4Kbcd − 3Kc3 + 4cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Similarly, concerning the occurrence of a pitchfork bifurcation, we consider � � � � � x = x + K(a − 2bx + 3cx2 − 4dx3), K(−2b + 6cx − 12dx2) = 0, x > 0, a > 0, b > 0, c > 0, d > 0, (13) and count the number of stable equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' More details are not reported here due to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We summarize all the obtained results in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1The method of triangular decomposition can be viewed as an extension of the method of Gaussian elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The main idea of both methods is to transform a system into a triangular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, the triangular decomposition method is available for polynomial systems, while the Gaussian elimination method is just for linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Refer to [30, 16, 12, 29] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 9 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 2 has at most two stable equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Specifically, there exists just one stable equilibrium if R1 < 0, R2 > 0 or R1 > 0, R2 < 0, and there exist two stable equilibria if R1 < 0, R2 < 0, R3 > 0, R4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, there is a period-doubling bifurcation if R2 = 0, R5 > 0, and there is a pitchfork bifurcation if R1 = 0, R2 > 0, R6 > 0 or R1 = 0, R2 > 0, R4 < 0, R6 > 0, where R6 = 48abd2 − 18ac2d − 8b2cd + 3bc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' To the best of our knowledge, the stability results regarding the parameters a, b, c, d, K reported in Theorem 1 are new although the special case of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05 has been discussed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The two parameters K, a play more ambitious roles than others in practice for K controls the speed of adjusting the monopolist’s output and a is the difference between the initial product price of the market without any supply and the initial marginal cost of the firm without any production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' By fixing b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05, we depict the (a, K) parameter plane in Figure 2, where the region for the existence of one stable equilibrium is colored in yellow, while the region for the existence of two stable equilibria is colored in blue-gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 2 behaves differently from typical oligopolies with gradient mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' As shown by Figure 2, for instance, even if the adjustment speed K is quite large, there always exist some values of a such that Model 2 is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, for a fixed value of K greater than around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7, Model 2 undergoes from instability to stability and then back to instability twice as the parameter a changes from low to high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 2: The two-dimensional (a, K) parameter plane of Model 2 with the other parameters fixed: b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The region for the existence of one stable equilibrium is colored in yellow, while that of two stable equilibria is colored in blue-gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 10 6- pitchfork bifurcation 5 curves (R1=0) 4- K 3 period-doubling bifurcation curves (R2=0) 2 1 0 0 1 2 3 4 5 6 a R=0 R=04 Periodic Solutions From an economic point of view, it is realistic to assume that a boundedly rational firm can not learn the pattern behind output and profits if periodic dynamics take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this regard, we investigate the existence and stability of periodic solutions with relatively lower orders in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Let I be an interval of real numbers, and let F : I → R be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If x ∈ I, suppose that F 0(x) represents x and F n+1(x) denotes F(F n(x)) for n ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' A point p ∈ I is said to be a periodic point with period n or order n if p = F n(p), and p ̸= F k(p) for any 1 ≤ k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If p is a point with period n, we call p �→ F 1(p) �→ · · · �→ F n(p) = p a n-cycle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, a point y ∈ I with period k is said to be asymptotically stable if there exists δ such that |F k(x) − y| < |x − y| for all x ∈ (y − δ, y + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The following lemma can be found in [15], which provides an algebraic criterion to verify the stability of a periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Assume that y ∈ I is a periodic point of F with period k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If F is differentiable at the points y, F(y), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' , F k−1(y), then y is asymptotically stable if ����� k−1 � i=0 d dxF(yi) ����� < 1, where yi = F i(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 Model 1 We start by considering the existence of periodic orbits with order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Assume that there is a 2-cycle orbit x �→ y �→ x, where �→ stands for the iteration map (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, we have y = x + f(e − x3), x = y + f(e − y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (14) Obviously, x ̸= y should be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Otherwise, x �→ y �→ x will degenerate into an equilib- rium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Then, the problem of determining the existence of 2-cycles is transformed into determining the existence of real solutions of � � � � � � � � � � � y = x + f(e − x3), x = y + f(e − y3), x ̸= y, x > 0, y > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (15) Since the above system involves two variables x and y, the approach used in Section 3 (feasible only for univariate systems) might not be directly employed herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, we can transform system (15) equivalently into univariate systems based on its triangular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Specifically, the triangular decomposition method permits us to decompose the equation part (14) into the following two triangular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' T11 = [y − x, x3 − e], T12 = [y + fx3 − x − ef, f3x6 − 3f2x4 − 2ef3x3 + 3fx2 + 3ef2x + e2f3 − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Since the first polynomial in T11 is y − x, which implies that x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, the zeros of T11 are not of our concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We only focus on T12, where the first polynomial y + fx3 − x − ef has degree one with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, one can directly solve y = −fx3 + x + ef and substitute it into relative inequalities of system (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In short, system (15) can be equivalently transformed into the following univariate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' � � � � � f3x6 − 3f2x4 − 2ef3x3 + 3fx2 + 3ef2x + e2f3 − 2 = 0, − fx3 + x + ef > 0, x > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' After that, the approach in Section 3 can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The results show that the above system has two real solutions if and only if 8/27 < e2f3 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It is evident that these two real solutions belong to the 11 same 2-cycle orbit because x, y are symmetric and can be replaced with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, there exists at most one 2-cycle orbit in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to Lemma 3, to determine whether the discovered 2-cycle is stable, we consider (15) together with the condition ���� d(x + f(e − x3)) dx × d(y + f(e − y3)) dy ���� < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', ��(1 − 3fx2)(1 − 3fy2) �� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The technique introduced in Remark 2 is needed to transform the system into a univariate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to our calculations, the unique 2-cycle orbit is stable if and only if 729e4f6 − 3294e2f3 + 1664 > 0 or equivalently 8/27 < e2f3 < (61 − 11 √ 17)/27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We collect the aforementioned results in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 1 has at most one 2-cycle orbit, which exists if 8/27 < e2f3 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, this unique 2-cycle is stable if 8/27 < e2f3 < 61 − 11 √ 17 27 , or approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2962962963 < e2f3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5794754859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The measurement of the magnitude of periodic orbits is economically interesting for it characterizes the size of fluctuations in dynamic economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For a n-cycle orbit p1 �→ p2 �→ · · · pn �→ p1, a direct definition of the magnitude measure is d = |p1 − p2| + |p2 − p3| + · · · + |pn−1 − pn| + |pn − p1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, to obtain better mathematical properties, we square each item and define the magnitude measure to be d = (p1 − p2)2 + (p2 − p3)2 + · · · + (pn−1 − pn)2 + (pn − p1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For a 2-cycle orbit x �→ y �→ x in Model 1, the magnitude measure becomes d = (x − y)2 + (y − x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, we have � � � � � d − (x − y)2 − (y − x)2 = 0, − y + x + f(e − x3) = 0, − x + y + f(e − y3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Using the method of triangular decomposition, we decompose the solutions of the above system into zeros of the following two triangular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' T21 = [ y − x, x3 − e, d ], T22 = [ y + x3f − ef − x, (d2f3 + 4df2 + 4f)x2 + (−6def3 − 12ef2)x + 36e2f3 − 2f2d2 − 8fd − 8, f3d3 − 12f2d2 − 60fd + 216e2f3 − 64 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The first polynomial y − x in T21 implies that x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, T21 is not of concern since it corresponds to equilibria rather than 2-cycle orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We focus on the last polynomial f3d3 − 12f2d2 − 60fd + 216e2f3 − 64 in T22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' By solving d from this polynomial, we obtain three solutions: d1 = 2 f �3H 2 + 6 H + 2 � , d2, d3 = 2 f � −3H 4 − 3 H + 2 ± i √ 3 2 �3H 2 − 6 H �� , 12 where H = 3� 8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Here, only the real solution d1 is meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, the magnitude measure of the unique 2-cycle orbit in Model 1 can be expressed as d = 2 f � 3 3� 8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 2 + 6 3� 8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the rest of this section, similar calculations as above are repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We omit these computation details due to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Concerning 3-cycle orbits in Model 1, we need to count real solutions of � � � � � � � � � � � � � � � y = x + f(e − x3), z = y + f(e − y3), x = z + f(e − z3), x ̸= y, x ̸= z, x > 0, y > 0, z > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Based on a series of computations, we derive the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 1 has no 3-cycle orbits for all possible parameter values such that e, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For a 4-cycle orbit x �→ y �→ z �→ w �→ x, we have the system � � � � � � � � � � � � � � � � � � � � � y = x + f(e − x3), z = y + f(e − y3), w = z + f(e − z3), x = w + f(e − w3), x ̸= y, x ̸= z, x ̸= w, x > 0, y > 0, z > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, the following condition is required to guarantee that the considered 4-cycle is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' ���� d(x + f(e − x3)) dx × d(y + f(e − y3)) dy × d(z + f(e − z3)) dz × d(w + f(e − w3)) dw ���� < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', ��(1 − 3fx2)(1 − 3fy2)(1 − 3fz2)(1 − 3fw2) �� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' As the polynomials involved in the conditions of the existence and stability of 4-cycle orbits are extremely complicated, we report below the obtained results in an approximate style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 1 has at most one 4-cycle orbit, which exists if 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5794754859 < e2f3 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='237575627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, this unique 4-cycle is stable if 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5794754859 < e2f3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6673871142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 3 (a) depicts the phase diagram of the unique 4-cycle in Model 1 with e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 and f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Since e2f3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='62208 ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5794754859, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6673871142), this unique 4-cycle in Model 1 is asymptotically stable according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Actually, the horizontal coordinates of A, B, C, D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', x, y, z, w, are the four points in the 4-cycle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For the sake of simplicity, we connect A, B, C, D with lines and use the simplified phase diagram as Figure 3 (b) to demonstrate periodic solutions in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 13 (a) phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 x(t-1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 x(t) A B D C (b) simplified phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 3: The unique stable 4-cycle in Model 1 with e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 and f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, by using the same approach as we computed the magnitude of the 2-cycle orbit, we conclude that if a 4-cycle x �→ y �→ z �→ w �→ x exists in Model 1, its magnitude measure equals to d = 4 f � 3 3� 8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 2 + 6 3� 8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 + 2 � , which is twice as large as that of the 2-cycle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The parameter plane of Model 1 is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One can see that the parameter region for the stability of the unique equilibrium (2-cycle or 4-cycle orbit) constitutes a connected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, the three regions for the stability of the equilibrium, 2-cycle, and 4-cycle adjoin without any gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the next subsection, one will find that the topological structure of the parameter space of Model 2 is much more complex than that of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 4: The parameter plane of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The light blue, yellow, and light orange regions are the parameter regions for the stability of the 4-cycle orbit, the 2-cycle orbit, and the equilibrium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 5 depicts the two-dimensional bifurcation diagram of Model 1 for (e, f) ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6]×[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For additional information regarding two-dimensional bifurcation diagrams, readers can refer to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 14 c A 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 x Z y 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' x(t-1)21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2In the numerical simulations of Figure 5, we set the initial state to be x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Parameter points corresponding to periodic orbits with different orders are marked in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, parameter points are colored in dark red if the order is just one (equilibria) and are marked in black if the order is greater than or equal to 24 (complex trajectories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the case that the order is greater than 24, the black points may be viewed as the parameter values where complex dynamics such as chaos take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, we also use black to mark those parameter points where the trajectories diverge to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One can see that Figure 5 confirms the theoretical results presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Figure 5, the transitions between different types of periodic orbits can also be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One can see that the equilibrium loses its stability through a series of period-doubling bifurcations as the value of e or f increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, along the line of e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0, the unique stable equilibrium bifurcates into a stable 2-cycle orbit at f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6665, which further bifurcates into a 4-cycle orbit at f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' There is a stable 8-cycle orbit when f ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8744, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8826).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Finally, chaotic dynamics take place if the value of f is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Additional details can be found in the one-dimensional bifurcation diagram presented in Figure 6, where we fix e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 and choose x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 to be the initial state of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 5: The two-dimensional bifurcation diagram of Model 1 for (e, f) ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We choose x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 to be the initial state of the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 Model 2 The formulation (3) of Model 2 involves five parameters, which might be particularly complex for symbolic computations of searching periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In what follows, we keep K as the only pa- rameter and assume that a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' This setting is meaningful and has been discussed by several economists, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', Puu [26], Al-Hdaibat and others [1], Matsumoto and Szidarovszky [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 eFigure 6: The one-dimensional bifurcation diagram of Model 1 with respect to f by fixing e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We choose x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 to be the initial state of the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Let x �→ y �→ x be a 2-cycle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Hence, we have � � � � � � � � � � � y = x + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2x3), x = y + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8y + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8y2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2y3), x ̸= y, x > 0, y > 0, K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (16) Furthermore, the following condition is required if the stability of the 2-cycle is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' |S(x) · S(y)| < 1, where S(x) = d(x + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2x3) dx = 1 − K(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (17) According to our computations, the following theorem is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, the possible number of 2-cycle orbits is zero (no real solutions in system (16)) or three (six real solutions in system (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' There exist three 2-cycle orbits if K > 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, two of them are stable if 5/3 < K < (5 √ 5 − 5)/3, or approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='666666667 < K < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='060113296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' To measure the magnitude of a 2-cycle orbit x �→ y �→ x, we also use d = (x − y)2 + (y − x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The method of triangular decomposition permits us to decompose the solutions of � � � � � d = (x − y)2 + (y − x)2, y = x + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2x3), x = y + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8y + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8y2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2y3) 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 finto zeros of the following triangular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' T31 = [ y − 3, x − 3, d ], T32 = [ y − x, x2 − 6x + 6, d ], T33 = [ y + x − 6, Kx2 − 6Kx + 6K − 10, Kd − 24K − 80 ], T34 = [ 5y + x3K − 9Kx2 + (24K − 5)x − 18K, K2x4 − 12K2x3 + (51K2 − 5K)x2 + (−90K2 + 30K)x + 54K2 − 45K + 25, Kd − 6K + 10 ], where the last two polynomials Kd − 24K − 80 and Kd − 6K + 10 in T33 and T34 are of our concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We conclude that d = (24K + 80)/K or d = (6K − 10)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One can see that two of the three 2-cycle orbits possess the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For a 3-cycle orbit x �→ y �→ z �→ x, we consider the system � � � � � � � � � � � � � � � y = x + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2x3), z = y + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8y + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8y2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2y3), x = z + K(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8z + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8z2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2z3), x ̸= y, x ̸= z, x > 0, y > 0, z > 0, K > 0, (18) as well as the stability condition |S(x) · S(y) · S(z)| < 1, (19) where S(x) is given in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Based on a series of calculations, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, all possible cases for the number of (stable) 3-cycle orbits are listed in Table 2, where m1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='417401607, m2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='434714456, m3 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='302953127, m4 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='303122765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Readers can refer to Remark 3 to understand how these mi are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Table 2: Numbers of (stable) 3-cycle orbits in Model 2 K ∈ (0, m1) (m1, m2) (m2, m3) (m3, m4) (m4, +∞) 3-cycles 0 4 4 8 8 Stable 3-cycles 0 2 0 2 0 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' As aforementioned, the border polynomial plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, one can derive that the properties of the border polynomial reported in Lemma 2 will retain if we use the squarefree part of the border polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The squarefree part SP of the border polynomial of (18)+(19) is simpler, which is given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Theorem 6, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' , m4 are the real roots of SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' A rigorous style of writing Theorem 6 is to express the conditions using factors in SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, this would be quite tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Since only one parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', K, is involved in SP, the regions divided by zeros of SP are indeed intervals and can be approximately described as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, it should be noticed that the values of m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' , m4 can be made arbitrarily accurate if we want because the exact expression of SP has already been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 7 depicts all the 3-cycle orbits in Model 2 with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='303 ∈ (m3, m4), where the 6 unstable cycles are marked in red and the 2 stable cycles are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' It is worth noting that two of the unstable 3-cycle orbits in red almost coincide with the stable ones in blue, but they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We should underline that this dynamic phenomenon, derived by symbolic computations, may be too subtle to observe through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 17 0 1 2 3 4 5 6 x(t-1) 0 1 2 3 4 5 6 x(t) Figure 7: The 3-cycle orbits in Model 2 with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The 6 unstable cycles are marked in red, while the 2 stable ones are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, if measuring the magnitude of the 3-cycle orbit x �→ y �→ z �→ x with d = (x − y)2 + (y − z)2 + (z − x)2, then we have K4d4 + (−54K4 − 90K3)d3 + (972K4 + 2700K3 + 1800K2)d2 + (−6696K4 − 19440K3 − 5400K2 + 27000K)d + 15552K4 + 38880K3 − 32400K2 − 162000K + 270000 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The above condition on K and d is plotted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 8: The magnitude d of the possible 3-cycle orbits in Model 2 as the variation of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Similarly, we analyze the 4-cycle and 5-cycle orbits in Model 2, and report the obtained results in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, all possible cases for the number of (stable) 4-cycle orbits are given in Table 3, where m1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='060113296, m2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='146719591, m3 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='579725065, m4 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='581385365, m5 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='062775154, m6 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='070194019, m7 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='279225134, m8 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='279260335, m9 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='319881360, m10 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='319889702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 18 Readers can refer to Remark 3 to understand how these mi are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Table 3: Numbers of (stable) 4-cycle orbits in Model 2 K ∈ (0, m1) (m1, m2) (m2, m3) (m3, m4) (m4, m5) (m5, m6) 4-cycles 0 2 2 6 6 10 Stable 4-cycles 0 2 0 2 0 2 K ∈ (m6, m7) (m7, m8) (m8, m9) (m9, m10) (m10, +∞) 4-cycles 10 14 14 18 18 Stable 4-cycles 0 2 0 2 0 In Figure 9, we show all the 4-cycle orbits in Model 2 with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='319885 ∈ (m9, m10), where the 16 unstable cycles are marked in red and the 2 stable ones are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If we measure the magnitude of the 4-cycle orbit x �→ y �→ z �→ w �→ x with d = (x−y)2 +(y −z)2 +(z −w)2 +(w −x)2, then d must satisfy one of the following equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Kd − 12K + 20 = 0, Kd − 48K − 160 = 0, K2d2 + (−36K2 − 60K)d + 288K2 + 960K + 1600 = 0, C4(K, d) = 0, where C4(K, d) is a complex polynomial given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 9: The 4-cycle orbits in Model 2 with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='319885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The 16 unstable cycles are marked in red, while the 2 stable ones are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, all possible cases for the number of (stable) 5-cycle orbits are listed in Table 4, where m1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='323208379, m2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='326320457, m3 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='509741151, m4 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='510528490, m5 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='632885028, m6 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='633089005, m7 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='997641294, m8 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='997736262, m9 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='113029799, m10 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='113069634, m11 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='197332995, m12 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='197354147, m13 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='219425160, m14 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='219440784, m15 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='269613400, m16 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='269618202, m17 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='288059620, m18 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='288062995, m19 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='314977518, m20 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='314978815, m21 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='324008184, m22 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='324008826, m23 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='332961824, m24 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='332961850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 19 5 4 x(t) 2 1 2 3 4 5 x(t-1)9Readers can refer to Remark 3 to understand how these mi are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Table 4: Numbers of (stable) 5-cycle orbits in Model 2 K ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m1) (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m2) (m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m3) (m3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m4) (m4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m5) 5-cycles 0 4 4 8 8 Stable 5-cycles 0 2 0 2 0 K ∈ (m5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m6) (m6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m7) (m7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m8) (m8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m9) (m9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m10) 5-cycles 12 12 16 16 20 Stable 5-cycles 2 0 2 0 2 K ∈ (m10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m11) (m11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m12) (m12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m13) (m13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m14) (m14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m15) 5-cycles 20 24 24 28 28 Stable 5-cycles 0 2 0 2 0 K ∈ (m15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m16) (m16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m17) (m17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m18) (m18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m19) (m19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m20) 5-cycles 32 32 36 36 40 Stable 5-cycles 2 0 2 0 2 K ∈ (m20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m21) (m21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m22) (m22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m23) (m23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' m24) (m24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' +∞) 5-cycles 40 44 44 48 48 Stable 5-cycles 0 2 0 2 0 In Figure 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' we plot all possible 5-cycle orbits in Model 2 with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='33296183 ∈ (m23, m24), where the 46 unstable cycles are marked in red and the 2 stable ones are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 10: The 5-cycle orbits in Model 2 with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='33296183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The 46 unstable cycles are marked in red, while the 2 stable ones are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' By Theorems 6, 7 and 8, one can see the parameter space of Model 2 is quite different from that of Model 1 in the sense that the stability regions for the 3-cycle, 4-cycle and 5-cycle orbits are disconnected sets formed by many disjoint portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, the topological structures of the regions for stable periodic orbits in Model 2 are much more complex than those in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' This may be because the inverse demand function of Model 2 has an inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, the following observations of Model 2 are similar to Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Theorem 5 shows that the stability region for the 2-cycles is a connected interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, the right boundary of the stability region for the 2-cycles is the same as the left boundary of the stability region for the 4-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' When a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05, by Theorem 1 we know that Model 2 has stable equilibria if K ∈ (0, 5/3), which adjoins the stability region for the 2-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, in Model 2, the stability regions for cycles with distinct periods may not intersect with each other, which means that multistability might only arise among 20 5 4 x(t) 3 2 0 1 2 3 4 5 x(t-1)96periodic orbits with the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 11 depicts the two-dimensional bifurcation diagram of Model 2 for (a, K) ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We fix the parameters b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05, and set the initial state to be x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Similarly, we use different colors to mark parameter points corresponding to trajectories with different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Parameter points are marked in black if the corresponding orbits have orders greater than 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, we also use black to mark the parameter points where the trajectories diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' One can see that Figure 11 confirms the theoretical results reported in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, Fig- ure 11 generated by numerical simulations is not accurate compared to Figure 2 based on symbolic computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 11: The two-dimensional bifurcation diagram of Model 2 for (a, K) ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We fix the parameters b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05, and choose x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 to be the initial state of the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 12 depicts the one-dimensional bifurcation diagrams of Model 2 with respect to K by fixing a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The bifurcation diagrams are different if the selected initial states of the iterations are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, in Figure 12 (a) and (b), the initial states are selected to be x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 and x(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The difference may be because two stable equilibria exist when K is relatively small and distinct initial states approach distinct equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' As shown by Figure 12 (a), the trajectory converges to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='058 when K < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1996 and converges to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='384 when K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Figure 12, the occurrence of period-doubling bifurcations can also be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 13 depicts the one-dimensional bifurcation diagrams of Model 2 with respect to a by fixing K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Figure 13 (a) and (b), the initial states of the iterations are selected to be x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 and x(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Similarly, the two bifurcation diagrams are different because of the selection of distinct initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, pitchfork bifurcations can be observed in Figure 13, where the number of stable equilibria changes from one to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, two stable equilibria may coexist (see the blue-gray region in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The equilibrium selection problem is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The final outcome of the iterations depends not only on the values of 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 - 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 a(a) x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (b) x(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 12: The one-dimensional bifurcation diagrams of Model 2 with respect to K by fixing a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (a) x(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (b) x(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Figure 13: The one-dimensional bifurcation diagrams of Model 2 with respect to a by fixing K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3 X 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 0 2 3 1 4 5 K6 5 4 X 3 - 2 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0 athe parameters but also on the starting conditions of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to our numerical simulations of Model 2, the basins of attraction of coexisting equilibria have complicated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, by fixing K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5, a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='5, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05, we have two stable equilibria E1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='19 and E2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The basin of E1 is B(E1) = (0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='168) ∪ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='518, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='577) ∪ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='745, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='781) ∪ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='786, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='789), while that of E2 is B(E2) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='168, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='518) ∪ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='577, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='745) ∪ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='781, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='786).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, when the initial state x(0) > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='786, the trajectory will not converge to any of the two stable equilibria but diverge to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Take K = 1 and a = 4 as the other example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If the other parameters keep unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='4, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='6, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='05, there are two stable equilibria E1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='99 and E2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Our simulations show that the basins of these two equilibria are B(E1) = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='807) ∪ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='192) ∪ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='431, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='647) ∪ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='653, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='659), and B(E2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='807, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='0) ∪ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='192, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='431) ∪ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='647, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='653), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The escape set is (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='659, +∞), where the trajectory diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In short, in Model 2, the basins of the two stable equilibria are disconnected sets and have complex topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 5 Chaotic Dynamics In the bifurcation diagrams (Figures 6 and 12), one can observe that the dynamics of the two considered models transition to chaos through period-doubling bifurcations as the adjustment speed increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' From an economic point of view, if chaos appears, the pattern behind output and profits is nearly impossible to learn even for completely rational players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, it is extremely hard for a firm to handle a chaotic economy, where no market rules could be discovered and followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this section, we rigorously prove the existence of chaos for the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The following famous lemma was first derived by Li and Yorke [15], which is mathematically deep and facilitates the exploration of complicated dynamics arising in one-dimensional discrete dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Let I be an interval of real numbers, and let F : I → R be a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Assume that there exists a point x ∈ I such that F 3(x) ≤ x < F(x) < F 2(x) or F 3(x) ≥ x > F(x) > F 2(x), (20) then the following two statements are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' }, there is a point pk ∈ I with period k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', F k(pk) = pk, and F i(pk) ̸= pk for 1 ≤ i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' There is an uncountable set S ⊂ I (containing no periodic points), which satisfies the following conditions: (a) for any p, q ∈ S with p ̸= q, lim sup n→∞ |F n(p) − F n(q)| > 0, (21) and lim inf n→∞ |F n(p) − F n(q)| = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (22) (b) for every point p ∈ S and every periodic point q ∈ I, lim sup n→∞ |F n(p) − F n(q)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (23) 23 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' (22) means that every trajectory in S can wander arbitrarily close to every other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, by (21) we know that no matter how close two distinct trajectories in S may come to each other, they must eventually wander away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, by (23) it is clear that every trajectory in S goes away from any periodic orbit in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' If the two statements in the above lemma are both satisfied, we say that there exist chaotic dynamics or chaos in the sense of Li-Yorke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, we can conclude that “period three implies chaos” for one-dimensional discrete dynam- ical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Section 4, we have rigorously derived the existence of 3-cycle orbits in Model 2 if K > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='417401607, which proves that chaos would arise for an uncountable set of initial states in the sense of Li-Yorke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' But in Model 1, we have proved that there are no solutions with period three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, it can not be concluded that there exist no chaotic trajectories since the existence of period three is not a necessary but only a sufficient condition of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In [20], Marotto indicated that the existence of snapback repellers also implies chaos for general n-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, Li and Chen [14] pointed out that Marotto’s original definition of snapback repeller may result in an insufficiency, and proposed the Marotto-Li-Chen Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, we give the following lemma for one-dimensional systems by simplifying the Marotto-Li-Chen Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Readers can refer to [11] for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Let I be an interval of real numbers, and let F : I → R be a differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Assume that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' x ∈ I is an equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', F(x) = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' there exists a close interval S ⊂ I such that x is an inner point of S, and the derivative of F has the absolute value greater than 1 at every point p ∈ S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', |F ′(p)| > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' for some integer m > 1, there exists a point y ∈ S such that y ̸= x, F m(y) = x, and F ′(F k(y)) ̸= 0 for all 1 ≤ k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Then the system x(t + 1) = F(x(t)) is chaotic in the sense of Li-Yorke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For Model 1, we have F(x) = x + f(e − x3) and F ′(x) = 1 − 3fx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Then |F ′(x)| > 1 and x > 0 imply that x > � 2 3f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Thus, if we can find x, y with x ̸= y such that both |F ′(x)| > 1 and |F ′(y)| > 1 are satisfied, then there must exist one closed interval S containing x, y as inner points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In such a case, it is obvious that |F ′(p)| > 1 for every point p ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Naturally, we start from m = 2 to verify the conditions of Lemma 5 by counting real solutions of the following system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' � � � � � � � � � � � � � � � � � � � � � � � � � � � x = x + f(e − x3), x = F 2(y) = y + f(e − y3) + f(e − (y + f(e − y3))3), |1 − 3fx2| > 1, |1 − 3fy2| > 1, |1 − 3f(y + f(e − y3))2| ̸= 0, x ̸= y, x > 0, y > 0, e > 0, f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The technique introduced in Remark 2 should be conducted first to transform the above system into a univariate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to our calculations, the above system has at least one real solution if and only if 8/27 < e2f3 < 64/27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Therefore, we conclude that Model 1 is chaotic in the sense of Li-Yorke provided that 8/27 < e2f3 < 64/27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 6 Concluding Remarks It is known that a monopoly may exhibit complex dynamics such as periodic orbits and chaos al- though it is the simplest oligopoly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this study, we investigated two monopoly models with gradient mechanisms, where the monopolists are knowledgeable firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The two models are distinct mainly in 24 their inverse demand functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Model 1 uses the inverse demand function of Naimzada and Ricchiuti [25], while Model 2 employs that of Puu [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Different from widely applied numerical methods such as numerical simulations and bifurcation continuation approaches, symbolic methods were applied in this paper to analyze the local stability, periodic solutions, and even chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Numerical methods have some shortcomings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', the computations may encounter the problem of instability, which makes the results completely useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In comparison, symbolic computations are exact, thus the obtained results can be used to rigorously prove economic theorems in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' By reproving the already-known results (Proposition 1) of the local stability and bifurcations of Model 1, we explained in detail how our symbolic approach works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Afterward, the analysis of the stability and bifurcations of Model 2 was conducted based on this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We acquired the complete conditions of the local stability and bifurcations of Model 2 for the first time (see Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Figure 2, it was observed that Model 2 behaves quite differently from typical oligopoly models with gradient mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For example, even if the adjustment speed K is quite large, there always exist some values of a (the difference between the initial commodity price and the initial marginal cost) such that Model 2 has a stable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Moreover, Model 2 may go from instability to stability and then back to instability twice as the value of a increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' From an economic point of view, the study of periodic solutions is of practical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Under the assumption of bounded rationality, firms can not learn the pattern behind output and profits if periodic dynamics take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For the two models, we explored the periodic solutions with lower orders as well as their local stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Differences between the two models were found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=', 3-cycle orbits exist in Model 2 but not in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 1, the parameter region for the stability of the periodic solution with a fixed order constitutes a connected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In Model 2, however, the stability regions for the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In other words, the topological structures of the regions for stable periodic orbits in Model 2 are much more complex than those in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The above differences may be because the inverse demand function of Model 2 has an inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' According to the numerical simulations of Model 2, we found that the basins of the two stable equilibria are disconnected sets and also have complex topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For a n-cycle orbit p1 �→ p2 �→ · · · pn �→ p1, we defined the magnitude measure to be d = (p1 − p2)2 + (p2 − p3)2 + · · · + (pn−1 − pn)2 + (pn − p1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' For the two considered models, we analytically investigated the formulae for the magnitude of periodic orbits with lower orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Furthermore, it is extremely hard for a firm to handle an economy when chaos appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In such a case, no market rules can be discovered and followed, and the pattern behind output and profits is nearly impossible to learn even for completely rational players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the bifurcation diagrams of the two models, it seems that chaos occurs when the adjustment speed is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' We clarified this observation analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' By virtue of the fact “period three implies chaos”, we derived that Model 2 is chaotic in the sense of Li-Yorke by proving the existence of 3-cycle orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' However, there are no 3-cycles in Model 1, but the Marotto-Li-Chen Theorem permitted us to prove the existence of chaos by finding snapback repellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this paper, we take the assumption of knowledgeable players, which means the enterprise has full information regarding the inverse demand function and can compute its marginal profit at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In the real world, however, it is more reasonable to assume players to be limited rather than knowledgeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' In this case, the enterprise does not know the form of the inverse demand function, but possesses the values of output and price only in the past periods and estimates its marginal profit with a simple difference formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' The investigation of the dynamics of limited firms might be an important direction for our future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Acknowledgments The authors wish to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Bo Huang for the beneficial discussions and are grateful to the anony- mous referees for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 25 This work has been supported by Philosophy and Social Science Foundation of Guangdong under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' GD21CLJ01, Major Research and Cultivation Project of Dongguan City University under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 2021YZDYB04Z and 2022YZD05R, National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 11601023, and Beijing Natural Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' 1212005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Declaration of competing interest The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' Appendix SP = (972K8 + 19440K7 + 127575K6 + 162000K5 − 1552500K4 − 6412500K3 − 5062500K2 + 23437500K + 67187500)(8503056K12 + 191318760K11 + 1523464200K10 + 3754532250K9 − 14134854375K8 − 101982543750K7 − 146939062500K6 + 399469218750K5 + 1522072265625K4 + 261457031250K3 − 4576816406250K2 − 1938867187500K + 13981445312500),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' C4(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content=' d) = K8d8 + (−126K8 − 210K7)d7 + (6660K8 + 21300K7 + 17800K6)d6 + (−192024K8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='− 874800K7 − 1382400K6 − 731000K5)d5 + (3285360K8 + 18688320K7 + 41115600K6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='+ 39438000K5 + 13350000K4)d4 + (−33957792K8 − 221940000K7 − 588016800K6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='− 728172000K5 − 379740000K4 − 45500000K3)d3 + (206172864K8 + 1453101120K7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='+ 4191652800K6 + 5433912000K5 + 2183760000K4 − 1105200000K3 − 478000000K2)d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='+ (−672686208K8 − 4870886400K7 − 14246409600K6 − 16185744000K5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='+ 2054160000K4 + 13262400000K3 − 7632000000K2 − 11520000000K)d + 906992640K8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='+ 6500113920K7 + 18223833600K6 + 13351392000K5 − 25284960000K4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf'} +page_content='− 27302400000K3 + 65376000000K2 + 30720000000K − 102400000000.' 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a/C9E4T4oBgHgl3EQfeg2i/content/tmp_files/2301.05100v1.pdf.txt b/C9E4T4oBgHgl3EQfeg2i/content/tmp_files/2301.05100v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d08312cdbee8ae1479e48536a69e39fea9f7f081 --- /dev/null +++ b/C9E4T4oBgHgl3EQfeg2i/content/tmp_files/2301.05100v1.pdf.txt @@ -0,0 +1,2587 @@ +Draft version January 13, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Cosmological-Scale HI Distribution Around Galaxies and AGN +Probed with the HETDEX and SDSS Spectroscopic Data +Dongsheng Sun,1, 2 Ken Mawatari,3, 1 Masami Ouchi,3, 1, 4 Yoshiaki Ono,1 Hidenobu Yajima,5 Yechi Zhang,1, 2, 4 +Makito Abe,5 William P. Bowman,6 Erin Mentuch Cooper,7, 8 Dustin Davis,7 Daniel J. Farrow,9, 10 +Karl Gebhardt,7 Gary J. Hill,8, 7 Chenxu Liu,11, 7 and Donald P. Schneider12, 13 +1Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan +2Department of Astronomy, Graduate School of Science, the University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan +3National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +4Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo, 5-1-5 Kashiwanoha, +Kashiwa, Chiba, 277-8583, Japan +5Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan +6Department of Astronomy, Yale University, New Haven, CT 06520 +7Department of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA +8McDonald Observatory, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA +9University Observatory, Fakult¨at f¨ur Physik, Ludwig-Maximilians University Munich, Scheinerstrasse 1, 81679 Munich, Germany +10Max-Planck Institut f¨ur extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany +11South-Western Institute for Astronomy Research, Yunnan University, Kunming, Yunnan, 650500, People’s Republic of China +12Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA +13Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA +ABSTRACT +We present cosmological-scale 3-dimensional (3D) neutral hydrogen (Hi) tomographic maps at z = +2−3 over a total of 837 deg2 in two blank fields that are developed with Lyα forest absorptions of 14,736 +background Sloan Digital Sky Survey (SDSS) quasars at z=2.08-3.67. Using the tomographic maps, +we investigate the large-scale (≳ 10 h−1cMpc) average Hi radial profiles and two-direction profiles of +the line-of-sight (LoS) and transverse (TS) directions around galaxies and AGN at z = 2 − 3 identified +by the Hobby-Eberly Telescope Dark Energy eXperiment (HETDEX) and SDSS surveys, respectively. +The peak of the Hi radial profile around galaxies is lower than the one around AGN, suggesting that +the dark-matter halos of galaxies are less massive on average than those of AGN. The LoS profile of +AGN is narrower than the TS profile, indicating the Kaiser effect. There exist ionized outskirts at +≳ 30 h−1cMpc beyond Hi rich structures of galaxies and AGN found in the LoS profiles that can +be explained by the influence of high energy photons propagating over a long distance. Our findings +indicate that the Hi radial profile of AGN has transitions from proximity zones (≲ a few h−1cMpc) +to the Hi rich structures (∼ 1 − 30 h−1cMpc) and the ionized outskirts (≳ 30 h−1cMpc). Although +there is no significant dependence of AGN types (type-1 vs. type-2) on the Hi profiles, the peaks of +the radial profiles anti-correlate with AGN luminosities, suggesting that AGN’s ionization effects are +stronger than the gas mass differences. +Keywords: galaxies: formation — galaxies: evolution — galaxies: high-redshift — intergalactic medium +1. INTRODUCTION +Galaxy formation in the Universe is closely related +to the neutral hydrogen (Hi) gas in the intergalactic +Corresponding author: Dongsheng Sun +sunds@icrr.u-tokyo.ac.jp +medium (IGM). Within the modern paradigm of galaxy +formation, galaxies form and evolve in the filament +structure of Hi gas (e.g., Meiksin 2009; Mo et al. 2010). +Cosmological hydrodynamics simulations suggest that +the picture of galaxy formation and evolution is asso- +ciated with large-scale baryonic gas exchange between +the galaxy and the IGM (fox 2017; van de Voort 2017). +arXiv:2301.05100v1 [astro-ph.GA] 12 Jan 2023 + +2 +Sun et al. +Enormous rivers of cold gas (∼ 104 K) flow into the +galaxy and trigger the star formation. (e.g., Dekel et al. +2009; Kereˇs et al. 2005) The cold gas is heated by star +formation and then ejected by the powerful galactic- +scale outflows due to feedback caused by stellar winds +and supernovae. +The circulation of gas is one of the keys to under- +standing galaxy formation and evolution. The interplay +of gravitational and feedback-driven processes can have +surprisingly large effects on the large scale behavior of +the IGM. Some of the radiation produced by massive +stars and black hole accretion disks can escape from +the dense gaseous environments and propagate out of +galaxies and photoionize the Hi gas in the circumgalac- +tic medium (CGM) and even in the IGM (National +Academies of Sciences, Engineering 2021; Mukae et al. +2020). +Great progress has been achieved in exploring the Hi +distribution around galaxies and active galactic nuclei +(AGN). The cross-correlation of the Hi in the IGM and +galaxies has been detected by Lyα absorption features +in the spectra of background quasars (e.g., Rauch 1998; +Faucher-Gigu`ere et al. 2008a; Prochaska et al. 2013) +and bright star-forming galaxies (Steidel et al. 2010; +Mawatari et al. 2016; Thomas et al. 2017). The Keck +Baryon Structure Survey (KBSS: Rudie et al. 2012; Ra- +kic et al. 2012; Turner et al. 2014), the Very Large +Telescope LBG Redshift Survey (VLRS: Crighton et al. +2011; Tummuangpak et al. 2014), and other spectro- +scopic programs (e.g., Adelberger et al. 2003, 2005) have +investigated the detailed properties of the Hi distri- +bution around galaxies. These observations target Hi +gas around galaxies on the scale of the circumgalactic +medium (CGM). Recently, 3-dimensional (3D) Hi to- +mography mapping, a powerful technique to reconstruct +the large scale structure of Hi gas, has been developed +by Lee et al. (2014, 2016, 2018). Hi tomography map- +ping is originally proposed by Pichon et al. (2001) and +Caucci et al. (2008) with the aim of reconstructing the +3D matter distribution from the Hi absorption of mul- +tiple sightlines. By this technique, the COSMOS Lyα +Mapping and Tomography Observations (CLAMATO) +survey (Lee et al. 2014, 2018) has revealed Hi large +scale structures with spatial resolutions of 2.5 h−1 co- +moving Megaparsec (cMpc). This survey demonstrates +the power of 3D Hi tomography mapping in a number +of applications, including the study of a protocluster at +z = 2.44 (Lee et al. 2016) and the identification of cos- +mic voids (Krolewski et al. 2018). Due to an interpola- +tion algorithm (Section 4.3) used in the reconstruction +of the 3D Hi tomography map, we are able to estimate +the Hi distribution along lines-of-sight where there are +no available background sources. Based on the 3D Hi to- +mography map of the CLAMATO survey, Momose et al. +(2021) have reported measurements the IGM Hi–galaxy +cross-correlation function (CCF) for several galaxy pop- +ulations. Due to the limited volume of the CLAMATO +3D IGM tomography data, Momose et al. (2021) can- +not construct the CCFs at scales over 24 h−1cMpc in +the direction of transverse to the line-of-sight. Mukae +et al. (2020) have investigated a larger field than the +one of Momose et al. (2021) using 3D Hi tomography +mapping and report that a huge ionized structure of +Hi gas associated with an extreme QSO overdensity re- +gion in the EGS field. Mukae et al. (2020) interpret the +large ionized structure as the overlap of multiple prox- +imity zones which are photoionized regions created by +the enhanced ultraviolet background (UVB) of quasars. +However, Mukae et al. (2020) found only one example of +a huge ionized bubble, and no others have been reported +in the literature. +Dispite the great effort made by previous studies, +the limited volume of previous work prevents us from +understanding how ubiquitous or rare these large ion- +ized structures are. +In order to answer this ques- +tion, we must investigate the statistical Hi distribu- +tions around galaxies and AGN at much larger spatial +scales (≳ 10 h−1cMpc). Although Momose et al. (2021) +derived CCFs for different populations: Lyα emitters +(LAEs), Hα emitters (HAEs), [Oiii] emitters (O3Es), +active galactic nuclei (AGN), and submillimeter galaxies +(SMGs), on a scale of more than 20 h−1cMpc, the lim- +ited sample size results in large uncertainties in the CCF +at large scales and prevents definitive conclusions to be +made regarding the statistical Hi distributions around +galaxies and AGN. +Another open question is the luminosity and AGN +type dependence of the large scale Hi distribution +around AGN. Font-Ribera et al. (2013) have estimated +the Hi distribution around AGN using the Sloan Dig- +ital Sky Survey (SDSS; York et al. 2000) data release +9 quasar catalog (DR9Q; (Pˆaris et al. 2011)) and find +no dependence of the Hi distribution on AGN luminos- +ity. In this study, we investigate the luminosity depen- +dence using the SDSS data release 14 quasar (DR14Q; +Pˆaris et al. 2018) catalog, which includes sources ∼ 2 +magnitude fainter than those used by Font-Ribera et al. +(2013). +In the AGN unification model (Antonucci & +Miller 1985; see also Spinoglio & Fern´andez-Ontiveros +2021), which provides a physical picture that a hot ac- +cretion disk of super-massive blackhole is obscured by a +dusty torus, the type-1 and type-2 classes are produced +by different accretion disk viewing angles. In this pic- +ture, the type-1 (type-2) AGN is biased to AGN with + +Cosmological-Scale Hi Distribution Around Galaxies and AGN +3 +a wide (narrow) opening angle. In the case of type-1 +AGN, one can directly observe the accretion disks and +the broad line region, while for type-2 AGN, only the +narrow line region is observable. Previous studies have +identified the proximity effect that the IGM of type-1 +AGN is statistically more ionized due to the local en- +hancement of the UV background on the line-of-sight +passing near the AGN (Faucher-Gigu`ere et al. 2008b). +Based on the unification model, the type-2 AGN ob- +scured on the line of sight statistically radiates in trans- +verse direction. The investigation of the AGN type de- +pendence on the surrounding Hi can reveal the large +scale Hi distribution influenced by the direction of radi- +ation from the AGN. +To investigate the Hi distributions around galaxies +and AGN on large scales, over tens of h−1cMpc, we +need conduct a new study in a field with length of any +side larger than 100 h−1cMpc. +We reconstruct a 3D +Hi tomography maps of Hi distribution at z ∼ 2 − 3 +in a total area of 837 deg2. +We use ≳ 15, 000 back- +ground sightlines from SDSS quasars (Pˆaris et al. 2018; +Lyke et al. 2020) for the Hi tomography map recon- +struction and have a large number of unbiased galaxies +and AGN from the Hobby Eberly Telescope Dark En- +ergy eXperiment (HETDEX; Gebhardt et al. 2021) and +SDSS surveys for the investigations of the large scale Hi +distributions around galaxies and AGN. +This paper is organized as follows. Section 2 describes +the details of the HETDEX survey and our spectroscopic +data. Our foreground and background samples of galax- +ies and AGN are presented in Section 3. The technique +of creating the Hi tomography mapping and the recon- +structed Hi tomography map are described in Section 4, +and the observational results of Hi distributions around +galaxies and AGN are given in Section 5. In this section, +we also interpret our results in the context of previous +studies, and investigate the dependence of out tomog- +raphy maps on AGN type and luminosity. We adopt +a cosmological parameter set of (Ωm, ΩΛ, h) = (0.29, +0.71, 0.7) in this study. +2. DATA +2.1. HETDEX Spectra +HETDEX provides an un-targeted, wide-area, integral +field spectroscopic survey, and aims to determine the +evolution of dark energy in the redshift range 1.88 − +3.52 using ∼ 1 million Lyman-α emitters (LAEs) over +540 deg2 in the northern and equatorial fields that are +referred to as “Spring” and “Fall” fields, respectively. +The total survey volume is ∼ 10.9 comoving Gpc3. +The HETDEX spectroscopic data are gathered us- +ing the 10 m Hobby-Eberly Telescope (HET; Ramsey +et al. 1994; Hill et al. 2021) to collect light for the Visi- +ble Integral-field Replicable Unit Spectrograph (VIRUS; +Hill et al. 2018, 2021) with 78 integral field unit (IFUs; +Kelz et al. 2014) fiber arrays. VIRUS covers a wave- +length, with resolving power ranging from 750 − 950. +Each IFU has 448 fibers with a 1′′.5 diameter. The 78 +IFUs are spread over the 22 arcmin field of view, with +a 1/4.6 fill factor. +Here we make use of the data re- +lease 2 of the HETDEX (HDR2; Cooper et al. 2023) +over the Fall and Spring fields. In this study, we inves- +tigate the fields where HETDEX survey data are taken +between 2017 January and 2020 June. The effective area +is 11542 arcmin2. The estimated depth of an emission +line at S/N= 5 reaches 3 − 4 × 10−17 erg cm−2 s−1. +2.2. Subaru HSC Imaging +The HETDEX-HSC imaging survey was carried out +in a total time allocation of 3 nights in 2015 − 2018 +(semesters S15A, S17A, and S18A; PI: A. Schulze) and +2019 − 2020 (semester S19B; PI: S. Mukae) over a ∼250 +deg2 area in the Spring field, accomplishing a 5σ limit- +ing magnitude of r = 25.1 mag. The SSP-HSC program +has obtained deep multi-color imaging data on the 300 +deg2 sky, half of which overlaps with the HETDEX foot- +prints. In this study, we use the r-band imaging data +from the public data release 2 (PDR2) of SSP-HSC. The +5σ depth of the SSP-HSC PDR2 r-band imaging data +is typically 27.7 mag for the 3′′.0 diameter aperture. +The data reduction of HETDEX-HSC survey and SSP- +HSC program are processed with HSC pipeline software, +hscPipe (Bosch et al. 2018) version 6.7. +Because the spectral coverage width of the HETDEX +survey is narrow, only 2000 ˚A, most sources appear as +single-line emitters. Furthermore, since the Oii doublet +is not resolved, we rely on the equivalent width (EW) to +distinguish Lyα from Oii. The high-z Lyα emission is +typically stronger than low-z [Oii] lines, due to the in- +trinsic line strengths and the cosmological effects. The +continuum estimate from the HETDEX spectra reach +about g= 25.5 (Davis et al. 2021; Cooper et al. 2023) +and we improve on this using the deep HSC imaging. +We estimate EW using continua measured from two sets +of images taken by HSC r-band imaging survey for HET- +DEX (HETDEX-HSC survey) and the Subaru Strategic +Program (SSP-HSC; Aihara et al. 2018). Davis et al. +and Cooper et al. find that our contamination of Oii +emitters in the LAE sample to be below 2%. +2.3. SDSS-IV eBOSS Spectra +We use quasar data from eBOSS (Dawson et al. 2016), +which is publically available in the SDSS Data Release +14 and 16 quasar catalog (DR14Q, DR16Q; Pˆaris et al. + +4 +Sun et al. +2018; Lyke et al. 2020). The cosmology survey, eBOSS, +is part of SDSS-IV. The eBOSS quasar targets are se- +lected by the XDQSOz method (Bovy et al. 2012) and +the color cut +mopt − mW ISE ≥ (g − i) + 3, +(1) +where mopt is a weighted stacked magnitude in the g, r +and i bands and mW ISE is a weighted stacked magni- +tude in the W1 and W2 bands of the Wide-Field In- +frared Survey (WISE; Wright et al. 2010). The aim of +the eBOSS is to accomplish precision angular-diameter +distance measurements and the Hubble parameter deter- +mination at z ∼ 0.6 − 3.5 using different tracers of the +underlying density fields over 7500 deg2. Its final goal is +to obtain spectra of ∼ 2.5 million luminous red galaxies, +∼ 1.95 million emission line galaxies, ∼ 450,000 QSOs at +0.9 ≤ z ≤ 2.2, and the Lyman-α forest of 60,000 QSOs +at z > 2 over four years of operation. +The eBOSS program is conducted with twin SDSS +spectrographs (Smee et al. 2013), which are fed by 1,000 +fibers connected from the focal plane of the 2.5m Sloan +telescope (Gunn et al. 2006) at Apache Point Observa- +tory. SDSS spectrographs have a fixed spectral band- +pass of 3600 − 10000 ˚A over the 7 deg2 field of view. +The spectral resolution varies from 1300 at the blue end +to 2600 at the red end, where one pixel corresponds to +1.8 − 5.2 ˚A. +3. SAMPLES +Our study aims to map the statistical distribution of +Hi gas on a cosmological scale around foreground galax- +ies and AGN by the 3D Hi tomography mapping tech- +nique with background sources at z = 2−3. We use the +foreground galaxies, foreground AGN, and background +sources presented in Sections 3.1, 3.2, and 3.3, respec- +tively. +Two of the goals of this study are to explore the de- +pendence of luminosity and AGN type on the Hi distri- +bution. To examine statistical results, we need a large +number of bright AGN and type-2 AGN. Compared to +moderately bright AGN and type-1 AGN, bright AGN +and type-2 AGN are relatively rare. To obtain a suffi- +ciently large samples of bright AGN and type-2 AGN, we +expand the Spring and Fall fields of the HETDEX sur- +vey, from which we are able to investigate the statistical +luminosity and AGN type dependence of the HI distribu- +tion around AGN (Section 3.2). The northern extended +Spring field flanking the HETDEX survey fields, referred +to as the “ExSpring field”, covers over 738 deg2, while +the equatorial extended Fall field flanking the HETDEX +survey fields, here after “ExFall field”, covers 99 deg2. +The total area of our 3D Hi tomography mapping field +is 837 deg2 in the ExSpring and ExFall fields that is re- +ferred to as “our study field”. Our analysis is conducted +in our study field where the foreground galaxies+AGN +and the background sources overlap on the sky. As an +example, we present the foreground galaxies+AGN in +the ExFall field at z = 2.0 − 2.2 in Figure 1. We also +present the sky distribution of the background sources +within the ExFall field in Figure 2. The rest of the fore- +ground and background sources are shown in the Ap- +pendix. +3.1. Foreground Galaxy Sample +We make a sample of foreground galaxies from the +data of the HETDEX spectra (Section 2.1) and the Sub- +aru HSC images (Section 2.2). With these data, Zhang +et al. (2021) have build a catalog of LAEs that have +the rest-frame equivalent widths (EW0) of EW0 > 20 +˚A and the HETDEXs Emission Line eXplorer (ELiXer) +probabilities (Davis et al. 2021; Davis et al. 2023) larger +than 1. This EW0 cut is similar to previous LAE studies +(e.g., Gronwall et al. 2007; Konno et al. 2016). This cat- +alog of LAEs is composed of 15959 objects. Because the +LAE catalog of Zhang et al. (2021) consists of galaxies, +type-1 AGN, and type-2 AGN, we isolate galaxies from +the sources of the LAE catalog with the limited observa- +tional quantities, Lyα and UV magnitude (MUV), that +can be obtained from the HETDEX and Subaru/HSC +data. Because type-1 AGN have broad-line Lyα emis- +sion, we remove sources with broad-line Lyα whose full +width half maximum (FWHM) of the Lyα emission lines +are greater than 1000 km s−1. To remove clear type-2 +AGN from the LAE catalog, we apply a UV magnitude +cut of MUV < −22 mag that is the bright end of the UV +luminosity function dominated by galaxies (Zhang et al. +2021). We then select sources in our study field, and +apply the redshift cut of z = 2.0 − 3.0 (as measured by +the principle component analysis of multiple lines; Pˆaris +et al. 2018) to match the redshift range over which we +construct Hi tomography map. These redshifts are mea- +sured with Lyα emission (Zhang et al. 2021), because +Lyα is the only emission available for all of the sources. +By these selections, we obtain 14130 galaxies from the +LAE catalog. These 14130 galaxies are referred to as the +“Galaxy” sample. +3.2. Foreground AGN Samples +In this subsection, we describe how we select fore- +ground AGN from two sources, (a) the combination of +the HETDEX spectra and the HSC imaging data and +(b) the SDSS DR14Q catalog. +The type-1 AGN are +identified with the sources of (a) and (b), while the type- +2 AGN are drawn from the source of (b). + +Cosmological-Scale Hi Distribution Around Galaxies and AGN +5 +Figure 1. +Sky distribution of the foreground AGN and galaxies at z = 2.0 − 2.2 in the ExFall field. The squares present +the positions of All-AGN sample sources. Pink (magenta) squares represent the sources of the T1-AGN (T2-AGN) sample. +The cyan and blue dots show the positions of the Galaxy and T1-AGN(H) sample sources, respectively. The black dashed line +indicates the border of the Hi tomography map in the Exfall field. +Figure 2. Sky distribution of background AGN in the ExFall field. The gray crosses indicate background AGN that are used +to reconstruct our Hi tomography map. The back dashed line has the same meaning as that in Figure 1. +Table 1. Sample size of foreground samples at z = 2 − 3 +Name of sample +ExFall +ExSpring +Total +Survey +Criteria +Galaxy +3431 +11436 +14867 +HETDEX +EW0 > 20 ˚A, FWHMLyα < 1000 km/s, Muv> −22 mag +T1-AGN(H) +438 +1349 +1787 +HETDEX +EW0 > 20 ˚A, FWHMLyα > 1000 km/s +T1-AGN +2393 +12300 +14693 +SDSS +FWHMLyα > 1000 km/s +T2-AGN +436 +1633 +2069 +SDSS +FWHMLyα < 1000 km/s +Table 2. Sample size of background sample at z = 2.08 − 3.67 +Name of sample +ExFall +ExSpring +Total +Survey +Criteria +background AGN +2181 +12555 +14736 +SDSS +⟨S/N⟩Lyαforest > 1.4 +With the source (a) that is the same as the one stated +in Section 3.1, Zhang et al. (2021) have constructed +the LAE catalog. We use the catalog of Zhang et al. +(2021) to select LAEs at z ∼ 2 − 3 that fall in our study +field. Applying a Lyα line width criterion of FWHM +> 1000 km s−1 with the HETDEX spectra, we identify +broad-line AGN, i.e. type-1 AGN, from the LAEs. We +thus obtain 1829 type-1 AGN that are referred to as +T1-AGN(H). +We use the width of Lyα emission line for the selection +of type-1 AGN. This is because no other emission lines +characterising AGN, e.g. Civ, are available for all of the +LAEs due to the limited wavelength coverage and the +sensitivity of HETDEX. Similarly, the redshifts of T1- + +Dec.[deg] +2 +0 +2 +35 +30 +25 +20 +15 +10 +5 +R.A.[deg]Dec.[deg] +1 +35 +30 +25 +20 +15 +10 +5 +R.A.[deg]6 +Sun et al. +AGN(H) objects are measured with Lyα emission whose +redshifts may be shifted from the systemic redshifts by +up to a few 100 km s−1 (See Section 3.1). We do not +select type-2 AGN from the source of (a), because we +cannot identify type-2 AGN easily with the given data +set of source (a). +From the source (b), we obtain the other samples of +foreground AGN. We first choose objects with a classi- +fication of QSOs of the SDSS DR14Q, and remove ob- +jects outside the redshift range of z = 2.0 − 3.0 in our +study field. We obtain 23721 AGN. For 16762 out of +23721 AGN, Lyα FWHM measurements are available +from Rakshit et al. (2020). +The other AGN without +FWHM measurement are removed due to the poor qual- +ity of the Lyα line. We thus use these 16762 AGN with +good quality of the Lyα line to compose our AGN sam- +ple, referred to as All-AGN sample. +To investigate the type dependence, we classify these +16762 AGN into type-1 and type-2 AGN. In the same +manner as the T1-AGN(H) sample construction, we use +Lyα line width measurements of Rakshit et al. (2020) +for the type-1 and type-2 AGN classification. For the +16762 AGN, we apply the criterion of Lyα FWHM > +1000 km s−1 (Villarroel & Korn 2014; Panessa & Bassani +2002) to select type-1 AGN, and obtain 14693 type-1 +AGN. Following Villarroel & Korn (2014); Panessa & +Bassani (2002), we classify type-2 AGN by the criterion +of Lyα FWHM < 1000 km s−1 and obtain 2069 type- +2 AGN (c.f. Alexandroff et al. 2013; Zakamska et al. +2003). These type-1 and type-2 AGN are referred to as +T1-AGN and T2-AGN, respectively. +Table 1 presents the summary of foreground samples. +We obtain 14693 and 1829 type-1 AGN, which referred +to as T1-AGN and T1-AGN(H), from the SDSS and +HETDEX surveys, respectively. We select 2069 type-2 +AGN that are referred to as T2-AGN from the SDSS +survey. +3.3. Background Source Sample +In this subsection, we describe how the background +sources are selected. We select the background sources +with the SDSS DR16Q catalog, following the three steps +below. +In the first step, we extract QSOs in our study field +from the SDSS DR16Q catalog. We then select QSOs +falling in the range of redshifts from 2.08 to 3.67. The +lower and upper limits of the redshift range are deter- +mined by the Lyα forest. Our goal is to probe Hi ab- +sorbers at z = 2.0−3.0 with the Lyα forest. Because the +Lyα forest is observed in the rest-frame 1040 − 1185 ˚A +of the background sources, we obtain the lower and up- +per limits of the redshifts, 2.08 and 3.67, by 1216 × (1 + +2.0)/1185−1 = 2.08 and 1216×(1+3.0)/1040−1 = 3.67, +respectively. By this step, we have selected 26899 back- +ground source candidates. +In the second step, we choose background source can- +didates with good quality. We calculate the average sig- +nal to noise ratio, ⟨S/N⟩, in the wavelength range of the +Lyα forest for the 26899 background source candidates, +and select 15573 candidates with ⟨S/N⟩ greater than 1.4. +To maximize the special resolution of the tomography +map, we set the threshold, ⟨S/N⟩ > 1.4, smaller than +the value used by Mukae et al. (2020). This threshold +is more conservative than the value, 1.2, used in Lee +et al. (2018). In the third step, we remove damped Lyα +absorbers (DLAs) and broad absorption lines (BALs) +from the Lyα forest of the 15573 candidates, because the +DLAs and BALs cause an overestimation of the absorp- +tion of the Lyα forest. We identify and remove DLAs +using the catalog of Chabanier et al. (2022), which is +based on the SDSS DR16Q (Lyke et al. 2020). We mask +out the wavelength ranges contaminated by the DLAs of +the Chabanier et al. (2022) catalog (see Section 4.1 for +the procedures). We conduct visual inspection for the +15573 candidates to remove 115 BALs. In Figure 3, we +show the spectrum with BALs identified by visual in- +spection. In this way, we obtain 15458 (= 15573 − 115) +sources whose spectra are free from DLAs and BALs, +which we refer to as the background source sample. Ta- +ble 2 lists the number of background sources in each +field. +Figure 3. Spectrum of background AGN with BALs. The +black line represents the spectrum of a background source. +The vertical dashed lines present the central wavelengths +of the metal absorptions. +The yellow hatches show the +wavelength ranges of the BALs. +The gray hatches indi- +cate the wavelength ranges not used for the reconstruction +of Hi tomography maps. The SDSS ID of this spectrum is +106584616, whose redshift is 3.067837. +4. HI TOMOGRAPHY AND MAPPING + +8 +Flux density +6 +2 +4500 +000S +5500 +6000 +6500 +Wavelength [A]Cosmological-Scale Hi Distribution Around Galaxies and AGN +7 +In this section we describe the process to construct Hi +tomography maps with the spectra of the background +sources. For Hi tomography, we need to obtain intrin- +sic continua of the background sources. Section 4.2 ex- +plains masking the biasing absorption features in the +background sources, while Section 4.3 determines the +intrinsic continua of the background source spectra. In +Section 4.3, we construct Hi tomography maps with the +intrinsic continuum spectra. +4.1. DLA and Intrinsic Absorption Masking +Because a DLA is an absorption system with a high +neutral hydrogen column density NHI > 2 × 1020 cm−2, +the intervening DLA completely absorbs a large por- +tion of the Lyα forest over ∆v ∼ 103 km s−1, which +gives bias in the estimates of the intrinsic continua of +the background sources. For the spectra of the back- +ground sources, we mask out the DLAs identified in +Section 3.3. We determine the range of wavelengths for +masking with the IDL code of Lee et al. (2012). The +wavelength range corresponds to the equivalent width +of each DLA (Draine 2011): +W ∼ λα +� e2 +mec2 NHIfαλα +�γαλα +c +��1/2 +. +(2) +In the formula, λα is the rest-frame wavelength of the +hydrogen Lyα line (i.e. +1216 ˚A), while c, e, me, fα, +NHi, and γα are the speed of light, the electron charge, +the electron mass, the Lyα oscillator strength, the Hi +column density of the DLA, and the sum of the Einstein +A coefficients. We mask out these wavelength ranges of +the background source spectra. In Figure 5, the masked +DLA is indicated by yellow hatches. +We also mask out the intrinsic absorption lines of the +metal absorption lines, which are the other sources of +bias. We mask SIv λ1062, Nii λ1084, Ni λ1134, and +Ciii λ1176 (Lee et al. 2012), which are shown by the +dashed lines in Figure 5. Because the spectral resolu- +tions of SDSS DR14Q are ∆λ = 1.8 − 5.2 ˚A, we adopt +the masking size of 10 ˚A in the observed frame. +4.2. Intrinsic Continuum Determination +In order to obtain the intrinsic continuum of the back- +ground source (Section 3.3) in the Lyα forest wavelength +range (LAF-WR; 1040−1185 ˚A), we conduct mean-flux +regulated principle component analysis (MF-PCA) fit- +ting with the IDL code (Lee et al. 2012) for the back- +ground sources after the masking (Section 4.1). +There are two steps in the MF-PCA fitting process. +The first step is to predict the shape of the intrinsic +continuum of the background sources in the LAF-WR. +We conduct least-squares principle component analysis +(PCA) fitting (Suzuki et al. 2005; Lee et al. 2012) to the +background source spectrum in the rest frame 1216 − +1600 ˚A : +fPCA(λ) = µ(λ) + +8 +� +j=1 +cjξj(λ), +(3) +where λ is the rest-frame wavelength. The values of cj +are the free parameters for the weights. The function of +µ(λ) is the average spectrum calculated from the 50 lo- +cal QSO spectra in Suzuki et al. (2005). The function of +ξj(λ) represents the jth principle component (or ‘eigen- +spectrum’) out of the 8 principle components taken from +the PCA template shown in Figure 4. +In the second step, we predict the intrinsic continuum +of the background source in the LAF-WR. Because the +PCA template is obtained with the local QSO spectra, +the best-fit fPCA in the LAF-WR does not include cos- +mic evolution on the average transmission rate of the +Lyα forest. On average, the best-fit fPCA in the LAF- +WR should agree with the cosmic mean-flux evolution +(Faucher-Gigu`ere et al. 2008c): +⟨F(z)⟩ = exp[−0.001845(1 + z)3.924], +(4) +where z is the redshift of the absorber. We use fPCA and +a correction function of a + bλ to estimate the intrinsic +continuum fintrinsic(λ) for large-scale power along the +line of sight with the equation: +fintrinsic(λ) = fPCA(λ) × (a + bλ), +(5) +where a and b are the free parameters. +Because the +ratio of fobs(λ)/fintrinsic(λ) should agree with the cosmic +average ⟨F(z)⟩ for z = (λ/1216)−1 in the LAF-WR, we +conduct least-squares-fitting to find the values of a and +b providing the best fit between the mean ratio and the +cosmic average. The red line shown by the bottom panel +of Figure 5 presents a MF-PCA fitted continuum derived +from the spectrum of one of our background sources. +By the MF-PCA fitting, we have obtained the esti- +mates of fintrinsic(λ) for 14736 out of the 15458 back- +ground sources. We find the other background sources +show poor fitting results found by visual inspection. We +do not use these background sources in the following +analyses. +Figure 6 shows an example of poor fitting +result due to the unknown absorption. We adopt con- +tinuum fitting errors of ∼ 7%, ∼ 6%, and ∼ 4% for Lyα +forests with mean S/N values of < 4, 4 − 10, and > 10, +respectively (Lee et al. 2012). +4.3. HI Tomography Map Reconstruction +We reconstruct our Hi tomography maps by a proce- +dure similar to Lee et al. (2018). We define Lyα forest + +8 +Sun et al. +Figure 4. Principle components and mean flux taken from +Suzuki et al. (2005). The top panel shows the normalized +mean flux of 50 local QSOs in rest-frame wavelength. The +bottom 8 panels show the 1st − 8th principle components +that are used in the PCA fitting in our study. Each principle +component is normalized to the mean flux. +fluctuations δF at each pixel on the spectrum by +δF = fobs/fintrinsic +⟨F(z)⟩ +− 1 +(6) +, where fobs and fintrinsic are the observed spectrum +and estimated intrinsic continuum, respectively. ⟨F(z)⟩ +is the cosmic average transmission. We calculate δF with +our background source spectra. The top panel of Figure +5 shows the ‘spectrum’ of δF derived from the fobs and +fintrinsic in the bottom panel. +For the pixels in the +wavelength ranges of masking (Section 4.1), we do not +use δF in our further analyses. We thus obtain δF in +876,560 pixels. +For the the HI tomography map of the Extended Fall +field, we define the cells of the Hi tomography map in the +three-dimensional comoving space. We choose a volume +of 30◦ × 3.3◦ in the longitudinal and latitudinal dimen- +Figure 5. Example of a background source spectrum that +was used for the reconstruction of the Hi tomography map. +Bottom panel: Estimation of intrinsic continuum. The thin +black line is the spectrum of a background source taken from +the SDSS survey. +The red and magenta lines are the re- +sults of MF-PCA and PCA fitting, respectively. The vertical +dashed lines present the central wavelengths of the metal ab- +sorptions. The gray hatches represent the wavelength ranges +that are not used for the Hi tomography map reconstruc- +tions. The yellow hatch indicates the wavelength ranges of +DLA. Top panel: Spectrum of δF extracted from the bottom +panel in the LAF-WR. The vertical yellow and gray hatches +are the same as those in the bottom panel. The black and +pink lines show the spectrum of δF and the error of δF at the +corresponding wavelength extracted from the bottom panel. +The horizontal line indicates the cosmic average of Lyα forest +transmission. +Figure 6. Same as the bottom panel of Figure 5, but for +the background spectrum with a poor fitting result. +The +red and magenta lines are the results of MF-PCA and PCA +continuum fitting, respectively. The yellow hatch indicates +the wavelength range of unknown absorption. +sions, respectively, in the redshift range of 2.0 < z < 3.0. +The comoving size of our Hi tomography map is 2257 +h−1cMpc × 233 h−1cMpc × 811 h−1cMpc in the right + +5 +Mean flux +0 +2 +1st Component +0 +.... +0.2 +2nd Component +0.0 +0.1 +3rd Component +0.0 +0.1 +0.05 +0.00 +4th Component +0.05 +0.1 +0.0 +5th Component +0.1 +0.1 +0.0 +6th Component +0.1 +0.2 +0.0 +7th Component +0.2 +0.1 +0.0 +-0.1 +8th Component +1000 +1100 +1200 +1300 +1400 +1500 +1600 +Rest-frame wavelength [A]0.6 +0.3 +AF0.0 +-0.3 +-0.6 +12 +Flux density +080 +4000 +4500 +000S +5500 +6000 +Wavelength [A]6 +Flux density +2 +0 +3500 +4000 +4500 +5000 +5500 +Wavelength [A]Cosmological-Scale Hi Distribution Around Galaxies and AGN +9 +ascension (R.A.), declination (Dec), and z directions, +respectively in the same manner as Mukae et al. (2020). +Our Hi tomography map has 451 × 46 × 162 cells, and +one cell is a cubic with a size of 5.0 h−1cMpc on a side, +where the line-of-sight distance is estimated under the +assumption of the Hubble flow. +We conduct a Wiener filtering scheme for reconstruct- +ing the sightlines that do not have background sources. +We use the calculation code developed by Stark et al. +(2015). The solution for each cell of the reconstructed +sightline is obtained by +δrec +F += CMD · (CDD + N)−1 · δF, +(7) +where CMD, CDD, and N are the map-data, data-data, +and noise covariances, respectively. We assume Gaus- +sian covariances between two points r1 and r2: +CMD = CDD = C(r1, r2), +(8) +C(r1, r2) = σ2 +F exp +� +−(∆r∥)2 +2L2 +∥ +� +exp +� +−(∆r⊥)2 +2L2 +⊥ +� +, +(9) +where ∆r∥ and ∆r⊥ are the distances between r1 and +r2 in the directions of parallel and transverse to the line +of sight, respectively. The values of L⊥ and L∥ are the +correlation lengths for vertical and parallel to the line- +of-sight (LoS) direction, respectively, and defined with +L⊥ = L∥ = 15 h−1cMpc. The value of σ2 +F is the normal- +ization factor that is σ2 +F = 0.05. Stark et al. (2015) de- +velop this Gaussian form to obtain a reasonable estimate +of the true correlation function of the Lyα forest. We +perform the Wiener filtering reconstruction with the val- +ues of δF at the 898390 pixels, using the aforementioned +parameters of the Stark et al. (2015) algorithm with a +stopping tolerance of 10−3 for the pre-conditioned con- +jugation gradient solver. As noted by Lee et al. (2016), +the boundary effect that leads to an additional error +on δF occurs at the positions that are near the bound- +aries of an Hi tomography map. The boundary effect +is caused by the background sightlines not covering the +region that contribute to the calculation of the δF values +for cells near the Hi tomography map boundaries. To +avoid the boundary effect, we extend a distance of 40 +h−1cMpc for each side of the Hi tomography map of the +ExFall field. The resulting map is shown in Figure 7. +For the HI tomography map reconstruction of the Ex- +tended Spring field (hereafter ExSpring field), we per- +form almost the same procedure as the one of the Ex- +Fall field. The area of the ExSpring field is more than +6 times larger than that of the ExFall field. We sep- +arate the ExSpring field into 8 × 3 = 24 footprints to +save calculation time. Each footprint covers an area of +10◦ × 5◦ in the R.A. and Dec directions, respectively. +We reconstruct the Hi tomography map one by one for +the footprints of the ExSpring field. +To weaken the boundary effect, we extend a distance +of 40 h−1cMpc for each side of the footprints. The ex- +tensions mean that every two adjacent footprints has an +overlapping region of 80 h−1cMpc width. The width of +the overlapping regions is a conservative value to weaken +the boundary effect since it is much larger than the res- +olution, 15 h−1cMpc, of our Hi tomography maps. By +the 40 h−1cMpc extension, we reduce the uncertainty +in the δF value for the edge of each footprint caused by +boundary effect to ±0.01. This value corresponds to the +1/10 of the typical error for each cell of the Hi tomogra- +phy map (Mukae et al. 2020) The remaining additional +error caused by boundary effect is negligible compared +to the statistical uncertainties in the HI distributions ob- +tained in Section 5. Then we follow the reconstruction +procedure for the ExFall field to reconstruct HI tomog- +raphy maps of the footprints and cut off all the cells +within 40 h−1cMpc to the borders that are affected by +the boundary effect. Finally we obtain the Hi tomogra- +phy map of the ExSpring field with a special volume of +3475 h−1cMpc × 1058 h−1cMpc × 811 h−1cMpc in the +R.A., Dec, and z directions, respectively (Figure 8). +5. RESULTS AND DISCUSSIONS +5.1. Average HI Profiles around AGN: Validations of +our AGN Samples +In this section we present the Hi profile, δF as a func- +tion of distance, with the All-AGN sample sources, us- +ing the reconstructed Hi tomography maps. We com- +pare the Hi profile of the All-AGN sample to the one of +the previous study (Font-Ribera et al. 2013). We also +present the comparison of the Hi profiles between T1- +AGN(H) and T1-AGN samples that are made with the +HETDEX and SDSS data. In this study, we only discuss +the structures having size ≳ 15 h−1cMpc corresponding +to the resolution of our 3D Hi tomography maps. +For the Hi profiles with the All-AGN sample, we +extract δF values around the 16978 All-AGN sample +sources in the Hi tomography map. +We cut the Hi +tomography map centered at the positions of the All- +AGN sample sources, and stack the δF values to make a +two dimensional (2D) map of the average δF distribution +around the sources that is referred to as a 2D Hi profile +of the All-AGN sample sources. The two dimensions of +the 2D Hi profile correspond to the transverse distance +DTrans and the LoS Hubble distance. The velocity corre- +sponding to the LoS Hubble distance is referred to as the +LoS velocity. Here we define the Lyα forest absorption +fluctuation +AF ≡ −δF +(10) + +10 +Sun et al. +Figure 7. 3D Hi tomography map of the ExFall field. The color contours represent the values of δF from negative (red) to +positive (blue). The spatial volume of the Hi tomography map is 2257×233×811 h−3cMpc3. The redshift range is z = 2.0−3.0. +that is an indicator of the amount of the Hi absorption. +Figure 9 shows the 2D Hi profile with values of AF +(δF) for All-AGN sample. The solid black lines denote +the contours of AF. In each cell of the 2D Hi profile, +we define the 1σ error with the standard deviation of +AF values of the 100 mock 2D Hi profiles. Each mock +2D Hi profile is obtained in the same manner as the +real 2D Hi profile, but with random positions of sources +whose number is the same as the one of All-AGN sample +sources. +In Figure 9, the dotted black lines indicate +the contours of the 6σ, 9σ and 12σ confidence levels, +respectively. We find the 19.5σ level detection of AF at +the source position (0,0). The AF value at the source +position indicates the averaging value over the ranges of +(−7.5 h−1cMpc, +7.5 h−1cMpc) in both the LoS and +transverse directions. The 19.5σ level detection at the +source position is suggestive that rich Hi gas exists near +the All-AGN sources on average The 2D Hi profile is +more extended in the transverse direction than along +the line of sight. We discuss this difference in Section +5.2. +We then define a 3D distance, D, under the assump- +tion of the Hubble flow in the LoS direction. We derive +AF as a function of D that is referred to as ”Hi radial +profile”, averaging AF values of the 2D Hi profile over +the 3D distance. Figure 10 shows the Hi radial profile +of the All-AGN sample. We find that the AF values de- +crease towards a large distance. This trend is consistent +with the one found by Ravoux et al. (2020) with the +SDSS quasars. +Ravoux et al. (2020) have obtained the average Hi +absorption distribution around the AGN taken from the +SDSS data release 16 quasar (SDSS DR16Q) catalog in +the field of Strip 82. The criteria of the target selection +for the SDSS DR16Q and SDSS DR14Q sources are the +same. The luminosity distribution of AGN for Ravoux +et al. (2020) is almost the same as that of our All-AGN +sample sources that are taken from the SDSS DR14Q +catalog. We derive the average radial Hi profile of the +Ravoux et al. (2020) AGN sources by the same method +as for our All-AGN sample, using the 3D Hi tomogra- +phy map reconstructed by Ravoux et al. (2020). +We +compare the radial Hi profile of the All-AGN sample +with the one derived from the 3D Hi tomography map +of Ravoux et al. (2020). The comparison is shown in +Figure 10. Our result agrees with that of Ravoux et al. +(2020) within the error range at scale D > 10 h−1 cMpc. +The peak values of AF are comparable, AF ≃ 0.02. The + +0.300 +0.214 +0.129 +Dec[cMpc] +0.0429 +2250 +300 +2000 +-0.0429 +750 +1500 +200 +1250 +-0.129 +1000 +750 +100 +750 +500 +RA +[cMpc +500 +-0.214 +250 +250 +0 +z[cMpc] +-0.300Cosmological-Scale Hi Distribution Around Galaxies and AGN +11 +Figure 8. Same as Figure 7, but for the ExSpring field. The spatial volume of the Hi tomography map is 3475 × 1058 × 811 +h−3cMpc3. +slight difference between the peak values of our and +Ravoux et al.’s results can be explained by the differ- +ent approaches of the estimation for the intrinsic con- +tinuum adopted by Ravoux et al. and us. Ravoux et al. +conduct power law fitting, which is different from the +MF-PCA fitting that we used, for the intrinsic contin- +uum in the wavelength range of the Lyα forest. Given +the low (∼ 15 h−1) spatial resolution of both our Hi to- +mography map and that of Ravoux et al. (2020), neither +studies are able to search for the proximity effect mak- +ing a photoionization region around AGN (D’Odorico +et al. 2008). From the comparison shown by Figure 10, +we conclude that the Hi distribution derived from our +Hi tomography map is reliable. +To check the reliability of the HETDEX survey results, +we use the reliable result of the SDSS AGN to compare +with the result derived by the HETDEX AGN. +We select type-1 AGN from the HETDEX’s T1- +AGN(H) and SDSS’s T1-AGN samples to make sub- +samples of T1-AGN(H) and T1-AGN whose rest-frame +1350 ˚A luminosity (L1350) distributions are the same. +For T1-AGN, the measurements directly from the SDSS +spectra (Lspec +1350) are available (Rakshit et al. 2020). For +T1-AGN(H), we do not have Lspec +1350 measurements from +the HETDEX spectra, we estimate it using HSC r-band +imaging. +Since the central wavelength of the r-band +imaging is rest-frame ∼ 1700˚A, we calibrate the conver- +sion between r-band luminosity, Lphot +UV , and Lspec +1350. We +examine the 283 type-1 AGN sources that appear in +both the SDSS and HETDEX surveys (and, thus, have +both Lspec +1350 measurements from SDSS and r-band lumi- +nosities from HSC) to calibrate the relationship. The +results are displayed in Figure 11. The Lphot +UV are always +smaller than those of Lspec +1350 (Rakshit et al. 2020). Due +to the blue UV slope of the spectra for the AGN both +categorized in the T1-AGN(H) and T1-AGN samples, +the luminosity of the rest-frame 1350 ˚A always shows +a larger value than the one of rest-frame 1700 ˚A. We +conduct linear fitting to the data points of Figure 11, +and obtain the best-fit linear function. With the best-fit +linear function, we estimate Lspec +1350 values for the HET- +DEX’s T1-AGN(H) sample sources. +We show the Lspec +1350 distributions of all the T1-AGN(H) +and T1-AGN sample sources in the upper panel of Fig- +ure 12. We make the sub-samples of T1-AGN and T1- +AGN(H) that consist of the sources in the overlapping +area of Lspec +1350 distributions. We present the Lspec +1350 distri- +butions of the T1-AGN and T1-AGN(H) sub-samples in + +0.300 +0.214 +Dec [cMpc] +0.129 +11000 +0.0429 +900 +800 +-0.0429 +700 +35Q0 +600 +3250 +3000 +-0.129 +2750 +500 +2500 +2250 +400 +2000 +1750 +-0.214 +1500 +300 +RA[cMpcl +1250 +200 +1000 +750 +750 +-0.300 +10Q +500 +500 +250 +250 +0 +z[cMpc]12 +Sun et al. +Figure 9. +2D Hi profile of the All-AGN sample sources. The color map indicates the AF (δF) values of each cell of the 2D +Hi profile. The solid lines denote constant AF (δF) values in steps of 0.01 (−0.01) starting at 0.01 (−0.01). The dotted lines +correspond to multiples of 3σ starting at 6σ. +Figure 10. +Hi radial profile of the All-AGN and Ravoux +et al. (2020) AGN samples. The black and gray data points +and error bars show the Hi radial profiles of our All-AGN +sample sources and the AGN of Ravoux et al. 2020, respec- +tively. The horizontal dashed line shows the cosmic average +Hi absorption, AF = 0 (δF = 0). +the bottom panel of Figure 12. We obtain 540 and 4338 +type-1 AGN for the sub-samples of T1-AGN(H) and T1- +AGN, respectively, whose Lspec +1350 distributions are shown +in the bottom panel of Figure 11. +We derive the Hi radial profiles for the sub-samples +of T1-AGN(H) and T1-AGN sample sources, as shown +in Figure 13. The Hi radial profiles of T1-AGN(H) and +T1-AGN sub-sample sources are in good agreement. +Figure 11. Relations of Lphot +UV +against Lspec +1350 for the sources +both categorized in the T1-AGN(H) and T1-AGN samples. +The Lphot +UV +and Lspec +1350 are measured from the HSC r-band +imaging and SDSS spectra (Rakshit et al. 2020), respectively. +The gray points show the distribution of Lspec +1350 − Lphot +UV +re- +lations for the sources both categorized in the T1-AGN(H) +and T1-AGN samples. The black dashed line indicates the +relation where Lspec +1350 = Lphot +UV . The red dashed line represents +the linear best fit of the blue points. +5.2. AGN Average Line-of-Sight and Transverse Hi +Profiles + +0.04 +0.04 +Ravoux+20 AGN +0.03 +0.03 +All-AGN +AF +0.02 +-0.02 +0.010F +0.01 +0.00 +0.00 +0.01 +0.01 +0 +10 20 3040506070 +D [h-1cMpc]best fit +46 +[erg s +. +45 +logL +44 +45 +46 +logL +spec +[erg s-1] +1350LoS Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +50 +0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN +13 +Figure 12. Top panel: Lspec +1350 distributions of the T1-AGN +and T1-AGN(H) samples with blue and red histograms, re- +spectively. Bottom panel: Same as the top panel, but for the +T1-AGN and T1-AGN(H) sub-sample sources. +Figure 13. +Hi radial profiles of the T1-AGN and T1- +AGN(H) sub-samples. The blue and red triangles show the +values of AF as a function of distance, D, for the T1-AGN +and T1-AGN(H) sample sources, respectively. +The hori- +zontal dashed line shows the cosmic average Hi absorption, +AF = 0. The right y-axis shows the corresponding δF values. +Based on the 2D Hi profile of the All-AGN sample +(Figure 9), we find that the Hi distributions of the All- +AGN sample sources are more extended in the trans- +verse direction. In this section, we present the Hi radial +profiles of All-AGN sample in the LoS and transverse +directions and compare these two Hi radial profiles. +To derive the Hi radial profile of the All-AGN sample +with the absolute LoS distance, which is referred to as +the LoS Hi radial profile (Figure 15), we average AF val- +ues of the 2D Hi profiles of All-AGN over DTrans < 7.5 +h−1cMpc (from −7.5 h−1cMpc to +7.5 h−1cMpc in the +transverse direction) that corresponds to the spatial res- +olution of the 2D Hi profile map, 15 h−1cMpc. Among +the 16,978 All-AGN sample sources, 10,884 sources are +used as both background and foreground sources. In this +case, the Hi absorption (AF) of these 10,884 sources at +the LoS velocity ≲ −5250 km s−1 is estimated mainly +from their own spectrum. As the discussion in Youles +et al. (2022), the redshift uncertainty of the SDSS AGN +causes the overestimation of intrinsic continuum and the +underestimation of AF around the metal emission lines +such as Ciii λ1176. This leads to a systemics toward +positive AF in the Hi radial profile of LoS velocity (LoS +distance) at the LoS velocity ≲ 5250 km s−1 (Figure 14). +The Hi radial profile of LoS velocity (LoS distance) is de- +rived by averaging AF values over DTrans < 7.5 h−1cMpc +as a function of the negative and positive LoS velocity +(LoS distance). In this study, we only use the values of +AF at the LoS distance > −52.5h−1cMpc (LoS velocity +> −5250 km s−1) to derive the LoS Hi radial profile +of the All-AGN sample (Figure 15). The scale, LoS dis- +tance > −52.5h−1cMpc (LoS velocity > −5250 km s−1), +is determined by the maximum wavelength of the Lyα +forest we used, the smoothing scale of the Wiener filter- +ing scheme, and the AGN redshift uncertainty, assumed +by Youles et al. (2022). After removing the AF values af- +fected the systemics in the 2D Hi profile, we present the +LoS Hi radial profile of the All-AGN sample in Figure +15. +We estimate the Hi radial profiles of DTrans, which is +referred to as the Transverse Hi radial profile,by averag- +ing the AF values over the LoS velocity of (−750, +750) +km s−1 whose velocity width corresponds to 15 h−1 +cMpc in the Hubble-flow distance. The Hi radial profile +of DTrans is also shown in Figure 15. +We compare the LoS and Transverse Hi radial profile. +The AF value decrease more rapidly in the LoS direc- +tion than those in the Transverse direction (Figure 15). +This difference may be explained by an effect similar to +the Kaiser effect (Kaiser 1987), doppler shifts in AGN +redshifts caused by the large-scale coherent motions of +the gas towards the AGN. The LoS Hi radial profile is +negative, AF ∼ −0.002±0.0008, at the large scale, ≳ 30 +h−1cMpc. In Section 5.5, we discuss the negative AF +values of LoS Hi radial profiles at large scale and com- +pare our observational result to the models of a previous +study, Font-Ribera et al. (2013). +5.3. Source Dependences of the AGN Average HI +Profiles +In this section, we present 2D and Hi radial profiles +of the AGN sub-samples to investigate how the average +Hi density depends on luminosity and AGN type. +5.3.1. AGN Luminosity Dependence + +Fraction +0.15 +T1-AGN +0.10 +T1-AGN(H) +0.05 +0.00 +Fraction +0.10 +0.05 +0.00 +42 43 4445 46 470.04 +0.04 +T1-AGN +0.03 +0.03 +T1-AGN(H) +AF +0.02 +0.02 +TT +0.018F +0.01 +0.00 +0.00 +-0.01 +0.01 +0 +10 20 30 40 506070 +D [h-1cMpc]14 +Sun et al. +Figure 14. Hi radial profiles of LoS velocity (LoS distance) +for the All-AGN sample. The black solid line shows the AF +values as a function of LoS velocity (LoS distance) for the +All-AGN sample. The vertical dashed line presents the posi- +tion of LoS velosity = 0 km s−1 (LoS distance = 0 h−1cMpc). +The horizontal dashed indicates the cosmic average Hi ab- +sorption, AF = 0. The gray shaded area shows the range of +the AF not used to derive LoS Hi radial profile. +Figure 15. LoS and Transverse Hi radial profiles the All- +AGN sample. The black and gray lines show the AF (δF) +values as a function of LoS distance and DT rans, respectively. +The horizontal dashed line indicates AF (δF) = 0 (= 0). +We study the AGN-luminosity dependence of the +average Hi profiles. +Figure 16 presents the Lspec +1350 +distribution of All-AGN. We make 3 sub-samples +of All-AGN that are All-AGN-L3, All-AGN-L2 and +All-AGN-L1. +The luminosity ranges of the sub- +samples are 43.70 +< +log(Lspec +1350/[erg s−1]) +< +45.41, +45.41 < log(Lspec +1350/[erg s−1]) < 45.75, and 45.75 < +log(Lspec +1350/[erg s−1]) < 47.35, respectively. The luminos- +ity ranges of the 3 sub-samples are defined in a way that +the numbers of the AGN are same 5695 in each subsam- +ples. We derive the 2D Hi profiles of the sub-samples +in the same manner as Section 5.1, and present the pro- +files in Figures 17. In these 2D Hi profiles, The bright- +est sub-sample of All-AGN-L1 (the faintest sub-sample +of All-AGN-L3) shows the weakest (the strongest) Hi +absorptions around the source position, D = 0. +We then extract the Hi radial profiles from the 2D +Hi profiles of the All-AGN sub-samples, and present +the Hi radial profiles in Figure 18. In this figure, we +find that the peak values of AF for the All-AGN sub- +samples is anti-correlates with AGN luminosities. The +peak AF values near the source position drops from +the faintest All-AGN-L3 subsample to the brightest All- +AGN-L1 subsample. +The gas densities around bright +AGN are higher than (or comparable to) those around +faint AGN, this result would suggest that the ioniza- +tion fraction of the hydrogen gas around bright AGN is +higher than the one around faint AGN on average. +We also present the LoS and Transverse Hi radial pro- +files of the All-AGN sub-samples derived by the same +method as that for the All-AGN sample in Figure 19. +Similar to what we found in the comparison of the Hi +radial profiles for the All-AGN sub-samples, the peak +values of the LoS and Transverse Hi profiles also de- +crease from the faintest sub-sample, All-AGN L3, to the +brightest sub-sample, All-AGN L1. For the LoS (Trans- +verse) Hi radial profiles at the scales beyond 25 h−1 +cMpc, we do not find any significant differences in the +comparison of the LoS (Transverse) Hi radial profiles for +the All-AGN sub-samples. +Figure 16. logLspec +1350 distribution of the bright and All-AGN +sample sources. The vertical dashed lines indicate the board- +ers of Lspec +1350 where log(Lspec +1350/[erg s−1]) = 45.41 and 45.75, re- +spectively. These three borders separate the All-AGN sam- +ple into 3 sub-samples of All-AGN-L3, All-AGN-L2, and All- +AGN-L1, respectively. +5.3.2. AGN Type Dependence + +LoS distance [h-1cMpc] +75 +-50-25 +0 +25 1 50 +75 +0.03 +0.03 +All-AGN LoS +0.02 +0.02 +A +F0.01 +0.01 +H +0.00 +0.00 +-7500-5000-2500 +0 +2500 5000 7500 +LoS velocity [km s-1]Los Velocity [km s-1] +0 +2500 +5000 +7500 +0.03 +0.03 +All-AGN LoS +All-AGN Trans +0.02 +0.02 +AF +0.01 +0.01 +0.00 +0.00 +0.01 +25 +50 +75 +[ +D [h-1cMpc]0.10 +All-AGN-L3 +Fraction +All-AGN-L2 +All-AGN-L1 +-- +0.05 +0.00 +- +44 +45 +46 +47Cosmological-Scale Hi Distribution Around Galaxies and AGN +15 +Figure 17. Same as Figure 9, but for the All-AGN-L3 (top), +All-AGN-L2 (middle) and All-AGN-L1 (bottom) samples. +Figure 18. +Same as Figure 13, but for the All-AGN-L3 +(red), All-AGN-L2 (gray) and All-AGN-L1 (black) samples. +We investigate the dependence of Hi profiles on type-1 +and type-2 AGN. To remove the effects of the AGN lumi- +Figure 19. LoS and Transverse Hi radial profiles of the All- +AGN-L3, All-AGN-L2, and All-AGN-L1 sub-samples. The +top figure (bottom figure) presents the LoS (Transverse) Hi +radial profiles of the All-AGN-L3, All-AGN-L2, and All- +AGN-L1 sub-samples, shown by the red, gray, and black +lines, respectively. +The meaning of the horizontal dashed +lines both in the top and bottom figures are the same as the +one in Figure 10. +nosity dependence (Section 5.3.1), we make sub-samples +of T1-AGN and T2-AGN with the same Lspec +1350 distribu- +tion by the same manner as the one we conduct for the +selection of T1-AGN and T1-AGN(H) sub-samples in +Section 5.1. +The top panel of Figure 20 presents the +Lspec +1350 distributions of T1-AGN and T2-AGN samples, +while the bottom panel of Figure 20 shows those of the +T1-AGN and T2-AGN sub-samples. The sub-samples +of T1-AGN and T2-AGN are composed of 10329 type- +1 AGN and 1462 type-2 AGN, respectively. We derive +the 2D Hi profiles from the T1-AGN and T2-AGN sub- +samples. The profiles are presented in Figure 21. We + +LoS Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]LoS Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +50 +-0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]Los Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +50 +-0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]0.04 +0.04 +A1l-AGN-L3 +0.03 +0.03 +All-AGN-L2 +AF +0.02 +All-AGN-L1 +0.02 +0.010F +0.01 +0.00 +0.00 +-0.01 +0.01 +0 +10 20 3040506070 +D [h-1cMpc]Los Velocity [km s-1] +0 +2500 +5000 +7500 +0.03 +0.03 +All-AGN-L3 LoS +All-AGN-L2 LoS +0.02 +All-AGN-L1 LoS +-0.02 +AF 0.01 +0.01 +0.00 +0.00 +0.01 +25 +50 +75 +D [h-1cMpc]0.03 +0.03 +All-AGN-L3 Trans +All-AGN-L2 Trans +0.02 +All-AGN-L1 Trans +0.02 +AF 0.01 +0.01 +OF +0.00 +0.00 +-0.01 +25 +50 +75 +D [h-1cMpc]16 +Sun et al. +find 17.7 and 7.9 σ detections at the source center po- +sition (0,0) of the T1-AGN and T2-AGN sub-samples, +respectively. +We calculate the Hi radial profiles from +the 2D Hi profiles of the T1-AGN and T2-AGN sub- +samples. In Figure 22, we compare the Hi radial profiles +of the T1-AGN and T2-AGN sub-samples. No notable +difference is found within 1σ error. The peak value of +AF of the T2-AGN subsample is within 1σ error of the +peak value of the T1-AGN subsample near the source +position. +To compare the Hi distributions of type-1 and type-2 +AGN in the LoS and transverse directions, we derive the +LoS and Transverse Hi radial profiles of the T1-AGN +and T2-AGN sub-samples and present the profiles in +Figure 23. Similar to the trend of the Hi radial profiles, +the peak values of the LoS and Transverse Hi radial +profiles for T1-AGN and T2-AGN sub-samples are not +significantly different. The comparable peak values of +the LoS and Transverse Hi radial profiles suggest that +the selectively different orientation and opening angles +of the dusty tori of the type-1 and type-2 AGN do not +significantly affect the Hi distribution at the scale ≲ 15 +h−1cMpc. +For the Hi radial profiles at the scale > 15 h−1cMpc, +we find that the AF value for the LoS Hi radial pro- +file of the T1-AGN sub-sample is greater than those of +the T2-AGN sub-sample over the 1σ error bar at the +scale around 25 h−1cMpc. +This result may hint that +the type-2 AGN have a stronger power of ionization at +25 h−1cMpc than the type-1 AGN. The interpretation +of ionization at large-scales is in Section 5.5. +Figure 20. Same as Figure 12, but for the T1-AGN (blue) +and T2-AGN (red) samples. +Figure 21. +Same as Figure 9, but for the T1-AGN (top +figure) and T2-AGN (bottom figure) sub-samples. +Figure 22. Same as Figure 13, but for the T1-AGN (blue) +and T2-AGN (red) sub-samples and the Galaxy (gray) sam- +ple. +5.4. Average HI Profiles around Galaxy +We derive the 2D Hi profile at the positions of the +Galaxy sample sources in the same manner as the one +of the All-AGN sample sources. Figure 24 presents the +2D Hi profile of the Galaxy sample sources. There is +a clear 10.5σ detection at the source position of (0,0). +Similarly, we calculate the Hi radial profile from the +2D Hi profile of the Galaxy sample (Figure 25). The Hi +radial profile of the Galaxy sample shows a trend similar +to those of the All-AGN sample. Both for the Galaxy + +raction +0.15 +T1-AGN +0.10 +T2-AGN +0.05 +0.00 +Fraction +0.10 +0.05 +0.00 +424344454647LoS Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +50 +0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]Los Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]0.04 +0.04 +T1-AGN +0.03 +0.03 +T2-AGN +AF +0.02 +Galaxy +-0.02 +0.010F +0.01 +0.00 +0.00 +-0.01 +0.01 +0 +10 203040506070 +D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN +17 +Figure 23. Same as Figure 19, but for the T1-AGN and +T2-AGN sub-samples. +and All-AGN samples, the Hi radial profile decreases +towards the large scales, reaching AF ∼ 0. +In Figure 24, we find that the Hi distributions in the +LoS and transverse directions are different. A similar +difference between the values of AF in LoS and trans- +verse directions of 2D Hi profiles is claimed by Mukae +et al. (2020). To investigate the difference between the +Hi distributions in LoS and transverse directions for the +Galaxy sample, we present the LoS and Transverse Hi +radial profiles of the Galaxy sample in Figure 27. We +find that the LoS and Transverse Hi radial profiles of the +Galaxy sample show different gradient of the decreasing +AF at the scale D ∼ 3.75−50 h−1cMpc. This difference +can be explained by the gas version of the Kaiser effect +that we discussed in Section 5.2. In the LoS Hi radial +profile of the Galaxy sample, we find that the AF val- +ues are negative on the scale of D = 25 − 70 h−1cMpc, +which is similar to the negative AF values we found on +the large scale of the LoS Hi radial profile for the All- +AGN sample. We discuss these negative AF values on +the LoS Hi radial profile of the Galaxy sample in Section +5.5. +Figure 24. 2D Hi profile of the Galaxy sample sources. The +color map indicates the AF (δF) values of each cell of the 2D +Hi profile. The dotted lines show confidence level contours +of 3σ and 6σ. The solid line presents the contour where AF += 0.01 (δF = −0.01). +5.4.1. Galaxy-AGN Dependence +We derive 2D Hi profiles for the T1-AGN(H) sample +constructed from the HETDEX data. Figure 24 and 25 +show the 2D Hi profiles of the Galaxy and T1-AGN(H) +samples. We find 7.6σ detection around the source posi- +tion for the T1-AGN(H) sample. Figure 26 presents the +Hi radial profiles of the Galaxy and T1-AGN(H) samples +derived from the 2D Hi profiles. We also compare the +Hi radial profiles of the Galaxy sample with those of T1- +AGN and T2-AGN in Figure 22. In the Hi radial profiles +of the Galaxy and T1-AGN(H) samples, the AF values +increase toward the source position D = 0. In Figure +26 (22), we find that the AF values of T1-AGN(H) (T1- +AGN and T2-AGN) are larger than those of the galaxies +at ≲ 20 h−1 cMpc. These AF excesses of the AGN may +be explained by the hosting dark matter halos of the +AGN being more massive than those of the galaxies. +Momose et al. (2021) also investigate the Hi radial pro- +file around AGN, and find Hi absorption decrement at +the source center (≲ 5 h−1Mpc). They argure that this +trend can be explained by the proximity effect. On the +other hand, their result is different from ours that the +AF values monotonically increase with decreasing dis- +tance. This difference between our and Momose et al.’s +results is produced by the fact that our results for ≲ 10 +h−1 cMpc are largely affected by the Hi absorption at +∼ 10 h−1 cMpc due to the coarse resolution of our Hi +tomography map, 15 h−1 cMpc, in contrast with 2.5 h−1 +cMpc for the resolution of Momose et al. (2021). +We then derive the LoS and Transverse radial Hi pro- +file of the T1-AGN(H) sample. The results of the profiles + +0 +2500 +5000 +7500 +0.03 +0.03 +T1-AGN LoS +T2-AGN LoS +0.02 +0.02 +AF 0.01 +0.01 +0.00 +0.00 +-0.01 +0 +25 +50 +75 +D [h-1cMpc]0.03 +0.03 +T1-AGN Trans +T2-AGN Trans +0.02 +0.02 +AF +0.01 +0.01 +0.00 +0.00 +0.01 +25 +50 +0 +75 +D [h-1cMpc]Los Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +-0.010.01 +AF +2500 +25 +0.030.03 +0 +0 +204060 +DTrans [h-1cMpc]18 +Sun et al. +are shown in Figure 27. Similar to the LoS and Trans- +verse Hi radial profiles of the All-AGN and Galaxy sam- +ples, the gas version of the Kaiser effect and the nega- +tive AF in the LoS direction on the scale beyond D = 25 +h−1cMpc are also found in those of the T1-AGN(H) sam- +ple. +Figure 25. +Same as Figure 17, but for the T1-AGN(H) +sample. +Figure 26. Same as Figure 18, but for Galaxy (gray) and +T1-AGN(H) (black) samples. +5.5. Comparison with Theoretical Models +There are theoretical models of Hi radial profiles +around AGN that are made by Font-Ribera et al. (2013). +Font-Ribera et al. (2013) present their Hi radial profiles +with the LoS distance in the form of cross-correlation +function (CCF). +We first calculate theoretical CCFs of All-AGN, fol- +lowing the definition of the CCF presented in Font- +Ribera et al. (2013). Font-Ribera et al. (2013) assume +the linear cross-power spectrum of the QSOs and Lyα +forest, +PqF(k, z) = bq(z)[1+βq(z)µ2 +k]bF(z)[1+βF(z)µ2 +k]PL(k, z), +(11) +Figure 27. +Same as Figure 15, but for the Galaxy and +T1-AGN(H) samples. +where PL(k, z) is the linear matter power spectrum. +Here µk is the cosine of the angle between the Fourier +mode and the LoS (Kaiser 1987). The values of bq and bF +(βq and βF) are the bias factors (redshift space distortion +parameters) of the QSO and Lyα density, respectively. +The redshift distortion parameter of QSO obeys the +relation βq = f(Ω)/bq, where f(Ω) is the logarithmic +derivative of the linear growth factor (Kaiser 1987), +bq = 3.8±0.3 (White et al. 2012). We use the condition +of Lyα forest, bF(1 + βF) = −0.336 for bF ∝ (1 + z)2.9, +that is determined by observations of Lyα forest at +z ≃ 2.25 (Slosar et al. 2011). Font-Ribera et al. (2013) +estimate the CCF of QSOs by the Fourier transform of +PqF (Hamilton 1992): +ξ(r) = ξ0(r)P0(µ) + ξ2(r)P2(µ) + ξ4(r)P4(µ), +(12) +where µ is the cosine of angle between the position r +and the LoS in the redshift space. The values of P0, +P2, and P4 are the Legendre polynomials, P0 = 1, P2 = +(3µ2 − 1), and P4 = (35µ4 − 30µ2 + 3)/8, respectively. +The functions of ξ0, ξ2, and ξ4 are: +ξ0(r) = bqbF[1 + (βq + βF)/3 + βqβF/5]ζ(r), +(13) +ξ2(r) = bqbF[2/3(βq+βF)+4/7βqβF][ζ(r)− ¯ζ(r)], (14) +ξ4(r) = 8/35bqbFβqβF[ζ(r) − 5/2¯ζ(r) − 7/2¯¯ζ(r)]. (15) +The function ζ(r) is the standard CDM linear correla- +tion function in real space (Bardeen et al. 1986; Hamil- +ton et al. 1991). The functions ¯ζ(r) and ¯¯ζ(r) are given +by: +¯ζ(r) ≡ 3r−3 +� r +0 +ζ(s)s2ds, +(16) + +LoS Velocity [km s-1] +LoS Hubble distance +7500 +75 +0.01 +0.01 +[h-1cMpc] +5000 +0.0110.01 +AF +2500 +25 +0.030.03 +0 +204060 +DTrans [h-1cMpc]0.04 +0.04 +Galaxy +0.03 +0.03 +T1-AGN(H) +AF +0.02 +0.02 +0.018F +0.01 +0.00 +0.00 +0.01 +0.01 +0 +10 203040506070 +D [h-1cMpc]0 +2500 +5000 +7500 +0.03 +0.03 +Galaxy LoS +Galaxy Trans +0.02 +T1-AGN(H) LoS +-0.02 +T1-AGN(H) Trans +AF +0.01 +0.01 +OF +0.00 +0.00 +0.01 +25 +50 +75 +D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN +19 +¯¯ζ(r) ≡ 5r−5 +� r +0 +ζ(s)s4ds. +(17) +Here we define +ξ′(r) ≡ −ξ(r). +(18) +In Figure 28, we present Dξ′ as a function of the LoS +distance for the model of Font-Ribera et al. (2013) that +is calculated under the assumption of the mean over- +density of the 15 h−1cMpc corresponding to the spatial +resolution of our observational results. +To compare our observational measurements with the +model CCF of Font-Ribera et al. (2013), we calculate +the value of ξ′ for our All-AGN sample. The value of ξ′ +in each cell ξ′cell is calculated by +ξ′ +cell = +� +i∈cell ωiAFi +� +i∈cell ωi +, +(19) +where ωi is the weight determined by the observational +errors and the intrinsic variance of the Lyα forest. The +value of ωi is obtained by +ωi = +� +σ2 +F(zi) + +1 +⟨S/N⟩2 × ⟨F(zi)⟩2 +�−1 +, +(20) +where σF(zi) is the intrinsic variance of the Lyα forest. +The value of ⟨F(zi)⟩ is the cosmic average Lyα transmis- +sion (Eq.4). We adopt ⟨S/N⟩ = 1.4 that is the criterion +of the background source selection (Section 3.3). The +intrinsic variance, σF(zi), of the Lyα forest taken from +Font-Ribera et al. (2013) is: +σ2 +F(zi) = 0.065[(1 + zi)/3.25]3.8. +(21) +We calculate ξ′ with our All-AGN sample via the +Equations 19, 20, and 21, using the binning sizes same +as those in Font-Ribera et al. (2013). +We present ξ′ +multiplied by D with the black squares in Figure 28. +(explanation of Momose+21) For reference, we also de- +rive the ξ′ for our Galaxy sample shown by the blue +triangles. +In Figure 28, we find that the Dξ′ profile of our All- +AGN sample show a trend similar to the one of the +model predicted by Font-Ribera et al. (2013). The ob- +servational Dξ′ profile of our All-AGN sample shows +a good agreement with the model Dξ′ profile of Font- +Ribera et al. (2013) at the scale of D > 30 h−1cMpc. Al- +though the model Dξ′ profile of Font-Ribera et al. (2013) +is slightly higher than the Dξ′ profiles of the observa- +tions at ≳ 60 h−1cMpc, the general trend of the negative +Dξ′ profiles at ≳ 30 h−1cMpc are the same. Font-Ribera +et al. (2013) suggests that the negative Dξ +′ values at the +large scale of ≳ 30 h−1cMpc are explained by the ion- +ization. In the model of ionization, Font-Ribera et al. +(2013) assume the spectrum of the AGN at D = 0 with +Lν ∝ ν−α, where α = 1.5 (1.0) for the frequency ν over +(below) the Lyman limit. The luminosity of λ = 1420 +˚A is normalized as Lν = 3.1 × 1030 erg/s/Hz, which +is taken from the mean luminosity of the SDSS data re- +lease 9 quasars. No assumptions of AGN type have been +made in the models of Font-Ribera+13. Based on the +model of ionization, Font-Ribera et al. (2013) calculate +ξ for the homogeneous gas radiated by AGN, and obtain +the function +ξ = 0.0065(20 h−1cMpc/D)2. +(22) +With the ξ function, we calculate Dξ +′ that is presented +with the cyan dashed curve in Figure 28. +The cyan +dashed curve shows the plateau at D ≥ 40 h−1cMpc +with negative Dξ +′ values that is comparable with the +model Dξ′ profile of Font-Ribera et al. (2013). It indi- +cates that the negative Dξ +′ values are originated from +the ionization of radiation including the hard radiation. +Similarly, the negative Dξ +′ values of our All-AGN at the +large scale towards ≳ 40 h−1cMpc may be explained by +the ionization of radiation. +To distinguish the large- +scale negative Dξ +′ values, which are referred to as the +‘ionized outskirts’, from the proximity zone created by +the proximity effect, we plot the observational CCF of +AGN obtained by Momose et al. (2021) in Figure 28. +The AGN CCF obtained by Momose et al. shows a de- +creasing Hi absorption toward source position (D = 0 +h−1cMpc) caused by the proximity effect. Our findings +indicate that the Hi radial profile of AGN has transi- +tions from proximity zones (≲ a few h−1cMpc) to the +Hi structures (∼ 1 − 30 h−1cMpc) and the ionized out- +skirts (≳ 30 h−1cMpc). The hard radiation may pass +through the Hi structure due to the small cross-section +and ionizes the Hi gas in the regions of ionized out- +skirts. Because of the low recombination rate, the Hi +gas remains ionized in the ionized outskirt. +Interestingly, the Dξ′ profile of our Galaxy sample also +shows negative Dξ +′ values towards ≳ 30 h−1cMpc which +is similar to those of the model and our All-AGN sam- +ple. This result may suggest that the Hi gas at large +scale (≳ 20 h−1cMpc) around galaxies has been ion- +ized. The ionizing source causing the structure of neg- +ative Dξ +′ values at the large scale may not be a single +galaxy, but a group of galaxies within a radius of a few +cMpc. Regions around galaxies are special as galaxies +are clustered together. Galaxies in this work are bright +with MUV < −22 mag. The galaxies can be hosted by +massive haloes, and are likely to distribute at overden- +sity regions. The overdensity region suggests that each +galaxy can be surrounded by several satellite galaxies. +Although it is difficult for a galaxy to ionize the Hi gas + +20 +Sun et al. +on a scale of ≳ 20 h−1cMpc, a group galaxies may have +enough ionizing photons to ionize the Hi on this scale. +Figure 28. Comparison between our All-AGN and Galaxy +results and the models of Font-Ribera et al. (2013) in the LoS +CCF (ξ +′) multiplied by distance (D). The black and blue +points are the results derived from the All-AGN and Galaxy +samples sources, respectively. The orange curve is the LoS +CCF of QSOs with the Lyα forest derived by Font-Ribera +et al. (2013). The cyan dashed curve shows the ionization +of radiation effect taken from Font-Ribera et al. (2013). The +pink line presents the CCF of AGN obtained by Momose +et al. (2021). The gray shade presents the range of the Hi +structure. Two white areas show the regions of proximity +zone and ionized outskirt. The horizontal gray line indicates +the cosmic average where Dξ +′ = 0. +6. SUMMARY +We reconstruct two 3D Hi tomography maps based +on the Lyα forests in the spectra of 14763 background +QSOs from the SDSS survey with no signatures of +damped Lyα system or broad absorption lines. +The +maps cover the extended Fall and Spring fields defined +by the HETDEX survey. The spatial volume of the re- +constructed 3D Hi tomography maps are 2257×233×811 +h−3cMpc3 and 3475 × 1058 × 811 h−3cMpc3. We inves- +tigate Hi distribution around galaxies and AGN with +samples made from HETDEX and SDSS survey results +in our study field. Our results are summarized below. +• We derive the 2D Hi and Hi radial profiles of the +All-AGN sample consisted of SDSS AGN. We find +that the 2D Hi profile is more extended in the +transverse direction than along the line of sight. In +the Hi radial profile All-AGN sample, the values +of Hi absorption, AF, decrease toward the large +scale, touching to AF ∼ 0. +• We compare the Hi radial profiles derived from +the T1-AGN and T1-AGN(H) sub-samples, whose +Lspec +1350 distributions are the same. +We find that +the Hi radial profile of the T1-AGN sub-sample +agrees with that of the T1-AGN(H) sub-sample. +This agreement suggests that the systematic un- +certainty between the SDSS and the HETDEX +survey results is negligible. +• We examine the dependence of the Hi profile on +AGN luminosity by deriving the 2D Hi, Hi ra- +dial, LoS Hi radial, and Transverse Hi radial pro- +files of the All-AGN-L3 (the faintest), All-AGN- +L2, and All-AGN-L1 (the brightest) sub-samples. +We find that the Hi absorption is the greatest in +the lowest-luminosity AGN sub-sample, and that +the Hi absorption becomes weaker with increasing +AGN luminosity This result suggests that, on av- +erage, if the density of Hi gas around the bright +AGN is greater than (or comparable to) those of +the faint AGN, the ionization fraction of Hi gas +around bright AGN is higher than that around +faint AGN. +• We investigate the AGN type dependence of Hi +distribution around type-1 and type-2 AGN by the +2D Hi, Hi radial, LoS Hi radial, and Transverse +Hi radial profiles extracted from the T1-AGN and +T2-AGN sub-samples with the same Lspec +1350 distri- +butions. The comparison between the Hi radial +profiles of T1-AGN and T2-AGN sub-samples in- +dicates that the Hi absorption around the T2- +AGN sub-sample is comparable to the one of the +T1-AGN sub-sample on average. This trend sug- +gests that, the selectively different opening angle +and orientation of the dusty torus for type-1 and +type-2 AGN do not have a significant impact on +the Mpc-scale Hi distribution. +• We compare the Hi distributions around galax- +ies and type-1 AGN with the 2D Hi, Hi radial, +LoS Hi radial, and Transverse Hi radial profiles +derived from the Galaxy and T1-AGN(H) sample +sources. +The Hi absorption values, AF, around +the T1-AGN(H) sample are larger than those of +the Galaxy sample on average. This result may +be caused by the dark matter halos of type-1 AGN +having a larger mass than the one of galaxies on +average. +• We find that the Hi radial profiles of the LoS dis- +tance for the Galaxy and All-AGN samples show +negative AF values, which means weak Hi absorp- +tion, at the scale over ∼ 30 h−1cMpc. We extract + +LoS velocity [km s-1] +0 +2000 +4000 +6000 +8000 +0.50 +HI structure +Ionized Outskirt +0.25 +DS +0.00 +Zone +-0.25 +Momose+21 +CCF LoS model +Ionization model +-0.50 +Galaxy Los +All-AGN LoS +0 +102030 +4050 +60 +70 +80 +D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN +21 +the Dξ′ profile of our Galaxy and All-AGN sam- +ples to compare with the model CCF of AGN from +Font-Ribera et al. (2013). The general trend of the +negative Dξ′ at ≳ 30 h−1cMpc is the same as the +model CCF. This results suggest that the Hi ra- +dial profile of AGN has transitions from proximity +zones (≲ a few h−1cMpc) to the Hi rich struc- +tures (∼ 1−30 h−1cMpc) and the ionized outskirts +(≳ 30 h−1cMpc). +ACKNOWLEDGEMENTS +We thank Nobunari Kashikawa, Khee-Gan Lee, Akio +Inoue, Rikako Ishimoto, Shengli Tang, Yongming Liang, +Rieko Momose, and Koki Kakiichi for giving us helpful +comments. +HETDEX is led by the University of Texas at Austin +McDonald Observatory and Department of Astron- +omy with participation from the Ludwig-Maximilians- +Universit¨at +M¨unchen, +Max-Planck-Institut +f¨ur +Ex- +traterrestrische Physik (MPE), Leibniz-Institut f¨ur As- +trophysik Potsdam (AIP), Texas A&M University, +Pennsylvania State University, Institut f¨ur Astrophysik +G¨ottingen, The University of Oxford, Max-Planck- +Institut f¨ur Astrophysik (MPA), The University of +Tokyo and Missouri University of Science and Tech- +nology. +In addition to Institutional support, HET- +DEX is funded by the National Science Foundation +(grant AST-0926815), the State of Texas, the US Air +Force (AFRL FA9451-04-2- 0355), and generous sup- +port from private individuals and foundations. The ob- +servations were obtained with the Hobby-Eberly Tele- +scope (HET), which is a joint project of the University +of Texas at Austin, the Pennsylvania State University, +Ludwig-Maximilians-Universit¨at M¨unchen, and Georg- +August-Universit¨at G¨ottingen. The HET is named in +honor of its principal benefactors, William P. Hobby and +Robert E. Eberly. The authors acknowledge the Texas +Advanced Computing Center (TACC) at The University +of Texas at Austin for providing high performance com- +puting, visualization, and storage resources that have +contributed to the research results reported within this +paper. URL: http://www.tacc.utexas.edu +VIRUS is a joint project of the University of Texas +at Austin, Leibniz-Institut f¨ur Astrophysik Potsdam +(AIP), Texas A&M University (TAMU), Max-Planck- +Institut f¨ur Extraterrestrische Physik (MPE), Ludwig- +Maximilians-Universit¨at Muenchen, Pennsylvania State +University, Institut fur Astrophysik G¨ottingen, Univer- +sity of Oxford, and the Max-Planck-Institut f¨ur As- +trophysik (MPA). In addition to Institutional support, +VIRUS was partially funded by the National Science +Foundation, the State of Texas, and generous support +from private individuals and foundations. +This work is supported in part by MEXT/JSPS KAK- +ENHI Grant Number 21H04489 (HY), JST FOREST +Program, Grant Number JP-MJFR202Z (HY). +K. M. acknowledges financial support from the Japan +Society for the Promotion of Science (JSPS) through +KAKENHI grant No. 20K14516. +This paper is supported by World Premier Inter- +national Research Center Initiative (WPI Initiative), +MEXT, Japan, the joint research program of the In- +stitute of Cosmic Ray Research (ICRR), the Univer- +sity of Tokyo, and KAKENHI (19H00697, 20H00180, +and 21H04467) Grant-in-Aid for Scientific Research (A) +through the Japan Society for the Promotion of Science. + +22 +Sun et al. +REFERENCES +2017, Astrophysics and Space Science Library, Vol. 430, +Gas Accretion onto Galaxies +Adelberger, K. L., Shapley, A. E., Steidel, C. C., et al. +2005, ApJ, 629, 636, doi: 10.1086/431753 +Adelberger, K. L., Steidel, C. C., Shapley, A. E., & Pettini, +M. 2003, ApJ, 584, 45, doi: 10.1086/345660 +Aihara, H., Arimoto, N., Armstrong, R., et al. 2018, PASJ, +70, S4, doi: 10.1093/pasj/psx066 +Alexandroff, R., Strauss, M. A., Greene, J. 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The different panels denote the coverages over different redshift ranges shown at the top +left of each panel. + +z = 2.2 - 2.4 +2 +- +2 +35 +30 +25 +20 +15 +10 +nz = 2.4 - 2.6 +Dec.[deg] +2 +- +2 +35 +30 +25 +20 +15 +10 +5z = 2.6 - 2.8 +2 +- +2 +35 +30 +25 +20 +15 +10 +5z = 2.8 - 3.0 +Dec.[deg] +2 +品 +0 +品 ++ +- ++.品 +中 +- +2 +35 +30 +25 +20 +15 +10 +5 +R.A.[deg]Cosmological-Scale Hi Distribution Around Galaxies and AGN +25 +Figure 30. +Same as Figure 1, but for the foreground sources in the ExSpring field. + +Z=2 +2.0 - 2.2 +60 +口 +55 +电 +50 +45 +160 +170 +180 +190 +200 +210 +220 +230 +240Z=2 +2.2 - 2.4 +60 +55 +50 +45 +160 +170 +180 +190 +200 +210 +220 +230 +240Z = 2.4 - 2.6 +60 +50 +45 +160 +170 +180 +190 +200 +210 +220 +230 +240 +R.A.[deg]26 +Sun et al. +Figure 31. Continued from Figure 30. +Figure 32. +Same as Figure 2, but for the background sources in the ExSpring field. + +Z=2 +2.6 - 2.8 +60 +品 +- +日 +. +55 +50 +45 +160 +170 +180 +190 +200 +210 +220 +230 +240Z= +2.8 -3.0 +60 +品 +.. +- +55 ++ +T +50 +45 +160 +170 +180 +190 +200 +210 +220 +230 +240 +R.A.[deg]60 +Dec.[deg] +55 +50 +45 +160 +170 +180 +190 +200 +210 +220 +230 +240 +R.A.[deg] \ No newline at end of file diff --git a/CtE4T4oBgHgl3EQfeg0W/vector_store/index.faiss b/CtE4T4oBgHgl3EQfeg0W/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d538fc6a1df8dd98adfbb2e203802badced4cbec --- /dev/null +++ b/CtE4T4oBgHgl3EQfeg0W/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d5f3518662c9b7113cb100ae371ae56599c9a591846a903e8485378137886c2e +size 2621485 diff --git a/DNFKT4oBgHgl3EQfYy5L/content/tmp_files/2301.11800v1.pdf.txt b/DNFKT4oBgHgl3EQfYy5L/content/tmp_files/2301.11800v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..26f3fd1cb55eab8eb3cfc2e6e83f9f54ce804818 --- /dev/null +++ b/DNFKT4oBgHgl3EQfYy5L/content/tmp_files/2301.11800v1.pdf.txt @@ -0,0 +1,2762 @@ +arXiv:2301.11800v1 [math.CV] 25 Jan 2023 +COMMUTING TOEPLITZ OPERATORS AND MOMENT MAPS +ON CARTAN DOMAINS OF TYPE III. +DAVID CUEVAS-ESTRADA AND RAUL QUIROGA-BARRANCO +Abstract. Let DIII +n +and Sn be the Cartan domains of type III that con- +sist of the symmetric n × n complex matrices Z that satisfy ZZ < In and +Im(Z) > 0, respectively. +For these domains, we study weighted Bergman +spaces and Toeplitz operators acting on them. We consider the Abelian groups +T, R+ and Symm(n, R) (symmetric n × n real matrices), and their actions on +the Cartan domains of type III. We call the corresponding actions Abelian +Elliptic, Abelian Hyperbolic and Parabolic. The moment maps of these three +actions are computed and functions of them (moment map symbols) are used +to construct commutative C∗-algebras generated by Toeplitz operators. This +leads to a natural generalization of known results for the unit disk. We also +compute spectral integral formulas for the Toeplitz operators corresponding to +the Abelian Elliptic and Parabolic cases. +1. Introduction +Bounded symmetric domains, weighted Bergman spaces on such domains and +Toeplitz operators acting on Bergman spaces constitute three fundamental objects +in operator theory. The reason is that they are specific enough to make explicit +computations that lead to interesting results, and at the same time they are com- +plicated enough so that such results are non-trivial and enlightening. +For some years now, operator theory analysts have found plenty of examples of +commutative C∗-algebras generated by Toeplitz operators when the corresponding +set of symbols is suitably restricted. The first such example was considered in [11], +where it was proved that Toeplitz operators on the unit disk D with radial symbols +are diagonal with respect to the orthogonal monomial basis. Clearly, a symbol on +D is radial if it is invariant under the natural T-action. We note that the T-action +on the unit disk D realizes, up to conjugacy, all the elliptic M¨obius transformations. +The introduction in [11] of Toeplitz operators with radial symbols was followed +by a series of developments found in [3, 4, 5]. +These references considered all +three fundamental types of M¨obius transformations on the unit disk D: elliptic, +hyperbolic and parabolic. +It was proved that symbols that are invariant under +the corresponding groups of M¨obius transformations yield Toeplitz operators that +generate commutative C∗-algebras. Then, it was found in [7] that, under suitable +smoothness conditions, these constructions yield the only commutative C∗-algebras +generated by Toeplitz operators acting on every weighted Bergman space on the +unit disk D. +2020 Mathematics Subject Classification. Primary 47B35 30H20; Secondary 53D20. +Key words and phrases. Toeplitz operators, Bergman spaces, Cartan domains, Lie groups, +K¨ahler manifolds, moment maps. +1 + +2 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +The next step was to study the behavior in the case of higher dimensional +bounded symmetric domains, and the unit ball Bn in Cn was the first natural +example to consider. It was found in [16, 17] that there exists exactly, up to con- +jugacy, n + 2 maximal Abelian subgroups (MASGs) of biholomorphisms each one +of which yields invariant symbols whose Toeplitz operators generate commutative +C∗-algebras. This is a natural generalization of the situation observed for the unit +disk D, since in this case we have n = 1 from which it follows the existence of three +MASGs. Nevertheless, some simplicity is lost because the number of MASGs grows +with the dimension of the unit ball Bn. +After these works, many other results have been found where a suitable symmetry +of the symbols yields commuting Toeplitz operators. Such symmetry is in most +cases a consequence of the invariance with respect to a certain biholomorphism +group. +This has been observed for every bounded symmetric domain on every +weighted Bergman space. We refer to [1] for a very general collection of related +results. +In a parallel line of development, symplectic geometry has been found to play +an special role in the construction of symbols whose Toeplitz operators generate +commutative C∗-algebras. It was proved in [14] that, for the unit ball Bn and on +any of its weighted Bergman spaces, every single Abelian connected group of bi- +holomorphisms provides symbols with mutually commuting Toeplitz operators. For +such a group H acting on Bn this is achieved by considering the so-called moment +map symbols for H instead of H-invariant symbols. We refer to Section 3 for the +details of the definitions and properties involved. However, we mention here that +the moment map of an action is a mapping defined on the corresponding bounded +symmetric domain using its symplectic manifold structure, and the moment map +symbols are functions of such moment maps. Another example of the use of moment +map symbols is given by the results found in [15], where the bounded symmetric +domain considered is the Cartan domain of type IV. +The goal of this work is to apply these ideas to study Toeplitz operators with +moment map symbols acting on the weighted Bergman spaces of Cartan domains of +type III. We recall that such domains are realized by the so-called generalized unit +disk DIII +n +and Siegel’s generalized upper half-plane Sn (see Section 2). In fact, as +we show in Section 4, to these domains we can associate three biholomorphic actions +that naturally generalize the three actions described above for the unit disk. Hence, +we call these actions on either DIII +n +or Sn the Elliptic, Hyperbolic and Parabolic +Actions (see subsection 4.1). +These come from the groups U(n), GL(n, R) and +Symm(n, R), respectively, of which only the last one is Abelian for every n. Hence, +we introduce actions that we call Abelian Elliptic and Abelian Hyperbolic (see +Definition 4.2). As noted in Remark 4.3 all three Abelian actions can be seen as +coming from the corresponding centers of the original groups involved. +We present in Section 2 all the theory needed to understand the Riemannian +and symplectic geometry background used in the rest of the work. In particular, +we compute in subsection 3.2 the Bergman metric and the K¨ahler form for both +DIII +n +and Sn. We use this to compute in subsection 4.2 the moment maps for our +three distinguished actions: Abelian Elliptic, Abelian Hyperbolic and Parabolic. +We introduce in Section 5 Toeplitz operators with special symbols. First, we +consider invariant symbols in subsection 5.1 and we recall some known commutative +C∗-algebras generated by Toeplitz operators for our setup. Second, we introduce + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +3 +moment map symbols in Definition 5.3, and with the use of our moment map +computations we obtain explicit formulas for moment map symbols for our three +distinguished Abelian actions. +We obtain the following general description (see +Proposition 5.4 for the precise statements) +• Abelian Elliptic symbols: Z �−→ f +� +tr(ZZ) +� +. +• Abelian Hyperbolic symbols: Z �−→ f +� +tr(Im(Z)−1Re(Z)) +� +. +• Parabolic symbols: Z �−→ f(Im(Z)). +From a quick comparison with the notions considered in the current literature, we +observe that these three types of symbols are natural, almost canonical, generaliza- +tions from the unit disk D to the domains DIII +n +and Sn of the symbols obtained +from the elliptic, hyperbolic and parabolic actions on D. +We prove in Theorem 5.8 that the three types of symbols above yield Toeplitz op- +erators that generate commutative C∗-algebras on every weighted Bergman space. +Our method of proof is based on the fact that these moment map symbols have an +additional invariance: they are invariant under the group from which the Abelian +group is the center. This allows to use the results from subsection 5.1. On the +other hand, it is interesting to observe the importance of having only three types +of symbols in Theorem 5.8 as a generalization of the corresponding result for the +unit disk. This is explained in Remark 5.9. +Finally, we obtain in Section 6 integral formulas for the Toeplitz operators with +moment map symbols that provide simultaneous diagonalization for them. This is +done for the Abelian Elliptic and Parabolic Actions; we leave the Abelian Hyper- +bolic case as an important project to develop. The relevant results are Theorems 6.3 +and 6.8. The simplicity of the formulas presented in Theorem 6.3 highlights the +importance of using symplectic geometry to solve these operator theory problems. +Likewise, Theorem 6.8 has very natural formulas that involve a Fourier-Laplace +transform obtained in Theorem 6.7. +2. The Cartan domains of type III and their analysis +We recall the basic geometric and analytic properties of the Cartan domains of +type III. +2.1. Bounded and unbounded realizations. In the rest of this work, and for F +either R or C, we will denote by Mat(n, F) the space of n × n matrices over F and +by Symm(n, F) its subspace of symmetric matrices. As usual, GL(n, F) will denote +the Lie group of invertible elements of Mat(n, F). +Definition 2.1. The n-dimensional Cartan domain of type III is the complex +domain given by DIII +n += {Z ∈ Symm(n, C) | In − ZZ > 0}. +The domain DIII +n +is clearly bounded. On the other hand, there is a natural +unbounded domain associated to DIII +n +. +Definition 2.2. The n-dimensional Siegel domain is the complex domain given by +Sn = {Z ∈ Symm(n, C) | Im(Z) > 0}. +We note that DIII +1 +and S1 are precisely the unit disk D and the upper half-plane +H, respectively, in the complex plane C. For this reason, the domains DIII +n +and Sn +are also known as the generalized unit disk and generalized upper-half plane, respec- +tively. Furthermore, these domains are related in a way similar to the well known +1-dimensional case. For the next result we refer to [9, Exercise C, Chapter VIII]. + +4 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +Proposition 2.3. The map ϕ : Sn → DIII +n +given by +Z �→ (In + iZ)(In − iZ)−1, +is a biholomorphism from Sn onto DIII +n +. +Because of the previous result, the domain Sn is also known as the unbounded +realization of the n-dimensional Cartan domain of type III. +2.2. Biholomorphism groups. In this section we describe the groups of biholo- +morphisms of the domains DIII +n +and Sn introduced above. We start by considering +the matrices +In,n = +� +In +0 +0 +−In +� +, +Jn = +� +0 +−In +In +0 +� +. +These naturally yield the next Lie groups. +Sp(n, C) = {M ∈ Mat(2n, C) | M ⊤JnM = Jn}, +Sp(n, R) = {M ∈ Mat(2n, R) | M ⊤JnM = Jn}, +U(n, n) = {M ∈ Mat(2n, C) | M ∗In,nM = In,n}. +We recall the notion of a bounded symmetric domain. +Definition 2.4. A domain D ⊂ CN is called symmetric if for every z ∈ D there +exists a biholomorphism ϕz : D → D such that ϕz(w) = w if and only if w = z. If +D is also bounded, then D is called a bounded symmetric domain. If D satisfies +tD = D, for every t ∈ T, then the domain D is called circled. +Through suitable actions of the groups introduced above, one can prove that +the domains DIII +n +and Sn are symmetric. +For the next result we refer to [13, +Paragraph (2.3)] (see also [9]). From now on, for any given matrix M ∈ Mat(2n, C) +a decomposition of the form +M = +�A +B +C +D +� +, +will always be taken so that A, B, C, D have size n × n. +Proposition 2.5. The action via generalized M¨obius transformations given by +Sp(n, C) ∩ U(n, n) × DIII +n +−→ DIII +n +�A +B +C +D +� +· Z �−→ (AZ + B)(CZ + D)−1, +realizes the biholomorphism group of DIII +n +. Furthermore, DIII +n +is a circled bounded +symmetric domain and it is given as the quotient +DIII +n +≃ Sp(n, C) ∩ U(n, n)/U(n), +where U(n) embedded in Sp(n, C) ∩ U(n, n) by +A �−→ +� +A +0 +0 +A +� +corresponds to the group of biholomorphisms of DIII +n +that fix the origin. +Similarly, we have the next description of the biholomorphism group of the do- +main Sn. We now refer to [9, Exercise C, Chapter VIII]. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +5 +Proposition 2.6. The action via generalized M¨obius transformations given by +Sp(n, R) × Sn −→ Sn +� +A +B +C +D +� +· Z �−→ (AZ + B)(CZ + D)−1, +realizes the biholomorphism group of Sn. Furthermore, Sn is a symmetric domain +and it is given as the quotient +Sn = Sp(n, R)/U(n), +where U(n) embedded in Sp(n, R) by +A �−→ +� +Re(A) +Im(A) +−Im(A) +Re(A) +� +corresponds to the group of biholomorphisms of Sn that fix the matrix iIn. +Remark 2.7. By Proposition 2.3 it follows that the biholomorphism groups of +DIII +n +and Sn are isomorphic. In fact, it is easy to prove that Sp(n, C) ∩ U(n, n) +and Sp(n, R) are conjugated (see [13]). +2.3. Bergman spaces and Toeplitz operators. From now on, D will denote +either of the domains DIII +n +or Sn, and dZ the Lebesgue measure on Symm(n, C). +A number of invariants can be associated to any symmetric domain. The simplest +one is the dimension, which for D is n(n + 1)/2. For other invariants we refer to +[19] for further details on their definitions and here we simply state their known +values with some remarks. +• The rank is defined as the dimension of maximal linearly embedded poly- +disks. For D the rank is n. +• The multiplicities are defined as the main invariants that describe the Jor- +dan triple system associated to the symmetric domain. For D the multi- +plicities are a = 1, b = 0. The vanishing of the latter implies that DIII +n +has +a tubular realization which is in fact given by Sn. For this we observe that +Sn = Symm(n, R) ⊕ iPos(n, R), +where Pos(n, R) denotes the cone of positive definite n × n real matrices. +In the rest of this work we will denote Ωn = Pos(n, R). +• For a tubular domain, the genus is given as p = 2d/r, where d and r are +the dimension and the rank of the domain, respectively. Hence, for D we +have p = n + 1. +We will make use of the multi-gamma function (see [19, Definition 2.4.2]) that +we will consider only for Cartan domains of type III. Such function is associated +to the cone part of a tubular realization of a tube type symmetric domain. In our +case, it is defined by +ΓΩn(λ) = (2π) +n(n−1) +4 +n +� +j=1 +Γ +� +λ − j − 1 +2 +� +, +for every λ > (n−1)/2. It is well known (see [10, 19]) that the volume of a bounded +symmetric domain can be expressed in terms of the multi-gamma functions. In this + +6 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +case we have (see [10]) +Vol(DIII +n +) = π +n(n+1) +2 +ΓΩn +� n+1 +2 +� +ΓΩn(n + 1) +. +Hence, we consider the normalized measure on Symm(n, C) +dv(Z) = +ΓΩn(n + 1) +π +n(n+1) +2 +ΓΩn +� n+1 +2 +� dZ. +In particular, dv(Z) is a probability measure on DIII +n +. +Definition 2.8. The (weightless) Bergman space A2(D) is the subspace of L2(D, v) +that consists of holomorphic functions. In other words, we have +A2(D) = {f ∈ L2(D, v) | f is holomorphic }. +It is a well known fact that A2(D) is a closed subspace of L2(D, v) (see [9, 19]). +We will denote by BD : L2(D, v) → A2(D) the corresponding orthogonal projection. +It is called the (weightless) Bergman projection. Moreover, it is also well known +that A2(D) is a reproducing kernel Hilbert space (see [9, Chapter VIII]) in the +sense that the evaluation map +evZ : A2(D) −→ C +f �−→ f(Z), +is continuous for every Z ∈ D. +This implies the existence of a unique smooth +function KD : D × D → C, holomorphic in the first variable and anti-holomorphic +in the second variable, satisfying KD(Z, ·) ∈ A2(D) for every Z ∈ D and for which +the Bergman projection is given by +BD(f)(Z) = +� +D +f(W)KD(Z, W) dv(W). +for every f ∈ L2(D, v) and Z ∈ D. The function KD is called the (weightless) +Bergman kernel of D. +The Bergman kernels of symmetric domains have closed known expressions. In +particular, it follows from Examples 2.4.17 and 2.9.15 in [19] that the Bergman +kernels of DIII +n +and Sn are given by the expressions +KDIII +n (Z, W) = det(In − ZW)−(n+1), +(2.1) +KSn(Z, W) = det(−i(Z − W))−(n+1), +(2.2) +respectively. We note that a linear biholomorphism has to be applied in order to +obtain the above expression for KSn from the one found in [19]. More precisely, our +unbounded realization of DIII +n +is obtained from the one considered in [19] through +the map Z �→ −iZ. +The next standard construction is to use powers of the Bergman kernel to obtain +weighted measures. The following formula, which holds for every λ > n, is useful +to normalize such weighted measures (see [19, Lemma 2.9.18]) +� +DIII +n +det(In − ZZ)λ−n−1 dZ = π +n(n+1) +2 +ΓΩn +� +λ − n+1 +2 +� +ΓΩn(λ) +. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +7 +Hence, we consider for every λ > n the measure +dvλ(Z) = +ΓΩn (λ) +π +n(n+1) +2 +ΓΩn +� +λ − n+1 +2 +� det(In − ZZ)λ−n−1 dZ +which is a probability measure on DIII +n +, and we also consider the normalized mea- +sure +d�vλ(Z) = +ΓΩn (λ) +π +n(n+1) +2 +ΓΩn +� +λ − n+1 +2 +� det(−i(Z − Z))λ−n−1 dZ. +on the domain Sn. +Definition 2.9. For λ > n, the weighted Bergman spaces on DIII +n +and Sn with +weight λ are given by +A2 +λ(DIII +n +) = {f ∈ L2(DIII +n +, vλ) | f is holomorphic }, +A2 +λ(Sn) = {f ∈ L2(Sn, �vλ) | f is holomorphic }, +respectively. We will denote by A2 +λ(D) the corresponding weighted Bergman space +when D is DIII +n +or Sn. +Note that for λ = n+1, we obtain A2 +λ(DIII +n +) = A2(DIII +n +) and A2 +λ(Sn) = A2(Sn), +which are the weightless Bergman spaces. +As before, it is well known that every weighted Bergman space is closed in +the corresponding L2 space in such a way that it is a reproducing kernel Hilbert +space. +In particular, for D either DIII +n +or Sn there exists a smooth function +KD,λ : D × D → C, holomorphic and anti-holomorphic in the first and second +variable (respectively), such that the orthogonal projection onto A2 +λ(D) is given by +BD,λ(f)(Z) = +� +D +f(W)KD,λ(Z, W) dνλ(W), +for every f ∈ L2(D, νλ) and Z ∈ D, where νλ denotes either vλ or �vλ according +to whether D is DIII +n +or Sn. +This projection is called the weighted Bergman +projection. It follows by Propositions 2.4.22 and 2.9.24 from [19] that the weighted +Bergman kernels for these domains are given by the following expressions +KDIII +n +,λ(Z, W) = det(In − ZW)−λ, +KSn,λ(Z, W) = det(−i(Z − W))−λ, +for every λ > n. In particular, we have KD,λ(Z, W) = KD(Z, W) +λ +n+1 for every +Z, W ∈ D. +The previous constructions allow us to define our main object of study. +Definition 2.10. For every weight λ > n and a ∈ L∞(D), the Toeplitz operator +with symbol a is the bounded operator T (λ) +a +acting on A2 +λ(D) that is given by +T (λ) +a += BD,λ ◦ Ma. +It is interesting to note that the Bergman spaces A2 +λ(DIII +n +) and A2 +λ(Sn) are uni- +tarily equivalent, thus simplifying some computations. This unitary equivalence is +stated without proof in the next result. Its proof is a straightforward generalization +of the arguments provided to obtain Theorem 4.9 in Chapter IV from [18]. + +8 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +Theorem 2.11. The map ϕ given in Proposition 2.3 induces the unitary operator +given by +Uϕ : A2 +λ(DIII +n +) −→ A2 +λ(Sn) +f �−→ JC(ϕ) +λ +n+1 f ◦ ϕ, +where JC(ϕ) = det(dϕC) denotes the complex Jacobian. +3. Geometry of Cartan domains of type III +3.1. Symplectic and K¨ahler geometry. We discuss here some basic material +from symplectic geometry, which will be essential for the main results of this work. +Definition 3.1. A symplectic manifold is a pair (M, ω), where M is a smooth +manifold and ω is a closed 2-form which yields a non-degenerate bilinear form at +every point. +Some of the most important examples of symplectic manifolds come from com- +plex differential geometry. We recall that a manifold M is complex if their charts +map onto open sets of complex vector spaces so that the changes of coordinates are +holomorphic. For such a manifold M, this yields a complex structure Jz on every +tangent space TzM, for every z ∈ M. In turn, this provides a tensor field J known +as the complex structure tensor of M. In particular, we have J2 = −I the identity +tensor acting on the fibers of the tangent bundle T M. We refer to [13] for further +details. +The next definition describes well behaved Riemannian metrics with respect to +these constructions. +Definition 3.2. Let M be a complex manifold with complex structure tensor J +and a given Riemannian metric g. We say that M is a Hermitian manifold if it +satisfies +gz(Jzu, Jzv) = gz(u, v) +for every z ∈ M and u, v ∈ TzM. +We now proceed to relate Hermitian manifolds to symplectic geometry. We will +explain the main constructions and refer to [13] for further details. Let us start +by considering a complex manifold M with complex structure tensor J. Then, the +tangent bundle can be complexified to a complex tangent bundle denoted by T CM, +and the action of J on T M can also be complexified to obtain a tensor JC acting +on T CM. Such complexifications are performed fiberwise. +Since (JC +z )2 = −I, for every z ∈ M, if we define the spaces +T 1,0 +z +M = {v ∈ T C +z M | JC +z v = iv}, +T 0,1 +z +M = {v ∈ T C +z M | JC +z v = −iv}, +then we have T C +z M = T 1,0 +z +M ⊕ T 0,1 +z +M. These spaces are known as the subspaces of +holomorphic and anti-holomorphic tangent vectors. If (z1, . . . , zn) is a holomorphic +chart with real components obtained from the decomposition zj = xj + iyj, then +the usual Wirtinger differential operators are given by +∂ +∂zj += 1 +2 +� +∂ +∂xj +− i ∂ +∂yj +� +, +∂ +∂zj += 1 +2 +� +∂ +∂xj ++ i ∂ +∂yj +� +, + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +9 +for every j = 1, . . . , n. The first set of operators define at every point in the domain +of the chart a basis for the corresponding fibers of T 1,0M. Similarly, the second set +of operators define a basis for the fibers of T 0,1M. The corresponding dual basis +are given by +dzj = dxj + i dyj, +dzj = dxj − i dyj, +where j = 1, . . . , n. +Let us now consider a Riemannian metric g on M for which M is a Hermitian +manifold. +We can complexify g to a complex bilinear tensor gC defined on the +complexified tangent bundle T CM. This yields a positive definite Hermitian form +T 1,0 +z +M × T 1,0 +z +M −→ C, +(u, v) �−→ gC +z (u, v), +for every z ∈ M. In local coordinates, this can be written as +n +� +j,k=1 +gjk(z)dzj ⊗ dzk. +For this reason, we will denote this field of complex Hermitian forms with the same +symbol g. To more easily distinguish between the two of them, we will refer to +the original g as the Riemannian metric of M and we will call the previous field of +Hermitian forms the Hermitian metric of M. +The previous setup and constructions allow to introduce the next important +geometric object. +Definition 3.3. For a Hermitian manifold M with Hermitian metric g as con- +structed above, the associated 2-form is given by +ω = g(J(·), ·) = −2Im(g) +where the first occurrence of g is the Riemannian metric and the second one is the +corresponding Hermitian metric. The Hermitian manifold M is called K¨ahler if its +associated 2-form is closed. In this case, ω is called the K¨ahler form of M. +It is straightforward to check that the associated 2-form of any Hermitian mani- +fold is non-degenerate. Hence, every K¨ahler manifold is a symplectic manifold, and +in this case the K¨ahler form is its symplectic form. +One can alternatively provide a K¨ahler structure on a complex manifold by +introducing a field of Hermitian bilinear forms. This is the content of the next +result which is a particular case of Proposition 1 in page 18 from [13]. +Proposition 3.4. Let M be a complex manifold and let g be a tensor field of positive +definite Hermitian bilinear forms on T 1,0M. Assume that for every holomorphic +coordinate chart (z1, . . . , zn), in a family of charts covering M, there is some real +valued function F such that +g = +n +� +j,k=1 +∂2F +∂zj∂zk +dzj ⊗ dzk +in the domain of the given chart. Then, the tensor 2Re(g) is a Riemannian metric +that yields a K¨ahler structure on M whose Hermitian metric is given by g. + +10 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +3.2. The Bergman metric and its K¨ahler form. We now use the results pre- +viously obtained to construct a K¨ahler structure on the Cartan domains of type +III. The next fundamental theorem is a particular case of the discussion in the first +part of Chapter 4 in [13] (see also [9, 18]). Note that, from now on, we will use the +canonical complex linear coordinates of Symm(n, C). +Theorem 3.5. Let D be either of DIII +n +or Sn and let KD(Z, W) be the reproducing +Bergman kernel of D. Then, the tensor given by +� +1≤l≤m≤n +1≤j≤k≤n +∂2 log KD(Z, Z) +∂zlm∂zjk +dzlm ⊗ dzjk, +is a field of positive definite Hermitian forms that yields a structure of K¨ahler man- +ifold on D. Furthermore, both the corresponding Riemannian metric and associated +K¨ahler form are invariant under the group of biholomorphisms. +We use Theorem 3.5 to introduce K¨ahler structures on DIII +n +and Sn by nor- +malizing the tensor considered in its statement. These normalization will simplify +some formulas below. +Definition 3.6. Let D be either of DIII +n +or Sn and KD(Z, W) the Bergman kernel +of D. The Bergman metric of D is the field of Hermitian forms given by +gD = cD +� +1≤l≤m≤n +1≤j≤k≤n +∂2 log KD(Z, Z) +∂zlm∂zjk +dzlm ⊗ dzjk, +where cDIII +n += +1 +n+1 and cSn = +4 +n+1. +The next two results are very well known properties of the Wirtinger differ- +ential operators that will be useful in this work. We state them for the sake of +completeness. +Lemma 3.7. For any smooth function f : CN −→ C we have +df = +N +� +j=1 +� ∂f +∂zj +dzj + ∂f +∂zj +dzj +� +. +Lemma 3.8 (Chain rule for Wirtinger derivatives). Let g : Cn → Cm and f : +Cm → C be smooth functions. Then, we have +∂(f ◦ g) +∂zj += +m +� +k=1 +� ∂f +∂zk +◦ g ∂gk +∂zj ++ ∂f +∂zk +◦ g ∂gk +∂zj +� +, +∂(f ◦ g) +∂zj += +m +� +k=1 +� ∂f +∂zk +◦ g ∂gk +∂zj ++ ∂f +∂zk +◦ g ∂gk +∂zj +� +. +The following elementary computation will be used latter on. We provide its +proof for the sake of completeness. +Lemma 3.9. The differential of det : Mat(n, C) → C is given by +d(det)A = tr +� +adj(A)dA +� +, +for every A ∈ Mat(n, C), where adj(A) (adjugate of A) is the transpose of the +cofactor matrix of A. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +11 +Proof. If A = (alm) ∈ Mat(n, C) and clm is the cofactor of alm, then the cofactor +expansion of the determinant along the k-th column is given by +det A = +n +� +l=1 +clkalk = +n +� +l=1 +� +adj(A)T � +lkalk. +It follows that +∂ det +∂ajk +(A) = cjk = +� +adj(A)T � +jk +and we obtain the differential +d(det)A = +n +� +j,k=1 +∂ det +∂ajk +(A) dajk = +n +� +j,k=1 +cjk dajk += +n +� +j,k=1 +� +adj(A)T � +jk dajk = +n +� +j,k=1 +� +adj(A) +� +kj dajk += +n +� +k=1 +(adj(A) dA)kk = tr(adj(A) dA). +□ +We now obtain explicit formulas for the Bergman metrics of the Cartan domains +of type III. Note that we have provided coordinate free expressions. This will be +useful for our computations in the rest of this work. +Theorem 3.10. The Bergman metrics on DIII +n +and Sn are respectively given by +gDIII +n +Z +(U, V ) = tr +� +(In − ZZ)−1U(In − ZZ)−1V +� +, +gSn +Z (U, V ) = tr +� +Im(Z)−1UIm(Z)−1V +� +, +for every U, V ∈ Symm(n, C). In particular, the K¨ahler forms of DIII +n +and Sn are +respectively given by +ωDIII +n +Z +(U, V ) = i tr +� +(In − ZZ)−1U(In − ZZ)−1V +� +− i tr +� +(In − ZZ)−1U(In − ZZ)−1V +� +, +ωSn +Z (U, V ) = 2 tr +� +Im(Z)−1Re(U)Im(Z)−1Im(V ) +� +− 2 tr +� +Im(Z)−1Im(U)Im(Z)−1Re(V ) +� +, +for every U, V ∈ Symm(n, C). +Proof. In this proof we will consider the complex vector spaces Symm(n, C) and +Mat(n, C) whose coordinates will be denoted in both cases by zjk, even though they +have different meanings for such spaces. However, from the context where these +coordinates are used it will be easy to identify the actual meaning. +We start by computing the Bergman metric on DIII +n +. First, we observe that we +have the following partial derivative +∂(In − ZZ) +∂zjk +(Z) = −Z ∂Z +∂zjk += −ZEjk, +where Ejk is the n × n symmetric matrix that has 1 in the entries (j, k) and (k, j) +and 0 elsewhere. Note that these matrices are the basis with respect to which we +are considering the canonical coordinates in Symm(n, C). + +12 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +Next, using the previous computation, applying Lemmas 3.9 and 3.8, Equa- +tion (2.1) and using the fact that det is holomorphic, we obtain +1 +n + 1 +∂ +∂zjk +log KDIII +n (Z, Z) = += +1 +det(In − ZZ) +n +� +l,m=1 +∂ det +∂zlm +(In − ZZ)(ZEjk)lm += +1 +det(In − ZZ) +n +� +l,m=1 +(adj(In − ZZ)T )lm(ZEjk)lm += +1 +det(In − ZZ)tr +� +adj(In − ZZ)ZEjk +� += tr +� +(In − ZZ)−1ZEjk +� +. +Now, we will use the easy to prove relations +(In − ZZ)−1Z = Z(In − ZZ)−1, +Z(In − ZZ)−1 = (In − ZZ)−1Z, +which hold for every Z ∈ DIII +n +. Using the identities obtained so far we compute +1 +n + 1 +∂2 +∂zlm∂zjk +log KDIII +n (Z, Z) = +∂ +∂zlm +tr +� +(In − ZZ)−1ZEjk +� += tr +� +(In − ZZ)−1ElmZ(In − ZZ)−1ZEjk +� ++ tr +� +(In − ZZ)−1ElmEjk +� += tr +� +(In − ZZ)−1Elm(In − ZZ)−1ZZEjk +� ++ tr +� +(In − ZZ)−1ElmEjk +� += tr +� +(In − ZZ)−1Elm(In − ZZ)−1(ZZ + (In − ZZ))Ejk +� += tr +� +(In − ZZ)−1Elm(In − ZZ)−1Ejk +� +. +This implies that the metric gDIII +n +Z +satisfies the required identity on the basic el- +ements of the vector space Symm(n, C), thus proving the result for the Bergman +metric of DIII +n +. The corresponding computation of the Bergman metric for Sn is +obtained similarly. +From Definition 3.3 the K¨ahler form of DIII +n +is given by +ωDIII +n +Z +(U, V ) = −2Im +� +gDIII +n +Z +(U, V ) +� += i +� +gDIII +n +Z +(U, V ) − gDIII +n +Z +(U, V ) +� +, +which yields the stated formula from our computation of the Bergman metric of +DIII +n +. +Finally, for the K¨ahler form of Sn we compute +ωSn +Z (U, V ) = −2Im +� +gSn +Z (U, V ) +� += −2Im +� +tr +� +Im(Z)−1(Re(U) + iIm(U))Im(Z)−1(Re(V ) − iIm(V )) +�� +, +from which the stated formula is easily obtained. +□ + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +13 +3.3. Moment maps. Now we turn back our attention to symplectic geometry. It +will provide the main geometric tools and objects that we will apply to the study +of Toeplitz operators. We refer to [12] for the symplectic geometry facts stated +without proof. +In the rest of this subsection (M, ω) will denote a fixed symplectic manifold. A +diffeomorphism ϕ : M → M is called a symplectomorphism if ϕ∗(ω) = ω. In other +words, a symplectomorphism is a diffeomorphism preserving the symplectic form. +If H is a Lie group with a smooth action on M, then we say that the H-action +is symplectic if the map +M −→ M +z �−→ h · z +is a symplectomorphism for every h ∈ H. +There are two important types of vector fields on M. From now on, we will +denote by X(M) the Lie algebra of vector fields over M. A field X ∈ X(M) is +called a symplectic vector field if and only if the 1-form ω(X, ·) is closed, and it is +called a Hamiltonian vector field if and only if the form ω(X, ·) is exact. We will +denote by X(M, ω) the space of symplectic vector fields on M. It is a well known +fact that X(M, ω) is a Lie subalgebra of X(M). +For any smooth function f : M → R, the non-degeneracy of ω implies the +existence of a unique element Xf ∈ X(M) such that +df = ω(Xf, ·). +In this case, Xf is called the Hamiltonian vector field associated to f. +Symplectic vector fields can be characterized by symplectomorphisms. +More +precisely, it is well known that an element X ∈ X(M) belongs to X(M, ω) if and +only if the local flow generated by X acts by (locally defined) symplectomorphisms. +An important converse to the previous fact relates symplectic actions to sym- +plectic vector fields as follows. Let us consider a symplectic action of a Lie group H +on M. Then, for every X ∈ h (the Lie algebra of H), we define the induced vector +field on M by +X♯ +z = d +ds +��� +s=0 exp(sX) · z. +for every z ∈ M, where exp : h → H is the exponential map of H. Then, the fact +that the H-action is symplectic implies that X♯ ∈ X(M, ω) for every X ∈ h. +In the previous discussion, we have shown two different constructions that map +into the space X(M, ω) of symplectic vector fields. Hence, a natural problem to +consider is the existence of a map h → C∞(M) that makes the following diagram +commute +C∞(M) +� +h +�① +① +① +① +① +① +① +① +① +� X(M, ω) +where the vertical arrow is the map f �→ Xf and the horizontal arrow is the map +X �→ X♯. The existence of such diagonal map yields the notion of a moment map +for the H-action. The precise definition requires some additional conditions. We +recall that Ad = AdH : H → GL(h) denotes the adjoint representation of the Lie + +14 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +group H, and that Ad∗ denotes the dual representation on h∗. In particular, we +have Ad∗(h) = Ad(h−1)⊤ for every h ∈ H. +Definition 3.11. Let (M, ω) be a symplectic manifold and let H be a Lie group +acting by symplectomorphisms on M. A moment map for the H-action is a smooth +map µ : M → h∗, where h∗ is the vector space dual of h, that satisfies the following +properties. +(1) For every X ∈ h consider the map µX : M → R given by µX(z) = ⟨µ(z), X⟩. +Then, the Hamiltonian vector field associated to µX is X♯, for every X ∈ h. +In other words, it holds +dµX = ω(X#, ·), +for every X ∈ h. +(2) The map µ is H-equivariant. In other words, we have +µ(h · z) = Ad∗(h)(µ(z)), +for every z ∈ M and h ∈ H. +Remark 3.12. If H is an Abelian group, then its adjoint representation satisfies +Ad(h) = Ih for every h ∈ H. Hence, in this case, condition 2. in Definition 3.11 +reduces to +µ(h · z) = µ(z), +for every h ∈ H and z ∈ M. In other words, this requires the smooth map to be +H-invariant. +4. Three Abelian biholomorphism groups and their moment maps +In this section we study three special types of subgroups of biholomorphisms +acting on Cartan domains of type III. For the corresponding Abelian groups, we +compute the moment maps. We will see later on that these moment maps are a +powerful tool to find commutative C∗-algebras generated by Toeplitz operators. +4.1. Elliptic, Hyperbolic, and Parabolic Actions. The Cartan domains DIII +n +and their unbounded realizations Sn carry three interesting actions of subgroups of +biholomorphisms. As we will see, these actions generalize the three different types +of M¨obius transformations found for the unit disk D and the upper half plane H. +Proposition 2.5 provides the action +U(n) × DIII +n +−→ DIII +n +U · Z = UZU ⊤, +which yields the subgroup of biholomorphisms that fixes the origin. Up to con- +jugacy, this characterizes the subgroups that fix some point in the domain DIII +n +. +This is so because of the homogeneity of this domain. We will call this the Elliptic +Action on DIII +n +. +Next we observe that there is a canonical homomorphism of Lie groups given by +GL(n, R) −→ Sp(n, R) +A �−→ +� +A +0 +0 +(A−1)⊤ +� +. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +15 +A straightforward computation shows that this assignment is indeed a homomor- +phism whose image lies in Sp(n, R). Hence, this homomorphism and Proposition 2.6 +provide the action +GL(n, R) × Sn −→ Sn +A · Z = AZA⊤. +It is easily seen that this action realizes the subgroup of biholomorphisms that fixes +the origin, a boundary point of the domain Sn. For this reason, we will call this +the Hyperbolic Action on Sn. +Finally, we have a canonical homomorphism of Lie groups given by +Symm(n, R) −→ Sp(n, R) +S �−→ +�In +S +0 +In +� +, +where Symm(n, R) is endowed with the Lie group structure with operation given +by the sum of matrices. Again, it is straightforward to show that this map is indeed +a homomorphism into the group Sp(n, R). We now have that this homomorphism +together with Proposition 2.6 provide the action +Symm(n, R) × Sn −→ Sn +S · Z = Z + S. +This action realizes the subgroup of biholomorphisms of the tube type domain Sn +that correspond to translations on the real vector space part. Since this action +clearly generalizes the translation action on the real part on the upper half-plane +H, we will call this action on Sn the Parabolic Action. +In fact, all three actions introduced above generalize the behavior observed in +the 1-dimensional case. +This is stated in the following well known result. +We +recall that two biholomorphisms are conjugated if they are so under some other +biholomorphism.This result justifies our choice of notation for the actions considered +above. +Corollary 4.1. Let us denote by D either D or H. If ϕ is a biholomorphism of D, +then the following equivalences hold. +(1) The M¨obius transformation ϕ is elliptic if and only if it is conjugated to a +transformation that belongs to the action T × D → D given by z �→ tz. +(2) The M¨obius transformation ϕ is hyperbolic if and only if it is conjugated to +a transformation that belongs to the action R+ × H → H given by z �→ rz. +(3) The M¨obius transformation ϕ is parabolic if and only if it is conjugated to +a transformation that belongs to the action R × H → H given by z �→ z + s. +We note that the Elliptic and Hyperbolic Actions are given by actions of Abelian +groups if and only if n = 1. Nevertheless, the Parabolic Action is given by an +Abelian Lie group in any dimension. For these reason, we introduce in the next +definition actions of Abelian groups associated to the Elliptic and Hyperbolic cases. +Definition 4.2. The Abelian Elliptic Action on DIII +n +is defined by +T × DIII +n +−→ DIII +n +t · Z = t2Z. + +16 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +The Abelian Hyperbolic Action on Sn is defined by +R+ × Sn −→ Sn +r · Z = r2Z. +Remark 4.3. We note that the Abelian Elliptic and Abelian Hyperbolic actions are +obtained by considering the center of the groups corresponding to the non-Abelian +actions. More precisely, we have the centers +Z(U(n)) = TIn, +Z(GL(n, R)) = R+In ∪ (−R+In), +and the actions in Definition 4.2 are the restriction of the previously defined actions +to these center groups. On the other hand, Symm(n, R) is already Abelian so that +it coincides with its center, in other words we have +Z(Symm(n, R)) = Symm(n, R). +Hence, the most obvious definition of “Abelian Parabolic Action” would yield what +we already have defined as the Parabolic Action. We also observe that these three +actions of Abelian groups of biholomorphisms, the Abelian Elliptic, Abelian Hyper- +bolic and Parabolic, are natural generalizations of the actions described in Corol- +lary 4.1. +4.2. Moment maps of the Abelian actions. We will now compute moment +maps for all three Abelian actions introduced in this section. We refer to Defini- +tion 4.2 and Remark 4.3. It follows from Theorem 3.5 that every biholomorphism +of either of the domains DIII +n +and Sn preserves the corresponding K¨ahler form. +Hence, all the groups considered above act by symplectomorphisms. In particular, +the notion of moment map given in Definition 3.11 can be applied to such actions. +4.2.1. Moment map of the Abelian Elliptic Action. The group in this case is T acting +on DIII +n +. The Lie algebra of this group is R. The latter is canonically isomorphic +to its dual R∗, so we will compute a moment map as a function DIII +n +→ R. +For every element t ∈ R the corresponding induced vector field on DIII +n +is +given by +t♯ +Z = d +ds +��� +s=0 exp(st) · Z = d +ds +��� +s=0 exp(2ist)Z = 2itZ, +for every Z ∈ DIII +n +. Note that we have used the fact that the (Lie group) exponen- +tial map R → T satisfies t �→ exp(it). +Proposition 4.4. The function given by +µT : DIII +n +−→ R +µT(Z) = −2tr +� +(In − ZZ)−1� +, +is a moment map for the Abelian Elliptic Action on DIII +n +. +Proof. We start by computing ωDIII +n +Z +(t♯ +Z, ·) for every t ∈ R and Z ∈ DIII +n +. For this +first computation we use the above formula for t♯ and the expression for the K¨ahler + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +17 +form of DIII +n +obtained in Theorem 3.10. We have +ωDIII +n +Z +(t♯ +Z, V ) = i tr +� +(In − ZZ)−12itZ(In − ZZ)−1V +� +− i tr +� +(In − ZZ)−12itZ(In − ZZ)−1V +� +, += − 2t tr +� +(In − ZZ)−1Z(In − ZZ)−1V +� +− 2t tr +� +(In − ZZ)−1Z(In − ZZ)−1V +� +, +for every V ∈ Symm(n, C). +On the other hand, we consider the function µt : DIII +n +→ R defined by +µt(Z) = ⟨µT(Z), t⟩ = tµT(Z), +and we compute its differential as follows +d(µt)Z(V ) = += − 2t d +ds +��� +s=0 tr +�� +In − (Z + sV )(Z + sV ) +�−1� += − 2t tr +�� +In − ZZ +�−1� +V Z + ZV +�� +In − ZZ +�−1� += − 2t tr +�� +In − ZZ +�−1V +� +In − ZZ +�−1Z +� +− 2t tr +� +Z +� +In − ZZ +�−1V +� +In − ZZ +�−1� +where we applied in the last identity the commutation relations between Z, (In − +ZZ)−1 and their conjugates used in the proof of Theorem 3.10. We conclude that +d(µt)Z(V ) = ωDIII +n +Z +(t♯ +Z, V ) +for every V ∈ Symm(n, C) and Z ∈ DIII +n +. It follows that the first condition in +Definition 3.11 is satisfied by the map in the statement. It remains to prove the +T-invariance of this map, but this is established through the identities +µT(t · Z) = µT(t2Z) = −2 tr +� +(In − t2Zt2Z)−1� += −2 tr +� +(In − ZZ)−1� += µT(Z) +that hold for every t ∈ T and Z ∈ DIII +n +. +□ +Remark 4.5. For the case n = 1, the Abelian Elliptic Action yields the T-action +on the unit disk D given by t · z = t2z. With this assumption, Proposition 4.4 +provides the moment map +µT(z) = −2 +1 +1 − |z|2 . +We observe that for actions of Abelian groups we can add to a given moment map an +arbitrary, but fixed, constant to obtain another moment map (see Definition 3.11). +Hence, the map given by +µ(z) = µT(z) + 2 = −2 +|z|2 +1 − |z|2 , +is a moment map as well for our T-action on D. This recovers, up to the multi- +plicative constant 2, the moment map obtained in [14, Proposition 4.1] for n = 1. + +18 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +This referenced result computes the moment map for the natural action of the n- +dimensional torus on the n-dimensional unit ball. We note that the factor 2 comes +from the reparameterization involved in using the action t · z = t2z instead of the +action t · z = tz. +4.2.2. Moment map of the Abelian Hyperbolic Action. We now have the group R+ +acting on Sn. The Lie algebra of this group is R itself, which is canonically isomor- +phic to its dual. Hence, the moment map will be computed as a function Sn → R. +For every t ∈ R the induced vector field on Sn is obtained as follows. This +computation uses the fact that the (Lie group) exponential map is given in this +case by t �→ exp(t). +t♯ +Z = d +ds +��� +s=0 exp(st) · Z = d +ds +��� +s=0 exp(2st)Z = 2tZ, +for every Z ∈ Sn. +Proposition 4.6. The function given by +µR+ : Sn −→ R +µR+(Z) = −4tr +� +Im(Z)−1Re(Z) +� +is a moment map for the Abelian Hyperbolic Action on Sn. +Proof. We compute ωSn +Z (t♯ +Z, ·), for every t ∈ R and Z ∈ Sn. For this we use the +previous computations and the expression of the K¨ahler form of Sn obtained in +Theorem 3.10. We have in this case +ωSn +Z (t♯ +Z, V ) = 2 tr +� +Im(Z)−1Re(2tZ)Im(Z)−1Im(V ) +� +− 2 tr +� +Im(Z)−1Im(2tZ)Im(Z)−1Re(V ) +� += 4t tr +� +Im(Z)−1Re(Z)Im(Z)−1Im(V ) +� +− 4t tr +� +Im(Z)−1Re(V ) +� +, +for every V ∈ Symm(n, C). +On the other hand, we consider the function µt : Sn → R given by +µt(Z) = ⟨µR+(Z), t⟩ = tµR+(Z), +for which we compute the differential as follows +d(µt)Z(V ) = += −4t d +ds +��� +s=0 tr +� +Im(Z + sV )−1Re(Z + sV ) +� += −4t d +ds +��� +s=0 tr +�� +Im(Z) + sIm(V ) +�−1� +Re(Z) + sRe(V ) +�� += 4t tr +� +Im(Z)−1Im(V )Im(Z)−1Re(Z) +� +− 4t tr +� +Im(Z)−1Re(V ) +� +, +for every V ∈ Symm(n, C). From this we conclude that +d(µt)Z(V ) = ωSn +Z (t♯ +Z, V ), +for every V ∈ Symm(n, C) and Z ∈ Sn. Hence, by Definition 3.11 it remains to +show that µR+ is R+-invariant, and this is verified in the next computation +µR+(r · Z) = µR+(r2Z) = −4tr +� +Im(r2Z)−1Re(r2Z) +� += −4tr +� +Im(Z)−1Re(Z) +� += µR+(Z), + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +19 +which holds for every r ∈ R+ and Z ∈ Sn. +□ +Remark 4.7. For n = 1, the Abelian Hyperbolic Action yields the R+-action +on the upper half-plane H given by r · z = r2z. +Under this restriction, from +Proposition 4.6 we obtain the moment map +µR+(z) = −4Re(z) +Im(z). +This recover, up to a constant factor, the moment map obtained in [14, Proposi- +tion 4.3] for n = 1. In this case the factor comes from two sources. Firstly, we +use the action r · z = r2z, instead of the action r · z = rz used in [14]. Secondly, +our formula for the K¨ahler form for S1 = H differs by a constant factor from the +corresponding formula found in [14]. +4.2.3. Moment map of the Parabolic Action. Finally, we consider the group Symm(n, R) +acting on Sn. Since Symm(n, R) is a vector group, it follows that it coincides with +its Lie algebra and its exponential map is the identity. There is a canonical isomor- +phism between Symm(n, R) and its dual space given by the positive definite inner +product +⟨A, B⟩ = tr(AB), +defined for A, B ∈ Symm(n, R). +For every S ∈ Symm(n, R) the corresponding induced vector field on Sn satisfies +for every Z ∈ Symm(n, C) +S♯ +Z = d +ds +��� +s=0 exp(sS) · Z = d +ds +��� +s=0 (Z + sS) = S, +which is the constant vector with value S. +Proposition 4.8. The function given by +µSymm(n,R) : Sn −→ Symm(n, R) +µSymm(n,R)(Z) = −2Im(Z)−1, +is a moment map for the Parabolic Action on Sn. +Proof. For every S ∈ Symm(n, R) and Z ∈ Sn, using the above computations and +Theorem 3.10 we obtain +ωSn +Z (S♯ +Z, V ) = 2 tr +� +Im(Z)−1Re(S)Im(Z)−1Im(V ) +� +− 2 tr +� +Im(Z)−1Im(S)Im(Z)−1Re(V ) +� += 2 tr +� +Im(Z)−1S Im(Z)−1Im(V ) +� +, +for every Z ∈ Sn. +On the other hand, we consider for every S ∈ Symm(n, R) the map µS : Sn → +Symm(n, R) defined by +µS(Z) = −2tr +� +Im(Z)−1S +� +, +for which we compute +d(µS)Z(V ) = −2 d +ds +��� +s=0 tr +� +Im(Z + sV )−1S +� += 2 tr +� +Im(Z)−1Im(V )Im(Z)−1S +� +, + +20 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +for every V ∈ Symm(n, C) and Z ∈ Sn. This immediately yields +d(µS)Z(V ) = ωSn +Z (S♯ +Z, V ), +for every V ∈ Symm(n, C) and Z ∈ Sn. By Definition 3.11 it remains to establish +the Symm(n, R)-invariance of µSymm(n,R), and this achieved by noting that +µSymm(n,R)(S · Z) = µSymm(n,R)(Z + S) = −2 +� +Im(Z + S)−1� += −2 +� +Im(Z)−1� += µSymm(n,R)(Z) +for every Z ∈ Sn and S ∈ Symm(n, R). +□ +Remark 4.9. For n = 1, the Parabolic Action yields the R-action on the upper +half-plane H given by s · z = z + s. And in this situation, Proposition 4.8 provides +the moment map +µR(z) = −2 +1 +Im(z). +As in the previous cases, this recovers, up to a constant factor, the moment map +obtained in [14, Proposition 4.2] for n = 1. As in the case of Remark 4.7 the factor +comes from a different normalization of the K¨ahler form on this work and [14]. +5. Toeplitz operators with special symbols +We will now describe Toeplitz operators with special symbols using two related +alternatives: symbols invariant under biholomorphism groups and symbols depend- +ing on the moment maps of such groups. Both cases yield, under suitable conditions, +commutative C∗-algebras generated by Toeplitz operators. +First we introduce a general notation. As before, in the rest of this work D +denotes either of the domains DIII +n +or Sn. For A ⊂ L∞(D) a set of essentially +bounded symbols, we denote by T (λ)(A) the C∗-algebra generated by the Toeplitz +operators T (λ) +a +where a ∈ A. +5.1. Invariant symbols. Let H be a closed subgroup of biholomorphisms of D. +We will denote by L∞(D)H the subspace of L∞(D) consisting of H-invariant sym- +bols. In other words, we have +L∞(D)H = {a ∈ L∞(D) : h · a = a, for all h ∈ H}, +where, for a given a ∈ L∞(D) and h ∈ H, we define +(h · a)(Z) = a(h−1 · Z), +for almost every Z ∈ D. +Symmetric pairs associated to symmetric domains can be used to obtain commu- +tative C∗-algebras generated by Toeplitz operators by considering invariant sym- +bols. The definitions and precise statements can be found in [1]. In this work, we +will use the fact that the pairs (Sp(n, R), GL(n, R)) and (Sp(n, C) ∩ U(n, n), U(n)) +are symmetric in order to obtain the following consequence of Theorem 5.1 from +[1]. +Theorem 5.1 ([1]). For every λ > n, the C∗-algebras T (λ)(L∞(DIII +n +)U(n)) and +T (λ)(L∞(Sn)GL(n,R)) acting on the weighted Bergman spaces A2 +λ(DIII +n +) and A2 +λ(Sn), +respectively, are commutative. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +21 +With the notation from subsection 4.1, Theorem 5.1 states that for the Elliptic +and Hyperbolic actions on DIII +n +and Sn, respectively, the symbols invariant under +such actions yield Toeplitz operators that generate commutative C∗-algebras. +The Parabolic Action provides the same sort of conclusion. This follows from +the next consequence of [1, Theorem 5.8]. We note that in this case the group +Symm(n, R) does not yield a symmetric pair in the group Sp(n, R) of biholomor- +phisms of Sn. +Theorem 5.2 ([1]). For every λ > n, the C∗-algebra T (λ)(L∞(Sn)Symm(n,R)) +acting on the weighted Bergman space A2 +λ(Sn) is commutative. +5.2. Moment map symbols. Following [14, 15] we define the notion of moment +map symbol for the setup of this work. +Definition 5.3. Let D be either of the domains DIII +n +or Sn and H a closed +subgroup of the biholomorphism group of D. If µH : D → h∗ is a moment map +for the action of H on D, then a moment map symbol for H or a µH-symbol is a +symbol a ∈ L∞(D) that can be written in the form a = f ◦µH for some measurable +function f. We denote by L∞(D)µH the space of all essentially bounded measurable +µH-symbols on D. +We have computed moment maps for the Abelian Elliptic, Abelian Hyperbolic +and Parabolic actions in subsection 4.2. These computations allow us to provide +the following explicit description of moment map symbols for these three actions. +Proposition 5.4. Let a ∈ L∞(DIII +n +) and b ∈ L∞(Sn) be given. Then, the follow- +ing equivalences hold +(1) The measurable function a is a µT-symbol if and only if there exists a mea- +surable function f such that a(Z) = f +� +tr(ZZ) +� +, for almost every Z ∈ DIII +n +. +(2) The measurable function b is a µR+-symbol if and only if there exists a +measurable function f such that b(Z) = f +� +tr(Im(Z)−1Re(Z)) +� +, for almost +every Z ∈ Sn. +(3) The measurable function b is a µSymm(n,R)-symbol if and only if there exists +a measurable function f such that b(Z) = f(Im(Z)), for almost every Z ∈ +Sn. +Proof. We note that the claims on the symbols b ∈ L∞(Sn) are immediate conse- +quences of Definition 5.3 and Propositions 4.6 and 4.8. Hence, we consider the case +of moment maps for the Abelian Elliptic Action. +By Proposition 4.4 and Definition 5.3, a symbol a ∈ L∞(DIII +n +) is a µT-symbol if +and only if there is a measurable function g such that +a(Z) = g +� +tr +� +(In − ZZ)−1�� +, +for almost every Z ∈ DIII +n +. In the cone Pos(n, C) of positive definite n× n complex +matrices let us consider the open subsets given by +(0, In) = {Z ∈ Pos(n, C) | Z < In} +(In, ∞) = {W ∈ Pos(n, C) | In < W}. +It is straightforward to verify that the maps +F : (0, In) −→ (In, ∞) +G : (In, ∞) −→ (0, In) +Z �−→ (In − Z)−1 +W �−→ In − W −1 + +22 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +are well defined smooth maps, such that they are inverses of each other. In partic- +ular, these maps satisfy +(In − ZZ)−1 = F(ZZ), +ZZ = G +� +(In − ZZ)−1� +, +for every Z ∈ DIII +n +. The case of the Abelian Elliptic Action clearly follows from +these remarks. +□ +By definition, the moment map symbols for Abelian groups are invariant under +the corresponding actions. It turns out that the moment maps of the first two +actions are in fact invariant under larger groups, those considered in Theorem 5.1. +This is the content of the next two results. +Proposition 5.5. Let µT : DIII +n +→ R be the moment map for the T-action on DIII +n +given in Proposition 4.4. Then, µT is a U(n)-invariant function. In particular, we +have L∞(DIII +n +)µT ⊂ L∞(DIII +n +)U(n). +Proof. Recall that U(n) acts on DIII +n +by U · Z = UZU T . Using the expression of +µT obtained in Proposition 4.4, we have for every U ∈ U(n) +µT(g · Z) = −2tr +� +(In − UZU TUZU T))−1� += −2tr +� +(In − UZZU T )−1� += −2tr +� +(U(In − ZZ)U T )−1� += −2tr +� +U(In − ZZ)−1U −1� += −2tr +� +(In − ZZ)−1� += µT(Z), +for every Z ∈ DIII +n +. The last claim is now an immediate consequence of Defini- +tion 5.3. +□ +Proposition 5.6. Let µR+ : Sn → R be the moment map of the R+-action on +Sn given in Proposition 4.6. +Then, µR+ is a GL(n, R)-invariant function. +In +particular, we have L∞(Sn)µR+ ⊂ L∞(Sn)GL(n,R). +Proof. Recall that GL(n, R) acts on Sn by A · Z = AZAT . +We now use the +expression of µR+ obtained in Proposition 4.6, and for every A ∈ GL(n, R) we +compute +µR+(A · Z) = −4tr +� +(AIm(Z)A⊤)−1ARe(Z)A⊤� += −4tr +� +(A⊤)−1Im(Z)−1A−1ARe(Z)A⊤� += −4tr +� +Im(Z)−1Re(Z) +� += µR+(Z), +for every Z ∈ Sn. Again, the last claim now follows immediately. +□ +For the Parabolic Action, it turns out that Symm(n, R)-invariance and being a +µSymm(n,R)-symbol are equivalent. This is the content of the next result. +Proposition 5.7. For the moment map µSymm(n,R) : Sn → Symm(n, R) of the +Symm(n, R)-action on Sn given in Proposition 4.8, we have +L∞(Sn)µSymm(n,R) = L∞(Sn)Symm(n,R). + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +23 +Proof. The Symm(n, R)-action on Sn is given by the expression S · Z = Z + S. +Hence, for a given a ∈ L∞(Sn) we have the following sequence of equivalences +a is Symm(n, R)-invariant +⇐⇒ for every S ∈ Symm(n, R) : a(Z + S) = a(Z) for a.e. Z ∈ Sn +⇐⇒ for some measurable f : a(Z) = f(Im(Z)) for a.e. Z ∈ Sn, +and the result follows from the last case in Proposition 5.4. +□ +5.3. Commuting Toeplitz operators with moment maps symbols. We now +state one of our main results: for D either of the domains DIII +n +or Sn, there are three +Abelian groups of biholomorphisms of D to which we can associate commutative +C∗-algebras generated by Toeplitz operators. +Theorem 5.8. Let D be either of the domains DIII +n +or Sn. The Abelian Elliptic +Action, the Abelian Hyperbolic Action and the Parabolic Action on D yield three +Abelian groups of biholomorphisms of D which provide, for every λ > n, the fol- +lowing commutative C∗-algebras generated by Toeplitz operators. +Abelian Elliptic: The C∗-algebra T (λ)� +L∞(DIII +n +)µT� +, acting on A2 +λ(DIII +n +), +obtained from the moment map of the T-action on DIII +n +. +Abelian Hyperbolic: The C∗-algebra T (λ)� +L∞(Sn)µR+� +, acting on A2 +λ(Sn), +obtained from the moment map of the R+-action on Sn. +Parabolic: The C∗-algebra T (λ)� +L∞(Sn)µSymm(n,R)� +, acting on A2 +λ(Sn), ob- +tained from the moment map of the Symm(n, R)-action on Sn. +Proof. First, we note that Propositions 5.5 and 5.6 imply the inclusions +T (λ)(L∞(DIII +n +)µT) ⊂ T (λ)(L∞(DIII +n +)U(n)) +T (λ)(L∞(Sn)µR+) ⊂ T (λ)(L∞(Sn)GL(n,R)), +and so the cases of the Abelian Elliptic and Abelian Hyperbolic Actions follow from +Theorem 5.1. +For the Parabolic Action, we note that Proposition 5.7 yields the identity +T (λ)(L∞(Sn)µSymm(n,R)) = T (λ)(L∞(Sn)Symm(n,R)), +and now the result is a consequence of Theorem 5.2. +□ +Remark 5.9. It follows from the discussion in subsection 4.1 (see Corollary 4.1 and +Definition 4.2) that for n = 1 the three actions considered in Theorem 5.8 reduce to +the usual elliptic, hyperbolic and parabolic actions known from complex analysis. +These three actions have been previously used to obtain commutative C∗-algebras +generated by Toeplitz operators, notably in the results found in [3, 4, 5, 7] (see also +[11]). In fact, the commutative C∗-algebras generated by Toeplitz operators from +Theorem 5.8 reduce to those from these previous works when n = 1. +One of the main guiding lights in this line of study of Toeplitz operators has +been to find generalizations to higher dimensions of these commutative C∗-algebras +observed in the case of the unit disk. This was achieved for the unit ball Bn in Cn +through the use of maximal Abelian subgroups of the biholomorphism group of Bn +(see [16, 17]). However, the result for the unit ball Bn from these references lead to + +24 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +n + 2 different commutative C∗-algebras, which is in contrast with the simplicity +of only three Abelian groups for the case of the unit disk. +On the other hand, our Theorem 5.8 recovers for higher dimensions the simplic- +ity observed in the case of the unit disk. More precisely, we consider the generalized +unit disk DIII +n +and its unbounded realization Sn, Siegel’s generalized upper half- +plane. For these domains, Theorem 5.8 yields three commutative C∗-algebras gen- +erated by Toeplitz operators, acting on the Bergman spaces of DIII +n +and Sn, which +can be seen as natural extensions of the case of the unit disk. And this is achieved +while using only three Abelian groups for any dimension. This is possible due to +the fact that we have replaced invariant symbols with moment map symbols. This +highlights the importance of using symplectic geometry to study Toeplitz operators +acting on Bergman spaces in higher dimensions. +6. Spectral integral formulas for Toeplitz operator with moment +map symbols +In this final section we present explicit integral formulas that simultaneously +diagonalize Toeplitz operators. +This will be done for the Abelian Elliptic and +Parabolic Actions. +6.1. Toeplitz operators with Abelian Elliptic symbols. In this case, we are +dealing with symbols that belong to T (λ)(L∞(DIII +n +)µT) ⊂ T (λ)(L∞(DIII +n +)U(n)). In +particular, it is useful to consider the U(n)-action on the Bergman spaces over DIII +n +. +We recall some properties of such action and refer to [19, 2] for further details. +For every λ > n, there is a unitary representation given by +πλ : U(n) × A2 +λ(DIII +n +) −→ A2 +λ(DIII +n +) +πλ(A)(f) = f ◦ A−1. +This representation leaves invariant the subspace of (holomorphic) polynomials on +DIII +n +⊂ Symm(n, C), that we will denote by P(Symm(n, C)) = P, for simplicity. In +particular, for every λ > n, the decomposition of A2 +λ(DIII +n +) into irreducible U(n)- +submodules is the same as the one corresponding to the U(n)-action on P. Let us +denote by −→ +N n the set of integer n-tuples that satisfy α1 ≥ · · · ≥ αn ≥ 0. Then, +using the representation πλ, one can show that, for every λ > n, there is a Hilbert +direct sum decomposition +(6.1) +A2 +λ(DIII +n +) = +� +α∈− +→ +N n +Pα, +where +� +Pα� +α∈− +→ +N n is family of mutually non-isomorphic U(n)-submodules of P. For +the proof of this claim we refer to [19, Chapter 2] (see also [2, 8]). +We consider the polynomials given by +∆j(Z) = det(Zj) +where Zj is the upper-left corner j × j submatrix of Z. For every α ∈ −→ +N n we will +also consider the polynomial +∆α(Z) = +n +� +j=1 +∆j(Z)αj−αj+1 + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +25 +where we agree to define αn+1 = 0. These are known as the conical polynomials +for the representation of U(n) on P (see [19, Chapter 2]). +With the previous notation, the following result is an application of Proposi- +tion 4.7 and Theorem 4.11 from [2] to our current setup. +Theorem 6.1 ([2]). Let a ∈ L∞(DIII +n +)U(n) and λ > n be given. Then, the Toeplitz +operator T (λ) +a +acting on the Bergman space A2 +λ(DIII +n +) preserves the Hilbert direct +sum (6.1). Furthermore, we have +T (λ) +a +|Pα = ca,λ(α)IPα, +where the complex constant ca,λ(α) is given by +ca,λ(α) = +� +0 · · · > x1 > 0}, +which is only n-dimensional. In the rest of this work, we will denote by D(x) the +diagonal n × n matrix with diagonal elements given by x ∈ Rn. Also, for a given +x ∈ Rn ++ we will write √x = (√x1, . . . , √xn). +Theorem 6.2. Let a ∈ L∞(DIII +n +)U(n) and λ > n be given. Then, the complex +constants (ca,λ(α))α∈− +→ +N n such that +T (λ) +a +|Pα = ca,λ(α)IPα, +for every α ∈ −→ +N n, as obtained in Theorem 6.1, are given by +ca,λ(α) = +� +−−→ +(0,1)n +a(D(√x)) +� +n +� +j=1 +xj +�αn� +n +� +j=1 +(1 − xj) +�λ−n−1� � +j 0 such that +Ia,λ(α) = C +� +−−→ +(0,1)n +� +O(n) +a +�� +AD(x)A⊤ +� +∆α +� +AD(x)A⊤� +× +× ∆n +� +In − AD(x)A⊤�λ−n−1 � +j n be given. Let f be a measurable +function such that a(Z) = f +� +tr(ZZ) +� +, for almost every Z ∈ DIII +n +. +Then, the +complex constants (ca,λ(α))α∈− +→ +N n such that +T (λ) +a +|Pα = ca,λ(α)IPα, +for every α ∈ −→ +N n, as obtained in Theorem 6.1, are given by +ca,λ(α) = +� +−−→ +(0,1)n +f(∥x∥1) +� +n +� +j=1 +xj +�αn� +n +� +j=1 +(1 − xj) +�λ−n−1� � +j n, +the weighted measure �vλ decomposes as +d�vλ(Z) = Cλ,n det(2Y )λ−n−1 dX dY, + +28 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +with the coordinates Z = X + iY , (X ∈ Symm(n, R) and Y ∈ Ωn) and for the +positive constant +Cλ,n = +ΓΩn(λ) +π +n(n+1) +2 +ΓΩn +� +λ − n+1 +2 +�. +This yields the natural isometry +L2(Sn, �v) ≃ L2(Symm(n, R), dX) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ), +that we will use in the rest of this work. +Let us consider the unitary operator U = F ⊗ I defined on L2(Sn, �vλ), where F +is the Fourier transform on Symm(n, R). More precisely, we have +(F(f))(X) = +1 +(2π) +n(n+1) +4 +� +Symm(n,R) +e−itr(Xξ)f(ξ) dξ +for every f ∈ L1(Symm(n, R))∩L2(Symm(n, R)). In particular, we use as canonical +inner product on Symm(n, R) the one induced by the trace. We recall that, with +respect to such inner product, the cone Ωn is self-dual in the sense that +Ωn = {ξ ∈ Symm(n, R) | tr(ξX) > 0 for all X ∈ Ωn \ {0}}. +We will use this fundamental property (see [19]) to apply some well known formulas +associated to symmetric cones. +The next two results allow to describe the Bergman spaces after applying the +unitary map U. +Lemma 6.4. Let Hλ(Sn) = U(A2 +λ(Sn)) be the image of the Bergman space +A2 +λ(Sn) under the unitary map U. Then, the operator given by +Sλ : L2(Ωn) −→ L2(Symm(n, R)) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ) +(Sλ(f))(X, Y ) = (2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 χΩn(X)f(X) det(X) +λ +2 − n+1 +4 e−tr(XY ), +is an isometry onto Hλ(Sn). +Proof. From the basic properties of the Fourier transform, the Cauchy-Riemann +equations on Sn are transformed under U to the equations +� +Xjk + +∂ +∂Yjk +� +ϕ = 0, +which must hold for every 1 ≤ j ≤ k ≤ n. The general solution of these equations +is ϕ(X, Y ) = ψ(X)e−tr(XY ). Next, we need to consider the L2-integrability of these +solutions, and for this we evaluate +� +Sn +|ϕ(X, Y )|2Cλ,n det(2Y )λ−n−1 dX dY = += Cλ,n +� +Sn +|ψ(X)|2e−2tr(XY ) det(2Y )λ−n−1 dX dY += Cλ,n +� +Symm(n,R) +|ψ(X)|2 +� � +Ωn +e−2tr(XY ) det(2Y )λ−n−1 dY +� +dX. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +29 +For this to be finite it is necessary that supp(ψ) ⊂ Ωn. On the other hand, [19, +Equation 2.4.30] implies that (after some simple changes of variable) we have +� +Ωn +e−2tr(XY ) det(2Y )λ−n−1 dY = ΓΩn +� +λ − n+1 +2 +� +2 +n(n+1) +2 +det(X) +n+1 +2 +−λ +Hence, in the above solution of the Cauchy-Riemann equations we replace ψ(X) by +the function +ψ(X) = (2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 χΩn(X)f(X) det(X) +λ +2 − n+1 +4 +for a suitable function f. With these choices and the previous computations we +obtain +∥ϕ∥2 +Hλ(Sn) = Cλ,n +� +Ωn +(2π) +n(n+1) +2 +ΓΩn(λ) +|f(X)|2 det(X)λ− n+1 +2 × +× ΓΩn +� +λ − n+1 +2 +� +2 +n(n+1) +2 +det(X) +n+1 +2 +−λ dX += Cλ,n +π +n(n+1) +2 +ΓΩn +� +λ − n+1 +2 +� +ΓΩn(λ) +∥f∥2 +L2(Ωn) = ∥f∥2 +L2(Ωn), +where we have used the definition of the constant Cλ,n. The last set of identities +completes the proof by the definition of Sλ. +□ +Lemma 6.5. The adjoint operator of Sλ from Lemma 6.4 is a partial isometry +with initial space Hλ(Sn) and final space L2(Ωn). Furthermore, we have +(S∗ +λ(ϕ))(X) = +2 +n(n+1) +4 +ΓΩn(λ) +1 +2 +π +n(n+1) +4 +ΓΩn +� +λ − n+1 +2 +� det(X) +λ +2 − n+1 +4 × +× +� +Ωn +ϕ(X, Y )e−tr(XY ) det(2Y )λ−n−1 dY, +for every ϕ ∈ L2(Symm(n, R)) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ). +Proof. The first claim follows from Lemma 6.4. On the other hand, the expression +for S∗ +λ is a consequence of the following straightforward computation for f and ϕ +in the corresponding spaces +⟨Sλ(f),ϕ⟩ = += Cλ,n +� +Sn +(Sλ(f))(X, Y )ϕ(X, Y ) det(2Y )λ−n−1 dX dY += Cλ,n +(2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 +� +Ωn +f(X)× +× det(X) +λ +2 − n+1 +4 +� � +Ωn +ϕ(X, Y )e−tr(XY ) det(2Y )λ−n−1 +� +dX, +where we have used again the value of Cλ,n. +□ +The next result provides a formula for the Bergman projection after applying +the unitary map U. + +30 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +Lemma 6.6. Let Bλ = UBSn,λU ∗ be the orthogonal projection +L2(Symm(n, R)) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ) −→ Hλ(Sn). +Then, we have the identities +S∗ +λSλ = IL2(Ωn), +SλS∗ +λ = Bλ. +In particular, the orthogonal projection Bλ is given by +(Bλ(ϕ))(X, Y ) = +2 +n(n+1) +2 +ΓΩn +� +λ − n+1 +2 +�χΩn(X) det(X)λ− n+1 +2 e−tr(XY )× +× +� +Ωn +e−tr(Xη) det(2η)λ−n−1ϕ(X, η) dη. +Proof. By Lemma 6.4, the operator Sλ is a partial isometry with initial space +L2(Ωn) and final space Hλ(Sn). This implies the first two identities in the state- +ment. Hence, it remains to compute SλS∗ +λ explicitly, and this is done as follows for +every ϕ ∈ Hλ(Sn) +((SλS∗ +λ)(ϕ))(X, Y ) = += (2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 χΩn(X)(S∗ +λ(ϕ))(X) det(X) +λ +2 − n+1 +4 e−tr(XY ) += (2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 χΩn(X) +� +2 +n(n+1) +4 +ΓΩn(λ) +1 +2 +π +n(n+1) +4 +ΓΩn +� +λ − n+1 +2 +� det(X) +λ +2 − n+1 +4 × +× +� +Ωn +ϕ(X, η)e−tr(Xη) det(2η)λ−n−1 dη +� +, +which clearly simplifies to the required expression. +□ +The constructions considered so far allow us to introduce in the next result a +Fourier-Laplace transform from A2 +λ(Sn) onto L2(Ωn). We refer to [19, Proposi- +tion 2.4.26] for a similar related construction. +Theorem 6.7. With the current notation and for every λ > n, the operator Rλ = +S∗ +λU : L2 +λ(Sn, �vλ) → L2(Ωn) is a partial isometry with initial space A2 +λ(Sn) and +final space L2(Ωn). In particular, its adjoint +R∗ +λ : L2(Ωn) −→ L2(Sn, �vλ) +is an isometry onto A2 +λ(Sn). Furthermore, we have +(R∗ +λ(f))(Z) = +1 +ΓΛ(λ) +1 +2 +� +Ωn +f(ξ) det(ξ) +λ +2 − n+1 +4 eitr(ξZ) dξ, +for every f ∈ L2(Ωn) and Z ∈ Sn. +Proof. Since U is a unitary operator mapping A2 +λ(Sn) onto Hλ(Sn), it follows from +Lemma 6.5 that Rλ is a partial isometry with the indicated initial and final spaces. +From this we now conclude that R∗ +λ is an isometry from L2(Ωn) onto A2 +λ(Sn). + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +31 +It only remains to find the expression stated for R∗ +λ, which is achieved in the +following computation. For every f ∈ L2(Ωn) and Z ∈ Sn we have +(R∗ +λ(f))(Z) = ((U ∗Sλ)(f))(Z) = ((F−1 ⊗ I) ◦ Sλ(f))(Z) += (F−1 ⊗ I) +�(2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 χΩn(X)f(X) det(X) +λ +2 − n+1 +4 e−tr(XY ) +� += +1 +ΓΩn(λ) +1 +2 +� +Ωn +f(ξ) det(ξ) +λ +2 − n+1 +4 e−tr(ξY )eitr(Xξ) dξ += +1 +ΓΩn(λ) +1 +2 +� +Ωn +f(ξ) det(ξ) +λ +2 − n+1 +4 eitr(ξZ)dξ, +where Z = X + iY , with X, Y real matrices. +□ +We recall from Proposition 5.7 that +L∞(Sn)µSymm(n,R) = L∞(Sn)Symm(n,R). +In other words, the Symm(n, R)-invariant symbols and the moment map symbols +for the Symm(n, R)-action on Sn are the same. Hence, the next result provides +integral formulas that simultaneously diagonalizes Toeplitz operators with either +type of symbols. +Theorem 6.8. Let a ∈ L∞(Sn) be a Symm(n, R)-invariant symbol and λ > n +be given. +Then, for Rλ the operator from Theorem 6.7 we have a commutative +diagram +A2 +λ(Sn) +T (λ) +a +� +Rλ +� L2(Ωn) +Mγa,λ +� +A2 +λ(Sn) +Rλ +� L2(Ωn) +where γa,λ ∈ L∞(Ωn) is given by +γa,λ(X) = += 2 +n(n+1) +2 +det(X)λ− n+1 +2 +ΓΩn +� +λ − n+1 +2 +� +� +Ωn +a(Y )e−2tr(XY ) det(2Y )λ−n−1 dY, +for every X ∈ Ωn. +Proof. For our given Symm(n, R)-invariant symbol a ∈ L∞(Sn) we have T (λ) +a += +BSn,λ ◦ Ma. Then, Theorem 6.7 implies that +RλT (λ) +a +R∗ +λ = RλBSn,λMaBSn,λR∗ +λ += Rλ(R∗ +λRλ)Ma(R∗ +λRλ)R∗ +λ += RλMaR∗ +λ = S∗ +λUMaU ∗Sλ += S∗ +λMaSλ, +where we have used that UMaU ∗ = Ma, since U = F ⊗ I and a depends only on +Y = Im(Z). + +32 +CUEVAS-ESTRADA AND QUIROGA-BARRANCO +We now evaluate the last composition as follows for every f ∈ L2(Ωn) and +X ∈ Ωn +(S∗ +λMaSλ(f))(X) = += +2 +n(n+1) +4 +ΓΩn(λ) +1 +2 +π +n(n+1) +4 +ΓΩn +� +λ − n+1 +2 +� det(X) +λ +2 − n+1 +4 × +� +Ωn +� +a(Y )(2π) +n(n+1) +4 +ΓΩn(λ) +1 +2 f(X) det(X) +λ +2 − n+1 +4 e−tr(XY ) +� +× +e−tr(XY ) det(2Y )λ−n−1 dY += 2 +n(n+1) +2 +det(X)λ− n+1 +2 +ΓΩn +� +λ − n+1 +2 +� +f(X) +� +Ωn +a(Y )e−2tr(XY ) det(2Y )λ−n−1 dY, +which yields the required conclusion. +□ +Remark 6.9. Theorem 6.8 can be seen as a generalization of some of the results +found in [20]. More precisely, [20, Theorem 4.1] provides the diagonalization of +Toeplitz operators with the so-called cone component symbols, and such results +holds for every tubular domain. However, [20] considers only the weightless case. +On the other hand, we have considered only the tubular domain Sn, but our result +holds for arbitrarily weighted Bergman spaces and Toeplitz operators with symbols +that depend only on the cone coordinates. +Acknowledgment. This research was supported by a Conacyt scholarship, SNI- +Conacyt and Conacyt grants 280732 and 61517. +References +[1] Dawson, Matthew; ´Olafsson, Gestur and Quiroga-Barranco, Raul: +Commuting Toeplitz +operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic +discrete series. J. Funct. Anal. 268 (2015), no. 7, 1711–1732. +[2] Dawson, Matthew and Quiroga-Barranco, Raul: Radial Toeplitz operators on the weighted +Bergman spaces of Cartan domains. Representation theory and harmonic analysis on sym- +metric spaces, 97–114, Contemp. Math., 714, Amer. Math. Soc., Providence, RI, 2018. +[3] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Toeplitz operators on the unit ball in Cn +with radial symbols. J. Operator Theory 49 (2003), no. 2, 325–346. +[4] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Dynamics of properties of Toeplitz oper- +ators on the upper half-plane: hyperbolic case. Bol. Soc. Mat. Mexicana (3) 10 (2004), no. +1, 119–138. +[5] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Dynamics of properties of Toeplitz oper- +ators on the upper half-plane: parabolic case. J. Operator Theory 52 (2004), no. 1, 185–214. +[6] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Dynamics of properties of Toeplitz oper- +ators with radial symbols. Integral Equations Operator Theory 50 (2004), no. 2, 217–253. +[7] Grudsky, S., Quiroga-Barranco, R. and Vasilevski N.: Commutative C∗-algebras of Toeplitz +operators and quantization on the unit disk. J. Funct. Anal. 234 (2006), no. 1, 1–44. +[8] Johnson, Kenneth D.: On a ring of invariant polynomials on a Hermitian symmetric space. +J. Algebra 67 (1980), no. 1, 72–81. +[9] Helgason, Sigurdur: Differential geometry, Lie groups, and symmetric spaces. Corrected +reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical +Society, Providence, RI, 2001. +[10] Hua, L. K.: Harmonic analysis of functions of several complex variables in the classical +domains. Translated from the Russian by Leo Ebner and Adam Kor´anyi American Mathe- +matical Society, Providence, R.I. 1963. + +TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III +33 +[11] Korenblum, Boris and Zhu, Ke He: An application of Tauberian theorems to Toeplitz oper- +ators. J. Operator Theory 33 (1995), no. 2, 353–361. +[12] McDuff, Dusa and Salamon, Dietmar: Introduction to symplectic topology. Third edition. +Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2017. +[13] Mok, Ngaiming: Metric rigidity theorems on Hermitian locally symmetric manifolds. Series +in Pure Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. +[14] Quiroga-Barranco, Raul and Sanchez-Nungaray, Armando: Moment maps of Abelian groups +and commuting Toeplitz operators acting on the unit ball, Journal of Functional Analysis +281 (2021), no. 3, article 109039. +[15] Quiroga-Barranco, Raul and Seng, Monyrattanak: Commuting Toeplitz operators on Cartan +domains of type IV and moment maps. Complex Anal. Oper. Theory 16 (2022), no. 7, Paper +No. 102, 41 pp. +[16] Quiroga-Barranco, Raul and Vasilevski, Nikolai: Commutative C∗-algebras of Toeplitz oper- +ators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz +operators. Integral Equations Operator Theory 59 (2007), no. 3, 379–419. +[17] Quiroga-Barranco, Raul and Vasilevski, Nikolai: Commutative C∗-algebras of Toeplitz oper- +ators on the unit ball. II. Geometry of the level sets of symbols. Integral Equations Operator +Theory 60 (2008), no. 1, 89–132. +[18] Range, R. Michael: Holomorphic functions and integral representations in several complex +variables. Graduate Texts in Mathematics, 108. Springer-Verlag, New York, 1986. +[19] Upmeier, Harald: Toeplitz operators and index theory in several complex variables. Operator +Theory: Advances and Applications, 81. Birkh¨auser Verlag, Basel, 1996. +[20] Vasilevski, N. L.: Bergman space on tube domains and commuting Toeplitz operators. Pro- +ceedings of the Second ISAAC Congress, Vol. 2 (Fukuoka, 1999), 1523–1537, Int. Soc. Anal. +Appl. Comput., 8, Kluwer Acad. Publ., Dordrecht, 2000. +Centro de Investigaci´on en Matem´aticas, Guanajuato, Guanajuato, M´exico +Email address: david.cuevas@cimat.mx +Centro de Investigaci´on en Matem´aticas, Guanajuato, Guanajuato, M´exico +Email address: quiroga@cimat.mx + diff --git a/DNFKT4oBgHgl3EQfYy5L/content/tmp_files/load_file.txt b/DNFKT4oBgHgl3EQfYy5L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f9c22ff2f290b5804edc951257cd6566ac6e1ec --- /dev/null +++ b/DNFKT4oBgHgl3EQfYy5L/content/tmp_files/load_file.txt @@ -0,0 +1,972 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf,len=971 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11800v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='CV] 25 Jan 2023 COMMUTING TOEPLITZ OPERATORS AND MOMENT MAPS ON CARTAN DOMAINS OF TYPE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' DAVID CUEVAS-ESTRADA AND RAUL QUIROGA-BARRANCO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let DIII n and Sn be the Cartan domains of type III that con- sist of the symmetric n × n complex matrices Z that satisfy ZZ < In and Im(Z) > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For these domains, we study weighted Bergman spaces and Toeplitz operators acting on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We consider the Abelian groups T, R+ and Symm(n, R) (symmetric n × n real matrices), and their actions on the Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We call the corresponding actions Abelian Elliptic, Abelian Hyperbolic and Parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The moment maps of these three actions are computed and functions of them (moment map symbols) are used to construct commutative C∗-algebras generated by Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This leads to a natural generalization of known results for the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We also compute spectral integral formulas for the Toeplitz operators corresponding to the Abelian Elliptic and Parabolic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Introduction Bounded symmetric domains, weighted Bergman spaces on such domains and Toeplitz operators acting on Bergman spaces constitute three fundamental objects in operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The reason is that they are specific enough to make explicit computations that lead to interesting results, and at the same time they are com- plicated enough so that such results are non-trivial and enlightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For some years now, operator theory analysts have found plenty of examples of commutative C∗-algebras generated by Toeplitz operators when the corresponding set of symbols is suitably restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The first such example was considered in [11], where it was proved that Toeplitz operators on the unit disk D with radial symbols are diagonal with respect to the orthogonal monomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Clearly, a symbol on D is radial if it is invariant under the natural T-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that the T-action on the unit disk D realizes, up to conjugacy, all the elliptic M¨obius transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The introduction in [11] of Toeplitz operators with radial symbols was followed by a series of developments found in [3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These references considered all three fundamental types of M¨obius transformations on the unit disk D: elliptic, hyperbolic and parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It was proved that symbols that are invariant under the corresponding groups of M¨obius transformations yield Toeplitz operators that generate commutative C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, it was found in [7] that, under suitable smoothness conditions, these constructions yield the only commutative C∗-algebras generated by Toeplitz operators acting on every weighted Bergman space on the unit disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Primary 47B35 30H20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Secondary 53D20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Toeplitz operators, Bergman spaces, Cartan domains, Lie groups, K¨ahler manifolds, moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 1 2 CUEVAS-ESTRADA AND QUIROGA-BARRANCO The next step was to study the behavior in the case of higher dimensional bounded symmetric domains, and the unit ball Bn in Cn was the first natural example to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It was found in [16, 17] that there exists exactly, up to con- jugacy, n + 2 maximal Abelian subgroups (MASGs) of biholomorphisms each one of which yields invariant symbols whose Toeplitz operators generate commutative C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is a natural generalization of the situation observed for the unit disk D, since in this case we have n = 1 from which it follows the existence of three MASGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Nevertheless, some simplicity is lost because the number of MASGs grows with the dimension of the unit ball Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' After these works, many other results have been found where a suitable symmetry of the symbols yields commuting Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Such symmetry is in most cases a consequence of the invariance with respect to a certain biholomorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This has been observed for every bounded symmetric domain on every weighted Bergman space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We refer to [1] for a very general collection of related results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In a parallel line of development, symplectic geometry has been found to play an special role in the construction of symbols whose Toeplitz operators generate commutative C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It was proved in [14] that, for the unit ball Bn and on any of its weighted Bergman spaces, every single Abelian connected group of bi- holomorphisms provides symbols with mutually commuting Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For such a group H acting on Bn this is achieved by considering the so-called moment map symbols for H instead of H-invariant symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We refer to Section 3 for the details of the definitions and properties involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' However, we mention here that the moment map of an action is a mapping defined on the corresponding bounded symmetric domain using its symplectic manifold structure, and the moment map symbols are functions of such moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Another example of the use of moment map symbols is given by the results found in [15], where the bounded symmetric domain considered is the Cartan domain of type IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The goal of this work is to apply these ideas to study Toeplitz operators with moment map symbols acting on the weighted Bergman spaces of Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We recall that such domains are realized by the so-called generalized unit disk DIII n and Siegel’s generalized upper half-plane Sn (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In fact, as we show in Section 4, to these domains we can associate three biholomorphic actions that naturally generalize the three actions described above for the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, we call these actions on either DIII n or Sn the Elliptic, Hyperbolic and Parabolic Actions (see subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These come from the groups U(n), GL(n, R) and Symm(n, R), respectively, of which only the last one is Abelian for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, we introduce actions that we call Abelian Elliptic and Abelian Hyperbolic (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As noted in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 all three Abelian actions can be seen as coming from the corresponding centers of the original groups involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We present in Section 2 all the theory needed to understand the Riemannian and symplectic geometry background used in the rest of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, we compute in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2 the Bergman metric and the K¨ahler form for both DIII n and Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We use this to compute in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2 the moment maps for our three distinguished actions: Abelian Elliptic, Abelian Hyperbolic and Parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We introduce in Section 5 Toeplitz operators with special symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' First, we consider invariant symbols in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 and we recall some known commutative C∗-algebras generated by Toeplitz operators for our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Second, we introduce TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 3 moment map symbols in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3, and with the use of our moment map computations we obtain explicit formulas for moment map symbols for our three distinguished Abelian actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We obtain the following general description (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4 for the precise statements) Abelian Elliptic symbols: Z �−→ f � tr(ZZ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Abelian Hyperbolic symbols: Z �−→ f � tr(Im(Z)−1Re(Z)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Parabolic symbols: Z �−→ f(Im(Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' From a quick comparison with the notions considered in the current literature, we observe that these three types of symbols are natural, almost canonical, generaliza- tions from the unit disk D to the domains DIII n and Sn of the symbols obtained from the elliptic, hyperbolic and parabolic actions on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We prove in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 that the three types of symbols above yield Toeplitz op- erators that generate commutative C∗-algebras on every weighted Bergman space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Our method of proof is based on the fact that these moment map symbols have an additional invariance: they are invariant under the group from which the Abelian group is the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This allows to use the results from subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand, it is interesting to observe the importance of having only three types of symbols in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 as a generalization of the corresponding result for the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is explained in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Finally, we obtain in Section 6 integral formulas for the Toeplitz operators with moment map symbols that provide simultaneous diagonalization for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is done for the Abelian Elliptic and Parabolic Actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' we leave the Abelian Hyper- bolic case as an important project to develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The relevant results are Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The simplicity of the formulas presented in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 highlights the importance of using symplectic geometry to solve these operator theory problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Likewise, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 has very natural formulas that involve a Fourier-Laplace transform obtained in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Cartan domains of type III and their analysis We recall the basic geometric and analytic properties of the Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Bounded and unbounded realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In the rest of this work, and for F either R or C, we will denote by Mat(n, F) the space of n × n matrices over F and by Symm(n, F) its subspace of symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As usual, GL(n, F) will denote the Lie group of invertible elements of Mat(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The n-dimensional Cartan domain of type III is the complex domain given by DIII n = {Z ∈ Symm(n, C) | In − ZZ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The domain DIII n is clearly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand, there is a natural unbounded domain associated to DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The n-dimensional Siegel domain is the complex domain given by Sn = {Z ∈ Symm(n, C) | Im(Z) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that DIII 1 and S1 are precisely the unit disk D and the upper half-plane H, respectively, in the complex plane C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For this reason, the domains DIII n and Sn are also known as the generalized unit disk and generalized upper-half plane, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Furthermore, these domains are related in a way similar to the well known 1-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the next result we refer to [9, Exercise C, Chapter VIII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4 CUEVAS-ESTRADA AND QUIROGA-BARRANCO Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The map ϕ : Sn → DIII n given by Z �→ (In + iZ)(In − iZ)−1, is a biholomorphism from Sn onto DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Because of the previous result, the domain Sn is also known as the unbounded realization of the n-dimensional Cartan domain of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Biholomorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this section we describe the groups of biholo- morphisms of the domains DIII n and Sn introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We start by considering the matrices In,n = � In 0 0 −In � , Jn = � 0 −In In 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These naturally yield the next Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Sp(n, C) = {M ∈ Mat(2n, C) | M ⊤JnM = Jn}, Sp(n, R) = {M ∈ Mat(2n, R) | M ⊤JnM = Jn}, U(n, n) = {M ∈ Mat(2n, C) | M ∗In,nM = In,n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We recall the notion of a bounded symmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' A domain D ⊂ CN is called symmetric if for every z ∈ D there exists a biholomorphism ϕz : D → D such that ϕz(w) = w if and only if w = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If D is also bounded, then D is called a bounded symmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If D satisfies tD = D, for every t ∈ T, then the domain D is called circled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Through suitable actions of the groups introduced above, one can prove that the domains DIII n and Sn are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the next result we refer to [13, Paragraph (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3)] (see also [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' From now on, for any given matrix M ∈ Mat(2n, C) a decomposition of the form M = �A B C D � , will always be taken so that A, B, C, D have size n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The action via generalized M¨obius transformations given by Sp(n, C) ∩ U(n, n) × DIII n −→ DIII n �A B C D � Z �−→ (AZ + B)(CZ + D)−1, realizes the biholomorphism group of DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Furthermore, DIII n is a circled bounded symmetric domain and it is given as the quotient DIII n ≃ Sp(n, C) ∩ U(n, n)/U(n), where U(n) embedded in Sp(n, C) ∩ U(n, n) by A �−→ � A 0 0 A � corresponds to the group of biholomorphisms of DIII n that fix the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Similarly, we have the next description of the biholomorphism group of the do- main Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now refer to [9, Exercise C, Chapter VIII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 5 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The action via generalized M¨obius transformations given by Sp(n, R) × Sn −→ Sn � A B C D � Z �−→ (AZ + B)(CZ + D)−1, realizes the biholomorphism group of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Furthermore, Sn is a symmetric domain and it is given as the quotient Sn = Sp(n, R)/U(n), where U(n) embedded in Sp(n, R) by A �−→ � Re(A) Im(A) −Im(A) Re(A) � corresponds to the group of biholomorphisms of Sn that fix the matrix iIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 it follows that the biholomorphism groups of DIII n and Sn are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In fact, it is easy to prove that Sp(n, C) ∩ U(n, n) and Sp(n, R) are conjugated (see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Bergman spaces and Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' From now on, D will denote either of the domains DIII n or Sn, and dZ the Lebesgue measure on Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' A number of invariants can be associated to any symmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The simplest one is the dimension, which for D is n(n + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For other invariants we refer to [19] for further details on their definitions and here we simply state their known values with some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The rank is defined as the dimension of maximal linearly embedded poly- disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For D the rank is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The multiplicities are defined as the main invariants that describe the Jor- dan triple system associated to the symmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For D the multi- plicities are a = 1, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The vanishing of the latter implies that DIII n has a tubular realization which is in fact given by Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For this we observe that Sn = Symm(n, R) ⊕ iPos(n, R), where Pos(n, R) denotes the cone of positive definite n × n real matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In the rest of this work we will denote Ωn = Pos(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For a tubular domain, the genus is given as p = 2d/r, where d and r are the dimension and the rank of the domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, for D we have p = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will make use of the multi-gamma function (see [19, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2]) that we will consider only for Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Such function is associated to the cone part of a tubular realization of a tube type symmetric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In our case, it is defined by ΓΩn(λ) = (2π) n(n−1) 4 n � j=1 Γ � λ − j − 1 2 � , for every λ > (n−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is well known (see [10, 19]) that the volume of a bounded symmetric domain can be expressed in terms of the multi-gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this 6 CUEVAS-ESTRADA AND QUIROGA-BARRANCO case we have (see [10]) Vol(DIII n ) = π n(n+1) 2 ΓΩn � n+1 2 � ΓΩn(n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, we consider the normalized measure on Symm(n, C) dv(Z) = ΓΩn(n + 1) π n(n+1) 2 ΓΩn � n+1 2 � dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, dv(Z) is a probability measure on DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The (weightless) Bergman space A2(D) is the subspace of L2(D, v) that consists of holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In other words, we have A2(D) = {f ∈ L2(D, v) | f is holomorphic }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is a well known fact that A2(D) is a closed subspace of L2(D, v) (see [9, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will denote by BD : L2(D, v) → A2(D) the corresponding orthogonal projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is called the (weightless) Bergman projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moreover, it is also well known that A2(D) is a reproducing kernel Hilbert space (see [9, Chapter VIII]) in the sense that the evaluation map evZ : A2(D) −→ C f �−→ f(Z), is continuous for every Z ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This implies the existence of a unique smooth function KD : D × D → C, holomorphic in the first variable and anti-holomorphic in the second variable, satisfying KD(Z, ·) ∈ A2(D) for every Z ∈ D and for which the Bergman projection is given by BD(f)(Z) = � D f(W)KD(Z, W) dv(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' for every f ∈ L2(D, v) and Z ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The function KD is called the (weightless) Bergman kernel of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Bergman kernels of symmetric domains have closed known expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, it follows from Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='17 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='15 in [19] that the Bergman kernels of DIII n and Sn are given by the expressions KDIII n (Z, W) = det(In − ZW)−(n+1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1) KSn(Z, W) = det(−i(Z − W))−(n+1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that a linear biholomorphism has to be applied in order to obtain the above expression for KSn from the one found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' More precisely, our unbounded realization of DIII n is obtained from the one considered in [19] through the map Z �→ −iZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The next standard construction is to use powers of the Bergman kernel to obtain weighted measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The following formula, which holds for every λ > n, is useful to normalize such weighted measures (see [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='18]) � DIII n det(In − ZZ)λ−n−1 dZ = π n(n+1) 2 ΓΩn � λ − n+1 2 � ΓΩn(λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 7 Hence, we consider for every λ > n the measure dvλ(Z) = ΓΩn (λ) π n(n+1) 2 ΓΩn � λ − n+1 2 � det(In − ZZ)λ−n−1 dZ which is a probability measure on DIII n , and we also consider the normalized mea- sure d�vλ(Z) = ΓΩn (λ) π n(n+1) 2 ΓΩn � λ − n+1 2 � det(−i(Z − Z))λ−n−1 dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' on the domain Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For λ > n, the weighted Bergman spaces on DIII n and Sn with weight λ are given by A2 λ(DIII n ) = {f ∈ L2(DIII n , vλ) | f is holomorphic }, A2 λ(Sn) = {f ∈ L2(Sn, �vλ) | f is holomorphic }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will denote by A2 λ(D) the corresponding weighted Bergman space when D is DIII n or Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Note that for λ = n+1, we obtain A2 λ(DIII n ) = A2(DIII n ) and A2 λ(Sn) = A2(Sn), which are the weightless Bergman spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As before, it is well known that every weighted Bergman space is closed in the corresponding L2 space in such a way that it is a reproducing kernel Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, for D either DIII n or Sn there exists a smooth function KD,λ : D × D → C, holomorphic and anti-holomorphic in the first and second variable (respectively), such that the orthogonal projection onto A2 λ(D) is given by BD,λ(f)(Z) = � D f(W)KD,λ(Z, W) dνλ(W), for every f ∈ L2(D, νλ) and Z ∈ D, where νλ denotes either vλ or �vλ according to whether D is DIII n or Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This projection is called the weighted Bergman projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It follows by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='22 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='24 from [19] that the weighted Bergman kernels for these domains are given by the following expressions KDIII n ,λ(Z, W) = det(In − ZW)−λ, KSn,λ(Z, W) = det(−i(Z − W))−λ, for every λ > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, we have KD,λ(Z, W) = KD(Z, W) λ n+1 for every Z, W ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The previous constructions allow us to define our main object of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every weight λ > n and a ∈ L∞(D), the Toeplitz operator with symbol a is the bounded operator T (λ) a acting on A2 λ(D) that is given by T (λ) a = BD,λ ◦ Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is interesting to note that the Bergman spaces A2 λ(DIII n ) and A2 λ(Sn) are uni- tarily equivalent, thus simplifying some computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This unitary equivalence is stated without proof in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Its proof is a straightforward generalization of the arguments provided to obtain Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9 in Chapter IV from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 8 CUEVAS-ESTRADA AND QUIROGA-BARRANCO Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The map ϕ given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 induces the unitary operator given by Uϕ : A2 λ(DIII n ) −→ A2 λ(Sn) f �−→ JC(ϕ) λ n+1 f ◦ ϕ, where JC(ϕ) = det(dϕC) denotes the complex Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Geometry of Cartan domains of type III 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Symplectic and K¨ahler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We discuss here some basic material from symplectic geometry, which will be essential for the main results of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' A symplectic manifold is a pair (M, ω), where M is a smooth manifold and ω is a closed 2-form which yields a non-degenerate bilinear form at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Some of the most important examples of symplectic manifolds come from com- plex differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We recall that a manifold M is complex if their charts map onto open sets of complex vector spaces so that the changes of coordinates are holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For such a manifold M, this yields a complex structure Jz on every tangent space TzM, for every z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In turn, this provides a tensor field J known as the complex structure tensor of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, we have J2 = −I the identity tensor acting on the fibers of the tangent bundle T M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We refer to [13] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The next definition describes well behaved Riemannian metrics with respect to these constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let M be a complex manifold with complex structure tensor J and a given Riemannian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We say that M is a Hermitian manifold if it satisfies gz(Jzu, Jzv) = gz(u, v) for every z ∈ M and u, v ∈ TzM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now proceed to relate Hermitian manifolds to symplectic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will explain the main constructions and refer to [13] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let us start by considering a complex manifold M with complex structure tensor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the tangent bundle can be complexified to a complex tangent bundle denoted by T CM, and the action of J on T M can also be complexified to obtain a tensor JC acting on T CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Such complexifications are performed fiberwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Since (JC z )2 = −I, for every z ∈ M, if we define the spaces T 1,0 z M = {v ∈ T C z M | JC z v = iv}, T 0,1 z M = {v ∈ T C z M | JC z v = −iv}, then we have T C z M = T 1,0 z M ⊕ T 0,1 z M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These spaces are known as the subspaces of holomorphic and anti-holomorphic tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' , zn) is a holomorphic chart with real components obtained from the decomposition zj = xj + iyj, then the usual Wirtinger differential operators are given by ∂ ∂zj = 1 2 � ∂ ∂xj − i ∂ ∂yj � , ∂ ∂zj = 1 2 � ∂ ∂xj + i ∂ ∂yj � , TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 9 for every j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The first set of operators define at every point in the domain of the chart a basis for the corresponding fibers of T 1,0M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Similarly, the second set of operators define a basis for the fibers of T 0,1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The corresponding dual basis are given by dzj = dxj + i dyj, dzj = dxj − i dyj, where j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let us now consider a Riemannian metric g on M for which M is a Hermitian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We can complexify g to a complex bilinear tensor gC defined on the complexified tangent bundle T CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This yields a positive definite Hermitian form T 1,0 z M × T 1,0 z M −→ C, (u, v) �−→ gC z (u, v), for every z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In local coordinates, this can be written as n � j,k=1 gjk(z)dzj ⊗ dzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For this reason, we will denote this field of complex Hermitian forms with the same symbol g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' To more easily distinguish between the two of them, we will refer to the original g as the Riemannian metric of M and we will call the previous field of Hermitian forms the Hermitian metric of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The previous setup and constructions allow to introduce the next important geometric object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For a Hermitian manifold M with Hermitian metric g as con- structed above, the associated 2-form is given by ω = g(J(·), ·) = −2Im(g) where the first occurrence of g is the Riemannian metric and the second one is the corresponding Hermitian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Hermitian manifold M is called K¨ahler if its associated 2-form is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this case, ω is called the K¨ahler form of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is straightforward to check that the associated 2-form of any Hermitian mani- fold is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, every K¨ahler manifold is a symplectic manifold, and in this case the K¨ahler form is its symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' One can alternatively provide a K¨ahler structure on a complex manifold by introducing a field of Hermitian bilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is the content of the next result which is a particular case of Proposition 1 in page 18 from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let M be a complex manifold and let g be a tensor field of positive definite Hermitian bilinear forms on T 1,0M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Assume that for every holomorphic coordinate chart (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' , zn), in a family of charts covering M, there is some real valued function F such that g = n � j,k=1 ∂2F ∂zj∂zk dzj ⊗ dzk in the domain of the given chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the tensor 2Re(g) is a Riemannian metric that yields a K¨ahler structure on M whose Hermitian metric is given by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 10 CUEVAS-ESTRADA AND QUIROGA-BARRANCO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Bergman metric and its K¨ahler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now use the results pre- viously obtained to construct a K¨ahler structure on the Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The next fundamental theorem is a particular case of the discussion in the first part of Chapter 4 in [13] (see also [9, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Note that, from now on, we will use the canonical complex linear coordinates of Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let D be either of DIII n or Sn and let KD(Z, W) be the reproducing Bergman kernel of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the tensor given by � 1≤l≤m≤n 1≤j≤k≤n ∂2 log KD(Z, Z) ∂zlm∂zjk dzlm ⊗ dzjk, is a field of positive definite Hermitian forms that yields a structure of K¨ahler man- ifold on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Furthermore, both the corresponding Riemannian metric and associated K¨ahler form are invariant under the group of biholomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5 to introduce K¨ahler structures on DIII n and Sn by nor- malizing the tensor considered in its statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These normalization will simplify some formulas below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let D be either of DIII n or Sn and KD(Z, W) the Bergman kernel of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Bergman metric of D is the field of Hermitian forms given by gD = cD � 1≤l≤m≤n 1≤j≤k≤n ∂2 log KD(Z, Z) ∂zlm∂zjk dzlm ⊗ dzjk, where cDIII n = 1 n+1 and cSn = 4 n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The next two results are very well known properties of the Wirtinger differ- ential operators that will be useful in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We state them for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For any smooth function f : CN −→ C we have df = N � j=1 � ∂f ∂zj dzj + ∂f ∂zj dzj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 (Chain rule for Wirtinger derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let g : Cn → Cm and f : Cm → C be smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, we have ∂(f ◦ g) ∂zj = m � k=1 � ∂f ∂zk g ∂gk ∂zj + ∂f ∂zk g ∂gk ∂zj � , ∂(f ◦ g) ∂zj = m � k=1 � ∂f ∂zk g ∂gk ∂zj + ∂f ∂zk g ∂gk ∂zj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The following elementary computation will be used latter on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We provide its proof for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The differential of det : Mat(n, C) → C is given by d(det)A = tr � adj(A)dA � , for every A ∈ Mat(n, C), where adj(A) (adjugate of A) is the transpose of the cofactor matrix of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If A = (alm) ∈ Mat(n, C) and clm is the cofactor of alm, then the cofactor expansion of the determinant along the k-th column is given by det A = n � l=1 clkalk = n � l=1 � adj(A)T � lkalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It follows that ∂ det ∂ajk (A) = cjk = � adj(A)T � jk and we obtain the differential d(det)A = n � j,k=1 ∂ det ∂ajk (A) dajk = n � j,k=1 cjk dajk = n � j,k=1 � adj(A)T � jk dajk = n � j,k=1 � adj(A) � kj dajk = n � k=1 (adj(A) dA)kk = tr(adj(A) dA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ We now obtain explicit formulas for the Bergman metrics of the Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Note that we have provided coordinate free expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This will be useful for our computations in the rest of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Bergman metrics on DIII n and Sn are respectively given by gDIII n Z (U, V ) = tr � (In − ZZ)−1U(In − ZZ)−1V � , gSn Z (U, V ) = tr � Im(Z)−1UIm(Z)−1V � , for every U, V ∈ Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, the K¨ahler forms of DIII n and Sn are respectively given by ωDIII n Z (U, V ) = i tr � (In − ZZ)−1U(In − ZZ)−1V � − i tr � (In − ZZ)−1U(In − ZZ)−1V � , ωSn Z (U, V ) = 2 tr � Im(Z)−1Re(U)Im(Z)−1Im(V ) � − 2 tr � Im(Z)−1Im(U)Im(Z)−1Re(V ) � , for every U, V ∈ Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this proof we will consider the complex vector spaces Symm(n, C) and Mat(n, C) whose coordinates will be denoted in both cases by zjk, even though they have different meanings for such spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' However, from the context where these coordinates are used it will be easy to identify the actual meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We start by computing the Bergman metric on DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' First, we observe that we have the following partial derivative ∂(In − ZZ) ∂zjk (Z) = −Z ∂Z ∂zjk = −ZEjk, where Ejk is the n × n symmetric matrix that has 1 in the entries (j, k) and (k, j) and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Note that these matrices are the basis with respect to which we are considering the canonical coordinates in Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 12 CUEVAS-ESTRADA AND QUIROGA-BARRANCO Next, using the previous computation, applying Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8, Equa- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1) and using the fact that det is holomorphic, we obtain 1 n + 1 ∂ ∂zjk log KDIII n (Z, Z) = = 1 det(In − ZZ) n � l,m=1 ∂ det ∂zlm (In − ZZ)(ZEjk)lm = 1 det(In − ZZ) n � l,m=1 (adj(In − ZZ)T )lm(ZEjk)lm = 1 det(In − ZZ)tr � adj(In − ZZ)ZEjk � = tr � (In − ZZ)−1ZEjk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Now, we will use the easy to prove relations (In − ZZ)−1Z = Z(In − ZZ)−1, Z(In − ZZ)−1 = (In − ZZ)−1Z, which hold for every Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Using the identities obtained so far we compute 1 n + 1 ∂2 ∂zlm∂zjk log KDIII n (Z, Z) = ∂ ∂zlm tr � (In − ZZ)−1ZEjk � = tr � (In − ZZ)−1ElmZ(In − ZZ)−1ZEjk � + tr � (In − ZZ)−1ElmEjk � = tr � (In − ZZ)−1Elm(In − ZZ)−1ZZEjk � + tr � (In − ZZ)−1ElmEjk � = tr � (In − ZZ)−1Elm(In − ZZ)−1(ZZ + (In − ZZ))Ejk � = tr � (In − ZZ)−1Elm(In − ZZ)−1Ejk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This implies that the metric gDIII n Z satisfies the required identity on the basic el- ements of the vector space Symm(n, C), thus proving the result for the Bergman metric of DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The corresponding computation of the Bergman metric for Sn is obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' From Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 the K¨ahler form of DIII n is given by ωDIII n Z (U, V ) = −2Im � gDIII n Z (U, V ) � = i � gDIII n Z (U, V ) − gDIII n Z (U, V ) � , which yields the stated formula from our computation of the Bergman metric of DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Finally, for the K¨ahler form of Sn we compute ωSn Z (U, V ) = −2Im � gSn Z (U, V ) � = −2Im � tr � Im(Z)−1(Re(U) + iIm(U))Im(Z)−1(Re(V ) − iIm(V )) �� , from which the stated formula is easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Now we turn back our attention to symplectic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It will provide the main geometric tools and objects that we will apply to the study of Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We refer to [12] for the symplectic geometry facts stated without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In the rest of this subsection (M, ω) will denote a fixed symplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' A diffeomorphism ϕ : M → M is called a symplectomorphism if ϕ∗(ω) = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In other words, a symplectomorphism is a diffeomorphism preserving the symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If H is a Lie group with a smooth action on M, then we say that the H-action is symplectic if the map M −→ M z �−→ h · z is a symplectomorphism for every h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' There are two important types of vector fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' From now on, we will denote by X(M) the Lie algebra of vector fields over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' A field X ∈ X(M) is called a symplectic vector field if and only if the 1-form ω(X, ·) is closed, and it is called a Hamiltonian vector field if and only if the form ω(X, ·) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will denote by X(M, ω) the space of symplectic vector fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is a well known fact that X(M, ω) is a Lie subalgebra of X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For any smooth function f : M → R, the non-degeneracy of ω implies the existence of a unique element Xf ∈ X(M) such that df = ω(Xf, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this case, Xf is called the Hamiltonian vector field associated to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Symplectic vector fields can be characterized by symplectomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' More precisely, it is well known that an element X ∈ X(M) belongs to X(M, ω) if and only if the local flow generated by X acts by (locally defined) symplectomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' An important converse to the previous fact relates symplectic actions to sym- plectic vector fields as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let us consider a symplectic action of a Lie group H on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, for every X ∈ h (the Lie algebra of H), we define the induced vector field on M by X♯ z = d ds ��� s=0 exp(sX) · z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' for every z ∈ M, where exp : h → H is the exponential map of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the fact that the H-action is symplectic implies that X♯ ∈ X(M, ω) for every X ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In the previous discussion, we have shown two different constructions that map into the space X(M, ω) of symplectic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, a natural problem to consider is the existence of a map h → C∞(M) that makes the following diagram commute C∞(M) � h �① ① ① ① ① ① ① ① ① � X(M, ω) where the vertical arrow is the map f �→ Xf and the horizontal arrow is the map X �→ X♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The existence of such diagonal map yields the notion of a moment map for the H-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The precise definition requires some additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We recall that Ad = AdH : H → GL(h) denotes the adjoint representation of the Lie 14 CUEVAS-ESTRADA AND QUIROGA-BARRANCO group H, and that Ad∗ denotes the dual representation on h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, we have Ad∗(h) = Ad(h−1)⊤ for every h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let (M, ω) be a symplectic manifold and let H be a Lie group acting by symplectomorphisms on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' A moment map for the H-action is a smooth map µ : M → h∗, where h∗ is the vector space dual of h, that satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (1) For every X ∈ h consider the map µX : M → R given by µX(z) = ⟨µ(z), X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the Hamiltonian vector field associated to µX is X♯, for every X ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In other words, it holds dµX = ω(X#, ·), for every X ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (2) The map µ is H-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In other words, we have µ(h · z) = Ad∗(h)(µ(z)), for every z ∈ M and h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If H is an Abelian group, then its adjoint representation satisfies Ad(h) = Ih for every h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, in this case, condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11 reduces to µ(h · z) = µ(z), for every h ∈ H and z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In other words, this requires the smooth map to be H-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Three Abelian biholomorphism groups and their moment maps In this section we study three special types of subgroups of biholomorphisms acting on Cartan domains of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the corresponding Abelian groups, we compute the moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will see later on that these moment maps are a powerful tool to find commutative C∗-algebras generated by Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Elliptic, Hyperbolic, and Parabolic Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Cartan domains DIII n and their unbounded realizations Sn carry three interesting actions of subgroups of biholomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As we will see, these actions generalize the three different types of M¨obius transformations found for the unit disk D and the upper half plane H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5 provides the action U(n) × DIII n −→ DIII n U · Z = UZU ⊤, which yields the subgroup of biholomorphisms that fixes the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Up to con- jugacy, this characterizes the subgroups that fix some point in the domain DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is so because of the homogeneity of this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will call this the Elliptic Action on DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Next we observe that there is a canonical homomorphism of Lie groups given by GL(n, R) −→ Sp(n, R) A �−→ � A 0 0 (A−1)⊤ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 15 A straightforward computation shows that this assignment is indeed a homomor- phism whose image lies in Sp(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, this homomorphism and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6 provide the action GL(n, R) × Sn −→ Sn A · Z = AZA⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is easily seen that this action realizes the subgroup of biholomorphisms that fixes the origin, a boundary point of the domain Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For this reason, we will call this the Hyperbolic Action on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Finally, we have a canonical homomorphism of Lie groups given by Symm(n, R) −→ Sp(n, R) S �−→ �In S 0 In � , where Symm(n, R) is endowed with the Lie group structure with operation given by the sum of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Again, it is straightforward to show that this map is indeed a homomorphism into the group Sp(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now have that this homomorphism together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6 provide the action Symm(n, R) × Sn −→ Sn S · Z = Z + S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This action realizes the subgroup of biholomorphisms of the tube type domain Sn that correspond to translations on the real vector space part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Since this action clearly generalizes the translation action on the real part on the upper half-plane H, we will call this action on Sn the Parabolic Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In fact, all three actions introduced above generalize the behavior observed in the 1-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is stated in the following well known result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We recall that two biholomorphisms are conjugated if they are so under some other biholomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='This result justifies our choice of notation for the actions considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let us denote by D either D or H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If ϕ is a biholomorphism of D, then the following equivalences hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (1) The M¨obius transformation ϕ is elliptic if and only if it is conjugated to a transformation that belongs to the action T × D → D given by z �→ tz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (2) The M¨obius transformation ϕ is hyperbolic if and only if it is conjugated to a transformation that belongs to the action R+ × H → H given by z �→ rz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (3) The M¨obius transformation ϕ is parabolic if and only if it is conjugated to a transformation that belongs to the action R × H → H given by z �→ z + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that the Elliptic and Hyperbolic Actions are given by actions of Abelian groups if and only if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Nevertheless, the Parabolic Action is given by an Abelian Lie group in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For these reason, we introduce in the next definition actions of Abelian groups associated to the Elliptic and Hyperbolic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Abelian Elliptic Action on DIII n is defined by T × DIII n −→ DIII n t · Z = t2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 16 CUEVAS-ESTRADA AND QUIROGA-BARRANCO The Abelian Hyperbolic Action on Sn is defined by R+ × Sn −→ Sn r · Z = r2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that the Abelian Elliptic and Abelian Hyperbolic actions are obtained by considering the center of the groups corresponding to the non-Abelian actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' More precisely, we have the centers Z(U(n)) = TIn, Z(GL(n, R)) = R+In ∪ (−R+In), and the actions in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2 are the restriction of the previously defined actions to these center groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand, Symm(n, R) is already Abelian so that it coincides with its center, in other words we have Z(Symm(n, R)) = Symm(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, the most obvious definition of “Abelian Parabolic Action” would yield what we already have defined as the Parabolic Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We also observe that these three actions of Abelian groups of biholomorphisms, the Abelian Elliptic, Abelian Hyper- bolic and Parabolic, are natural generalizations of the actions described in Corol- lary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moment maps of the Abelian actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will now compute moment maps for all three Abelian actions introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We refer to Defini- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5 that every biholomorphism of either of the domains DIII n and Sn preserves the corresponding K¨ahler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, all the groups considered above act by symplectomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, the notion of moment map given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11 can be applied to such actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moment map of the Abelian Elliptic Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The group in this case is T acting on DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Lie algebra of this group is R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The latter is canonically isomorphic to its dual R∗, so we will compute a moment map as a function DIII n → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every element t ∈ R the corresponding induced vector field on DIII n is given by t♯ Z = d ds ��� s=0 exp(st) · Z = d ds ��� s=0 exp(2ist)Z = 2itZ, for every Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Note that we have used the fact that the (Lie group) exponen- tial map R → T satisfies t �→ exp(it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The function given by µT : DIII n −→ R µT(Z) = −2tr � (In − ZZ)−1� , is a moment map for the Abelian Elliptic Action on DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We start by computing ωDIII n Z (t♯ Z, ·) for every t ∈ R and Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For this first computation we use the above formula for t♯ and the expression for the K¨ahler TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 17 form of DIII n obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We have ωDIII n Z (t♯ Z, V ) = i tr � (In − ZZ)−12itZ(In − ZZ)−1V � − i tr � (In − ZZ)−12itZ(In − ZZ)−1V � , = − 2t tr � (In − ZZ)−1Z(In − ZZ)−1V � − 2t tr � (In − ZZ)−1Z(In − ZZ)−1V � , for every V ∈ Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' we consider the function µt : DIII n → R defined by µt(Z) = ⟨µT(Z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' t⟩ = tµT(Z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' and we compute its differential as follows d(µt)Z(V ) = = − 2t d ds ��� s=0 tr �� In − (Z + sV )(Z + sV ) �−1� = − 2t tr �� In − ZZ �−1� V Z + ZV �� In − ZZ �−1� = − 2t tr �� In − ZZ �−1V � In − ZZ �−1Z � − 2t tr � Z � In − ZZ �−1V � In − ZZ �−1� where we applied in the last identity the commutation relations between Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (In − ZZ)−1 and their conjugates used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We conclude that d(µt)Z(V ) = ωDIII n Z (t♯ Z, V ) for every V ∈ Symm(n, C) and Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It follows that the first condition in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11 is satisfied by the map in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It remains to prove the T-invariance of this map, but this is established through the identities µT(t · Z) = µT(t2Z) = −2 tr � (In − t2Zt2Z)−1� = −2 tr � (In − ZZ)−1� = µT(Z) that hold for every t ∈ T and Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the case n = 1, the Abelian Elliptic Action yields the T-action on the unit disk D given by t · z = t2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' With this assumption, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4 provides the moment map µT(z) = −2 1 1 − |z|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We observe that for actions of Abelian groups we can add to a given moment map an arbitrary, but fixed, constant to obtain another moment map (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, the map given by µ(z) = µT(z) + 2 = −2 |z|2 1 − |z|2 , is a moment map as well for our T-action on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This recovers, up to the multi- plicative constant 2, the moment map obtained in [14, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1] for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 18 CUEVAS-ESTRADA AND QUIROGA-BARRANCO This referenced result computes the moment map for the natural action of the n- dimensional torus on the n-dimensional unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that the factor 2 comes from the reparameterization involved in using the action t · z = t2z instead of the action t · z = tz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moment map of the Abelian Hyperbolic Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now have the group R+ acting on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Lie algebra of this group is R itself, which is canonically isomor- phic to its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, the moment map will be computed as a function Sn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every t ∈ R the induced vector field on Sn is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This computation uses the fact that the (Lie group) exponential map is given in this case by t �→ exp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' t♯ Z = d ds ��� s=0 exp(st) · Z = d ds ��� s=0 exp(2st)Z = 2tZ, for every Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The function given by µR+ : Sn −→ R µR+(Z) = −4tr � Im(Z)−1Re(Z) � is a moment map for the Abelian Hyperbolic Action on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We compute ωSn Z (t♯ Z, ·), for every t ∈ R and Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For this we use the previous computations and the expression of the K¨ahler form of Sn obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We have in this case ωSn Z (t♯ Z, V ) = 2 tr � Im(Z)−1Re(2tZ)Im(Z)−1Im(V ) � − 2 tr � Im(Z)−1Im(2tZ)Im(Z)−1Re(V ) � = 4t tr � Im(Z)−1Re(Z)Im(Z)−1Im(V ) � − 4t tr � Im(Z)−1Re(V ) � , for every V ∈ Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand, we consider the function µt : Sn → R given by µt(Z) = ⟨µR+(Z), t⟩ = tµR+(Z), for which we compute the differential as follows d(µt)Z(V ) = = −4t d ds ��� s=0 tr � Im(Z + sV )−1Re(Z + sV ) � = −4t d ds ��� s=0 tr �� Im(Z) + sIm(V ) �−1� Re(Z) + sRe(V ) �� = 4t tr � Im(Z)−1Im(V )Im(Z)−1Re(Z) � − 4t tr � Im(Z)−1Re(V ) � , for every V ∈ Symm(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' From this we conclude that d(µt)Z(V ) = ωSn Z (t♯ Z, V ), for every V ∈ Symm(n, C) and Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11 it remains to show that µR+ is R+-invariant, and this is verified in the next computation µR+(r · Z) = µR+(r2Z) = −4tr � Im(r2Z)−1Re(r2Z) � = −4tr � Im(Z)−1Re(Z) � = µR+(Z), TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 19 which holds for every r ∈ R+ and Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For n = 1, the Abelian Hyperbolic Action yields the R+-action on the upper half-plane H given by r · z = r2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Under this restriction, from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6 we obtain the moment map µR+(z) = −4Re(z) Im(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This recover, up to a constant factor, the moment map obtained in [14, Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3] for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this case the factor comes from two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Firstly, we use the action r · z = r2z, instead of the action r · z = rz used in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Secondly, our formula for the K¨ahler form for S1 = H differs by a constant factor from the corresponding formula found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moment map of the Parabolic Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Finally, we consider the group Symm(n, R) acting on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Since Symm(n, R) is a vector group, it follows that it coincides with its Lie algebra and its exponential map is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' There is a canonical isomor- phism between Symm(n, R) and its dual space given by the positive definite inner product ⟨A, B⟩ = tr(AB), defined for A, B ∈ Symm(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every S ∈ Symm(n, R) the corresponding induced vector field on Sn satisfies for every Z ∈ Symm(n, C) S♯ Z = d ds ��� s=0 exp(sS) · Z = d ds ��� s=0 (Z + sS) = S, which is the constant vector with value S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The function given by µSymm(n,R) : Sn −→ Symm(n, R) µSymm(n,R)(Z) = −2Im(Z)−1, is a moment map for the Parabolic Action on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every S ∈ Symm(n, R) and Z ∈ Sn, using the above computations and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='10 we obtain ωSn Z (S♯ Z, V ) = 2 tr � Im(Z)−1Re(S)Im(Z)−1Im(V ) � − 2 tr � Im(Z)−1Im(S)Im(Z)−1Re(V ) � = 2 tr � Im(Z)−1S Im(Z)−1Im(V ) � , for every Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand, we consider for every S ∈ Symm(n, R) the map µS : Sn → Symm(n, R) defined by µS(Z) = −2tr � Im(Z)−1S � , for which we compute d(µS)Z(V ) = −2 d ds ��� s=0 tr � Im(Z + sV )−1S � = 2 tr � Im(Z)−1Im(V )Im(Z)−1S � , 20 CUEVAS-ESTRADA AND QUIROGA-BARRANCO for every V ∈ Symm(n, C) and Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This immediately yields d(µS)Z(V ) = ωSn Z (S♯ Z, V ), for every V ∈ Symm(n, C) and Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11 it remains to establish the Symm(n, R)-invariance of µSymm(n,R), and this achieved by noting that µSymm(n,R)(S · Z) = µSymm(n,R)(Z + S) = −2 � Im(Z + S)−1� = −2 � Im(Z)−1� = µSymm(n,R)(Z) for every Z ∈ Sn and S ∈ Symm(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For n = 1, the Parabolic Action yields the R-action on the upper half-plane H given by s · z = z + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' And in this situation, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 provides the moment map µR(z) = −2 1 Im(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As in the previous cases, this recovers, up to a constant factor, the moment map obtained in [14, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2] for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As in the case of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7 the factor comes from a different normalization of the K¨ahler form on this work and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Toeplitz operators with special symbols We will now describe Toeplitz operators with special symbols using two related alternatives: symbols invariant under biholomorphism groups and symbols depend- ing on the moment maps of such groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Both cases yield, under suitable conditions, commutative C∗-algebras generated by Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' First we introduce a general notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' As before, in the rest of this work D denotes either of the domains DIII n or Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For A ⊂ L∞(D) a set of essentially bounded symbols, we denote by T (λ)(A) the C∗-algebra generated by the Toeplitz operators T (λ) a where a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Invariant symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let H be a closed subgroup of biholomorphisms of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We will denote by L∞(D)H the subspace of L∞(D) consisting of H-invariant sym- bols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In other words, we have L∞(D)H = {a ∈ L∞(D) : h · a = a, for all h ∈ H}, where, for a given a ∈ L∞(D) and h ∈ H, we define (h · a)(Z) = a(h−1 · Z), for almost every Z ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Symmetric pairs associated to symmetric domains can be used to obtain commu- tative C∗-algebras generated by Toeplitz operators by considering invariant sym- bols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The definitions and precise statements can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this work, we will use the fact that the pairs (Sp(n, R), GL(n, R)) and (Sp(n, C) ∩ U(n, n), U(n)) are symmetric in order to obtain the following consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every λ > n, the C∗-algebras T (λ)(L∞(DIII n )U(n)) and T (λ)(L∞(Sn)GL(n,R)) acting on the weighted Bergman spaces A2 λ(DIII n ) and A2 λ(Sn), respectively, are commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 21 With the notation from subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 states that for the Elliptic and Hyperbolic actions on DIII n and Sn, respectively, the symbols invariant under such actions yield Toeplitz operators that generate commutative C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Parabolic Action provides the same sort of conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This follows from the next consequence of [1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that in this case the group Symm(n, R) does not yield a symmetric pair in the group Sp(n, R) of biholomor- phisms of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every λ > n, the C∗-algebra T (λ)(L∞(Sn)Symm(n,R)) acting on the weighted Bergman space A2 λ(Sn) is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Moment map symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Following [14, 15] we define the notion of moment map symbol for the setup of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let D be either of the domains DIII n or Sn and H a closed subgroup of the biholomorphism group of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' If µH : D → h∗ is a moment map for the action of H on D, then a moment map symbol for H or a µH-symbol is a symbol a ∈ L∞(D) that can be written in the form a = f ◦µH for some measurable function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We denote by L∞(D)µH the space of all essentially bounded measurable µH-symbols on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We have computed moment maps for the Abelian Elliptic, Abelian Hyperbolic and Parabolic actions in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These computations allow us to provide the following explicit description of moment map symbols for these three actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let a ∈ L∞(DIII n ) and b ∈ L∞(Sn) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the follow- ing equivalences hold (1) The measurable function a is a µT-symbol if and only if there exists a mea- surable function f such that a(Z) = f � tr(ZZ) � , for almost every Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (2) The measurable function b is a µR+-symbol if and only if there exists a measurable function f such that b(Z) = f � tr(Im(Z)−1Re(Z)) � , for almost every Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' (3) The measurable function b is a µSymm(n,R)-symbol if and only if there exists a measurable function f such that b(Z) = f(Im(Z)), for almost every Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We note that the claims on the symbols b ∈ L∞(Sn) are immediate conse- quences of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3 and Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, we consider the case of moment maps for the Abelian Elliptic Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4 and Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3, a symbol a ∈ L∞(DIII n ) is a µT-symbol if and only if there is a measurable function g such that a(Z) = g � tr � (In − ZZ)−1�� , for almost every Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In the cone Pos(n, C) of positive definite n× n complex matrices let us consider the open subsets given by (0, In) = {Z ∈ Pos(n, C) | Z < In} (In, ∞) = {W ∈ Pos(n, C) | In < W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It is straightforward to verify that the maps F : (0, In) −→ (In, ∞) G : (In, ∞) −→ (0, In) Z �−→ (In − Z)−1 W �−→ In − W −1 22 CUEVAS-ESTRADA AND QUIROGA-BARRANCO are well defined smooth maps, such that they are inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In partic- ular, these maps satisfy (In − ZZ)−1 = F(ZZ), ZZ = G � (In − ZZ)−1� , for every Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The case of the Abelian Elliptic Action clearly follows from these remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ By definition, the moment map symbols for Abelian groups are invariant under the corresponding actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It turns out that the moment maps of the first two actions are in fact invariant under larger groups, those considered in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is the content of the next two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let µT : DIII n → R be the moment map for the T-action on DIII n given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, µT is a U(n)-invariant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, we have L∞(DIII n )µT ⊂ L∞(DIII n )U(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Recall that U(n) acts on DIII n by U · Z = UZU T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Using the expression of µT obtained in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4, we have for every U ∈ U(n) µT(g · Z) = −2tr � (In − UZU TUZU T))−1� = −2tr � (In − UZZU T )−1� = −2tr � (U(In − ZZ)U T )−1� = −2tr � U(In − ZZ)−1U −1� = −2tr � (In − ZZ)−1� = µT(Z), for every Z ∈ DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The last claim is now an immediate consequence of Defini- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let µR+ : Sn → R be the moment map of the R+-action on Sn given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, µR+ is a GL(n, R)-invariant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, we have L∞(Sn)µR+ ⊂ L∞(Sn)GL(n,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Recall that GL(n, R) acts on Sn by A · Z = AZAT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now use the expression of µR+ obtained in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6, and for every A ∈ GL(n, R) we compute µR+(A · Z) = −4tr � (AIm(Z)A⊤)−1ARe(Z)A⊤� = −4tr � (A⊤)−1Im(Z)−1A−1ARe(Z)A⊤� = −4tr � Im(Z)−1Re(Z) � = µR+(Z), for every Z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Again, the last claim now follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ For the Parabolic Action, it turns out that Symm(n, R)-invariance and being a µSymm(n,R)-symbol are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is the content of the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the moment map µSymm(n,R) : Sn → Symm(n, R) of the Symm(n, R)-action on Sn given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8, we have L∞(Sn)µSymm(n,R) = L∞(Sn)Symm(n,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Symm(n, R)-action on Sn is given by the expression S · Z = Z + S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Hence, for a given a ∈ L∞(Sn) we have the following sequence of equivalences a is Symm(n, R)-invariant ⇐⇒ for every S ∈ Symm(n, R) : a(Z + S) = a(Z) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Z ∈ Sn ⇐⇒ for some measurable f : a(Z) = f(Im(Z)) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Z ∈ Sn, and the result follows from the last case in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Commuting Toeplitz operators with moment maps symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We now state one of our main results: for D either of the domains DIII n or Sn, there are three Abelian groups of biholomorphisms of D to which we can associate commutative C∗-algebras generated by Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let D be either of the domains DIII n or Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' The Abelian Elliptic Action, the Abelian Hyperbolic Action and the Parabolic Action on D yield three Abelian groups of biholomorphisms of D which provide, for every λ > n, the fol- lowing commutative C∗-algebras generated by Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Abelian Elliptic: The C∗-algebra T (λ)� L∞(DIII n )µT� , acting on A2 λ(DIII n ), obtained from the moment map of the T-action on DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Abelian Hyperbolic: The C∗-algebra T (λ)� L∞(Sn)µR+� , acting on A2 λ(Sn), obtained from the moment map of the R+-action on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Parabolic: The C∗-algebra T (λ)� L∞(Sn)µSymm(n,R)� , acting on A2 λ(Sn), ob- tained from the moment map of the Symm(n, R)-action on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' First, we note that Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='6 imply the inclusions T (λ)(L∞(DIII n )µT) ⊂ T (λ)(L∞(DIII n )U(n)) T (λ)(L∞(Sn)µR+) ⊂ T (λ)(L∞(Sn)GL(n,R)), and so the cases of the Abelian Elliptic and Abelian Hyperbolic Actions follow from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the Parabolic Action, we note that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7 yields the identity T (λ)(L∞(Sn)µSymm(n,R)) = T (λ)(L∞(Sn)Symm(n,R)), and now the result is a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' It follows from the discussion in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='2) that for n = 1 the three actions considered in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 reduce to the usual elliptic, hyperbolic and parabolic actions known from complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These three actions have been previously used to obtain commutative C∗-algebras generated by Toeplitz operators, notably in the results found in [3, 4, 5, 7] (see also [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In fact, the commutative C∗-algebras generated by Toeplitz operators from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 reduce to those from these previous works when n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' One of the main guiding lights in this line of study of Toeplitz operators has been to find generalizations to higher dimensions of these commutative C∗-algebras observed in the case of the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This was achieved for the unit ball Bn in Cn through the use of maximal Abelian subgroups of the biholomorphism group of Bn (see [16, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' However, the result for the unit ball Bn from these references lead to 24 CUEVAS-ESTRADA AND QUIROGA-BARRANCO n + 2 different commutative C∗-algebras, which is in contrast with the simplicity of only three Abelian groups for the case of the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' On the other hand, our Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 recovers for higher dimensions the simplic- ity observed in the case of the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' More precisely, we consider the generalized unit disk DIII n and its unbounded realization Sn, Siegel’s generalized upper half- plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For these domains, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='8 yields three commutative C∗-algebras gen- erated by Toeplitz operators, acting on the Bergman spaces of DIII n and Sn, which can be seen as natural extensions of the case of the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' And this is achieved while using only three Abelian groups for any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This is possible due to the fact that we have replaced invariant symbols with moment map symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This highlights the importance of using symplectic geometry to study Toeplitz operators acting on Bergman spaces in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Spectral integral formulas for Toeplitz operator with moment map symbols In this final section we present explicit integral formulas that simultaneously diagonalize Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This will be done for the Abelian Elliptic and Parabolic Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Toeplitz operators with Abelian Elliptic symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In this case, we are dealing with symbols that belong to T (λ)(L∞(DIII n )µT) ⊂ T (λ)(L∞(DIII n )U(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, it is useful to consider the U(n)-action on the Bergman spaces over DIII n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We recall some properties of such action and refer to [19, 2] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every λ > n, there is a unitary representation given by πλ : U(n) × A2 λ(DIII n ) −→ A2 λ(DIII n ) πλ(A)(f) = f ◦ A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' This representation leaves invariant the subspace of (holomorphic) polynomials on DIII n ⊂ Symm(n, C), that we will denote by P(Symm(n, C)) = P, for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' In particular, for every λ > n, the decomposition of A2 λ(DIII n ) into irreducible U(n)- submodules is the same as the one corresponding to the U(n)-action on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let us denote by −→ N n the set of integer n-tuples that satisfy α1 ≥ · · · ≥ αn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, using the representation πλ, one can show that, for every λ > n, there is a Hilbert direct sum decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1) A2 λ(DIII n ) = � α∈− → N n Pα, where � Pα� α∈− → N n is family of mutually non-isomorphic U(n)-submodules of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For the proof of this claim we refer to [19, Chapter 2] (see also [2, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' We consider the polynomials given by ∆j(Z) = det(Zj) where Zj is the upper-left corner j × j submatrix of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' For every α ∈ −→ N n we will also consider the polynomial ∆α(Z) = n � j=1 ∆j(Z)αj−αj+1 TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III 25 where we agree to define αn+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' These are known as the conical polynomials for the representation of U(n) on P (see [19, Chapter 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' With the previous notation, the following result is an application of Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='7 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='11 from [2] to our current setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1 ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Let a ∈ L∞(DIII n )U(n) and λ > n be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Then, the Toeplitz operator T (λ) a acting on the Bergman space A2 λ(DIII n ) preserves the Hilbert direct sum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFKT4oBgHgl3EQfYy5L/content/2301.11800v1.pdf'} +page_content=' Furthermore, we have T (λ) a |Pα = ca,λ(α)IPα, where the complex constant ca,λ(α) is given by ca,λ(α) = � 0 D, and +• ∆ ≤ D + 2 when the capacity of special vertices is at least D1/2−ǫ, and +• ∆ ≤ D + D1/2−ǫ when the capacity of special vertices is less than D1/2−ǫ, +then χ′ +s(G) ≤ 5D − D1/2−ǫ + 2. +Proof. Let G be a counterexample with the fewest number of vertices of degree at least two. +Since G is 2-degenerate, G must have a set S of special vertices. Let u ∈ S and Wu be the set +whose vertices share 2-neighbors with u such that the maximum number of common 2-neighbors +of u and vertices of Wu is the capacity of special vertices of G. Let Wu = {w1, . . . , ws} and u1, u2 +be the two neighbors of u with degree more than 2. For each wi ∈ Wu, let Wi = {vi,1, . . . , vi,ti} +be the common 2-neighbors of wi and u. Then N(u) = {u1, u2} ∪ �s +i=1{vi,1, . . . , vi,ti}. We assume +that t1 ≥ t2 ≥ . . . ≥ ts ≥ 1. Then t1 is the capacity of special vertices of G. It is not hard to see +that u has no neighbors of degree one and wi has degree at least two for each i. +For edge uv, let N2(uv) be the set of edges xy such that x or y is adjacent to u or v. By definition, +uv should have a color different from the colors on edges in N2(uv) in a valid strong edge-coloring. +Case 1. t1 < D1/2−ǫ. In this case, we have ∆ ≤ D + D1/2−ǫ. +Let G′ be the graph obtained from G−{uv1,1, . . . , uvs,ts} by adding up to D +1 +2 −ǫ pendant neighbors +to each of {w1, . . . , ws} so that wi has degree at most D+D1/2−ǫ and wi has at least D +1 +2 −ǫ pendant +neighbors. +Then the graph G′ has fewer vertices of degree at least 2 and can be colored with +5D − D1/2−ǫ + 2 colors. We modify the coloring of G′ to obtain a coloring of G according to the +following algorithm. +(1) Keep the colors of edges that appear in both G and G′, but if wivi,j for some i, j in G has +the same color as uu1 or uu2, then we switch color of wivi,j with a color on other pendant +edges incident to wi in G′. +(2) For each i, if a color c appears on both a pendant edge incident to wi in G′ and an edge +incident to u1 or u2 (not including uu1 and uu2), then we switch the color of wivi,j for some +j with the color c. +(3) After (2), if a color c appears on pendant edges of two or more vertices in Wu in G′, then +we switch the color of wivi,j for some j with c for each such vertex wi ∈ Wu. +(4) After (2) and (3), we color the edges uv1,1, . . . , uvs,ts in reverse order with colors available +to them. +2 + +Now we show that the above algorithm gives a valid strong edge-coloring of G. To do that, we +only need to show that each of the edges in {uv1,1, . . . , uvs,ts} can be colored. Consider uvi,j for +1 ≤ i ≤ s and 1 ≤ j ≤ ti. It needs to get a color not on edges in N2(uvi,j). Note that N2(uvi,j) +contains the edges incident to u1, u2, wi and the edges incident to the 2-neighbors of u; So the +number of colored edges in N2(uvi,j), denoted as n2(uvi,j), is at most +d(u1) + d(u2) + d(wi) + d(u) − 3 + d(u) − 2 − ti − +i−1 +� +p=1 +tp − (j − 1). +We assume that n2(uvi,j) ≥ 5D − D1/2−ǫ + 2, for otherwise, uvi,j can be colored. Because of the +way the edges being colored, we have some repeated colors on edges incident to the 2-neighbors of +u, namely v1,1w1, v1,2w1, . . . , vs,tsws. The number of colors on N2(uvi,j) and edges incident to wi for +i ∈ [s] (with repetition) is n2(uvi,j)+D1/2−ǫ·(s−1). Thus n2(uvi,j)+D +1 +2 −ǫ(s−1)−(5D−D +1 +2−ǫ+2) +colors are repeated. As each wi may allow only one edge (for example, ti = 1) whose color is the +same as other edges in N2(uvi,j), at least n2(uvi,j)+D +1 +2 −ǫ(s−1)−(5D−D +1 +2 −ǫ+2) +D +1 +2 −ǫ +edges have the same +colors as others. Since t1 < D1/2−ǫ and s ≥ d(u)−2 +t1 +≥ +d(u)−2 +D1/2−ǫ , the number of different colors in +N2(uvi,j) is at most +n2(uvi,j) − n2(uvi,j) + D +1 +2−ǫ(s − 1) − (5D − D +1 +2 −ǫ + 2) +D +1 +2−ǫ +≤n2(uvi,j)(1 − +1 +D +1 +2 −ǫ) − (s − 1) + 5D1/2+ǫ − 1 + +2 +D +1 +2−ǫ +≤(d(u1) + d(u2) + 2d(u) − 5 + d(wi) − t1)(1 − +1 +D +1 +2 −ǫ) − d(u) − 2 +D1/2−ǫ + 5D1/2+ǫ + +2 +D +1 +2−ǫ +≤3∆(1 − +1 +D +1 +2−ǫ ) + d(u)(2 − +3 +D +1 +2−ǫ ) + 5D1/2+ǫ − 5 + +9 +D +1 +2 −ǫ +≤3(D + D +1 +2 −ǫ)(1 − +1 +D +1 +2 −ǫ) + D(2 − +3 +D +1 +2 −ǫ) + 5D1/2+ǫ − 5 + +9 +D +1 +2−ǫ +≤5D + 3D +1 +2 −ǫ − D1/2+ǫ − 8 + +9 +D +1 +2 −ǫ ≤ 5D − D1/2−ǫ + 1. +The last inequality holds because D1/2+ǫ ≥ 4D1/2−ǫ and +9 +D +1 +2 −ǫ ≤ 9 when D ≥ 4 +1 +2ǫ . Therefore, +the edge uvi,j can be colored, which implies that we can color all the uncolored edges. +Case 2. t1 ≥ D1/2−ǫ. In this case, ∆ ≤ D + 2. +Let G′ be the graph obtained from G by deleting the edge uv1,1 and adding up to two pendant +edges notated {w1v′, w1v′′} so that w1 has at least three pendant edges and continues to have +maximum degree at most D + 2. +Note that for 1 ≤ i ≤ s and 1 ≤ j ≤ ts, +(1) |N2(uvi,j)| ≤ d(u1) + d(u2) + d(wi) + d(u) − 3+ d(u) − 2− ti ≤ 3∆ + 2D − 5− ti ≤ 5D + 1− ti. +We observe that G′ is 2-degenerate, has fewer vertices of degree at least 2, and contains at least +d(v) − D vertices of degree one if d(v) > D. So, G′ can be colored with 5D − D1/2−ǫ + 2 colors. +Keep the colors of the edges in both G and G′, we try to get a valid coloring of G. +3 + +Case 2.1. the color of edge v1,1w1 is different from ones on {uu1, uu2, uv1,2, . . . , uvs,ts}. By (1), +we have at most 5D + 1 − D1/2−ǫ edges in N2(uv1,1). Since there are 5D − D1/2−ǫ + 2 different +colors available, we can color uv1,1 with a color not on the edges in N2(uv1,1). +Case 2.2. the color of edge v1,1w1 is the same as the color of edge uu1 (or edge uu2). In this +case, we swap the color of v1,1w1 with the color of w1v′ or w1v′′. By (1) again, N2(uv1,1) has at +most 5D +1−D1/2−ǫ edges in N2(uv1,1). Since there are 5D −D1/2−ǫ +2 different colors available, +we can color uv1,1 with a color not on the edges in N2(uv1,1). +Case 2.3. the color of edge v1,1w1 is the same as the color of edge uvi,j for some 2 ≤ i ≤ s. In +this case, we uncolor the edges uvi,j, uv1,1, . . . , uv1,t1, and recolor them in the order. By (1), there +are at most 5D + 1 − ti − (t1 − 1) ≤ 5D − D1/2−ǫ + 1 colors on edges in N2(uvi,j), so uvi,j can be +colored. Similarly, for each j, there are at most 5D − D1/2−ǫ + 1 colors on edges in N2(uv1,j), so +uv1,j can be colored as well. +In any case, G can be colored with 5D − D1/2−ǫ + 2 colors. So χ′ +s(G) ≤ 5D − D1/2−ǫ + 2. +□ +Observe that our main result follows from Theorem 3 with ∆ = D. Our proof does not work for +corresponding result of list version. +References +[1] L. D. Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Math. 108 (1-3) (1992) +231–252. +[2] M. Bonamy, T. Perrett and L. Postle, Colouring graphs with sparse neighbourhoods: Bounds and applications, +J. Combin. Theory Ser. B 155 (2022) 278–317. +[3] H. Bruhn and F. Joos, A stronger bound for the strong chromatic index, Combin. Probab. Comput. 27 (1) (2018) +21–43. +[4] G. J. Chang and N. Narayanan, Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2) (2013) +119–126. +[5] I. Choi, J. Kim, A. V. Kostochka and A. Raspaud, Strong edge-colorings of sparse graphs with large maximum +degree, European J. Combin. 67 (2018) 21–39. +[6] F. R. K. Chung, A. Gy´arf´as, Z. Tuza and W. T. Trotter, The maximum number of edges in 2K2-free graphs of +bounded degree, Discrete Math. 81 (2) (1990) 129–135. +[7] K. Deng, G. Yu and X. Zhou, Recent progress on strong edge-coloring of graphs, Discrete Math. Algorithms +Appl. 11 (5) (2019) 1950062. +[8] P. Erd˝os and J. Neˇsetˇril, Irregularities of partitions, (G. Halasz, V.T. S´os (Eds.)), [Problem] (1989) 162-163. +[9] J.-L. Fouquet and J.-L. Jolivet, Strong edge-colorings of graphs and applications to multi-k-gons, Ars Combin. +16 (A) (1983) 141–150. +[10] P. Hor´ak, Q. He and W. T. Trotter, Induced matchings in cubic graphs, J. Graph Theory 17 (2) (1993) 151–160. +[11] M. Huang, M. Santana and G. Yu, Strong chromatic index of graphs with maximum degree four, Electron. J. +Combin. 25 (3) (2018) Paper #3.31. +[12] E. Hurley, R. de Joannis de Verclos and R. J. Kang, An improved procedure for colouring graphs of bounded +local density, Adv. in Combinatorics, (2022), 7, 33pp. +[13] R. Luo and G. Yu, A note on strong edge-colorings of 2-degenerate graphs, arXiv:1212.6092, December 25, 2012, +https://arxiv.org/abs/1212.6092. +[14] M. Molloy and B. Reed, A bound on the strong chromatic index of a graph, J. Combin. Theory Ser. B 69 (2) +(1997) 103–109. +[15] T. Wang, Strong chromatic index of k-degenerate graphs, Discrete Math. 330 (2014) 17–19. +[16] G. Yu, Strong edge-colorings for k-degenerate graphs, Graphs Combin. 31 (5) (2015) 1815–1818. +4 + diff --git a/I9FOT4oBgHgl3EQfxTS9/content/tmp_files/load_file.txt b/I9FOT4oBgHgl3EQfxTS9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..01608f8a55bebb4f80735c23b8dcadc0a75d9a6d --- /dev/null +++ b/I9FOT4oBgHgl3EQfxTS9/content/tmp_files/load_file.txt @@ -0,0 +1,247 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf,len=246 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='12924v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='CO] 30 Jan 2023 STRONG EDGE-COLORING OF 2-DEGENERATE GRAPHS GEXIN YU1 AND RACHEL YU2 1Department of Mathematics, William & Mary, Williamsburg, VA 23185, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' 2Jamestown High School, Williamsburg, VA 23185, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' A strong edge-coloring of a graph G is an edge-coloring in which every color class is an induced matching, and the strong chromatic index χ′ s(G) is the minimum number of colors needed in strong edge-colorings of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' A graph is 2-degenerate if every subgraph has minimum degree at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Choi, Kim, Kostochka, and Raspaud (2016) showed χ′ s(G) ≤ 5∆ + 1 if G is a 2-degenerate graph with maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In this article, we improve it to χ′ s(G) ≤ 5∆ − ∆1/2−ǫ + 2 when ∆ ≥ 41/(2ǫ) for any 0 < ǫ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Introduction A strong edge-coloring of a graph G is an edge-coloring in which every color class is an induced matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' that is, there are no 2-edge-colored triangles or paths of three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' The strong chromatic index χ′ s(G) is the minimum number of colors in a strong edge-coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' This notion was introduced by Fouquet and Jolivet [9] and one of the main open problems was proposed by Erd˝os and Neˇsetˇril [8] during a seminar in Prague: Conjecture 1 (Erd˝os and Neˇsetˇril, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' If G is a simple graph with maximum degree ∆, then χ′ s(G) ≤ 5∆2/4 if ∆ is even, and χ′ s(G) ≤ (5∆2 − 2∆ + 1)/4 if ∆ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' This conjecture is true for ∆ ≤ 3, see [1, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' For ∆ = 4, Huang, Santana and Yu [11] showed that χ′ s(G) ≤ 21, one more than the conjectured upper bound 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Chung, Gy´arf´as, Trotter, and Tuza (1990, [6]) confirmed the conjecture for 2K2-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Using probabilistic methods, Molloy and Reed [14], Bruhn and Joos [3], Bonamy, Perrett, and Postle [2], and recently Hurley, Verclos, and Kang [12] showed that χ′ s(G) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='772∆2 for sufficiently large ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Sparse graphs have also attracted a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Interested readers may see the survey paper [7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In this article, we are interested in k-degenerate graphs when k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' A graph is k-degenerate if every subgraph has minimum degree at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let G be a 2-degenerate graph with maximum degree ∆ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Chang and Narayanan [4] proved that χ′ s(G) ≤ 10∆ − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Luo and Yu [13] improved it to χ′ s(G) ≤ 8∆ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' For arbitrary values of k, Wang [15] improved the result of Yu [16] and showed the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' If G is a k-degenerate graph with maximum degree ∆ ≥ k, then χ′ s(G) ≤ (4k −2)∆− 2k2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' This implies that χ′ s(G) ≤ 6∆ − 7 for 2-degenerate graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Choi, Kim, Kostochka, and Raspaud [5] further improved it to χ′ s(G) ≤ 5∆ + 1 in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Many believe that the optimal bound should be 4∆ + C for some constant C, but no progress has yet been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' E-mail address: gyu@wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' 1 In this article, we show that for any 0 < ǫ ≤ 1/2, and ∆ ≥ 41/(2ǫ), χ′ s(G) ≤ 5∆ − ∆1/2−ǫ + 2 for 2-degenerate graph G with maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Main result and its proof A special vertex is a vertex with at most two neighbors of degree more than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Every 2- degenerate graph contains special vertices, which are the new 2−-vertices after we remove all vertices of degree at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let G be a 2-degenerate graph and S be the set of special vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' For each u ∈ S, there exists a set Wu of vertices such that each w ∈ Wu shares some 2-neighbors with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' The capacity of special vertices of G is the maximum number of common 2-neighbors that are shared by a vertex in S and other vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' A pendant edge is an edge incident with a leaf (a vertex of degree one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Below is the main result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' For any 0 < ǫ ≤ 1/2, let D be a positive integer when D ≥ 41/(2ǫ), for any 2- degenerate graph G with maximum degree ∆, if a vertex u is adjacent to at least d(u) − D leaves when it has d(u) > D, and ∆ ≤ D + 2 when the capacity of special vertices is at least D1/2−ǫ, and ∆ ≤ D + D1/2−ǫ when the capacity of special vertices is less than D1/2−ǫ, then χ′ s(G) ≤ 5D − D1/2−ǫ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let G be a counterexample with the fewest number of vertices of degree at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Since G is 2-degenerate, G must have a set S of special vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let u ∈ S and Wu be the set whose vertices share 2-neighbors with u such that the maximum number of common 2-neighbors of u and vertices of Wu is the capacity of special vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let Wu = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , ws} and u1, u2 be the two neighbors of u with degree more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' For each wi ∈ Wu, let Wi = {vi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , vi,ti} be the common 2-neighbors of wi and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Then N(u) = {u1, u2} ∪ �s i=1{vi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , vi,ti}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' We assume that t1 ≥ t2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' ≥ ts ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Then t1 is the capacity of special vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' It is not hard to see that u has no neighbors of degree one and wi has degree at least two for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' For edge uv, let N2(uv) be the set of edges xy such that x or y is adjacent to u or v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' By definition, uv should have a color different from the colors on edges in N2(uv) in a valid strong edge-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' t1 < D1/2−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In this case, we have ∆ ≤ D + D1/2−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let G′ be the graph obtained from G−{uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , uvs,ts} by adding up to D 1 2 −ǫ pendant neighbors to each of {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , ws} so that wi has degree at most D+D1/2−ǫ and wi has at least D 1 2 −ǫ pendant neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Then the graph G′ has fewer vertices of degree at least 2 and can be colored with 5D − D1/2−ǫ + 2 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' We modify the coloring of G′ to obtain a coloring of G according to the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' (1) Keep the colors of edges that appear in both G and G′, but if wivi,j for some i, j in G has the same color as uu1 or uu2, then we switch color of wivi,j with a color on other pendant edges incident to wi in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' (2) For each i, if a color c appears on both a pendant edge incident to wi in G′ and an edge incident to u1 or u2 (not including uu1 and uu2), then we switch the color of wivi,j for some j with the color c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' (3) After (2), if a color c appears on pendant edges of two or more vertices in Wu in G′, then we switch the color of wivi,j for some j with c for each such vertex wi ∈ Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' (4) After (2) and (3), we color the edges uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , uvs,ts in reverse order with colors available to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' 2 Now we show that the above algorithm gives a valid strong edge-coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' To do that, we only need to show that each of the edges in {uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , uvs,ts} can be colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Consider uvi,j for 1 ≤ i ≤ s and 1 ≤ j ≤ ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' It needs to get a color not on edges in N2(uvi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Note that N2(uvi,j) contains the edges incident to u1, u2, wi and the edges incident to the 2-neighbors of u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' So the number of colored edges in N2(uvi,j), denoted as n2(uvi,j), is at most d(u1) + d(u2) + d(wi) + d(u) − 3 + d(u) − 2 − ti − i−1 � p=1 tp − (j − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' We assume that n2(uvi,j) ≥ 5D − D1/2−ǫ + 2, for otherwise, uvi,j can be colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Because of the way the edges being colored, we have some repeated colors on edges incident to the 2-neighbors of u, namely v1,1w1, v1,2w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , vs,tsws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' The number of colors on N2(uvi,j) and edges incident to wi for i ∈ [s] (with repetition) is n2(uvi,j)+D1/2−ǫ·(s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Thus n2(uvi,j)+D 1 2 −ǫ(s−1)−(5D−D 1 2−ǫ+2) colors are repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' As each wi may allow only one edge (for example, ti = 1) whose color is the same as other edges in N2(uvi,j), at least n2(uvi,j)+D 1 2 −ǫ(s−1)−(5D−D 1 2 −ǫ+2) D 1 2 −ǫ edges have the same colors as others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Since t1 < D1/2−ǫ and s ≥ d(u)−2 t1 ≥ d(u)−2 D1/2−ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' the number of different colors in N2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='j) is at most n2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='j) − n2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='j) + D 1 2−ǫ(s − 1) − (5D − D 1 2 −ǫ + 2) D 1 2−ǫ ≤n2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='j)(1 − 1 D 1 2 −ǫ) − (s − 1) + 5D1/2+ǫ − 1 + 2 D 1 2−ǫ ≤(d(u1) + d(u2) + 2d(u) − 5 + d(wi) − t1)(1 − 1 D 1 2 −ǫ) − d(u) − 2 D1/2−ǫ + 5D1/2+ǫ + 2 D 1 2−ǫ ≤3∆(1 − 1 D 1 2−ǫ ) + d(u)(2 − 3 D 1 2−ǫ ) + 5D1/2+ǫ − 5 + 9 D 1 2 −ǫ ≤3(D + D 1 2 −ǫ)(1 − 1 D 1 2 −ǫ) + D(2 − 3 D 1 2 −ǫ) + 5D1/2+ǫ − 5 + 9 D 1 2−ǫ ≤5D + 3D 1 2 −ǫ − D1/2+ǫ − 8 + 9 D 1 2 −ǫ ≤ 5D − D1/2−ǫ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' The last inequality holds because D1/2+ǫ ≥ 4D1/2−ǫ and 9 D 1 2 −ǫ ≤ 9 when D ≥ 4 1 2ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Therefore, the edge uvi,j can be colored, which implies that we can color all the uncolored edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' t1 ≥ D1/2−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In this case, ∆ ≤ D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Let G′ be the graph obtained from G by deleting the edge uv1,1 and adding up to two pendant edges notated {w1v′, w1v′′} so that w1 has at least three pendant edges and continues to have maximum degree at most D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Note that for 1 ≤ i ≤ s and 1 ≤ j ≤ ts, (1) |N2(uvi,j)| ≤ d(u1) + d(u2) + d(wi) + d(u) − 3+ d(u) − 2− ti ≤ 3∆ + 2D − 5− ti ≤ 5D + 1− ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' We observe that G′ is 2-degenerate, has fewer vertices of degree at least 2, and contains at least d(v) − D vertices of degree one if d(v) > D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' So, G′ can be colored with 5D − D1/2−ǫ + 2 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Keep the colors of the edges in both G and G′, we try to get a valid coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' 3 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' the color of edge v1,1w1 is different from ones on {uu1, uu2, uv1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , uvs,ts}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' By (1), we have at most 5D + 1 − D1/2−ǫ edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Since there are 5D − D1/2−ǫ + 2 different colors available, we can color uv1,1 with a color not on the edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' the color of edge v1,1w1 is the same as the color of edge uu1 (or edge uu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In this case, we swap the color of v1,1w1 with the color of w1v′ or w1v′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' By (1) again, N2(uv1,1) has at most 5D +1−D1/2−ǫ edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Since there are 5D −D1/2−ǫ +2 different colors available, we can color uv1,1 with a color not on the edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' the color of edge v1,1w1 is the same as the color of edge uvi,j for some 2 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In this case, we uncolor the edges uvi,j, uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' , uv1,t1, and recolor them in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' By (1), there are at most 5D + 1 − ti − (t1 − 1) ≤ 5D − D1/2−ǫ + 1 colors on edges in N2(uvi,j), so uvi,j can be colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Similarly, for each j, there are at most 5D − D1/2−ǫ + 1 colors on edges in N2(uv1,j), so uv1,j can be colored as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' In any case, G can be colored with 5D − D1/2−ǫ + 2 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' So χ′ s(G) ≤ 5D − D1/2−ǫ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' □ Observe that our main result follows from Theorem 3 with ∆ = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Our proof does not work for corresponding result of list version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'} +page_content=' Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Math.' metadata={'source': 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that is specifically designed to obtain a +fast conformational clustering of long molecular dynamics simulation trajectories. In this approach +we combine two dimensionality reduction algorithms (cc analysis and encodermap) with a density- +based spatial clustering algorithm (HDBSCAN). The proposed scheme benefits from the strengths +of the three algorithms while avoiding most of the drawbacks of the individual methods. +Here +the cc analysis algorithm is for the first time applied to molecular simulation data. Encodermap +complements cc analysis by providing an efficient way to process and assign large amounts of data +to clusters. The main goal of the procedure is to maximize the number of assigned frames of a given +trajectory, while keeping a clear conformational identity of the clusters that are found. In practice +we achieve this by using an iterative clustering approach and a tunable root-mean-square-deviation- +based criterion in the final cluster assignment. This allows to find clusters of different densities as +well as different degrees of structural identity. With the help of four test systems we illustrate the +capability and performance of this clustering workflow: wild-type and thermostable mutant of the +Trp-cage protein (TC5b and TC10b), NTL9 and Protein B. Each of these systems poses individual +challenges to the scheme, which in total give a nice overview of the advantages, as well as potential +difficulties that can arise when using the proposed method. +I. +INTRODUCTION +With the ever-growing power of computers over the +last decades, researchers in the field of molecular dynam- +ics (MD) have gotten access to increasingly long trajec- +tories and thereby to increasingly large amounts of data. +The introduction of supercomputers which are specifi- +cally designed to generate MD trajectories (Anton [1] +and Anton 2 [2]) is only the latest high point in this +development. Furthermore, new sampling methods [3, 4] +as well as distributed computing projects, such as Fold- +ing@home [5], have contributed to a massive increase in +generated simulation trajectories. With this increasing +amount of data it is essential to have powerful analysis +tools to process and understand underlying systems and +processes. +There is a rapid increase in application of unsupervised +machine learning methods to analyze molecular simula- +tion data. +Two of the most used families of analysis +techniques are clustering and dimensionality reduction +(DR) algorithms. +They help to find low-dimensional +subspaces in which important aspects of the original +data are preserved and to group the data based on a +given similarity measure/metric and thereby gain a bet- +ter overview and understanding. +In practice, most of +the times clustering and DR methods are used in com- +bination. +The DR algorithms can be roughly divided +into: +linear methods (the most known are principal +component analysis (PCA) [6, 7] and time-lagged in- +∗ Copyright 2023 Hunkler, Peter. This article is distributed under +a Creative Commons Attribution (CC BY) License. +† kukharenko@mpip-mainz.mpg.de +‡ christine.peter@uni-konstanz.de +dependent component analysis (TICA) [8, 9]), nonlin- +ear methods (kernel and nonlinear PCA, multidimen- +sional scaling (MDS) [10, 11] and MDS-based methods +like sketch-map [12], isomap [13], diffusion maps [14, 15] +or UMAP [16], etc.) and autoencoder-based approaches +like (encodermap [17, 18], time-autoencoder [19], vari- +ational autoencoders [20] and Gaussian mixture varia- +tional autoencoders [21]). +In terms of clustering algo- +rithms, there are again a wide range of different methods: +K-Means [22, 23], spectral-clustering [24], DBSCAN [25], +density-peak clustering [26], CNN-clustering [27], root- +mean-square deviation (RMSD) based clustering [28], +neural-networks-based VAMPnets [29], etc. For a com- +prehensive overview of unsupervised ML methods com- +monly used to analyse MD simulation data we refer to +Ref. 30. +Even from this incomplete list of available methods +it should become obvious that there are a lot of differ- +ent clustering, as well as DR methods. All these meth- +ods have their individual strengths and weaknesses. But +there are still open challenges in the successful usage of +the listed methods for processing simulation data with +high spatial and temporal resolution. This is connected +either to the proper choice of hyper-parameters (such as +the number of dimensions for DR methods, the num- +ber of expected states for some clustering algorithms, +neural-networks architectures, different cut-offs, corre- +lation times, etc.), expensive optimisation steps or the +amount of data which could be processed simultaneously. +In this work we present a data processing scheme which +combines three different algorithms in one workflow to +create a powerful clustering machinery. It tackles a num- +ber of the mentioned challenges as it has a way to define +an appropriate lower dimensionality of the data, does not +require a priory information about the expected number +arXiv:2301.04492v1 [physics.chem-ph] 11 Jan 2023 + +2 +FIG. 1. +Data processing routine presented in this article. +of states and it is fast in processing extensive MD trajec- +tories with both a very high dimensionality and a large +number of observations. It is specifically designed to find +conformational clusters in long molecular simulation data +(Fig. 1). +We use two different DR algorithms, namely an al- +gorithm called “cc analysis” and the encodermap algo- +rithm. The cc analysis method belongs to the family of +the MDS-based techniques and was first introduced for +the analysis of crystallographic data [31, 32]. Here it is +used for the first time for projecting data of protein con- +formations. The dimensionality of the cc analysis-space +which is typically required is more than two (10 to 40 +for the systems shown in this work) and the amount of +data, which can be efficiently projected simultaneously is +limited by the available memory (about 50000 frames for +a 72 GB workstation). To process much longer trajecto- +ries and to obtain a two-dimensional representation we +use the second DR algorithm – encodermap [33]. Its loss +function however consist of two parts: the autoencoder +loss and a MDS-like distance loss, which introduces an +interpretability to the resulting 2D representation. More- +over, once the encodermap network is trained, the en- +coder function can be used to project data to the 2D +map in an extremely efficient way. We use encodermap to +project data into 2D and for a fast assignment of the addi- +tional members to the clusters defined in the cc analysis +space. Finally we use the HDBSCAN algorithm [34] to +cluster the data in the cc analysis space and then visu- +alize the resulting clusters in the 2D encodermap space. +HDBSCAN is a combination of density and hierarchi- +cal clustering, that can work efficiently with clusters of +varying density, ignores sparse regions, and requires a +minimum number of hyper parameters. We apply it in a +non-classical iterative way with varying RMSD-cutoffs to +extract the protein conformations of different similarities. +The combination of these three algorithms allows us +to leverage their different strengths, while avoiding the +drawbacks of the individual methods. Subsequently we +will show how the scheme performs on long MD trajecto- +ries of wild-type and mutated Trp-cage with native and +misfolded meta-stable states (208 µs and 3.2 µs long sim- +ulations); really extensive simulations of NTL9 (1877 µs); +and Protein B, where only a small percent of the simu- +lation data (5%) is in the folded state (104 µs). +II. +METHODS +A. +cc analysis +For dimensionality reduction, we use an cc analysis in- +troduced in Ref. 31, 32. This algorithm was originally +developed to analyse crystallographic data, where pres- +ence of noise and missing observations pose a challenge to +data processing in certain experimental situations. The +method separates the inter-data-set influences of ran- +dom error from those arising from systematic differences, +and reveals the relations between high-dimensional in- +put features by representing them as vectors in a low- +dimensional space. Due to this property we expected it +to be highly applicable to protein simulation data, where +one seeks to ignore the differences arising from random +fluctuations, and to separate the conformations based on +systematic differences. In the course of the manuscript +we show that this assumption proved to be correct. +The cc analysis algorithm belongs to the family of +MDS methods [10]. Its main distinction is that it min- +imizes the sum of squared differences between Pearson +correlation coefficients of pairs of high-dimensional de- +scriptors and the scalar product of the low-dimensional +vectors representing them (see Eq. +(1)). +The proce- +dure places the vectors into a unit sphere within a low- +dimensional space. Systematic differences between the +high-dimensional features lead to differences in the an- +gular directions of the vectors representing them, and +purely random differences of data points lead to different +vector lengths at the same angular direction. The algo- +rithm minimizes, e.g. iteratively using L-BFGS [35], the + +Full trajectory +Define high-D CVs +Encodermap +2D projection +Expand clusters +based on RMSD +and 2D projection +cc_analysis +HDBSCAN +Select random +subset +(up to 25000 frames) +Remove +For trajectories < 25000 frames +clustered +frames3 +expression +Φ(x) = +N−1 +� +i=1 +N +� +j=i+1 +(rij − xi · xj)2 +(1) +as a function of x, the column vector of the N low- +dimensional vectors {xk}. +Here rij is the correlation +coefficient between descriptors i and j in the high- +dimensional space and xi · xj denotes the dot product +of the unit vectors xi and xj representing the data in the +low-dimensional space; N is the number of observations, +e.g. protein conformations. The output of cc analysis is +the N low-dimensional vectors {xk}, and the eigenvalues +of the xxT matrix. +To understand why this is a sensible approach, one +can think about obtaining the least squares solution of +Eq. (1) algebraically by eigenanalysis of the matrix r = +{rij}. In that case one would have to solve +xxT = r +where r is the matrix of the correlation coefficients rij. +The n strongest eigenvalue/eigenvector pairs (eigenvec- +tors corresponding to the largest eigenvalues) could then +be used to reconstruct the N vectors xi, which are lo- +cated in a n-dimensional unit sphere. +The systematic +differences between the input data are thereby shown by +the different angular directions in this low-dimensional +sphere. +This approximation is meaningful because in +general the Pearson correlation coefficient can be written +as a dot product between two vectors (after subtraction +of the mean and dividing by the standard deviation to +scale the vectors to unit length) and is equal to the cosine +of the angle between them. Hence, in an ideal scenario, +�N +i,j xi · xj can exactly reproduce the high-dimensional +correlation coefficient matrix and Φ(x) in Eq. (1) would +be equal to zero. +The length of the vectors is less important than the +angle between them. The latter has an inherent meaning: +two high-dimensional feature vectors with a correlation +coefficient of zero between them would be projected to +unit vectors at 90◦ angles with respect to the origin, and +two feature vectors with a correlation coefficient of one +would have a corresponding angle of zero degrees. +Despite the generality of the cc analysis approach, by +now it was only applied to crystallographic data [36, 37]) +and protein sequence grouping [38]. Here we present a +first application of cc analysis for protein simulation data +analysis. +B. +Encodermap +To accelerate the processing of large datasets, e.g. from +extensive simulations, in addition to cc analysis, we make +use of one more dimensionality reduction technique – en- +codermap. +It was developed by Lemke and Peter [33] +and is used here for fast assignment of data points to +clusters as will be explained in details in Sec. II D. The +method combines the advantages of a neural network au- +toencoder [17] with a MDS contribution, here the loss +function from the sketch-map algorithm [12] (Fig. +2). +This combination is exceptionally efficient for projecting +large simulation data to the two-dimensional representa- +tions: the sketch-map loss function allows to concentrate +only on relevant dissimilarities between conformations +(ignoring thermal fluctuations and coping with the large +dissimilarity values caused by the data’s high dimension- +ality). Furthermore the autoencoder approach allows to +avoid complex minimisation steps of the sketch-map pro- +jection and to process huge amounts of data in a very +short time. +FIG. 2. +Schematic description of encodermap. It has an +architecture of the classic autoencoder consisting of two neu- +ral networks (encoder and decoder) with the same number of +layers and neurons in each layer connected through the bottle- +neck layer with two neurons. In addition to autoencoder loss +La(X, ˜ +X) encodermap loss has a term with the sketch-map +loss function Ls(X, x), which improves the quality of two- +dimensional projection obtained in the bottle-neck layer (see +Eq. (2)). +The encodermap loss function Lencodermap (Eq. (2)) is +a weighted sum of the autoencoder loss Lauto (Eq. (3)) +and the sketch-map loss function Lsketch (Eq. (4)), which +emphasizes mid-range distances by transforming all dis- +tances via a sigmoid function (Eq. (5)). +Lencodermap = kaLauto + ksLsketch + Reg, +(2) +Lauto = 1 +N +N +� +i=1 +D(Xi, ˜Xi), +(3) +Lsketch = 1 +N +N +� +i̸=j +[SIGh(D(Xi, Xj)) − SIGl(D(xi, xj))]2, +(4) +where ka, ks are adjustable weights, Reg is a regular- +ization used to prevent overfitting; N is a number of +data points to be projected; D(·, ·) is a distance be- +tween points, X is a high-dimensional input, x is a low- +dimensional projection (the bottleneck layer); SIGh and +SIGl are sigmoid functions of the form shown in Eq. (5). +SIGσ,a,b(D) = 1 − (1 + (2 +a +b − 1)(D +σ )a)− b +a , +(5) + +CVs +CVs +2D +projection +neural +neural +network +network +encoder +decoder +X4 +FIG. 3. +Application of HDBSCAN on a toy data set with +three clusters. i) Example for the computation of the MRD +for two points (red and blue). The red and blue circles in- +dicate the farthest distance to the 5 nearest neighbours for +both points. One can see that the distance between the red +and blue points (green line) is larger than both the radii of +the blue and the red circle. Therefore in this case the green +line distance is chosen as MRD. ii) The minimum spanning +tree based on the MRDs. iii) The cluster hierarchy. iv) The +condensed clustering. +where a, b and σ are parameters defining which distances +to preserve. +C. +Hierarchical Density-Based Spatial Clustering +of Applications with Noise (HDBSCAN) +The HDBSCAN [34, 39] can be approached from +two different sides: it can be described as a hierarchi- +cal implementation of a new formulation of the origi- +nal DBSCAN [25] algorithm called DBSCAN* by J. G. +B. Campello et al. [34] or it can be formulated as a ro- +bust version of single-linkage clustering with a sophisti- +cated method to obtain a flat clustering result, as done +by McInnes et al. [39]. Here we describe it through the +second approach. +In the first step the algorithm introduces the so-called +mutual reachability distance (MRD) (Eq. +(6)), which +transforms the space to make sparse points even sparser +but does not significantly change the distance between +already dense points. +Dmreach−k(xi, xj) = +max{corek(xi), corek(xj), D(xi, xj)}, +(6) +where x are points being clustered, k is a constant which +determines a number of nearest neighbouring points, +corek(x) is a function, which finds the maximum distance +between a point x and its k nearest neighbours; D(·, ·) is +a distance between two points. The maximum of three +distances is selected as the MRD (Fig. 3 i)). +In the next step the minimum spanning tree based on +the MRDs is build via Prim’s algorithm [40] (see Fig. 3 +ii)). This is done by starting with the lowest MRD in +the data set and connecting the two points by a straight +line. In the following steps always the next nearest data +point to the existing tree, which is not yet connected, is +added to the tree. +Once the minimum spanning tree is generated the clus- +ter hierarchy can be built. This is done by first, sorting +the edges of the tree by distance. Then the algorithm +iterates over the edges, always merging the clusters with +the smallest MRD. The result of this procedure can be +seen in Fig. 3 iii). +In order to extract a flat clustering form this hierarchy, +a final step is needed. In this step the cluster hierarchy +is condensed down, by defining a minimum cluster size +and checking at each splitting point if the new forming +cluster has at least the same amount of members as the +minimum cluster size. +If that is the case, then a new +cluster is accepted, if not then the data points splitting +off are considered noise. +The condensed tree of a toy +system can be seen in Fig. 3 iv). +D. +Introduction of a new clustering workflow +In this article we present a data processing routine +which we found to be extremely efficient for large molec- +ular dynamics simulation trajectories. +It relies on the +three algorithms introduced above. A schematic descrip- +tion is given in Fig. +1. In this workflow a given data +set is clustered iteratively until either a specified amount +of data points are assigned to clusters or a maximum +number of iterations have been reached. +Fig. +1 illustrates the sequence of data processing +steps along the clustering workflow. +In the first step +a high-dimensional collective variable (CV) is chosen. +For all systems that are shown in this article all pair- +wise distances between the Cα atoms were selected. Af- +ter a CV has been chosen, for trajectories containing +more than 25,000 frames, encodermap is trained on all +data. Thereby we obtain a function which can be used +to project data very efficiently to the newly generated +2D space. In parallel, a random subset from the entire +data set is generated. +The reason to use such a sub- +set is a limitation that comes with the cc analysis di- +mensionality reduction. As mentioned in Sec. II A the +cc analysis algorithm works with the correlation matrix. +This means that the Pearson correlation coefficients of +the selected CV (here the pairwise c-alpha distances) are +calculated for all unique pairs of frames, and used as in- +put to cc analysis. However the larger a data set is, the +larger the correlation coefficient matrix will be, until it +is no longer efficient to work with that matrix due to +very long computation times as well as memory issues. +Therefore a subset is created, by randomly selecting up +to 25,000 data points from the entire data set. This sub- +set is then used in the cc analysis dimensionality reduc- +tion to project the high dimensional CVs (between 190 +and 1081 dimensions for the systems in this article) to a + +ii) +dmreach +0.25 +云 +reachal +0.10 +Mutual +0.05 +iii) +iv) +0 +100 +0.25 +6 +5 +80 +0.20 +5 +points +4 +lue +10 +60 +of +val +m +Number +0.10 +^ 15 +40 +2 +0.05 +20 +20 +1 +0.00 +25 +0 +05 +lower dimensional subspace (20 to 30 dimensions for the +systems in this article). The choice of the appropriate +amount of reduced dimensions is done by searching for +a spectral gap among the cc analysis eigenvalues. Once +the cc analysis space has been identified, a clustering is +generated by applying the HDBSCAN algorithm to that +lower dimensional data. A detailed description on how +to choose the dimensionality for cc analysis and the pa- +rameters for HDBSCAN is given in the supporting infor- +mation (SI), Sec. S-I. +We use two different DR algorithms in the workflow +due to the following reasons. For once, the cc analysis +algorithm is used to project the smaller subsets to a +still comparably high-dimensional subspace, which holds +more information compared to the 2D projection of en- +codermap. This higher dimensional subspace is therefore +very well suited to be the clustering space. +Once the +data subset is clustered in the cc analysis space, the 2D +encodermap space is used to assign the points that were +not a part of the subset to the found clusters. The 2D +projection is very well suited to do a fast assignment of +additional points from the data set, as well as to serve for +visualization purposes. Additionally, encodermap is able +to project huge data sets very time-efficiently. +Hence, +the generated 2D projection of a given data set can be +used to avoid the main disadvantage of the cc analysis +algorithm in the way we use the algorithm here, which +is having to use subsets of the data due to memory is- +sues. In order to circumvent this disadvantage, we build +a convex hull in the 2D space for each cluster that was +found in the cc analysis space. If an unassigned point lies +inside a convex hull, the RMSD to the central conforma- +tion of that cluster is computed. In case the RMSD is +inside a given cutoff, the data point is considered to be +part of that cluster, else it is not assigned to the clus- +ter. This RMSD cutoff is chosen by taking the weighted +mean of all average internal cluster RMSDs 1 of the first +clustering iteration. We found that this procedure gen- +erates structurally quite well defined clusters with a low +internal cluster RMSD since the RMSD criterion is based +on well defined conformational states that emerged from +cc analysis combined with HDBSCAN. However there is +also the possibility to identify more fuzzy clusters that +only share a general structural motif by using a larger +RMSD cutoff for the assignment. An example of the iden- +tification of such fuzzy clusters is described in Sec. III B. +By introducing a RMSD criterion in the last step, we +force the clustering to only include structurally very sim- +ilar conformations in the respective clusters. There are of +course various other clustering algorithms, which group +MD data sets based on their RMSD values, e.g. an imple- +mentation [28] in the GROMACS software package [41]. +Such RMSD-based clustering algorithms have been used +in the MD community for at least 20 years now and they +1 By the average internal cluster RMSD we mean the average +RMSD of all conformations to the cluster centroid. +are a very obvious choice for conformational clusterings +of MD trajectories. They directly compare the positions +of specified atoms in various conformations of a molecule +and then group the individual conformations along the +trajectory using a cutoff value. However these methods +generally rely on the full RMSD matrix of a given data +set. For larger trajectories it becomes almost infeasible +to compute these matrices due to extremely long com- +putation times as well as memory issues that arise when +working with such large matrices. In our workflow we +can circumvent these issues by only having to compute +the RMSD between the coordinates of Cα atoms of the +central conformations of each cluster and the data points +that lie inside the convex hull of the respective clusters +in the 2D space. +In case a given system has less then about 50,000 +frames, it is in principle also possible to omit the en- +codermap part, since the cc analysis algorithm is able to +handle the entire data set. If this approach is chosen, +the user can either entirely skip the RMSD criterion, or +the members of clusters that are found in the cc analysis +space can still be accepted/rejected based on a RMSD +cutoff. An advantage of using both the cc analysis algo- +rithm and the encodermap algorithm together is the pos- +sibility to check the dimensionality reduction steps on the +fly. Since the clustering is done in one DR space, but vi- +sualized in the other, narrow and well defined clusters in +the 2D space indicate that the 2D map separates the dif- +ferent conformational clusters nicely and that therefore +the chosen encodermap parameters were well selected. +Our clustering scheme is not very dependent on the +quality of encodermap projection, as it is used only to as- +sign additional structures to already identified clusters. +Since the clustering itself is done in the higher dimen- +sional cc analysis space and the final cluster assignment +uses a RMSD cutoff. +The only requirement that the +scheme poses towards the 2D map is that similar con- +formations are located close to each other in the map. +This is achieved by the MDS-like distance loss part of +the overall loss function of encodermap. +Remaining points which were not assigned to any clus- +ter after one clustering iteration are then used as a new +pool of data, from which the new random subset is build. +This whole cycle is repeated until a certain amount of +data points are assigned to clusters or until a certain +number of clustering iterations are performed. To decide +on a stopping point for the iterative procedure we rely +on two possible convergence criteria: either a percentage +of assigned conformations or average cluster sizes found +at an iteration. If we observe a plateau in the percent- +age of unassigned data points during several successive +iterations the clustering procedure is stopped. +Due to +the design of our workflow, the average cluster size of +newly added clusters will decrease with each iteration. +Therefore, the average size of newly added clusters or +the convergence of this property during successive itera- +tions can also be used as a stopping criterion. Examples +are shown in SI, Sec. S-II, Fig. S2. + +6 +Trp-cage RE (TC5b) Trp-cage Anton (TC10b) +NTL9 +Protein B +Trajectory length in µs +3.2 +208 +1877 +104 +Number of frames +1,577,520 +1,044,000 +9,389,654 +520,250 +Input CVs dimensionality +190 +190 +703 +1081 +Number of cc analysis dimensions +20 +20 +20 +30 +Average iteration time +on our local workstation +(see SI, Sec. S-V) [min] +15 +18 +55 +12 +Average iteration time +over all used +CPU threads [min] +24 x 15 += 360 +24 x 18 += 432 +24 x 55 += 1320 +24 x 12 += 288 +Frames assigned to clusters +after 10 iterations +60% +33.1% +80.9% +20% +Total CPU time +over all iterations [min] +3600 +4320 +13200 +2880 +TABLE I: Proteins analysed in this study and performance overview of the clustering scheme. +III. +RESULTS AND DISCUSSION +A. +Description of the proteins’ trajectories used +for the analysis +In order to illustrate the capability and performance of +the proposed scheme, we chose four test systems: 40 tem- +perature replica exchange (RE) trajectories of the Trp- +cage protein (TC5b) analysed in the original encodermap +paper [33]; the other three systems are long trajectories of +Trp-cage (TC10b), NTL9 and Protein B simulated by the +Shaw group on the Anton supercomputer [42] and gen- +erously provided by them. The four systems are listed in +Table I. For all the systems we chose distances between +Cα atoms as the input collective variables. +The first protein we analyse in this work is the Trp- +cage system (TC5b) (Trp-cage RE). It is a comparatively +small protein (20 residues) which has a very stable native +state when simulated at room temperature. The combi- +nation of 40 temperature replica exchange trajectories +(temperature range from 300 to 570 K, 3.2 µs of simu- +lation time, 1,577,520 frames) give a very diverse mix- +ture of structures including trajectories where the sys- +tem is very stable and barely moves away from the na- +tive state, as well as highly disordered trajectories where +high-energy conformations are visited. This combination +of conformations makes the data set extremely diverse +and complicated for the analysis due to the high num- +ber of expected clusters with extremely varying size and +density. +Secondly we consider the K8A mutant of the ther- +mostable Trp-cage variant TC10b (Trp-cage Anton) sim- +ulated by Lindorff-Larsen et al. [42] (208 µs; 1,044,000 +frames). This simulation was run at 290 K and produced +a much more disordered trajectory compared to the low +temperature replica simulations of the TC5b system. De- +spite the fact that TC5b and the K8A mutant of TC10b +have slightly different amino acid sequences, we use the +same trained encodermap to project both systems in the +same 2D map (see Fig. 4 and Fig. 5), since both sys- +tems have the same number of residues and therefore the +same dimensionality of CVs. This offers the opportunity +to demonstrate that different systems can be compared +to each other very nicely when projected to the same 2D +space. +Next we probed our clustering scheme with extremely +long (1877 µs 2; 9,389,654 frames) simulations [42] of the +larger (39 amino acids) N-terminal fragment of ribosomal +protein L9 (NTL9) which has an incredibly stable native +state. Besides the possibility to show how the algorithm +deals with this extremely large data set, the system has +also been studied by several other researchers [29, 44]. +This allows us to compare our results to their findings. +Schwantes and Pande [44] reported on very low pop- +ulated states which involve register-shifts between the +residues that are involved in the formation of the beta +sheet structures of NTL9. This opens the opportunity +to show whether our clustering workflow is able to iden- +tify both very large states, as well as extremely lowly +populated states in the same data set. +Lastly we chose to analyse the protein B simulations +(104 µs; 520,250 frames) [42]. Compared to the afore- +2 We used the trajectories 0, 2 and 3 according to the nomenclature +of Ref. 42. We have not used trajectory 1 because the topology +file for this specific trajectory differs slightly form the other three +in terms of the order and the numbering of the atoms. This issue +has also been reported by other researchers [43]. + +中7 +FIG. 4. +Trp-Cage TC5b (40 temperature RE trajectories): Exemplary conformations of the most populated clusters found +in each of the areas indicated by coloured circles and their populations in percentages. The cluster representatives show the +average secondary structure over the entire cluster. The clusters are coloured randomly, the colours repeat. Therefore clusters +that have the same colour but are separated in the 2D space contain different conformations. The depicted clusters hold 36.5% +of all conformations. Most of the remaining 24% of conformations that have been assigned to clusters are slight variations of +the native structure and are not shown here due to visibility reasons. The cluster that is referred to by an arrow is one of the +fuzzy clusters that were generated by increasing the RMSD cutoff. Top right: a histogram of the 2D encodermap space. +mentioned proteins protein B does not have a single very +stable state, instead three helices can move quite easily +against each other. This leads to a broad conformational +space, where the energy barriers between the individual +states are very small. Therefore the individual confor- +mational states are not as easily separable and rather +fade/transition into each other. Taking into account the +long simulation time this system is very hard to cluster +conformationally. +To demonstrate how our clustering scheme works we +chose to apply it to these four systems that pose very +diverse challenges (e.g. an extremely large data set, both +highly and very lowly populated states in the same data, +differences in the amount of folded/unfolded conforma- +tions along the trajectories). For each of the systems we +initially conducted the same amount of clustering itera- +tions (10) and then evaluated the resulting clustering and +decided whether for a given system additional iterations +were needed. +B. +Trp-cage +a. +TC5b. +For the RE simulations of the Trp-cage +the clustering scheme was run over 10 iterations and as- +signed 60.5% of all conformations to clusters. +Fig. +4 +shows an encodermap projection of all 40 replicas with +some of the most populated clusters found after 10 it- +erations and representative conformations of these clus- +ters. +Similar conformations are grouped together and +rare structures are spread out across the map. For ex- +ample, the native conformation of Trp-cage RE (33.4% + +0.1% +0.1% +0.2% +0.2% +0.1% +0.2% +0.3% +0.1% +0.1% +<0.1% +0.1% +<0.1% +<0.1% +native; 33.4% +1.5% +<0.1%8 +FIG. 5. +The most populated clusters and respective conformations of Trp-Cage TC10b [42] projected to the same 2D +encodermap space as TC5b (Fig. 4).bTop right: a histogram of the 2D projection. +of all conformations) is shown in the bottom right of the +2D map in Fig. +4. On the bottom left conformations +with one turn near the middle of the backbone are lo- +cated. +The two parts of the backbone chain of these +conformations lie right next to each other and partially +form beta-sheet structures. +Using a larger cutoff distance in the RMSD-based as- +signment of structures to the clusters (the other clusters +were generated by applying a 1.8 ˚A RMSD cutoff to the +central conformation) we obtained larger and quite dif- +fuse clusters of extended conformations (one of these clus- +ters is shown in the left part of the projection in Fig. 4 +where it is referred to by an arrow). An appropriate size +of this RMSD cutoff was defined for each fuzzy cluster +individually by computing the mean value of the largest +20% of the RMSD values between the centroid and cluster +members of the cluster identified in the current iteration +(it is equal to 5.5 ˚A for the cluster shown here). Before we +identify fuzzy clusters, we first continuously assign struc- +tures based on a fixed RMSD cutoff (1.8 ˚A for TC5b) +until one of the stopping points defined in Sec. +II D is +reached (average cluster size for TC5b). Once this stop- +ping point is reached, the RMSD cutoff is adjusted in +the way explained above and fuzzy clusters are obtained. +Thereby one ensures that all conformations that can be +assigned to well-defined clusters are removed from con- +sideration before identifying fuzzy clusters. The usage of +such a varying cutoff can be very helpful in order to iden- +tify diffuse clusters, where the members share a certain +structural motif but do not converge to a very defined +conformation, just like the cluster shown here. +From the clustering results shown in Fig. +4 one can +see that the proposed clustering workflow manages to ef- +ficiently identify structurally very well defined clusters +for the TC5b system. Over 10 clustering iterations it as- +signed 60.5% of all conformations to 260 clusters. Besides +the highly populated native state (33.4%), the algorithm +also finds very ”rare” states, which contain only a very +small amount of conformations (≤0.1%) but show never- +theless a very defined structural identity. +b. +TC10b. +Fig. 5 shows the same analysis applied +to the trajectory of the K8A mutant of TC10b Trp-cage. + +8.2% +0.5% +0.1% +0.5% +0.1% +<0.1% +0.7% +0.02% +0.1% +0.2% +<0.1% +<0.1% +1.7% +0.7% +native; 12%9 +We used the encodermap which we trained on TC5b to +project the trajectories to the same 2D space. The iden- +tification of clusters however is of course entirely inde- +pendent and unique for both cases, since the clustering +is done in the higher dimensional cc analysis space. +Notably, the backbone conformation of the native state +of this mutant is extremely similar to the one in the TC5b +system. However this biggest cluster only contains 12% +of all conformations along the trajectory compared to +the 33.4% in the case of the TC5b system. If all clus- +ters whose central conformation are within a 2 ˚A RMSD +to the native conformation are combined, we get native +conformation percentage of 16.9%. This is in excellent +agreement with the native cluster sizes reported by Deng +et al. [45], Ghorbani et al. [46] who analysed the same +Trp-cage trajectories provided by Lindorff-Larsen et al. +[42]. Furthermore our 33.4% of assigned conformations +coincide very well with the reporting of Sidky et al. [47]. +They found a total of 31% of conformations distributed +over eight metastable macrostates and the remaining 69% +as one big ”molten globule” state. +The TC10b trajectory is more disordered, this can be +seen by the more homogeneous projection in 2D space +(upper right plot in Fig. +5) and the RMSD values to +the native conformation in SI, Sec. S-III, Fig. S3. This is +also the reason why the clustering scheme assigned only +33.4% of all conformations to clusters after 10 iterations. +If more frames should be assigned to clusters, more clus- +tering iterations can be performed, the RMSD cutoff can +be increased or both can be done simultaneously (for the +Protein B system we show the results of this approach +later in the article). +However the clusters in the very center of the map +(dark blue circle) are much more compact and collapsed +compared to the clusters that were found in the similar +area of Trp-cage RE’s 2D projection. Also some of the +clusters that were found in the very bottom of the left +hand side of the map in case of the replica trajectories +(light blue circle) were not found at all in the TC10b +trajectory. The very large and diffuse cluster on the left +side of the map is present in both systems as well. +c. +Clustering directly in 2D space of TC5b. +The +clustering discussed above was done in a 20 dimensional +space after applying the cc analysis algorithm and only +displayed at a 2D projection done with encodermap. In +order to demonstrate the advantages of our approach we +also directly clustered the 2D encodermap space using the +HDBSCAN. The encodermap space that we used for this +clustering is the same space that we used to visualize the +cc analysis clustering in Fig. 4 and Fig. 5. The results +of this clustering and a few chosen clusters can be seen +in Fig. 6. In total this clustering assigned 13.5% of all +conformations to 362 clusters. The biggest cluster that +was found is the native cluster, however it only contains +0.8% of all conformations compared to the 33.4% that +were found by clustering the cc analysis space. The clus- +tering in the 2D space identifies some structurally very +well defined clusters, such as the clusters 0, 1 and 3, but +FIG. 6. +2D encodermap space of TC5b clustered with HDB- +SCAN. Representations of chosen clusters that have the same +location in the 2D map as clusters found with the clustering +scheme in Fig. 4 are shown. +also a lot of very diffuse and inhomogeneous clusters. To +quantify this inhomogeneity we computed the average of +the internal cluster RMSDs. For the TC5b system our +clustering workflow resulted in an average cluster RMSD +of 1.34 ˚A and a weighted average RMSD of 1.03 ˚A, where +weights are defined as the fraction of each cluster to all +clustered data. The average RMSD for the direct cluster- +ing in the 2D space is 2.25 ˚A and the weighted average +RMSD is 2.73 ˚A. This clearly shows that the internal +cluster RMSD variance is on average much larger when +clustering directly in the 2D space. Furthermore the clus- +tering in the 2D space itself naturally highly depends on +the quality of the 2D map. +Other than the much clearer conformational identity +of the individual clusters (shown via internal cluster +RMSDs), our clustering scheme also manages to assign +60.5% of all conformations to different clusters. +Com- +pared to that the clustering in the 2D projection only +assigned 9-14% of all conformations depending on the +choice of clustering parameters. +d. +Comparison to other clustering approaches. +For a +further assessment of our clustering scheme we have also +applied a frequently used clustering routine to the TC5b +data. In Si, Sec. S-IV and Figs. S4 and S5 the results +of applying the k-means algorithm to an 11 dimensional +PCA projection of the same CVs (pairwise Cα distances +of TC5b) are shown. +In summary, the scheme identified both structurally +very defined as well as quite diffuse clusters in considered +systems. Even though the combination of the 40 RE tra- +jectories produces a very diverse data set, the clustering +scheme manages to assign a large amount of the confor- +mations to clusters (60%). Our clustering results for the +TC10b are in a very good agreement with the findings +of other researchers [45–47]. Furthermore the compar- +ison to a clustering in the 2D space clearly shows the +superiority of using more dimensions obtained with the +cc analysis algorithm in HDBSCAN over just relying on +a low-dimensional representation alone. + +Cluster 4 +Cluster 3 +Cluster 2 +Cluster 5 +Cluster 1 +Cluster O +Cluster 610 +FIG. 7. +The 2D encodermap projection of NTL9. The projection can be approximately divided into three parts: the upper part +with the most dense areas (where the native-like states are located); the lower left and right planes divided by an unpopulated +vertical gap. The left side includes various conformations with a singular beta sheet formed mostly between the beginning +and the end of the protein. In contrast on the right side lie mostly extended conformations with multiple helices along the +backbone. Exemplary conformations of some of the most populated clusters found in each of the marked areas in the map and +their populations are shown. All clusters in the yellow circle are extremely similar to the native cluster and can be summed up +to a total of 76% of all conformations. The structures that are shown here make up 78.4% of all conformations. Top right: +Histogram of the 2D encodermap space. +C. +NTL9 +Next we examined very long (1877 µs) simulations of +NTL9 [42]. With 9.38 million frames to cluster, this sys- +tem is an ideal candidate to demonstrate how the pro- +posed algorithm copes with large amounts of data. Af- +ter 10 iterations 81% of all conformations were assigned +to clusters. +Fig. +7 shows a 2D projection made with +encodermap, where points are colored according to the +clusters found after ten iterations of the scheme and a +histogram of the 2D space in the upper right corner. In +total we found 157 clusters and assigned them 81% of all +conformations over 10 clustering iterations. +A comparison of the timeseries of the RMSD values to +the folded state to the respective data of the Trp-cage +Anton simulations (SI, Sec. S-III, Fig. S3) reveals that +the two systems exhibit very different dynamics. While in +the Trp-cage case the RMSDs show the disordered nature +of the system, in the case of the NTL9 trajectories the +RMSDs are predominantly quite low and only spike up to +larger values for rather short time periods. This suggests +that the NTL9 system resides in a native-like state for +the majority of the simulated time. This is confirmed +during the very first iteration of the clustering scheme. +There we found two clusters which make up for 75.8% of +all conformations. +This example also nicely illustrates how the iterative +clustering approach can be efficient in identifying clus- +ters of very different size and density (highly populated +native states and low populated clusters). After finding + +1.3% +0.2% +0.2% +0.1% +1.4% +0.4% +>0.01% +native; 74.5% +0.3% +>0.01% +>0.01% +cumulative ~0.1%! +0.03% +0.03% +cumulative ~0.01% +>0.01%11 +FIG. 8. +Register-shifted states found in the NTL9 trajecto- +ries 0, 2 and 3. The residues which form the beta sheets in +the native state are colored based on their residue ID. +and removing the first two clusters (75.8% of the data) +the clustering algorithm becomes much more sensitive +towards the less dense areas in the CV-space in the fol- +lowing clustering iterations. +We compared our clustering results with other publi- +cations analyzing the NTL9 trajectories from Ref. [42]. +Mardt et al. [29] applied the VAMPnets to trajectory 0 +and found in total 89.1% of folded, native like confor- +mations. If we take the clusters we found by analysing +the trajectories 0, 2 and 3 and evaluate the conforma- +tions stemming from trajectory 0 (trajectory 0 resides +in the native-like state for a larger fraction of the simu- +lated time; see RMSD plots in SI, Sec. S-III, Fig. S3, the +amount of folded, native-like conformations we find is in +very good agreement with [29]. Furthermore Schwantes +and Pande [44] reported the finding of three “register- +shifted” states, which are very low populated and there- +fore very hard to find. “Register-shifted” refers to the +identity of the specific residues involved in forming the +beta sheet structure in the native-like states (residues 1- +6, 16-21 and 35-39). With our method we identified six +different register-shifted states in the NTL9 trajectories +0, 2 and 3 (see Fig. 8). +The states 0, 1 and 2 are the ones which were also +found in [44]. To our knowledge states 3, 4 and 5 have +not been reported yet. +In state 0 the central of the +three beta-sheet strands is shifted downwards, whereas +in state 2 the rightmost strand is shifted downwards. +In state 1 both the middle and the rightmost strands +are dislocated compared to the native state. State 3 is +similar to state 1 in the fact that both the middle and +the rightmost strands are shifted, however in state 3 the +rightmost strand is shifted upwards and not downwards +like in state 1. Among these six states state 4 is unique +since there the rightmost strand is turned by 180 degrees. +Finally state 5 differ from other states in having an extra +helix along the chain between the leftmost and the mid- +dle strand. Because of this additional helix the leftmost +strand is extremely shifted compared to the native state. +The identification of these register-shifted states high- +lights one asset of the proposed workflow. It is able to +find both very large states (native, 74.5%) as well as very +low populated clusters (<0.001%) in the same data set. +D. +Protein-B +The last system we analysed is Protein B. This sys- +tem does not have a very stable native state, instead +the three helices can move against each other relatively +freely. This can be seen in the timeseries of the RMSD to +the closest experimental homologue (1PRB) shown in SI, +Sec. S-III, Fig. S3. There are no extended periods where +the values are stable over some time, meaning there are +no large free-energy barriers separating the various acces- +sible conformations and thus the system constantly tran- +sitions into different conformations. This has also been +found in [42], where authors stated that they were un- +able to identify a free-energy barrier between folded and +unfolded states for Protein B (tested over many different +reaction coordinates). +Such a highly dynamic system is very challenging for a +conformational clustering. Here we want to show where +our algorithm has its limitations and what can be done +to get a satisfactory clustering result. Fig. +9 gives an +overview of some of the clusters found after ten iterations +of the scheme. These clusters include only 20% of the +Protein B trajectory and thus 80% of all conformations +are still unclustered. +In order to have more data assigned to clusters two pa- +rameters can be adjusted. First, the RMSD cutoff value +can be increased and thereby more conformations can be +assigned to the found clusters. In this specific case this +adjustment is justified, since due to the low free-energy +barriers between different states, the individual clusters +are not as sharply defined in terms of their conforma- +tions. In the 10 clustering iterations which are shown in +Fig. 9 we used a RMSD cutoff of 3.0 ˚A. In a second run +we increased it to 3.5 ˚A. This resulted in an assignment +of 31% of all conformations to generally more loosely de- +fined clusters. +A second approach is to increase the amount of clus- +tering iterations. For the first ten clustering iterations of +previously analysed systems, we tuned the clustering pa- +rameters manually. This includes the choice of the num- +ber of cc analysis dimensions, as well as the min samples +and min cluster size parameters of HDBSCAN. However +such a manual adjustment of the parameters is of course +not feasible for automating the script in order to perform +many more clustering iterations. +Since the amount of +cc analysis dimensions needs to be very rarely changed +once a suitable amount has been identified in the first +clustering iteration, the automation of the script only re- +lies on the choice of the HDBSCAN parameters. Once the +amount of clusters found in a single iteration falls below a +certain threshold (10 clusters in this case), the numerical + +Native State +State 4: ~0.001% +5 +State 1: r0.1% +State 3: 0.01% +State 0: ~0.1% +State 2; ~0.01% +State 5; ~0.1%12 +FIG. 9. +Protein B: Exemplary conformations of some of the most populated clusters found for the Protein B system after 10 +clustering iterations and their populations; Top right: Histogram of the 2D encodermap space. +values of the min samples and min cluster size parame- +ters of HDBSCAN are slightly decreased. This leads to +the detection of smaller clusters that have not been iden- +tified before. By applying this automation approach after +the first 10 iterations to Protein B and using a RMSD +cutoff of 3.5 ˚A, we could assign 44.3% of all conforma- +tions to clusters over 100 iterations, which took roughly +15 hours on our workstation. +IV. +DISCUSSION +The Trp-cage system (TC5b) is a relatively small pro- +tein which has a quite stable native conformation. The +combination of 40 temperature RE trajectories however +gives a very diverse data set including (under standard +conditions) very improbable high-energy conformations. +Over ten iterations the algorithm managed to assign +60.5% of all conformations to clusters, which took on av- +erage 360 min per iteration over all CPU threads (15 min +per iteration on a standard office machine with 24 CPU +threads). Table I shows the clustering performance for +the four systems discussed here. By switching the gen- +erally static RMSD cutoff to a varying cutoff we could +show that the algorithm can both generate conforma- +tionally very defined clusters as well as quite diffuse. +The conformations assigned to such loose clusters share +a general structural motif. The ability to identify both +of these cluster types is one of the advantages of the +proposed algorithm. Furthermore we demonstrate that +the clustering workflow is able to directly compare dif- +ferent systems (even if they slightly differ structurally), +by projecting them to the same 2D map using the en- +codermap algorithm. +This enables a direct and visual +comparison of the sampled phase-spaces of different tra- +jectories and their respective identified states. By com- +paring the clustering result where the clustering is done +in a 20-dimensional cc analysis space and then projected +to a two-dimensional space to a clustering where the +clusters are purely found in a 2D encodermap space, we +prove an advantage using more dimensions and combine +cc analysis with encodermap. The scheme created clus- + +0.1% +0.3% +0.1% +>0.1% +>0.1% +0.4% +0.3% +3.2% +1.5% +most populated; 5.2%13 +ters with a much clearer structural identity (lower RMSD +variance), while being much less dependent on the quality +of the 2D map. +We analysed long (9.38 million frames) trajectories of +NTL9 to show how the proposed scheme copes with very +large amounts of data. On average the algorithm needed +1320 min of computation time over all CPU threads per +iteration (55 min per iteration on our office machine). +Since this system also has one hugely populated native- +state, it is also a nice example to demonstrate an ad- +vantage of the iterative clustering. +After the clusters +with the native states are removed from consideration, +the algorithm becomes much more sensitive towards less +populated areas in the following iterations. +Applying +this approach we could identify three very low popu- +lated register-shifted states, which have been reported +before [44], and three not yet seen register-shifted states. +Lastly we looked at is Protein B, which is a highly +dynamic system. +To analyse this 1.04 million frames +trajectory it took on average 288 min of computation +time per iteration (12 min per iteration on our office +machine). +This system has no large free-energy barri- +ers separating the various conformations, which makes +it very difficult to cluster. This was confirmed by the +fact that after ten clustering iterations only 20% of all +conformations could be assigned to clusters. However by +increasing the RMSD cutoff from 3.0 ˚A to 3.5 ˚A we could +already increase the amount of assigned conformations to +31%, which of course resulted in slightly less structurally +defined clusters. It is also possible to automate the clus- +tering and run until a certain amount of conformations +are assigned to clusters or until a given number of itera- +tions is reached. In this specific case we ran the scheme +for 100 automated iterations (≈15 hours), during which +44.3% of the conformations were assigned to clusters. +For all considered systems the proposed workflow was +able to identify defined clusters at the cost of leaving +some amount of the trajectories unassigned. As we have +shown here, the rest of the structures does not belong to +any specific clusters and can be considered as unfolded +or transition states. We intentionally do not propose any +additional steps to assign or classify those conformations +as it is highly dependant on the intended application of +the data. For example in case the data is used to build +subsequent kinetic models the rest of the points can be +assigned to the nearest (e.g. in simulation time) cluster +using methods such as PCCA+ analysis [48], or defined +as a metastable transition state as in Ref. 47. It can also +be defined as noise and used as discussed in Ref. 49. +All performance data is shown in Table I and was ob- +tained by running the clustering scheme script on the +office workstation described in SI, Sec. S-V. The pro- +posed workflow is, however, highly parallelizable, since +the computationally most expensive step is the assign- +ment of additional data points to the initially identified +clusters in the small subset based on the convex hull and +the RMSD criterion. +If a large amount of CPU cores +are available, the 2D encodermap projection array can +be split by the amount of cores and the assignment can +thereby be run in parallel which leads to a significant +speed up. +The convex hull around the clusters identified in the +small subset is used to reduce the amount of RMSD com- +putations that have to be performed when assigning ad- +ditional conformations in each clustering iteration. This +however might in principle lead to the exclusion of data +points that might otherwise have been assigned to some +of the clusters. In order to get an idea of the magnitude +of this “loss” of potential cluster members, we computed +the RMSD of all data which was labeled as noise (623,000 +conformations; 39.5%) to each of the cluster centers of +TC5b (260 clusters). This computationally very expen- +sive task took an additional 5 hours on our working ma- +chine. We found that 42,000 conformations (2.7%) were +not assigned to the identified clusters due to the con- +vex hull criterion. When keeping in mind that the entire +10 iteration clustering process took 2.5 hours, the ”loss” +of 2.7% of unclustered data can be considered a worthy +trade-off. +Another point to consider is that due to the convex hull +criterion clusters can be split. If data points that would +be assigned to a certain cluster by reason of the RMSD +criterion lie outside of the convex hull, they could be iden- +tified as another cluster in one of the following clustering +iterations. In such cases it can make sense to merge these +clusters in hindsight, due to their very similar structural +identity. In order to showcase such a merge, we again +analysed TC5b. We computed the RMSDs between all +of the 260 central cluster conformations and merged all +clusters that had a RMSD of ≤ 1 ˚A. This resulted in a re- +duction to 201 clusters with only very marginal influence +on the average internal cluster RMSDs. +The code for the encodermap algorithm is avail- +able on the following github page https://github. +com/AG-Peter/encodermap. +The +cc analysis +code +can +be +found +under +https://strucbio.biologie. +uni-konstanz.de/xdswiki/index.php/Cc_analysis. +V. +CONCLUSION +We +developed +a +clustering +scheme +which +com- +bines two different dimensionality reduction algorithms +(cc analysis and encodermap) and the HDBSCAN in an +iterative approach to perform fast and accurate clus- +tering of molecular dynamics simulations’ trajectories. +The cc analysis dimensionality reduction method was +first applied to protein simulation data. +The method +projects collective variables to a usually relatively high- +dimensional (∼10-40 dim) unit sphere, separating noise +and fluctuations from important structural information. +Then the data can be efficiently clustered by density +based clustering methods, such as HDBSCAN. The it- +erative application of HDBSCAN allows to account for +the inhomogeneity in population and density of the pro- +jected points, which is very typical for protein simulation + +14 +data. As cc analysis relies on the calculations of correla- +tion matrices between each frame, this drastically limits +the amount of data one can project simultaneously. To +allow processing of long simulation trajectories we in- +cluded encodermap to the scheme. +In addition to the +obvious advantage of the two-dimensional visualisation +it is used – in combination with a RMSD-based accep- +tance criterion – for a fast structure-based assignment of +additional points to the clusters initially identified in the +higher dimensional projection done with cc analysis. To +demonstrate the accuracy and performance of the pro- +posed scheme we applied the clustering scheme to four +test systems: replica exchange simulations of Trp-cage +and three long trajectories of a Trp-cage mutant, NTL9 +and Protein B generated on the Anton supercomputer. +By applying the scheme to these four test systems we +could show that: the algorithm can efficiently handle +very large amounts of data, that it can be used to com- +pare the clusters of structurally different systems in one +2D map, and that it can also be applied to cluster sys- +tems which do not have very stable native states and +are therefore intrinsically very difficult to cluster confor- +mationally. +Furthermore the algorithm is able to find +clusters independent of their size. By varying a RMSD +cutoff both conformationally very well defined clusters, +as well as fuzzy clusters, whose members only share an +overall structural motive, can be identified. +VI. +SUPPORTING INFORMATION +Supporting Information (PDF) includes: +(S-I): Methods to chose parameters for cc analysis and +HDBSCAN. +(S-II): Stopping criteria for the clustering workflow. +(S-III): RMSD plots of trajectories for Trp-cage, Pro- +tein B and NTL9. +(S-IV): Comparison of the proposed clustering work- +flow to PCA and k-means clustering for Trp-cage (TC5b). +(S-V): Workstation specifications. +VII. +ACKNOWLEDGEMENTS +This work was supported by the DFG through +CRC 969. We also greatly appreciate the computing time +on bwHPC clusters which was used to produce the Trp- +cage TC5b trajectories. Furthermore we would like to +thank the D.E. Shaw research group for providing the +Trp-cage, NTL9 and Protein B trajectories. +[1] R. O. Dror, +C. Young, and D. E. 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Phys. +145, 164104 (2016). + diff --git a/M9E3T4oBgHgl3EQfYwr8/content/tmp_files/load_file.txt b/M9E3T4oBgHgl3EQfYwr8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc3629c60ac49888c19a2cb03972c799b10e325c --- /dev/null +++ b/M9E3T4oBgHgl3EQfYwr8/content/tmp_files/load_file.txt @@ -0,0 +1,1030 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf,len=1029 +page_content='Fast conformational clustering of extensive molecular dynamics simulation data∗ Simon Hunkler,1 Kay Diederichs,1 Oleksandra Kukharenko,2, † and Christine Peter1, ‡ 1Department of Chemistry, University of Konstanz 2Theory Department, Max Planck Institute for Polymer Research (Dated: January 12, 2023) We present an unsupervised data processing workflow that is specifically designed to obtain a fast conformational clustering of long molecular dynamics simulation trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In this approach we combine two dimensionality reduction algorithms (cc analysis and encodermap) with a density- based spatial clustering algorithm (HDBSCAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The proposed scheme benefits from the strengths of the three algorithms while avoiding most of the drawbacks of the individual methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Here the cc analysis algorithm is for the first time applied to molecular simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Encodermap complements cc analysis by providing an efficient way to process and assign large amounts of data to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The main goal of the procedure is to maximize the number of assigned frames of a given trajectory, while keeping a clear conformational identity of the clusters that are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In practice we achieve this by using an iterative clustering approach and a tunable root-mean-square-deviation- based criterion in the final cluster assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This allows to find clusters of different densities as well as different degrees of structural identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' With the help of four test systems we illustrate the capability and performance of this clustering workflow: wild-type and thermostable mutant of the Trp-cage protein (TC5b and TC10b), NTL9 and Protein B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Each of these systems poses individual challenges to the scheme, which in total give a nice overview of the advantages, as well as potential difficulties that can arise when using the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' INTRODUCTION With the ever-growing power of computers over the last decades, researchers in the field of molecular dynam- ics (MD) have gotten access to increasingly long trajec- tories and thereby to increasingly large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The introduction of supercomputers which are specifi- cally designed to generate MD trajectories (Anton [1] and Anton 2 [2]) is only the latest high point in this development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore, new sampling methods [3, 4] as well as distributed computing projects, such as Fold- ing@home [5], have contributed to a massive increase in generated simulation trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' With this increasing amount of data it is essential to have powerful analysis tools to process and understand underlying systems and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' There is a rapid increase in application of unsupervised machine learning methods to analyze molecular simula- tion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Two of the most used families of analysis techniques are clustering and dimensionality reduction (DR) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' They help to find low-dimensional subspaces in which important aspects of the original data are preserved and to group the data based on a given similarity measure/metric and thereby gain a bet- ter overview and understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In practice, most of the times clustering and DR methods are used in com- bination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The DR algorithms can be roughly divided into: linear methods (the most known are principal component analysis (PCA) [6, 7] and time-lagged in- ∗ Copyright 2023 Hunkler, Peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This article is distributed under a Creative Commons Attribution (CC BY) License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' † kukharenko@mpip-mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='de ‡ christine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='peter@uni-konstanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='de dependent component analysis (TICA) [8, 9]), nonlin- ear methods (kernel and nonlinear PCA, multidimen- sional scaling (MDS) [10, 11] and MDS-based methods like sketch-map [12], isomap [13], diffusion maps [14, 15] or UMAP [16], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=') and autoencoder-based approaches like (encodermap [17, 18], time-autoencoder [19], vari- ational autoencoders [20] and Gaussian mixture varia- tional autoencoders [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In terms of clustering algo- rithms, there are again a wide range of different methods: K-Means [22, 23], spectral-clustering [24], DBSCAN [25], density-peak clustering [26], CNN-clustering [27], root- mean-square deviation (RMSD) based clustering [28], neural-networks-based VAMPnets [29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For a com- prehensive overview of unsupervised ML methods com- monly used to analyse MD simulation data we refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Even from this incomplete list of available methods it should become obvious that there are a lot of differ- ent clustering, as well as DR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' All these meth- ods have their individual strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' But there are still open challenges in the successful usage of the listed methods for processing simulation data with high spatial and temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is connected either to the proper choice of hyper-parameters (such as the number of dimensions for DR methods, the num- ber of expected states for some clustering algorithms, neural-networks architectures, different cut-offs, corre- lation times, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' ), expensive optimisation steps or the amount of data which could be processed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In this work we present a data processing scheme which combines three different algorithms in one workflow to create a powerful clustering machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It tackles a num- ber of the mentioned challenges as it has a way to define an appropriate lower dimensionality of the data, does not require a priory information about the expected number arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='04492v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='chem-ph] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Data processing routine presented in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' of states and it is fast in processing extensive MD trajec- tories with both a very high dimensionality and a large number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It is specifically designed to find conformational clusters in long molecular simulation data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We use two different DR algorithms, namely an al- gorithm called “cc analysis” and the encodermap algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The cc analysis method belongs to the family of the MDS-based techniques and was first introduced for the analysis of crystallographic data [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Here it is used for the first time for projecting data of protein con- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The dimensionality of the cc analysis-space which is typically required is more than two (10 to 40 for the systems shown in this work) and the amount of data, which can be efficiently projected simultaneously is limited by the available memory (about 50000 frames for a 72 GB workstation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To process much longer trajecto- ries and to obtain a two-dimensional representation we use the second DR algorithm – encodermap [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Its loss function however consist of two parts: the autoencoder loss and a MDS-like distance loss, which introduces an interpretability to the resulting 2D representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' More- over, once the encodermap network is trained, the en- coder function can be used to project data to the 2D map in an extremely efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We use encodermap to project data into 2D and for a fast assignment of the addi- tional members to the clusters defined in the cc analysis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Finally we use the HDBSCAN algorithm [34] to cluster the data in the cc analysis space and then visu- alize the resulting clusters in the 2D encodermap space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' HDBSCAN is a combination of density and hierarchi- cal clustering, that can work efficiently with clusters of varying density, ignores sparse regions, and requires a minimum number of hyper parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We apply it in a non-classical iterative way with varying RMSD-cutoffs to extract the protein conformations of different similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The combination of these three algorithms allows us to leverage their different strengths, while avoiding the drawbacks of the individual methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Subsequently we will show how the scheme performs on long MD trajecto- ries of wild-type and mutated Trp-cage with native and misfolded meta-stable states (208 µs and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2 µs long sim- ulations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' really extensive simulations of NTL9 (1877 µs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' and Protein B, where only a small percent of the simu- lation data (5%) is in the folded state (104 µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' cc analysis For dimensionality reduction, we use an cc analysis in- troduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 31, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This algorithm was originally developed to analyse crystallographic data, where pres- ence of noise and missing observations pose a challenge to data processing in certain experimental situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The method separates the inter-data-set influences of ran- dom error from those arising from systematic differences, and reveals the relations between high-dimensional in- put features by representing them as vectors in a low- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Due to this property we expected it to be highly applicable to protein simulation data, where one seeks to ignore the differences arising from random fluctuations, and to separate the conformations based on systematic differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In the course of the manuscript we show that this assumption proved to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The cc analysis algorithm belongs to the family of MDS methods [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Its main distinction is that it min- imizes the sum of squared differences between Pearson correlation coefficients of pairs of high-dimensional de- scriptors and the scalar product of the low-dimensional vectors representing them (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The proce- dure places the vectors into a unit sphere within a low- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Systematic differences between the high-dimensional features lead to differences in the an- gular directions of the vectors representing them, and purely random differences of data points lead to different vector lengths at the same angular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The algo- rithm minimizes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' iteratively using L-BFGS [35], the Full trajectory Define high-D CVs Encodermap 2D projection Expand clusters based on RMSD and 2D projection cc_analysis HDBSCAN Select random subset (up to 25000 frames) Remove For trajectories < 25000 frames clustered frames3 expression Φ(x) = N−1 � i=1 N � j=i+1 (rij − xi · xj)2 (1) as a function of x, the column vector of the N low- dimensional vectors {xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Here rij is the correlation coefficient between descriptors i and j in the high- dimensional space and xi · xj denotes the dot product of the unit vectors xi and xj representing the data in the low-dimensional space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' N is the number of observations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' protein conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The output of cc analysis is the N low-dimensional vectors {xk}, and the eigenvalues of the xxT matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To understand why this is a sensible approach, one can think about obtaining the least squares solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (1) algebraically by eigenanalysis of the matrix r = {rij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In that case one would have to solve xxT = r where r is the matrix of the correlation coefficients rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The n strongest eigenvalue/eigenvector pairs (eigenvec- tors corresponding to the largest eigenvalues) could then be used to reconstruct the N vectors xi, which are lo- cated in a n-dimensional unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The systematic differences between the input data are thereby shown by the different angular directions in this low-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This approximation is meaningful because in general the Pearson correlation coefficient can be written as a dot product between two vectors (after subtraction of the mean and dividing by the standard deviation to scale the vectors to unit length) and is equal to the cosine of the angle between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Hence, in an ideal scenario, �N i,j xi · xj can exactly reproduce the high-dimensional correlation coefficient matrix and Φ(x) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (1) would be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The length of the vectors is less important than the angle between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The latter has an inherent meaning: two high-dimensional feature vectors with a correlation coefficient of zero between them would be projected to unit vectors at 90◦ angles with respect to the origin, and two feature vectors with a correlation coefficient of one would have a corresponding angle of zero degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Despite the generality of the cc analysis approach, by now it was only applied to crystallographic data [36, 37]) and protein sequence grouping [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Here we present a first application of cc analysis for protein simulation data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Encodermap To accelerate the processing of large datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' from extensive simulations, in addition to cc analysis, we make use of one more dimensionality reduction technique – en- codermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It was developed by Lemke and Peter [33] and is used here for fast assignment of data points to clusters as will be explained in details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The method combines the advantages of a neural network au- toencoder [17] with a MDS contribution, here the loss function from the sketch-map algorithm [12] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This combination is exceptionally efficient for projecting large simulation data to the two-dimensional representa- tions: the sketch-map loss function allows to concentrate only on relevant dissimilarities between conformations (ignoring thermal fluctuations and coping with the large dissimilarity values caused by the data’s high dimension- ality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore the autoencoder approach allows to avoid complex minimisation steps of the sketch-map pro- jection and to process huge amounts of data in a very short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Schematic description of encodermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It has an architecture of the classic autoencoder consisting of two neu- ral networks (encoder and decoder) with the same number of layers and neurons in each layer connected through the bottle- neck layer with two neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In addition to autoencoder loss La(X, ˜ X) encodermap loss has a term with the sketch-map loss function Ls(X, x), which improves the quality of two- dimensional projection obtained in the bottle-neck layer (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The encodermap loss function Lencodermap (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (2)) is a weighted sum of the autoencoder loss Lauto (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (3)) and the sketch-map loss function Lsketch (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (4)), which emphasizes mid-range distances by transforming all dis- tances via a sigmoid function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Lencodermap = kaLauto + ksLsketch + Reg, (2) Lauto = 1 N N � i=1 D(Xi, ˜Xi), (3) Lsketch = 1 N N � i̸=j [SIGh(D(Xi, Xj)) − SIGl(D(xi, xj))]2, (4) where ka, ks are adjustable weights, Reg is a regular- ization used to prevent overfitting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' N is a number of data points to be projected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' D(·, ·) is a distance be- tween points, X is a high-dimensional input, x is a low- dimensional projection (the bottleneck layer);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' SIGh and SIGl are sigmoid functions of the form shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' SIGσ,a,b(D) = 1 − (1 + (2 a b − 1)(D σ )a)− b a , (5) CVs CVs 2D projection neural neural network network encoder decoder X4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Application of HDBSCAN on a toy data set with three clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' i) Example for the computation of the MRD for two points (red and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The red and blue circles in- dicate the farthest distance to the 5 nearest neighbours for both points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' One can see that the distance between the red and blue points (green line) is larger than both the radii of the blue and the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Therefore in this case the green line distance is chosen as MRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' ii) The minimum spanning tree based on the MRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' iii) The cluster hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' iv) The condensed clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' where a, b and σ are parameters defining which distances to preserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) The HDBSCAN [34, 39] can be approached from two different sides: it can be described as a hierarchi- cal implementation of a new formulation of the origi- nal DBSCAN [25] algorithm called DBSCAN* by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Campello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [34] or it can be formulated as a ro- bust version of single-linkage clustering with a sophisti- cated method to obtain a flat clustering result, as done by McInnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Here we describe it through the second approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In the first step the algorithm introduces the so-called mutual reachability distance (MRD) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (6)), which transforms the space to make sparse points even sparser but does not significantly change the distance between already dense points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Dmreach−k(xi, xj) = max{corek(xi), corek(xj), D(xi, xj)}, (6) where x are points being clustered, k is a constant which determines a number of nearest neighbouring points, corek(x) is a function, which finds the maximum distance between a point x and its k nearest neighbours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' D(·, ·) is a distance between two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The maximum of three distances is selected as the MRD (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 3 i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In the next step the minimum spanning tree based on the MRDs is build via Prim’s algorithm [40] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 3 ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is done by starting with the lowest MRD in the data set and connecting the two points by a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In the following steps always the next nearest data point to the existing tree, which is not yet connected, is added to the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Once the minimum spanning tree is generated the clus- ter hierarchy can be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is done by first, sorting the edges of the tree by distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Then the algorithm iterates over the edges, always merging the clusters with the smallest MRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The result of this procedure can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 3 iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In order to extract a flat clustering form this hierarchy, a final step is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In this step the cluster hierarchy is condensed down, by defining a minimum cluster size and checking at each splitting point if the new forming cluster has at least the same amount of members as the minimum cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If that is the case, then a new cluster is accepted, if not then the data points splitting off are considered noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The condensed tree of a toy system can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 3 iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Introduction of a new clustering workflow In this article we present a data processing routine which we found to be extremely efficient for large molec- ular dynamics simulation trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It relies on the three algorithms introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' A schematic descrip- tion is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In this workflow a given data set is clustered iteratively until either a specified amount of data points are assigned to clusters or a maximum number of iterations have been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 1 illustrates the sequence of data processing steps along the clustering workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In the first step a high-dimensional collective variable (CV) is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For all systems that are shown in this article all pair- wise distances between the Cα atoms were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Af- ter a CV has been chosen, for trajectories containing more than 25,000 frames, encodermap is trained on all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Thereby we obtain a function which can be used to project data very efficiently to the newly generated 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In parallel, a random subset from the entire data set is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The reason to use such a sub- set is a limitation that comes with the cc analysis di- mensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' II A the cc analysis algorithm works with the correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This means that the Pearson correlation coefficients of the selected CV (here the pairwise c-alpha distances) are calculated for all unique pairs of frames, and used as in- put to cc analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However the larger a data set is, the larger the correlation coefficient matrix will be, until it is no longer efficient to work with that matrix due to very long computation times as well as memory issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Therefore a subset is created, by randomly selecting up to 25,000 data points from the entire data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This sub- set is then used in the cc analysis dimensionality reduc- tion to project the high dimensional CVs (between 190 and 1081 dimensions for the systems in this article) to a ii) dmreach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='25 云 reachal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='10 Mutual 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='05 iii) iv) 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='25 6 5 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='20 5 points 4 lue 10 60 of val m Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='10 ^ 15 40 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='05 20 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='00 25 0 05 lower dimensional subspace (20 to 30 dimensions for the systems in this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The choice of the appropriate amount of reduced dimensions is done by searching for a spectral gap among the cc analysis eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Once the cc analysis space has been identified, a clustering is generated by applying the HDBSCAN algorithm to that lower dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' A detailed description on how to choose the dimensionality for cc analysis and the pa- rameters for HDBSCAN is given in the supporting infor- mation (SI), Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We use two different DR algorithms in the workflow due to the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For once, the cc analysis algorithm is used to project the smaller subsets to a still comparably high-dimensional subspace, which holds more information compared to the 2D projection of en- codermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This higher dimensional subspace is therefore very well suited to be the clustering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Once the data subset is clustered in the cc analysis space, the 2D encodermap space is used to assign the points that were not a part of the subset to the found clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The 2D projection is very well suited to do a fast assignment of additional points from the data set, as well as to serve for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Additionally, encodermap is able to project huge data sets very time-efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Hence, the generated 2D projection of a given data set can be used to avoid the main disadvantage of the cc analysis algorithm in the way we use the algorithm here, which is having to use subsets of the data due to memory is- sues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In order to circumvent this disadvantage, we build a convex hull in the 2D space for each cluster that was found in the cc analysis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If an unassigned point lies inside a convex hull, the RMSD to the central conforma- tion of that cluster is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In case the RMSD is inside a given cutoff, the data point is considered to be part of that cluster, else it is not assigned to the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This RMSD cutoff is chosen by taking the weighted mean of all average internal cluster RMSDs 1 of the first clustering iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We found that this procedure gen- erates structurally quite well defined clusters with a low internal cluster RMSD since the RMSD criterion is based on well defined conformational states that emerged from cc analysis combined with HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However there is also the possibility to identify more fuzzy clusters that only share a general structural motif by using a larger RMSD cutoff for the assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' An example of the iden- tification of such fuzzy clusters is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' By introducing a RMSD criterion in the last step, we force the clustering to only include structurally very sim- ilar conformations in the respective clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' There are of course various other clustering algorithms, which group MD data sets based on their RMSD values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' an imple- mentation [28] in the GROMACS software package [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Such RMSD-based clustering algorithms have been used in the MD community for at least 20 years now and they 1 By the average internal cluster RMSD we mean the average RMSD of all conformations to the cluster centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' are a very obvious choice for conformational clusterings of MD trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' They directly compare the positions of specified atoms in various conformations of a molecule and then group the individual conformations along the trajectory using a cutoff value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However these methods generally rely on the full RMSD matrix of a given data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For larger trajectories it becomes almost infeasible to compute these matrices due to extremely long com- putation times as well as memory issues that arise when working with such large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In our workflow we can circumvent these issues by only having to compute the RMSD between the coordinates of Cα atoms of the central conformations of each cluster and the data points that lie inside the convex hull of the respective clusters in the 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In case a given system has less then about 50,000 frames, it is in principle also possible to omit the en- codermap part, since the cc analysis algorithm is able to handle the entire data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If this approach is chosen, the user can either entirely skip the RMSD criterion, or the members of clusters that are found in the cc analysis space can still be accepted/rejected based on a RMSD cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' An advantage of using both the cc analysis algo- rithm and the encodermap algorithm together is the pos- sibility to check the dimensionality reduction steps on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Since the clustering is done in one DR space, but vi- sualized in the other, narrow and well defined clusters in the 2D space indicate that the 2D map separates the dif- ferent conformational clusters nicely and that therefore the chosen encodermap parameters were well selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Our clustering scheme is not very dependent on the quality of encodermap projection, as it is used only to as- sign additional structures to already identified clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Since the clustering itself is done in the higher dimen- sional cc analysis space and the final cluster assignment uses a RMSD cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The only requirement that the scheme poses towards the 2D map is that similar con- formations are located close to each other in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is achieved by the MDS-like distance loss part of the overall loss function of encodermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Remaining points which were not assigned to any clus- ter after one clustering iteration are then used as a new pool of data, from which the new random subset is build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This whole cycle is repeated until a certain amount of data points are assigned to clusters or until a certain number of clustering iterations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To decide on a stopping point for the iterative procedure we rely on two possible convergence criteria: either a percentage of assigned conformations or average cluster sizes found at an iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If we observe a plateau in the percent- age of unassigned data points during several successive iterations the clustering procedure is stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Due to the design of our workflow, the average cluster size of newly added clusters will decrease with each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Therefore, the average size of newly added clusters or the convergence of this property during successive itera- tions can also be used as a stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Examples are shown in SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-II, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 6 Trp-cage RE (TC5b) Trp-cage Anton (TC10b) NTL9 Protein B Trajectory length in µs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2 208 1877 104 Number of frames 1,577,520 1,044,000 9,389,654 520,250 Input CVs dimensionality 190 190 703 1081 Number of cc analysis dimensions 20 20 20 30 Average iteration time on our local workstation (see SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-V) [min] 15 18 55 12 Average iteration time over all used CPU threads [min] 24 x 15 = 360 24 x 18 = 432 24 x 55 = 1320 24 x 12 = 288 Frames assigned to clusters after 10 iterations 60% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='9% 20% Total CPU time over all iterations [min] 3600 4320 13200 2880 TABLE I: Proteins analysed in this study and performance overview of the clustering scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Description of the proteins’ trajectories used for the analysis In order to illustrate the capability and performance of the proposed scheme, we chose four test systems: 40 tem- perature replica exchange (RE) trajectories of the Trp- cage protein (TC5b) analysed in the original encodermap paper [33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' the other three systems are long trajectories of Trp-cage (TC10b), NTL9 and Protein B simulated by the Shaw group on the Anton supercomputer [42] and gen- erously provided by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The four systems are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For all the systems we chose distances between Cα atoms as the input collective variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The first protein we analyse in this work is the Trp- cage system (TC5b) (Trp-cage RE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It is a comparatively small protein (20 residues) which has a very stable native state when simulated at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The combi- nation of 40 temperature replica exchange trajectories (temperature range from 300 to 570 K, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2 µs of simu- lation time, 1,577,520 frames) give a very diverse mix- ture of structures including trajectories where the sys- tem is very stable and barely moves away from the na- tive state, as well as highly disordered trajectories where high-energy conformations are visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This combination of conformations makes the data set extremely diverse and complicated for the analysis due to the high num- ber of expected clusters with extremely varying size and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Secondly we consider the K8A mutant of the ther- mostable Trp-cage variant TC10b (Trp-cage Anton) sim- ulated by Lindorff-Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [42] (208 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 1,044,000 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This simulation was run at 290 K and produced a much more disordered trajectory compared to the low temperature replica simulations of the TC5b system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' De- spite the fact that TC5b and the K8A mutant of TC10b have slightly different amino acid sequences, we use the same trained encodermap to project both systems in the same 2D map (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 5), since both sys- tems have the same number of residues and therefore the same dimensionality of CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This offers the opportunity to demonstrate that different systems can be compared to each other very nicely when projected to the same 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Next we probed our clustering scheme with extremely long (1877 µs 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 9,389,654 frames) simulations [42] of the larger (39 amino acids) N-terminal fragment of ribosomal protein L9 (NTL9) which has an incredibly stable native state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Besides the possibility to show how the algorithm deals with this extremely large data set, the system has also been studied by several other researchers [29, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This allows us to compare our results to their findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Schwantes and Pande [44] reported on very low pop- ulated states which involve register-shifts between the residues that are involved in the formation of the beta sheet structures of NTL9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This opens the opportunity to show whether our clustering workflow is able to iden- tify both very large states, as well as extremely lowly populated states in the same data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Lastly we chose to analyse the protein B simulations (104 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 520,250 frames) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Compared to the afore- 2 We used the trajectories 0, 2 and 3 according to the nomenclature of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We have not used trajectory 1 because the topology file for this specific trajectory differs slightly form the other three in terms of the order and the numbering of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This issue has also been reported by other researchers [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 中7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Trp-Cage TC5b (40 temperature RE trajectories): Exemplary conformations of the most populated clusters found in each of the areas indicated by coloured circles and their populations in percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The cluster representatives show the average secondary structure over the entire cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The clusters are coloured randomly, the colours repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Therefore clusters that have the same colour but are separated in the 2D space contain different conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The depicted clusters hold 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% of all conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Most of the remaining 24% of conformations that have been assigned to clusters are slight variations of the native structure and are not shown here due to visibility reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The cluster that is referred to by an arrow is one of the fuzzy clusters that were generated by increasing the RMSD cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Top right: a histogram of the 2D encodermap space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' mentioned proteins protein B does not have a single very stable state, instead three helices can move quite easily against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This leads to a broad conformational space, where the energy barriers between the individual states are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Therefore the individual confor- mational states are not as easily separable and rather fade/transition into each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Taking into account the long simulation time this system is very hard to cluster conformationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To demonstrate how our clustering scheme works we chose to apply it to these four systems that pose very diverse challenges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' an extremely large data set, both highly and very lowly populated states in the same data, differences in the amount of folded/unfolded conforma- tions along the trajectories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For each of the systems we initially conducted the same amount of clustering itera- tions (10) and then evaluated the resulting clustering and decided whether for a given system additional iterations were needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Trp-cage a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' TC5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For the RE simulations of the Trp-cage the clustering scheme was run over 10 iterations and as- signed 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% of all conformations to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4 shows an encodermap projection of all 40 replicas with some of the most populated clusters found after 10 it- erations and representative conformations of these clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Similar conformations are grouped together and rare structures are spread out across the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For ex- ample, the native conformation of Trp-cage RE (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% native;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1%8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The most populated clusters and respective conformations of Trp-Cage TC10b [42] projected to the same 2D encodermap space as TC5b (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='bTop right: a histogram of the 2D projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' of all conformations) is shown in the bottom right of the 2D map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' On the bottom left conformations with one turn near the middle of the backbone are lo- cated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The two parts of the backbone chain of these conformations lie right next to each other and partially form beta-sheet structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Using a larger cutoff distance in the RMSD-based as- signment of structures to the clusters (the other clusters were generated by applying a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='8 ˚A RMSD cutoff to the central conformation) we obtained larger and quite dif- fuse clusters of extended conformations (one of these clus- ters is shown in the left part of the projection in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4 where it is referred to by an arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' An appropriate size of this RMSD cutoff was defined for each fuzzy cluster individually by computing the mean value of the largest 20% of the RMSD values between the centroid and cluster members of the cluster identified in the current iteration (it is equal to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5 ˚A for the cluster shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Before we identify fuzzy clusters, we first continuously assign struc- tures based on a fixed RMSD cutoff (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='8 ˚A for TC5b) until one of the stopping points defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' II D is reached (average cluster size for TC5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Once this stop- ping point is reached, the RMSD cutoff is adjusted in the way explained above and fuzzy clusters are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Thereby one ensures that all conformations that can be assigned to well-defined clusters are removed from con- sideration before identifying fuzzy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The usage of such a varying cutoff can be very helpful in order to iden- tify diffuse clusters, where the members share a certain structural motif but do not converge to a very defined conformation, just like the cluster shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' From the clustering results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4 one can see that the proposed clustering workflow manages to ef- ficiently identify structurally very well defined clusters for the TC5b system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Over 10 clustering iterations it as- signed 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% of all conformations to 260 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Besides the highly populated native state (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4%), the algorithm also finds very ”rare” states, which contain only a very small amount of conformations (≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1%) but show never- theless a very defined structural identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' TC10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 5 shows the same analysis applied to the trajectory of the K8A mutant of TC10b Trp-cage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='02% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='7% native;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 12%9 We used the encodermap which we trained on TC5b to project the trajectories to the same 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The iden- tification of clusters however is of course entirely inde- pendent and unique for both cases, since the clustering is done in the higher dimensional cc analysis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Notably, the backbone conformation of the native state of this mutant is extremely similar to the one in the TC5b system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However this biggest cluster only contains 12% of all conformations along the trajectory compared to the 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% in the case of the TC5b system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If all clus- ters whose central conformation are within a 2 ˚A RMSD to the native conformation are combined, we get native conformation percentage of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is in excellent agreement with the native cluster sizes reported by Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [45], Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [46] who analysed the same Trp-cage trajectories provided by Lindorff-Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore our 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% of assigned conformations coincide very well with the reporting of Sidky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' They found a total of 31% of conformations distributed over eight metastable macrostates and the remaining 69% as one big ”molten globule” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The TC10b trajectory is more disordered, this can be seen by the more homogeneous projection in 2D space (upper right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 5) and the RMSD values to the native conformation in SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-III, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is also the reason why the clustering scheme assigned only 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% of all conformations to clusters after 10 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If more frames should be assigned to clusters, more clus- tering iterations can be performed, the RMSD cutoff can be increased or both can be done simultaneously (for the Protein B system we show the results of this approach later in the article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However the clusters in the very center of the map (dark blue circle) are much more compact and collapsed compared to the clusters that were found in the similar area of Trp-cage RE’s 2D projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Also some of the clusters that were found in the very bottom of the left hand side of the map in case of the replica trajectories (light blue circle) were not found at all in the TC10b trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The very large and diffuse cluster on the left side of the map is present in both systems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Clustering directly in 2D space of TC5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The clustering discussed above was done in a 20 dimensional space after applying the cc analysis algorithm and only displayed at a 2D projection done with encodermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In order to demonstrate the advantages of our approach we also directly clustered the 2D encodermap space using the HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The encodermap space that we used for this clustering is the same space that we used to visualize the cc analysis clustering in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The results of this clustering and a few chosen clusters can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In total this clustering assigned 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% of all conformations to 362 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The biggest cluster that was found is the native cluster, however it only contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='8% of all conformations compared to the 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% that were found by clustering the cc analysis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The clus- tering in the 2D space identifies some structurally very well defined clusters, such as the clusters 0, 1 and 3, but FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 2D encodermap space of TC5b clustered with HDB- SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Representations of chosen clusters that have the same location in the 2D map as clusters found with the clustering scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 4 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' also a lot of very diffuse and inhomogeneous clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To quantify this inhomogeneity we computed the average of the internal cluster RMSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For the TC5b system our clustering workflow resulted in an average cluster RMSD of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='34 ˚A and a weighted average RMSD of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='03 ˚A, where weights are defined as the fraction of each cluster to all clustered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The average RMSD for the direct cluster- ing in the 2D space is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='25 ˚A and the weighted average RMSD is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='73 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This clearly shows that the internal cluster RMSD variance is on average much larger when clustering directly in the 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore the clus- tering in the 2D space itself naturally highly depends on the quality of the 2D map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Other than the much clearer conformational identity of the individual clusters (shown via internal cluster RMSDs), our clustering scheme also manages to assign 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% of all conformations to different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Com- pared to that the clustering in the 2D projection only assigned 9-14% of all conformations depending on the choice of clustering parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Comparison to other clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For a further assessment of our clustering scheme we have also applied a frequently used clustering routine to the TC5b data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In Si, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-IV and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S4 and S5 the results of applying the k-means algorithm to an 11 dimensional PCA projection of the same CVs (pairwise Cα distances of TC5b) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In summary, the scheme identified both structurally very defined as well as quite diffuse clusters in considered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Even though the combination of the 40 RE tra- jectories produces a very diverse data set, the clustering scheme manages to assign a large amount of the confor- mations to clusters (60%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Our clustering results for the TC10b are in a very good agreement with the findings of other researchers [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore the compar- ison to a clustering in the 2D space clearly shows the superiority of using more dimensions obtained with the cc analysis algorithm in HDBSCAN over just relying on a low-dimensional representation alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Cluster 4 Cluster 3 Cluster 2 Cluster 5 Cluster 1 Cluster O Cluster 610 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The 2D encodermap projection of NTL9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The projection can be approximately divided into three parts: the upper part with the most dense areas (where the native-like states are located);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' the lower left and right planes divided by an unpopulated vertical gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The left side includes various conformations with a singular beta sheet formed mostly between the beginning and the end of the protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In contrast on the right side lie mostly extended conformations with multiple helices along the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Exemplary conformations of some of the most populated clusters found in each of the marked areas in the map and their populations are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' All clusters in the yellow circle are extremely similar to the native cluster and can be summed up to a total of 76% of all conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The structures that are shown here make up 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% of all conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Top right: Histogram of the 2D encodermap space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' NTL9 Next we examined very long (1877 µs) simulations of NTL9 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' With 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='38 million frames to cluster, this sys- tem is an ideal candidate to demonstrate how the pro- posed algorithm copes with large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Af- ter 10 iterations 81% of all conformations were assigned to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 7 shows a 2D projection made with encodermap, where points are colored according to the clusters found after ten iterations of the scheme and a histogram of the 2D space in the upper right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In total we found 157 clusters and assigned them 81% of all conformations over 10 clustering iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' A comparison of the timeseries of the RMSD values to the folded state to the respective data of the Trp-cage Anton simulations (SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-III, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S3) reveals that the two systems exhibit very different dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' While in the Trp-cage case the RMSDs show the disordered nature of the system, in the case of the NTL9 trajectories the RMSDs are predominantly quite low and only spike up to larger values for rather short time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This suggests that the NTL9 system resides in a native-like state for the majority of the simulated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This is confirmed during the very first iteration of the clustering scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' There we found two clusters which make up for 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='8% of all conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This example also nicely illustrates how the iterative clustering approach can be efficient in identifying clus- ters of very different size and density (highly populated native states and low populated clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' After finding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01% native;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01% >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01% cumulative ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='03% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='03% cumulative ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01% >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01%11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Register-shifted states found in the NTL9 trajecto- ries 0, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The residues which form the beta sheets in the native state are colored based on their residue ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' and removing the first two clusters (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='8% of the data) the clustering algorithm becomes much more sensitive towards the less dense areas in the CV-space in the fol- lowing clustering iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We compared our clustering results with other publi- cations analyzing the NTL9 trajectories from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Mardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [29] applied the VAMPnets to trajectory 0 and found in total 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% of folded, native like confor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If we take the clusters we found by analysing the trajectories 0, 2 and 3 and evaluate the conforma- tions stemming from trajectory 0 (trajectory 0 resides in the native-like state for a larger fraction of the simu- lated time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' see RMSD plots in SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-III, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S3, the amount of folded, native-like conformations we find is in very good agreement with [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore Schwantes and Pande [44] reported the finding of three “register- shifted” states, which are very low populated and there- fore very hard to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' “Register-shifted” refers to the identity of the specific residues involved in forming the beta sheet structure in the native-like states (residues 1- 6, 16-21 and 35-39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' With our method we identified six different register-shifted states in the NTL9 trajectories 0, 2 and 3 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The states 0, 1 and 2 are the ones which were also found in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To our knowledge states 3, 4 and 5 have not been reported yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In state 0 the central of the three beta-sheet strands is shifted downwards, whereas in state 2 the rightmost strand is shifted downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In state 1 both the middle and the rightmost strands are dislocated compared to the native state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' State 3 is similar to state 1 in the fact that both the middle and the rightmost strands are shifted, however in state 3 the rightmost strand is shifted upwards and not downwards like in state 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Among these six states state 4 is unique since there the rightmost strand is turned by 180 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Finally state 5 differ from other states in having an extra helix along the chain between the leftmost and the mid- dle strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Because of this additional helix the leftmost strand is extremely shifted compared to the native state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The identification of these register-shifted states high- lights one asset of the proposed workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It is able to find both very large states (native, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5%) as well as very low populated clusters (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='001%) in the same data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Protein-B The last system we analysed is Protein B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This sys- tem does not have a very stable native state, instead the three helices can move against each other relatively freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This can be seen in the timeseries of the RMSD to the closest experimental homologue (1PRB) shown in SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-III, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' There are no extended periods where the values are stable over some time, meaning there are no large free-energy barriers separating the various acces- sible conformations and thus the system constantly tran- sitions into different conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This has also been found in [42], where authors stated that they were un- able to identify a free-energy barrier between folded and unfolded states for Protein B (tested over many different reaction coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Such a highly dynamic system is very challenging for a conformational clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Here we want to show where our algorithm has its limitations and what can be done to get a satisfactory clustering result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 9 gives an overview of some of the clusters found after ten iterations of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' These clusters include only 20% of the Protein B trajectory and thus 80% of all conformations are still unclustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In order to have more data assigned to clusters two pa- rameters can be adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' First, the RMSD cutoff value can be increased and thereby more conformations can be assigned to the found clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In this specific case this adjustment is justified, since due to the low free-energy barriers between different states, the individual clusters are not as sharply defined in terms of their conforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In the 10 clustering iterations which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 9 we used a RMSD cutoff of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='0 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In a second run we increased it to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This resulted in an assignment of 31% of all conformations to generally more loosely de- fined clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' A second approach is to increase the amount of clus- tering iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For the first ten clustering iterations of previously analysed systems, we tuned the clustering pa- rameters manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This includes the choice of the num- ber of cc analysis dimensions, as well as the min samples and min cluster size parameters of HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However such a manual adjustment of the parameters is of course not feasible for automating the script in order to perform many more clustering iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Since the amount of cc analysis dimensions needs to be very rarely changed once a suitable amount has been identified in the first clustering iteration, the automation of the script only re- lies on the choice of the HDBSCAN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Once the amount of clusters found in a single iteration falls below a certain threshold (10 clusters in this case), the numerical Native State State 4: ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='001% 5 State 1: r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% State 3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01% State 0: ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% State 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='01% State 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1%12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Protein B: Exemplary conformations of some of the most populated clusters found for the Protein B system after 10 clustering iterations and their populations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Top right: Histogram of the 2D encodermap space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' values of the min samples and min cluster size parame- ters of HDBSCAN are slightly decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This leads to the detection of smaller clusters that have not been iden- tified before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' By applying this automation approach after the first 10 iterations to Protein B and using a RMSD cutoff of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5 ˚A, we could assign 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% of all conforma- tions to clusters over 100 iterations, which took roughly 15 hours on our workstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' DISCUSSION The Trp-cage system (TC5b) is a relatively small pro- tein which has a quite stable native conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The combination of 40 temperature RE trajectories however gives a very diverse data set including (under standard conditions) very improbable high-energy conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Over ten iterations the algorithm managed to assign 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% of all conformations to clusters, which took on av- erage 360 min per iteration over all CPU threads (15 min per iteration on a standard office machine with 24 CPU threads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Table I shows the clustering performance for the four systems discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' By switching the gen- erally static RMSD cutoff to a varying cutoff we could show that the algorithm can both generate conforma- tionally very defined clusters as well as quite diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The conformations assigned to such loose clusters share a general structural motif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The ability to identify both of these cluster types is one of the advantages of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore we demonstrate that the clustering workflow is able to directly compare dif- ferent systems (even if they slightly differ structurally), by projecting them to the same 2D map using the en- codermap algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This enables a direct and visual comparison of the sampled phase-spaces of different tra- jectories and their respective identified states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' By com- paring the clustering result where the clustering is done in a 20-dimensional cc analysis space and then projected to a two-dimensional space to a clustering where the clusters are purely found in a 2D encodermap space, we prove an advantage using more dimensions and combine cc analysis with encodermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The scheme created clus- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5% most populated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='2%13 ters with a much clearer structural identity (lower RMSD variance), while being much less dependent on the quality of the 2D map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We analysed long (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='38 million frames) trajectories of NTL9 to show how the proposed scheme copes with very large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' On average the algorithm needed 1320 min of computation time over all CPU threads per iteration (55 min per iteration on our office machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Since this system also has one hugely populated native- state, it is also a nice example to demonstrate an ad- vantage of the iterative clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' After the clusters with the native states are removed from consideration, the algorithm becomes much more sensitive towards less populated areas in the following iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Applying this approach we could identify three very low popu- lated register-shifted states, which have been reported before [44], and three not yet seen register-shifted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Lastly we looked at is Protein B, which is a highly dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To analyse this 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='04 million frames trajectory it took on average 288 min of computation time per iteration (12 min per iteration on our office machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This system has no large free-energy barri- ers separating the various conformations, which makes it very difficult to cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This was confirmed by the fact that after ten clustering iterations only 20% of all conformations could be assigned to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' However by increasing the RMSD cutoff from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='0 ˚A to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5 ˚A we could already increase the amount of assigned conformations to 31%, which of course resulted in slightly less structurally defined clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It is also possible to automate the clus- tering and run until a certain amount of conformations are assigned to clusters or until a given number of itera- tions is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In this specific case we ran the scheme for 100 automated iterations (≈15 hours), during which 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='3% of the conformations were assigned to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For all considered systems the proposed workflow was able to identify defined clusters at the cost of leaving some amount of the trajectories unassigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' As we have shown here, the rest of the structures does not belong to any specific clusters and can be considered as unfolded or transition states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We intentionally do not propose any additional steps to assign or classify those conformations as it is highly dependant on the intended application of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' For example in case the data is used to build subsequent kinetic models the rest of the points can be assigned to the nearest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' in simulation time) cluster using methods such as PCCA+ analysis [48], or defined as a metastable transition state as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' It can also be defined as noise and used as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' All performance data is shown in Table I and was ob- tained by running the clustering scheme script on the office workstation described in SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' S-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The pro- posed workflow is, however, highly parallelizable, since the computationally most expensive step is the assign- ment of additional data points to the initially identified clusters in the small subset based on the convex hull and the RMSD criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If a large amount of CPU cores are available, the 2D encodermap projection array can be split by the amount of cores and the assignment can thereby be run in parallel which leads to a significant speed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The convex hull around the clusters identified in the small subset is used to reduce the amount of RMSD com- putations that have to be performed when assigning ad- ditional conformations in each clustering iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This however might in principle lead to the exclusion of data points that might otherwise have been assigned to some of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In order to get an idea of the magnitude of this “loss” of potential cluster members, we computed the RMSD of all data which was labeled as noise (623,000 conformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5%) to each of the cluster centers of TC5b (260 clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This computationally very expen- sive task took an additional 5 hours on our working ma- chine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We found that 42,000 conformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='7%) were not assigned to the identified clusters due to the con- vex hull criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' When keeping in mind that the entire 10 iteration clustering process took 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='5 hours, the ”loss” of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='7% of unclustered data can be considered a worthy trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Another point to consider is that due to the convex hull criterion clusters can be split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' If data points that would be assigned to a certain cluster by reason of the RMSD criterion lie outside of the convex hull, they could be iden- tified as another cluster in one of the following clustering iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In such cases it can make sense to merge these clusters in hindsight, due to their very similar structural identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In order to showcase such a merge, we again analysed TC5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We computed the RMSDs between all of the 260 central cluster conformations and merged all clusters that had a RMSD of ≤ 1 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' This resulted in a re- duction to 201 clusters with only very marginal influence on the average internal cluster RMSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The code for the encodermap algorithm is avail- able on the following github page https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' com/AG-Peter/encodermap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The cc analysis code can be found under https://strucbio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='biologie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' uni-konstanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='de/xdswiki/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='php/Cc_analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' CONCLUSION We developed a clustering scheme which com- bines two different dimensionality reduction algorithms (cc analysis and encodermap) and the HDBSCAN in an iterative approach to perform fast and accurate clus- tering of molecular dynamics simulations’ trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The cc analysis dimensionality reduction method was first applied to protein simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The method projects collective variables to a usually relatively high- dimensional (∼10-40 dim) unit sphere, separating noise and fluctuations from important structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Then the data can be efficiently clustered by density based clustering methods, such as HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' The it- erative application of HDBSCAN allows to account for the inhomogeneity in population and density of the pro- jected points, which is very typical for protein simulation 14 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' As cc analysis relies on the calculations of correla- tion matrices between each frame, this drastically limits the amount of data one can project simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To allow processing of long simulation trajectories we in- cluded encodermap to the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' In addition to the obvious advantage of the two-dimensional visualisation it is used – in combination with a RMSD-based accep- tance criterion – for a fast structure-based assignment of additional points to the clusters initially identified in the higher dimensional projection done with cc analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' To demonstrate the accuracy and performance of the pro- posed scheme we applied the clustering scheme to four test systems: replica exchange simulations of Trp-cage and three long trajectories of a Trp-cage mutant, NTL9 and Protein B generated on the Anton supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' By applying the scheme to these four test systems we could show that: the algorithm can efficiently handle very large amounts of data, that it can be used to com- pare the clusters of structurally different systems in one 2D map, and that it can also be applied to cluster sys- tems which do not have very stable native states and are therefore intrinsically very difficult to cluster confor- mationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore the algorithm is able to find clusters independent of their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' By varying a RMSD cutoff both conformationally very well defined clusters, as well as fuzzy clusters, whose members only share an overall structural motive, can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' SUPPORTING INFORMATION Supporting Information (PDF) includes: (S-I): Methods to chose parameters for cc analysis and HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (S-II): Stopping criteria for the clustering workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (S-III): RMSD plots of trajectories for Trp-cage, Pro- tein B and NTL9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (S-IV): Comparison of the proposed clustering work- flow to PCA and k-means clustering for Trp-cage (TC5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' (S-V): Workstation specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was supported by the DFG through CRC 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' We also greatly appreciate the computing time on bwHPC clusters which was used to produce the Trp- cage TC5b trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Furthermore we would like to thank the D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' Shaw research group for providing the Trp-cage, NTL9 and Protein B trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfYwr8/content/2301.04492v1.pdf'} +page_content=' O.' metadata={'source': 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+Selective Conformal Inference with FCR Control +Yajie Baoa, Yuyang Huob, Haojie Rena and Changliang Zoub∗ +aSchool of Mathematical Sciences, Shanghai Jiao Tong University +Shanghai, P.R. China +bSchool of Statistics and Data Science, Nankai University +Tianjin, P.R. China +January 3, 2023 +Abstract +Conformal inference is a popular tool for constructing prediction intervals (PI). We consider here +the scenario of post-selection/selective conformal inference, that is PIs are reported only for individuals +selected from an unlabeled test data. To account for multiplicity, we develop a general split conformal +framework to construct selective PIs with the false coverage-statement rate (FCR) control. We first +investigate the Benjamini and Yekutieli (2005)’s FCR-adjusted method in the present setting, and show +that it is able to achieve FCR control but yields uniformly inflated PIs. We then propose a novel solution +to the problem, named as Selective COnditional conformal Predictions (SCOP), which entails performing +selection procedures on both calibration set and test set and construct marginal conformal PIs on the +selected sets by the aid of conditional empirical distribution obtained by the calibration set. Under +a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the +FCR. More importantly, we provide non-asymptotic miscoverage bounds for a general class of selection +procedures beyond exchangeablity and discuss the conditions under which the SCOP is able to control +the FCR. As special cases, the SCOP with quantile-based selection or conformal p-values-based multiple +testing procedures enjoys valid coverage guarantee under mild conditions. Numerical results confirm the +effectiveness and robustness of SCOP in FCR control and show that it achieves more narrowed PIs over +existing methods in many settings. +Keywords: Conditional empirical distribution; Distribution-free; Non-exchangeable conditions; Post- +selection inference; Prediction intervals; Split conformal. +∗Corresponding Author: nk.chlzou@gmail.com +1 +arXiv:2301.00584v1 [stat.ME] 2 Jan 2023 + +1 +Introduction +To improve the prediction performance in modern data, many sophisticated machine learning algorithms +including various “black-box” models are proposed. While often witnessing empirical success, quantifying +prediction uncertainty is one of the major issues for interpretable machine learning. Conformal inference +(Vovk et al., 1999, 2005) provides a powerful and flexible tool to quantify the uncertainty of predictions. +Consider a typical setting that we observe one labeled data set Dl = {(Xi, Yi)}2n +i=1 and a set of unla- +belled/test samples Du = {Xi}2n+m +i=2n+1 whose outcomes {Yi}2n+m +i=2n+1 are unobserved. Generally, suppose all +(Xi, Yi) ∈ X × Y are i.i.d from some unknown distribution, and µ(x) := Y | X = x as the prediction model +associated with (X, Y ), which is usually estimated by the labeled data Dl. For any Xj ∈ Du and a given +miscoverage level α, standard conformal prediction methods (Lei et al., 2018), yield a prediction interval (PI) +with distribution-free coverage guarantee, PIα(Xj), +P(Yj ∈ PIα(Xj)) ≥ 1 − α, +under independent and identically distributed (i.i.d) (or exchangeable data) assumptions. +With the development of big data, making predictive inference on all available data (Du) is either +unnecessary or inefficient in many applications. For example, in the recruitment decisions, only some selected +viable candidates can get into interview processes (Faliagka et al., 2012; Shehu and Saeed, 2016). In the drug +discovery trials, researchers select promising ones based on predicting candidates’ activity for further clinical +trials (Carracedo-Reboredo et al., 2021; Dara et al., 2021). Related applications also appear in financial +investment and scientific discovery (Jin and Candès, 2022). In such problems, the most common way is to +select a subset of interest with some rules through some statistical/machine learning algorithms at first, and +then perform statistical inference only on the selected samples. +Formally, letting ˆSu ⊆ {2n + 1, . . . , 2n + m} be the selected subset, our goal is to construct the PI of Yj +for each j ∈ ˆSu. As pointed by Benjamini and Yekutieli (2005), ignoring the multiplicity in construction of +post-selection intervals will result in distorted average coverage. Under the context of post-selection inference +in which confidence intervals for multiple selected parameters/variables are being reported, Benjamini and +Yekutieli (2005) pioneered the criterion, false coverage-statement rate (FCR), to take account for multiplicity. +The FCR, an analog of the false discovery rate (FDR), can readily be adapted to the present conformal +inference setting. It is defined as the expected ratio of number of reported PIs failing to cover their respective +true outcomes to the total number of reported PIs, say +FCR := E +� +|{j ∈ ˆSu : Yj ̸∈ PIj}| +max{| ˆSu|, 1} +� +, +(1) +where PIj is the PI for the selected sample j ∈ ˆSu. Benjamini and Yekutieli (2005) provided a selection- +agnostic method which adjusts the confidence level through multiplying α by a quantity which is related +to the proportion of selected candidates over all candidates and then constructed the marginal confidence +2 + +intervals at the adjusted level for each selected candidate. We will hereafter call it the FCR-adjusted method. +Accordingly, we may expect that the FCR-adjusted PIs enjoy valid FCR control properties. However, due +to the dependence structure among PI| ˆ +Su|α/m(Xj)’s, the results in Benjamini and Yekutieli (2005) are not +directly applicable in the setting of conformal inference. Please refer to Section 2.1 for detailed discussions +and rigorous theories. We notice that Weinstein and Ramdas (2020) also discussed the selective inference +problem under the framework of conformal prediction. The authors suggested to use the FCR-adjusted +method, however, they did not provide theoretical or empirical investigations. +While the FCR-adjusted approach can reach FCR control and is widely used, it is generally known to yield +uniformly inflated confidence intervals (Weinstein et al., 2013). This is because that when calculating the +noncoverage probabilities of confidence intervals, the adjusted confidence intervals do not take into account the +selection event. Along this line, Weinstein et al. (2013), Zhao and Cui (2020) and Zhao (2022) further proposed +some methods to narrow the adjusted confidence intervals by incorporating more selection information. Among +some others, Fithian et al. (2014), Lee et al. (2016) and Taylor and Tibshirani (2018) proposed constructing +conditional confidence intervals for each selected variables and showed that the selective error rate can +be controlled given that the selected set is equal to some deterministic subset. However, those methods +either require some tractable conditional distribution assumptions or are only applicable for some given +prediction algorithms, such as normality assumptions or LASSO model, which would limit their applicability +in the conformal inference. Fortunately, by the virtue of the availability of Dl, distribution/model-agnostic +conditional prediction intervals with theoretical guarantee can be achieved. +1.1 +Our contributions +In this paper, we develop a novel conformal framework to construct post-selection prediction intervals while +control the FCR, named as Selective COnditional conformal Predictions (SCOP). Our method stems from the +split conformal inference (Lei et al., 2018; Fithian and Lei, 2020), where the labeled data Dl is split into two +disjoint parts, one as the training set for obtaining a prediction model ˆµ(X), and the remaining one as the +calibration set for estimating the distribution of the discrepancy between the Y and ˆµ(X). Then, the key +ingredient of our proposal entails performing a pre-specified selective procedure on both the calibration set and +the test set and construct the marginal conformal PIs on the selected sets with the help of conditional empirical +distribution obtained by the calibration set. The proposed SCOP procedure is model- or distribution-agnostic, +in the sense that it could wrap around any prediction algorithms with commonly used selection procedures to +construct PIs. +The main contributions of the paper can be summarized as follows: +• Firstly, we investigate the FCR-adjusted method in the setting of conformal inference and show that +it is able to achieve FCR control under mild conditions, which lays a foundation for our subsequent +development of SCOP. +3 + +• Secondly, under a unified framework and exchangeable assumptions, we show that the SCOP can exactly +control the FCR at the target level. +• Thirdly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond +exchangeablity, termed as ranking-based procedures. This broadens the scopes of our SCOP in theoretical +guarantee and practical use. To address the non-exchangeability between the the post-selection test set +and calibration set, we introduce a virtual post-selection calibration set in our proof, and then quantify +the conditional miscoverage gap between the virtual calibration and the real calibration in SCOP. This +new technique may be of independent interest for conformal prediction for non-exchangeable data. +• Finally, we illustrate the easy coupling of the SCOP with commonly used prediction algorithms. +Numerical experiments indicate that it yields more accurate FCR control than existing methods, while +offers the narrowed prediction intervals. +1.2 +Connections to existing works +Post-selection inference. +Post-selection inference on a large number of variables has attracted considerable +research attention. Besides the references mentioned before, a relevant direction is the splitting-based strategy +for high-dimensional inference. The number of variables is firstly reduced to a manageable size using one +part of data, while confidence intervals or significance tests can be constructed by computing estimates in a +low-dimensional region with the other part of data and selected variables. See Wasserman and Roeder (2009), +Rinaldo et al. (2019), Du et al. (2021) and the references therein. One potential related work is Chen and +Bien (2020), in which the authors considered to construct confidence intervals for regression coefficients after +removing the potential outliers from the data. Our paradigm differs substantially with those works as we +focus on post-selection inference for sample selection rather than variable selection, and existing works on +variable selection is difficult to extend to the present problem due to the requirements on model or distribution +assumptions. +Conformal prediction. +The building block of our SCOP is the conformal inference framework, which has +been well studied in many settings, including non-parametric regression (Lei et al., 2013), quantile regression +(Romano et al., 2019), high-dimensional regression (Lei et al., 2018) and classification (Sadinle et al., 2019; +Romano et al., 2020), etc. More comprehensive reviews can be found in Shafer and Vovk (2008), Zeni et al. +(2020) and Angelopoulos and Bates (2021). Conventionally, conformal PIs enjoy distribution-free marginal +coverage guarantee with the assumption that the data are exchangeable. However, the exchangeability may be +violated in practice and would be more severe in the post-selection conformal inference because the selection +procedure might be determined by either the labelled data or the test data, or both. In such situations, one +particularly difficult issue is that the selected set ˆSu is random and has a complex dependence structure to +the labelled data and test data. Some conformal inference beyond exchangeability has attracted attention +4 + +(Tibshirani et al., 2019; Lei et al., 2021; Candès et al., 2021). In particular, Barber et al. (2022) proposed a +general framework to implement conformal inference when the algorithms cannot treat data exchangeable and +theoretically displayed the coverage deviations compared from exchangeability. However, how to decouple the +dependence to achieve FCR control in the present framework remains a challenge. +Taking a different but related perspective from multiple-testing, Bates et al. (2021) proposed a method to +construct conformal p-values with data splitting and apply it to detect outliers with finite-sample FDR control. +Zhang et al. (2022) extended that method and proposed a Jackknife implementation combined with automatic +model selection. Jin and Candès (2022) considered a scenario that one aims to select some individuals of +interest from the test sample and proposed a conformal p-value based method to control the FDR. Those +existing works are not concerned about the construction of PIs, which differs with our focus essentially. +1.3 +Organization and notations +The remainder of this paper is organized as follows. We introduce the FCR-adjusted prediction and SCOP +for valid FCR control in Section 2. Section 3 presents the theoretical properties of SCOP for ranking-based +procedures. Numerical results and real-data examples are presented in Sections 4. Section 5 concludes the +paper, and the technical proofs are relegated to the Supplementary Material. +Notations. For a positive integer n, we use [n] to denote the index set {1, 2, ..., n}. Let A = {Ai : i = 1, ..., n} +be a set of n real numbers, and S ⊆ [n] be an index subset. We use AS +(ℓ) to denote the ℓ-th smallest value +in {Ai : i ∈ S}. We use 1 {·} to denote the indicator function. For a real random sequences Xn and an +non-negative real deterministic sequence an, we write Xn = Op(an) if for any ϵ > 0, there exists some constant +C > 0 such that P(|Xn| > Can) ≤ ϵ. In our paper, the notations with subscript c or u refer to depending on +the calibration set or the test set respectively. +2 +Selective conditional conformal prediction +Denote the index sets for the labelled data Dl and the test data Du as L = {1, . . . , 2n} and U = {2n + +1, . . . , 2n + m}. The main prediction method studied in this paper is built upon the split conformal framework +(Vovk et al., 2005; Lei et al., 2018), which is also called “inductive conformal prediction”. That is we randomly +split Dl into two disjoint parts, the training set Dt and the calibration set Dc with n samples respectively. We +can firstly train a prediction model ˆµ(X) on the Dl, and then compute the empirical quantiles of the residuals +Ri = |Yi − ˆµ(Xi)| on the calibration set Dc. For Xj ∈ Du, the (1 − α)-marginal conformal PI is +PIM +j = +� +ˆµ(Xj) − QC(1 − α), ˆµ(Xj) + QC(1 − α) +� +, +(2) +where QC(1 − α) is the ⌈(1 − α)(n + 1)⌉-st smallest value in RC = {Ri = |Yi − ˆµ(Xi)| : i ∈ C}. Under the i.i.d. +(or more generally, exchangeable) assumption on Dc ∪ {(Xj, Yj)}, the marginal PI in (2) enjoys the coverage +guarantee, P +� +Yj /∈ PIM +j +� +≤ α. +5 + +Suppose g : X → R be one plausible score function, which can be user-specified or estimated by the +training data Dt. A particular selection procedure S can be applied to g(Xi) for i ∈ U to find the samples of +interest. For simplicity, denote Ti = g(Xi) and those Xi’s with smaller values of Ti tend to be chosen. Denote +the selected set as ˆSu = {i ∈ U : Ti ≤ ˆτ}, where ˆτ is the threshold. Different selective procedures S can be +chosen from different perspectives, and we summarize the selection threshold ˆτ into three types. +• (Fixed threshold) The ˆτ is user-specified or independent of the whole data. For example, ˆτ = t, where t +is either known as a priori or could possibly be obtained from an independent process of Dc ∪ Du. +• (Self-driven threshold) The ˆτ is only dependent on the scores {Ti : i ∈ U}. This type includes the +Top-K which choose the first K individuals, and the quantile of Ti values in the test set which a given +proportion of individuals with smallest Ti values in the test set, respectively (Fithian and Lei, 2020). +• (Calibration-assisted selection) The ˆτ relies on the calibration set. For example, ˆτ is some quantile of +true response Yi in calibration set, or the quantile based on both calibration and test set. In particular, +one may employ some multiple testing procedures to achieve error rate control, such like FDR control +based on the Benjamini–Hochberg (BH) procedure (Benjamini and Hochberg, 1995). Consequently, the +{Ti : i ∈ C} is required to approximate the distribution of {Ti : i ∈ U}. +Our goal is to construct conformal PIs for the selected subset ˆSu with the FCR control at α ∈ (0, 1). +2.1 +Adjusted conformal prediction +We firstly adapt the Benjamini and Yekutieli (2005)’s FCR-adjusted method to the present setting. Define +Mj +min := min +y +� +| ˆSTj←y +u +| : j ∈ ˆSTj←y +u +� +, +where ˆSTj←y +u +denotes the selected subset when replacing Tj with value y. The FCR-adjusted conformal PIs +are amount to marginally constructing larger 1 − α × Mj +min/m PIs instead of 1 − α level in (2), i.e., +PIAD +j += +� +ˆµ(Xj) − QC(1 − α × Mj +min), ˆµ(Xj) + QC(1 − α × Mj +min) +� +, j ∈ ˆSu. +(3) +Notice that given ˆµ(·) and ˆSu, PIAD +j +’s are not independent of each other because they all rely on the empirical +quantile obtained from Dc, and therefore the proofs in Benjamini and Yekutieli (2005) are not readily extended +to our setting. The following result demonstrates that the FCR-adjusted approach can successfully control +the FCR for any selection threshold that is independent of the calibration set given training set. +Proposition 2.1. Suppose that given Dt, {Ti : i ∈ C ∪ U} are independent random variables and the selection +threshold ˆτ is independent of Dc. Then the FCR value of the FCR-adjusted method in (3) satisfies FCRAD ≤ α. +For many plausible selection rules such as fixed-threshold selection, the Mj +min can be replaced by the +cardinality of the selected subset | �Su|. In practice, for ease of computation, one may prefer to use this +6 + +0.0 +0.1 +0.2 +0.0 +2.5 +5.0 +7.5 +10.0 +Ri +density +Marginal +Conditional +Test +Figure 1: The densities of Ri for i ∈ Dc (in blue), Ri for i ∈ ˆSc (in green) and the density of Rj for j ∈ ˆSu (in red), +respectively. There are 2n = 400 labeled data and m = 200 test data generated from a linear model with heterogeneous +noise, where the details of the model are in Section 4.1. The selection rule is ˆS = {k : ˆµ(Xk) ≤ −1}. +simplification, even though it does not have a theoretical guarantee for many data-dependent selection rules. +The FCR-adjusted method is known to be quite conservative (Weinstein et al., 2013), because it does not +incorporate the selection event into the calculation. Take the Top-K selection as an intuitive example. The +selected set ˆSu is fixed with | ˆSu| = K and the FCR can be written as +FCR = 1 +K +� +j∈U +P +� +j ∈ ˆSu, Yj ̸∈ PIj +� +. +(4) +Since the marginal PIAD +j +reaches the 1 − αK/m confidence level for any fixed K, the FCR-adjusted method +achieves the FCR control via +FCRAD = 1 +K +� +j∈U +P +� +j ∈ ˆSu, Yj ̸∈ PIAD +j +� +≤ 1 +K +� +j∈U +P +� +Yj ̸∈ PIAD +j +� +≤ α, +where the first inequality might be rather loose. A simple yet effective remedy is to use conditional calibration. +2.2 +Selective conditional conformal prediction (SCOP) +We start by making a decomposition of the FCR according to the contribution of each sample in the selected +set ˆSu, given as P +� +Yj ̸∈ PIj |j ∈ ˆSu +� +P +� +j ∈ ˆSu +� +. Notice that the FCR can naturally be controlled at level α if +the conditional control satisfies P(Yj ̸∈ PIj |j ∈ ˆSu) ≤ α, which sheds light on the construction of conditional +conformal PI. +In the regime of conformal inference, the conditional uncertainty of |Yj − ˆµ(Xj)| given j ∈ ˆSu can be reliably +approximated using the calibration set Dc, enabling us to construct model/distribution-agnostic conditional +7 + +PIs. To be specific, we conduct the selective algorithm S on the fitted score values {Ti = g(Xi) : i ∈ C} +and obtain the post-selection calibration set ˆSc = {i ∈ C : Ti ≤ ˆτ} with the same threshold ˆτ. Notice that +ˆSc is formed via the same selection criterion with ˆSu, and thus we utilize the residuals Ri for i ∈ ˆSc to +approximately characterize the conditional uncertainty of Rj for j ∈ ˆSu. To visualize the effect, we consider +a linear model with heterogeneous noise, where we use ordinary least-squares for predictions and select +ˆSu = {j ∈ U : ˆµ(Xj) ≤ −1}. In Figure 1, we display the densities of Ri for i ∈ Dc, Ri for i ∈ ˆSc and +the density of Rj for j ∈ ˆSu, respectively. The selection procedure significantly distorts the distribution of +residuals, but the conditional uncertainty on ˆSu can be well approximated by that on ˆSc. +The conditional conformal PI for j ∈ ˆSu can accordingly be constructed as +PISCOP +j += +� +ˆµ(Xj) − Q +ˆ +Sc(1 − α), ˆµ(Xj) + Q +ˆ +Sc(1 − α) +� +, +(5) +where Q +ˆ +Sc(1 − α) is the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest value in R +ˆ +Sc = {Ri : i ∈ ˆSc}. We refer this procedure +as Selective COnditional conformal Prediction (SCOP) and summarize it in Algorithm 1. +The following theorem shows that SCOP can control the FCR at α for exchangeable selective procedures. +Further, if the selection scores Ti are continuous (or almost surely distinct), we can obtain a lower bound for +the FCR value, guaranteeing that the SCOP is nearly exact in O(n−1). +Theorem 1. Suppose {Ti : i ∈ C ∪ U} are exchangeable random variables, and the threshold ˆτ is also +exchangeable with respective to the {Ti : i ∈ C ∪ U}. For each j ∈ U, the conditional miscoverage probability is +bounded by +P +� +Yj ̸∈ PISCOP +j +|j ∈ ˆSu +� +≤ α. +(6) +Further, the FCR value of the SCOP algorithm is controlled at FCRSCOP ≤ α. In addition, if Ti follows a +continuous distribution for i ∈ C ∪ U and P(| ˆSu| > 0) = 1, we also have +P +� +Yj ̸∈ PISCOP +j +|j ∈ ˆSu +� +≥ α − +1 +n + 1 +and FCRSCOP ≥ α − +1 +n+1. +Under the exchangeable assumption, the FCR results actually match the marginal miscoverage results of +original conformal PIs (Vovk et al., 2005). This theorem relies on exchangeability in two ways, i.e., the fitted +selection score {Ti : i ∈ C ∪ U} are exchangeable and the selection threshold ˆτ is assumed to keep the same +value by swapping Tj and Tk for any j, k ∈ C ∪ U. The former one is commonly used in conformal inference +and holds easily when the data are i.i.d given Dt (Lei et al., 2018). The later one imposes restrictions on +the selection procedures and can be fulfilled with some practical thresholds. The simplest case is the fixed +threshold. Another popular example is that ˆτ is some quantile of {Ti : i ∈ C ∪ U}. However, many selection +procedures may be excluded, such as the Top-K selection. In such cases, the threshold ˆτ is only determined +by the test data U, which does not treat the data points from calibration and test sets symmetrically. We will +next explore the effectiveness of the proposed SCOP for more general selection procedures. +8 + +Algorithm 1 Selective COnditional conformal Prediction (SCOP) +Input: Labeled set Dl, test set Du, selection procedure S, target FCR level α ∈ (0, 1). +Step 1 (Splitting and training) Split Dl into training set Dt and calibration set Dc with equal size n. Fit +prediction model ˆµ(·) and score function g (if needed) on the training set Dt. +Step 2 (Selection) Compute the scores: TC = {Ti = g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}. Apply +the selective procedure S to TC ∪ TU and obtain the threshold value ˆτ. Obtain the post-selection subsets: +ˆSu = {i ∈ U : Ti ≤ ˆτ} and ˆSc = {i ∈ C : Ti ≤ ˆτ}. +Step 3 (Calibration) Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}. Find the ⌈(1−α)(| ˆSc|+1)⌉-st +smallest value of RSc, Q +ˆ +Sc(1 − α). +Step 4 (Construction) Construct PI for each j ∈ ˆSu as PISCOP +j += [ˆµ(Xj)−Q +ˆ +Sc(1−α), ˆµ(Xj) +Q +ˆ +Sc(1−α)]. +Output: Prediction intervals {PISCOP +j +: j ∈ ˆSu}. +Remark 2.1. In predictive inference, several works considered to approximately construct the conditional PI +(Chernozhukov et al., 2021; Feldman et al., 2021), i.e., +P (Yj /∈ PI(Xj)|Xj = x) ≤ α, +for any x ∈ X. +(7) +However, it is well known that achieving “fully" conditional validity in (7) is impossible in distribution-free +regime (Lei et al., 2013; Foygel Barber et al., 2020). Our conditional miscoverage control in (6) is a weaker +guarantee compared with (7), since we only consider the selection events. For more discussion about these two +conditional guarantees, we refer to Appendix B in Weinstein and Ramdas (2020). The SCOP can leverage +the post-selection calibration set to approximate the selective conditional distribution of residuals, which +contributes to achieve a better conditional coverage. In addition, the conditional calibration of SCOP provides +an anti-conservative lower bound for FCR value in the continuous case. +3 +Ranking-based selection +In this section, we consider a general class of selection procedures named ranking-based selection and discuss +the conditions under which the proposed SCOP is able to control the FCR. In Sections 3.1 and 3.2, we discuss +the FCR control for the self-driven selection procedures and calibration-assisted ones, respectively. Then, in +Section 3.3, we demonstrate the effectiveness of the SCOP procedure when the selection procedures based on +conformal p-values are used. +We begin with some general assumptions and notations. For simplicity, we suppose Ti ∈ [0, 1] and the +selection algorithm S conducted on {Ti : i ∈ C ∪ U} outputs a ranking threshold ˆκ ∈ [m]. Say, we have the +selection threshold ˆτ = TU +(ˆκ) as the ˆκ-th smallest value in TU = {Tj : j ∈ U}. Then the selected subset of the +9 + +test set can be rewritten as +ˆSu = +� +j ∈ U : Tj ≤ TU +(ˆκ) +� +. +(8) +The ranking-based procedure in (8) incorporates many practical examples, such as Top-K selection, quantile- +based selection, step-up procedures (Fithian and Lei, 2020) and the well-known BH procedure1 (Benjamini +and Hochberg, 1995). +With the ranking based selection, we have | ˆSu| = ˆκ, which is usually random and coupled to each test +sample Xj ∈ Du. To decouple the dependence, we introduce Lemma 1 to control the FCR through conditioning +on the leave-one-out data set Du,−j, which is the test set Du without the sample j. Denote EDu,−j[·] and +PDu,−j(·) as the conditional expectation and probability given Du,−j. Let ˆκj←tu be the ranking threshold +obtained from the selection algorithm S by replacing Tj with some deterministic value tu ∈ [0, 1]. +Lemma 1. Suppose | ˆSu| > 0 almost surely and ˆκ = ˆκj←tu holds for any j ∈ ˆSu. If the conditional false +coverage probability satisfies +���PDu,−j +� +Yj ̸∈ PIj +��j ∈ ˆSu +� +− α +��� ≤ ∆(Du,−j), +(9) +where ∆(Du,−j) only depends on the data set Du,−j, then we have +| FCR −α| ≤ E +� +� 1 +| ˆSu| +� +j∈ ˆ +Su +∆(Du,−j) +� +� . +The leave-one-out technique often appears in the literature about FDR control under dependence (Heesen +and Janssen, 2015; Fithian and Lei, 2020; Luo et al., 2022). The key is to decompose the FCR into the +summation of conditional miscoverage probability of each candidate given other test samples, i.e, +FCR = E +� +�� +j∈U +1 +ˆκj←tu PDu,−j +� +Yj ̸∈ PIj |j ∈ ˆSu +� +PDu,−j +� +j ∈ ˆSu +� +� +� . +We may regard the term ∆(Du,−j) in (9) as the individual FCR contribution of the j-th candidate in ˆSu. The +detailed proof of Lemma 1 is deferred to Appendix C.1. +Next, we introduce two universal assumptions to find how the conditional false coverage probability in +(9) holds and further control the FCR of SCOP with the ranking-based selection. Denote the cumulative +distribution functions (CDF) of Ri and Ti as FT (·) and FR(·), respectively. Let F(R,T )(·, ·) be the joint CDF +of (Ri, Ti). +Assumption 1. The score function g depends only on the training set. Suppose {Ti : i ∈ C ∪ U} and +{Ri : i ∈ C ∪ U} are both i.i.d. continuous random variables. There exists some ρ ∈ (0, 1) such that +d +drF(R,T ) +� +F −1 +R (r), F −1 +T (t) +� +≥ ρt, +holds for any t ∈ (0, 1) and r ∈ (0, 1). +1The BH procedure is also an example of step-up procedures, see Fithian and Lei (2020) +10 + +Assumption 2. There exists some deterministic value tu ∈ [0, 1] such that ˆκj←tu = ˆκ holds for any j ∈ ˆSu, +and ˆκj←tu ≤ ˆκ + Iu holds for any j ∈ U \ ˆSu and some positive integer Iu ≤ m. +To facilitate our technical development, we impose mild distributional assumption on the joint CDF of +(Ri, Ti) in Assumption 1. It is worth noticing that the same selected set ˆSu and ˆSc can be obtained if one +applies the ranking-based selection procedure to the transformed scores {FT (Ti) : i ∈ U} instead of the scores +{Ti : i ∈ U}. Also, the transformed residuals {FR(Ri) : i ∈ C ∪ U} keep the original order as the residuals +{Ri : i ∈ C ∪ U} in the conformal coverage control. Therefore, without loss of generality, we can assume that +Ti +i.i.d. +∼ Unif([0, 1]) and Ri +i.i.d. +∼ Unif([0, 1]) for i ∈ C ∪ U in the theoretical analysis. Then the condition on +CDF in Assumption 1 will reduce to +d +drF(R,T )(r, t) ≥ ρt which appears quite weak. +For Assumption 2, we can verify that ˆκ = ˆκj←tu under the event {j ∈ ˆSu} in many cases, such as +the quantile-based selection procedure and BH procedure. For the selection procedures with fixed ranking +threshold, such as quantile-based selection and Top-K selection, Assumption 2 is clearly satisfied with tu = 0 +and Iu = 0. For the BH procedure based on the conformal p-values, taking tu as 0 for each j ∈ ˆSu leads a +smaller p-value pj. By the property of the BH procedure, for j ∈ ˆSu, assigning pj to a smaller value will not +change the rejection set (Fithian and Lei, 2020), and hence we have κj←0 = ˆκ for j ∈ ˆSu. In Section 3.3, we +also show that Iu = Op(log m) for the BH procedure. From now on, we write ˆκ(j) = ˆκj←tu for simplicity. +3.1 +FCR control for self-driven selection +When the self-driven selection procedures are used, the samples in the selected calibration set ˆSc and the +selected test set ˆSu are not exchangeable, but the original samples from the calibration set and the test set +are exchangeable. The following theorem provides delicate bounds for the conditional miscoverage gap of the +SCOP. +Theorem 2. Under Assumptions 1 and 2, for any absolute constant C ≥ 1, if 8C log n/(nTU\{j} +(ˆκ(j)) ) ≤ 1 holds +almost surely, the conditional miscoverage probability can be bounded by +PDu,−j +� +Yj ̸∈ PISCOP +j +���j ∈ ˆSu +� +≤ α + ∆(Du,−j), +and +PDu,−j +� +Yj ̸∈ PISCOP +j +���j ∈ ˆSu +� +≥ α − 2∆(Du,−j), +where +∆(Du,−j) = +8C log n +ρTU\{j} +(ˆκ(j)−Iu) +� +�6C log n +nTU\{j} +(ˆκ(j)) ++ +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) +� +� + +2 +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +� +TU\{j} +(ˆκ(j)−Iu) +. +(10) +Our theorem is closely connected to Barber et al. (2022)’s Theorem 2a. Both theorems involve assessing +how the deviations from the “idealized” exchangability would affect the actual miscoverage level. However, +11 + +the interpretations are very different. Barber et al. (2022) showed that under assumption that the test and +calibration samples are non-exchangeable, the miscoverage gap can be bounded by an error term regarding the +total variation between the two samples. Whereas in SCOP the deviation comes from the possible violation of +the similarity between the distributions of {Ri : i ∈ ˆSc} and {Rj : j ∈ ˆSu}. +Remark 3.1. The technical difficulty in proving Theorem 2 lies in coping with the dependence of ˆSc +and ˆSu. To address this problem, we introduce virtual post-selection test set and calibration set, ˆS(j) +u += +� +i ∈ U : Ti ≤ TU\{j} +(ˆκ(j)) +� +and ˆS(j) +c += +� +i ∈ C : Ti ≤ TU\{j} +(ˆκ(j)) +� +respectively. We denote the corresponding virtual +conformal PI constructed by ˆS(j) +c +as PIj( ˆS(j) +c ). For clarity, we rewrite the real conformal PI constructed by ˆSc in +Algorithm 1 as PIj( ˆSc) ≡ PISCOP +j +. Notice that, the threshold TU\{j} +(ˆκ(j)) and the virtual selected calibration set ˆS(j) +c +are independent of the test candidate j. Therefore, the test candidate j and the calibration candidate k are ex- +changeable in the set ˆS(j) +c +∪{j} under the selection conditions. It remains to control two conditional miscoverage +gaps: PDu,−j +� +j ̸∈ PIj( ˆS(j) +c )|j ∈ ˆS(j) +u +� +− α and PDu,−j +� +j ̸∈ PIj( ˆSc)|j ∈ ˆSu +� +− PDu,−j +� +j ̸∈ PIj( ˆS(j) +c )|j ∈ ˆS(j) +u +� +; +the former can be bounded as in conventional conformal inference. +Our theorem shows that a tight control of the deviation term ∆(Du,−j) in (10) leads to effective FCR +control. Next we carefully interpret the bound and present more explicit settings in which the FCR achieves +or is very close to the nominal level. Observe that controlling ∆(Du,−j) actually boils down to establishing +the upper bound of the difference TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) and the lower bound of the denominator TU\{j} +(ˆκ(j)−Iu). To +guarantee that the denominator will stay away from 0, we impose the following assumption on ˆκ. +Assumption 3. The ranking threshold satisfies ˆκ ≥ γm for some γ ∈ (0, 1). +The lower bound on ˆκ in Assumption 3 is mild and reasonable, since the FCR control will be extremely +difficult when | ˆSu|/n = ˆκ/n = o(1) for a small level α. Applying the well-known representation of spacing +between consecutive order statistics (c.f. Lemma D.1) to {Ti}i∈U\{j}, together with Assumption 3, we can +obtain the following FCR control result for self-driven selection procedures. +Theorem 3. Under Assumptions 1-3. If γ > Iu/m, the FCR value of SCOP with self-driven selection +procedures can be controlled at +FCRSCOP = α + O +� log2(n ∨ m) +ργ(γ − Iu/m) +�Iu +m + 1 +n +�� +. +In the asymptotic regime, FCRSCOP is exact if Iu = o(m), that is lim(n,m)→∞ FCRSCOP = α. Recalling +that for quantile-based selection and Top-K selection, we have Iu = 0, and thus Theorem 3 guarantees the +FCR of SCOP with such selection procedures can attain the target level in a nearly optimal rate (up to a +logarithmic factor). +12 + +3.2 +FCR control for calibration-assisted selection +For calibration-assisted selective procedures, the analysis is more complex because the ranking threshold ˆκ +depends also on the calibration set Dc. It implies that a more tractable ranking threshold is needed to decouple +the dependence on the selected samples and the calibration samples simultaneously. That is for any j ∈ ˆSu and +k ∈ C, let ˆκ(j,k) ≡ ˆκj←tu,k←tc be the ranking threshold by replacing Tj with tu and Tk with tc simultaneously. +The virtual post-selection calibration set is further defined as ˆS(j,k) +c += +� +i ∈ C \ {k} : Ti ≤ TU\{j} +(ˆκ(j,k)) +� +. +The following assumption, an analog of Assumption 2, is imposed to restrict the change in the ranking +threshold after replacing one calibration score. +Assumption 4. There exists some tc ∈ R and some positive integer Ic ≤ m such that ˆκ ≤ ˆκk←tc ≤ ˆκ + Ic +holds for any k ∈ C. +The following theorem is parallel with Theorem 2. +Theorem 4. Under Assumptions 1-4, for the calibration-assisted selection, the conditional miscoverage +probability of SCOP satisfies +PDu,−j +� +Yj ̸∈ PISCOP +j +��j ∈ ˆSu +� +≤ α + EDu,−j +� +max +k +∆(D(j,k)) +� +, +and +PDu,−j +� +Yj ̸∈ PISCOP +j +��j ∈ ˆSu +� +≥ α − 2EDu,−j +� +max +k +∆(D(j,k)) +� +, +where +∆(D(j,k)) := +2TU\{j} +(ˆκ(j,k)) +� +R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) +� +� +TU\{j} +(ˆκ(j,k)−Iu−Ic) +�2 ++ +4 +� +TU\{j} +(ˆκ(j,k)) − TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +TU\{j} +(ˆκ(j)) ++ +d(j,k) +| ˆS(j,k) +c +| + 1 +, +(11) +with d(j,k) = � +i∈ ˆ +S(j,k) +c +1 +� +TU\{j} +(ˆκ(j,k)−Ic−1) < Ti ≤ TU\{j} +(ˆκ(j,k)) +� +, U (j,k) = ⌈(1 − α)(| ˆS(j,k) +c +| + 2)⌉ + d(j,k) and L(j,k) = +⌈(1 − α)(| ˆS(j,k) +c +| + 2 − d(j,k))⌉ − 2. +Remark 3.2. We can see that all the terms in (11) are independent of the samples j ∈ ˆSu and k ∈ C. The +quantity d(j,k) measures the size difference between the real calibration set ˆSc and the virtual calibration set +ˆS(j,k) +c +. The term R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) represents the largest possible distance of the corresponding quantiles +in ˆSc and ˆS(j,k) +c +, which can be bounded in ˆS(j,k) +c +conditional on the data set Du,−j. The remaining parts in +maxk ∆(D(j,k)) rely on the difference between thresholds from TU\{j}. +Equipped with the conditional miscoverage gap in Theorem 4, we can obtain the FCR control results of +SCOP with calibration-assisted selection in the following theorem. +Theorem 5. Under Assumptions 1-4. If γ > (Ic + Iu)/m, the FCR value of SCOP for calibration-assisted +selection can be controlled at +FCRSCOP = α + O +� +log2(n ∨ m) +ρ(γ − (Ic + Iu)/m)2 +�Ic + Iu +m ++ 1 +n +�� +. +13 + +Similar to the results with self-driven selection, the SCOP can control the FCR around the target value +with a small gap. To decouple the dependence between the ranking threshold and calibration set, an addition +term Ic/m appears in Theorem 5, regarding to the effect of replacing one calibration sample. If Ic ∨Iu = o(m), +then we can take m = exp{o(n +1 +2 )} and have lim(n,m)→∞ FCRSCOP = α. +3.3 +Prediction-oriented selection with conformal p-values +We discuss the implementation of the SCOP with a special calibration-assisted selection procedure, the +selection via multiple testing based on conformal p-values. The concept of the conformal p-value was proposed +by Vovk et al. (2005). Similar to the conformal PI, the conformal p-values enjoy model/distribution-free +properties. Recently, there exists some works to apply conformal p-values to implement sample selection from +a multiple-testing perspective, such as Bates et al. (2021) and Jin and Candès (2022). +In particular, Jin and Candès (2022) investigated the prediction-oriented selection problem, aiming to +select samples whose unobserved outcomes exceed some specified values while control the proportion of falsely +selected units. This problem can be viewed as the following multiple hypothesis tests: for i ∈ U and some +b0 ∈ R, +H0,i : Yi ≥ b0 +v.s. +H1,i : Yi < b0. +By choosing a monotone function g0 : R+ → [0, 1], one could take the score function as g(x) = g0(ˆµ(x) − b0) +and compute the conformity scores as {Ti = g(Xi) : i ∈ C ∪ U}. +Denote the null set of calibration samples as C0 = {i ∈ C : Yi ≥ b0} and its size as n0 = |C0|. Given the +conformity scores {Ti : i ∈ C0} in the calibration set, the conformal p-value for each test data point can be +calculated by2 +pj := p(Xj) = 1 + |{i ∈ C0 : Ti ≤ Tj}| +n0 + 1 +, +for j ∈ U. +(12) +To control FDR at the target level β ∈ (0, 1), we may deploy BH procedure to {pj : j ∈ U} and obtain the +rejection set ˆSu. +Proposition 3.1. Let pU +(1) ≤ ... ≤ pU +(m) be order statistics of conformal p-values in the test set U. For any +i ∈ U, it holds that {pi ≤ pU +(ˆκ)} = {Ti ≤ TU +(ˆκ)}. +Proposition 3.1 indicates that using the conformal p-values in (12) to obtain ˆSu is equivalent to using the +conformity scores in TU with the same ranking threshold ˆκ, that is ˆSu = {i ∈ U : pi ≤ pU +(ˆκ)} ≡ {i ∈ U : Ti ≤ +TU +(ˆκ)}. Further, we can also obtain the post-selection calibration set by ˆSc = {i ∈ C : Ti ≤ TU +(ˆκ)}. Therefore, +we can frame the BH procedures with conformal p-values as a calibration-assisted selection in Section 3.2. +2Under our continuous assumption, we present the form of conformal p-value without ties in the conformity scores. For the +tie-breaking form, please refer to Bates et al. (2021) for more details. +14 + +Algorithm 2 SCOP under selection with conformal p-values +Input: Training data Dt, calibration data Dc, test data Du, threshold sequence {δ(r) : r ∈ [m]}. +Step 1 Fit prediction model ˆµ(·) and score function g(·) on Dt. Compute the score values TC = {Ti = +g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}. +Step 2 Compute the conformal p-values {pi : i ∈ U} according to (12) based on DC0. Apply the BH +procedure with target level β to TU and obtain (13). Obtain the post-selection subsets: ˆSu = {i ∈ U : Ti ≤ +TU +(ˆκ)} and ˆSc = {i ∈ C : Ti ≤ TU +(ˆκ)}. +Step 3 Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}. Find the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest +value of RSc, denoted by Q +ˆ +Sc(1 − α). +Step 4 Construct PI for each j ∈ ˆSu as PIj = [ˆµ(Xj) − Q +ˆ +Sc(1 − α), ˆµ(Xj) + Q +ˆ +Sc(1 − α)]. +Output: {PIj : j ∈ ˆSu}. +To study the FCR control with selection procedures based on conformal p-values, we consider a more +general class of step-up procedures introduced by Fithian and Lei (2020). Let 0 ≤ δ(1) ≤ · · · ≤ δ(m) ≤ 1 +denote an increasing sequence of thresholds, we choose the ranking threshold for step-up procedures as +ˆκ = max +� +r : pU +(r) ≤ δ(r) +� +, +(13) +where pU +(r) is the rth-smallest conformal p-value. Specially, the BH procedure takes δ(r) = rβ/m. We +summarize the SCOP with the step-up selection procedures in Algorithm 2. +To adapt Assumptions 2 and 4, we can simply take ˆκ(j) = ˆκj←0 and ˆκ(k) = ˆκk←1 by replacing Tj with 0 +for j ∈ U and Tk with 1 for k ∈ C, respectively. From Lemma 1 in Fithian and Lei (2020), we have ˆκ(j) = ˆκ +for any j ∈ ˆSu in the step-up procedures. The next proposition characterizes the magnitudes of ˆκ(j) − ˆκ for +any j ̸∈ ˆSu and ˆκ(k) − ˆκ for any k ∈ C. +Proposition 3.2. Suppose {Xi : i ∈ C ∪ U} are i.i.d. continuous random variables. Let Ω(r) = {ℓ ∈ [n0] : +δ(r) < +ℓ+1 +n0+1 ≤ δ(r + 1)}. For step-up procedures (13) using conformal p-values defined in (12) and any +absolute constant C > 1, +1. For any j ∈ ˆSu, we have ˆκ(j) = ˆκ. In addition, for any j ∈ U \ ˆSu, +ˆκ(j) − ˆκ ≤ 12C log m + 8Cm log m +n0 + 1 +max +⌈γm⌉−0.4 + 4(X(1) − 1)1X(2)≤−0.4.The +noise is ϵ ∼ N(0, 1) and independent of X too. +16 + +We fix the labeled data size 2n = 400 at first and equally split it into Dt and Dc. The regression model ˆµ(·) +is fitted on Dt by ordinary least-squares (OLS) for Scenario A, support vector machine (SVM) for Scenario B +and random forest (RF) for Scenario C, respectively. The SVM and RF are implemented by R packages ksvm +and randomForest with default parameters. In most cases, the selection score is chosen as the same as the +prediction value, i.e., Ti = ˆµ(Xi). +We want to select one subset via ˆSu = {i ∈ U : Ti ≤ ˆτ}, where ˆτ is the threshold. To illustrate the wide +applicability of the proposed method, several selection thresholds ˆτ are considered. +(1) T-cal(q): q%-quantile of true response Y in calibration set, that is ˆτ is q%-quantile of {Yi : i ∈ Dc}; +(2) T-test(q): q%-quantile of predicted response ˆµ(X) in test set, i.e. ˆτ = TU +(qm/100); +(3) T-exch(q): q%-quantile of predicted response ˆµ(X) in both calibration set and test set, that is ˆτ is the +q%-quantile of TC∪U = {Ti : Ti ∈ Dc ∪ Du}; +(4) T-cons(b0): a pre-determined constant value b0, i.e., ˆτ = b0; +(5) T-pos(b0,β): The prediction-oriented selection proposed by Jin and Candès (2022), where one would +like to select those test samples with response Y smaller than b0 while controlling the FDR level at +β = 0.2. Here the threshold ˆτ is computed by the BH procedure with conformal p-values in Section 4. +(6) T-top(K): Kth smallest value of predicted responses ˆµ(X) in test set, i.e. ˆτ = TU +(K). +(7) T-clu: One popular choice of thresholds is based on clustering, where the boundary value of two +possible groups is a natural cut point. To be specific, we want to find the threshold ˆτ that minimizes +the within-group sum of squares, i.e, +ˆτ = arg min +t∈TC∪U +� +i∈S1(t) +(Ti − ¯T +S1(t))2 + +� +k∈Sc +1(t) +(Tk − ¯T +Sc +1(t))2, +where S1(t) = {i ∈ C ∪ U : Ti ≤ t}, Sc +1(t) = C ∪ U/S1(t) and ¯T +S1(t) is the sample mean in S1(t). +Among all the considered threshold selections, only the T-exch(q), T-cons(b0) and T-clu satisfy the +exchangeability with respective to {Ti : i ∈ C ∪ U}. The threshold T-top(K) is actually a special case of +T-test(q) with K = [qm/100]. We apply SCOP to construct PIs for the selected individuals in ˆSu with +target FCR level α = 10%. Two benchmarks are included for comparison. One is to directly construct a +(1 − α)-marginal prediction interval as (2) for each selected sample based on the whole calibration set. We +refer this method as ordinary conformal prediction (OCP) and notice that it takes no account of the selection +effects. Another one is the FCR-adjusted conformal prediction (ACP), which builds 1 − α| ˆSu|/m level PI as +(3) for each selected individual. The performances are compared in terms of the FCR and average length (AL) +of constructed PIs among 1,000 repetitions. +17 + +method +SCOP +OCP +ACP +T−cal +T−test +T−exch +Scenario A +Scenario B +Scenario C +30 +60 +90 +30 +60 +90 +30 +60 +90 +5 +10 +15 +20 +25 +5 +10 +15 +20 +5 +10 +15 +q% +FCR(%) +T−cal +T−test +T−exch +Scenario A +Scenario B +Scenario C +30 +60 +90 +30 +60 +90 +30 +60 +90 +10.0 +12.5 +15.0 +17.5 +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +5 +6 +7 +8 +q% +Length +Figure 2: Empirical FCR (%) and average PI length for quantile based thresholds with varying quantile level q. The +black dashed line represents the target FCR level 10%. +We firstly fix the size of test data Du as m = 200 and consider the three quantile-based thresholds: +T-cal(q), T-test(q) and T-exch(q). Figure 2 displays the estimated FCR and AL of PIs through varying +the quantile level q% from 20% to 100%. Across all the settings, it is evident that the SCOP is able to deliver +quite accurate FCR control and have more narrowed PIs. As expected, the OCP yields the same lengths of +PIs under all the settings and can only control the FCR under q% = 100%, that is the situation all the test +data are included without selection. This can be understood since the OCP builds the marginal PIs using the +whole calibration set without consideration of the selection procedure and thus possesses the length of PI as +2QC(1 − α) in (2). The ACP results in considerably conservative FCR levels and accordingly it performs not +well in terms of the AL of PIs in most settings. This is not surprising as the ACP marginally constructs much +larger 1 − α| ˆSu|/m PIs to ensure the FCR control than the target level. +In Table 1, we present the results of the remaining four thresholds, including T-cons(b0), T-pos(b0,β), +T-top(K) and T-clu. Here, we fix the constant b0 for both T-cons(b0) and T-pos(b0,β) as the 30%-quantile +of the true response Yi’s, and choose the target FDR level β = 20% for T-pos(b0,β) and K = 60 for T-top(K). +It can be seen that the FCR levels of SCOP are close to the nominal level. The SCOP also achieves satisfactory +narrowed PIs under all the scenarios. The OCP leads to much different FCR levels but same average length +18 + +Table 1: Comparisons of Empirical FCR (%) and average length (AL) under different scenarios and thresholds +with target FCR α = 10%. The sample sizes of the calibration set and the test set are fixed as n = m = 200. +T-con(b0) +T-pos(b0, 20%) +T-top(60) +T-clu +SCOP +OCP +ACP +SCOP +OCP +ACP +SCOP +OCP +ACP +SCOP +OCP +ACP +Scenario A +FCR +10.02 +14.27 +4.87 +7.31 +13.56 +2.51 +9.75 +15.53 +4.85 +9.78 +10.16 +5.07 +AL +11.77 +9.91 +14.77 +15.52 +9.91 +22.53 +12.02 +9.91 +15.02 +10.15 +9.91 +12.86 +Scenario B +FCR +9.77 +17.41 +7.24 +9.61 +16.33 +7.28 +9.63 +17.70 +6.86 +9.75 +14.98 +7.41 +AL +5.86 +4.70 +6.43 +5.73 +4.70 +6.25 +5.95 +4.70 +6.53 +11.88 +4.70 +5.99 +Scenario C +FCR +9.74 +13.07 +3.62 +9.70 +12.06 +3.88 +10.03 +12.77 +3.92 +9.89 +12.66 +3.67 +AL +5.82 +5.27 +7.41 +5.72 +5.27 +7.19 +5.73 +5.27 +7.23 +5.77 +5.27 +7.33 +of PIs by different selection thresholds. This corroborates the insight that the OCP is unable to give a valid +coverage guarantee on the selected ones. In contrast, the FCRs of ACP are overly conservative and in turn its +PI lengths would be considerably inflated. +At last, we evaluate the effect of different sizes of calibration and test sets under Scenario C by varying +n and m from 100 to 200. Here four selection thresholds are included: T-test(30%), T-pos(−1.2,20%), +T-top(60) and T-clu. The results are reported in Table 2. We see that all the three methods tend to yield +narrowed PIs as the calibration size n increases. However, the SCOP method performs much better than +OCP and ACP in terms of FCR control across all the settings. This clearly demonstrates the efficiency of +our proposed SCOP, that is a data-driven method which enables FCR control with a wide range of selection +procedures and meanwhile tends to build a relatively narrowed PIs with 1 − α level. +4.2 +Real data applications +4.2.1 +Drug discovery +Early stages of drug discovery aim at finding those high binding affinity of a specific target from a pool of +drug-target pairs (Santos et al., 2017). It is important to provide reliable PIs for those drug-target pairs +which own high binding affinity predictions. After screening, an effective subset one may be interested in +can be selected for further clinical trials (Huang et al., 2022). In this example, we apply the proposed SCOP +to construct PIs of binding affinities for those promising drug-target pairs meanwhile achieve FCR control. +We consider the DAVIS dataset (Rogers and Hahn, 2010), which contains 25, 772 drug-target pairs. Each +pair includes the binding affinity, the structural information of drug compound and the amino acid sequence +19 + +Table 2: Empirical FCR (%) and AL values under Scenario C with different combinations of (n, m). The +target FCR level is α = 10%. +(n,m) +T-test(30%) +T-pos(−1.2,20%) +T-top(60) +T-clu +SCOP +OCP +ACP +SCOP +OCP +ACP +SCOP +OCP +ACP +SCOP +OCP +ACP +(100,100) +FCR +9.70 +14.54 +4.49 +9.31 +14.48 +4.40 +9.89 +8.95 +5.41 +9.73 +15.03 +4.34 +AL +7.22 +6.25 +8.69 +7.31 +6.25 +8.743 +6.12 +6.25 +7.31 +7.31 +6.25 +8.8 +(200,100) +FCR +9.89 +12.68 +3.69 +9.75 +12.48 +3.75 +10.00 +8.05 +4.62 +10.14 +13.24 +3.48 +AL +5.75 +5.26 +7.26 +5.77 +5.26 +7.23 +4.94 +5.26 +6.14 +5.82 +5.26 +7.39 +(100,200) +FCR +9.60 +14.55 +4.44 +9.23 +14.63 +4.39 +9.51 +14.55 +4.44 +9.66 +15.03 +4.26 +AL +7.32 +6.30 +8.72 +7.41 +6.30 +8.75 +7.34 +6.30 +8.72 +7.37 +6.30 +8.85 +(200,200) +FCR +9.88 +12.35 +3.83 +9.69 +12.18 +3.81 +9.80 +12.35 +3.83 +9.76 +12.71 +3.64 +AL +5.76 +5.29 +7.29 +5.75 +5.29 +7.27 +5.77 +5.29 +7.29 +5.81 +5.29 +7.40 +of target protein; the drugs and targets are firstly encoded into numerical features through Python library +DeepPurpose (Huang et al., 2020), and the responses are taken as the log-scale affinities. We randomly sample +2,000 observations as calibration set and another 2,000 ones as test set, and use the remaining ones as training +set to fit a small neural network model with 3 hidden layers and 5 epochs. +Our goal is to consider building PIs of drug-target pairs on the test set through selecting their predicted +affinities which exceed some specific threshold. Here we consider three different thresholds: 70%-quantile of +true responses in calibration set (T-cal(70%)), 70%-quantile of predicted affinities in test set (T-test(70%)), +and selecting those drug-target pairs with affinities larger than 9.21 while controlling FDR at 0.2 level +(T-pos(9.21,20%)). The target FCR level is α = 10%. +To evaluate the performance of SCOP , we also consider building PIs for those selected candidates by OCP +and ACP. Figure 3 shows the boxplots of FCP and average length of PIs based on three methods among 100 +runs. As illustrated, the SCOP has stable FCP close to the nominal level across all the threshold selections. +In comparison, both OCP and ACP result in conservative FCP levels. The lengths of PIs constructed by OCP +and ACP are much broader compared to those of the SCOP. In fact, the responses of log-scale affinities truly +range at (−5, 10), and thus those broader PIs would barely provide useful information for further clinical +trails. +4.2.2 +House price analysis +We apply the SCOP to implement the prediction of house prices of interest. As an economic indicator, better +understanding house prices can provide meaningful suggestions to researchers and decision makers in real +estate market (Anundsen and Jansen, 2013). In recent decades, business analysts use machine learning tools +to forecast house prices and determine the investment strategy (Park and Bae, 2015). In this example, we use +20 + +method +SCOP +OCP +ACP +0 +5 +10 +15 +T−cal(70%) +T−test(70%) +T−pos(9.21,20%) +FCP(%) +0 +5 +10 +15 +T−cal(70%) +T−test(70%) +T−pos(9.21,20%) +Length +Figure 3: Boxplots of the values of FCP (%) and PI length for the drug discovery example. The black dashed line +represents the target FCR level 10% and the red rhomboid dot denotes the average value. +our method to build PIs for those house prices which exceed certain thresholds. +We consider one house price prediction dataset from Kaggle3, which contains 4, 251 observations after +removing the missing data. The data records the house price and other covariates about house area, location, +building years and so on. We randomly sample 1, 500 observations and equally split them into three parts +as training, calibration and test sets respectively in each repetition. We firstly train a random forest model +to predict the house prices and consider three different thresholds to select those test observations with +high predicted house prices: T-test(70%), T-pos(0.6,20%) and T-clu. For example, the threshold T- +pos(0.6,20%) means that one would like to select those observations with house prices larger than 0.6 million +under the FDR control at 20% level. After selection, we construct PIs with α = 10%. Table 3 reports the +empirical FCR level and lengths of PIs among 500 replications. We observe that both SCOP and ACP achieve +valid FCR control, but our SCOP has more narrowed PIs compared to ACP. The FCRs of the OCP are +much inflated which implies that many test samples with truly high house prices cannot be covered. The two +examples demonstrate that the proposed SCOP works well for building PIs of selected samples in practical +applications. +3The data is available from https://www.kaggle.com/datasets/shree1992/housedata. +21 + +Table 3: Empirical FCRs (%) and average lengths of PIs for house price dataset with α = 10%. +T-test(70%) +T-pos(0.6,20%) +T-clu +SCOP +OCP +ACP +SCOP +OCP +ACP +SCOP +OCP +ACP +FCR +9.91 +21.75 +9.09 +9.78 +34.74 +7.27 +10.08 +39.71 +8.49 +AL +1.06 +0.67 +1.12 +1.58 +0.67 +2.64 +1.72 +0.67 +1.98 +5 +Concluding remarks +We have investigated the FCR control problem in the scenario of selective conformal inference. The validity +of FCR-adjusted method is verified in the predictive setting, and our proposed SCOP procedure is shown to +be widely applicable against both selection with exchangeable thresholds and non-exchangeable ranking-based +selection with rigorous theoretical guarantee. +To conclude, we point out several directions for future work. First, we focus mainly on using the residuals +as nonconformity scores for the construction of PI. In fact, our framework can readily be extended to +more general nonconformity scores such as the one based on quantile regression (Romano et al., 2019) or +distributional regression (Chernozhukov et al., 2021). Similar theoretical results are still valid for those more +general methods. Second, as split conformal introduces extra randomness from data splitting and reduces +the effectiveness of training models, we may consider the implementation via Jackknife and cross-validation +to refine the prediction intervals (Barber et al., 2021). The SCOP is applicable in such regimes, but certain +stability conditions posed on the predictive algorithms are necessary and theoretical guarantee of SCOP +requires further investigation. 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Let x(⌈n(1−α)⌉) is the ⌈n(1 − α)⌉-smallest value in {xi ∈ R : i ∈ [n]}. Then for any α ∈ (0, 1), +it holds that +1 +n +n +� +i=1 +1 +� +xi > x(⌈n(1−α)⌉) +� +≤ α. +If all values in {xi : i ∈ [n]} are distinct, it also holds that +1 +n +n +� +i=1 +1 +� +xi > x(⌈n(1−α)⌉) +� +≥ α − 1 +n, +Next lemma characterizes the change of order statistics after dropping one of the samples, which is very +useful in the theory of conformal inference (see Lemma 2 in Romano et al. (2019)). +Lemma A.2. For almost surely distinct random variables x1, ..., xn, let {x(r) : r ∈ [n]} be order statistics of +{xi : i ∈ [n]}, and {x[n]\{j} +(r) +: r ∈ [n − 1]} be the order statistics of {xi : i ∈ [n] \ {j}}, then we have: +(1) x(k) ≤ x[n]\{j} +(k) +≤ x(k+1). +(2) +� +xj ≤ x(k) +� += {xj ≤ x[n]\{j} +(k) +}. +Lemma A.3. Suppose all values in TC ∪ TU are almost surely distinct, ˆκ(j) = ˆκ holds for any j ∈ ˆSu and +ˆκ(j) ≤ ˆκ + Iu holds for any j ∈ U \ ˆSu, then we have +(1) For any j ∈ U, TU\{j} +(ˆκ(j)−Iu) ≤ TU +(ˆκ); +(2) For any j ∈ ˆSu, TU\{j} +(ˆκ(j)−1) ≤ TU +(ˆκ) ≤ TU\{j} +(ˆκ(j)) . +The proof of Lemma A.3 is deferred to Section D.1. The next two lemmas are corollaries of the well-known +spacing representation of consecutive random variables (c.f. Lemma D.1), and the proofs can be found in D.2 +and D.3. +Lemma A.4. Suppose Ui +i.i.d. +∼ Unif([0, 1]) and let U(1) ≤ · · · ≤ U(n) be the corresponding order statistics. For +any absolute constant C ≥ 1, it holds that +P +� +� +max +0≤ℓ≤n−1 +� +U(ℓ+1) − U(ℓ) +� +≥ +1 +1 − 2 +� +log(n∨m) +n+1 +log(n ∨ m) +n + 1 +� +� ≤ 2(n ∨ m)−C. +27 + +Lemma A.5. Let ˆSc(t) = {i ∈ C : Ti ≤ t}. If +d +drF(R,T )(r, t) ≥ ρt holds, then for any absolute constant C ≥ 1, +we have +P +� +� +max +0≤ℓ≤| ˆ +Sc(t)|−1 +� +R +ˆ +Sc(t) +(ℓ+1) − R +ˆ +Sc(t) +(ℓ) +� +≥ 1 +ρ +1 +1 − 2 +� +C log(n∨m) +| ˆ +Sc(t)|+1 +2C log(n ∨ m) +| ˆSc(t)| + 1 +� +� ≤ 2(n ∨ m)−C. +Lemma A.6 is used to bound the change in the size of the selected calibration set after changing threshold. +We defer the proof to Section D.4. Lemma A.7 is used to lower bound the size of selected calibration set with +arbitrary threshold t ∈ (0, 1), and Lemma A.8 is used to guarantee the threshold in SCOP will be far away +from 0 under Assumption 3. The proofs can be found in Section D.5 and D.6. +Lemma A.6. Let ˆSc(t2) = {i ∈ C : Ti ≤ t2} and Zi = 1 {t1 < Ti ≤ t2} − t2−t1 +t2 +for some 0 ≤ t1 < t2 ≤ 1. +Then we have +P +� +� +1 +| ˆSc(t2)| +������ +� +i∈ ˆ +Sc(t2) +Zi +������ +≥ 2 +� +eC log(n ∨ m) +| ˆSc(t2)| +� +t2 − t1 +t2 ++ 2eC log(n ∨ m) +| ˆSc(t2)| +� +� ≤ 2(n ∨ m)−C. +Lemma A.7. Let ˆSc(t) = {i ∈ C : Ti ≤ t}. For any C ≥ 1, if 8C log(n ∨ m)/(nt) ≤ 1, we have +P +� +| ˆSc(t)| ≥ n · t +2 +� +≤ (n ∨ m)−C. +Lemma A.8. For any fixed γ ∈ (0, 1) and any j ∈ U, if 8C log(n ∨ m)/(mγ) ≤ 1, we have +P +� +TU\{j} +⌈γm⌉ ≤ γ +2 +� +≤ 2(n ∨ m)−C. +B +Proof of the results in Section 2 +B.1 +Proof of Proposition 2.1 +Proof. For simplicity, we assume Dt is fixed. Let Av,r be the event: r PIs are constructed, and v of these do +not cover the corresponding true responses. Let NPIj denote the event that {Yj ̸∈ PIAD(Xj)}. It holds that +P (Av,r) = 1 +v +m +� +j=1 +P(Av,r, NPIj). +This claim is proved by the Lemma 1 in Benjamini and Yekutieli (2005). Note that ∪r +v=1Av,r is a disjoint +union of events such that | ˆSu| = r and |Nu| = v, where Nu := {j ∈ �Su : Yj ̸∈ PIAD +j +} is the set of constructed +PIs not covering their true labels. By the definition of FCR, we have +FCR = +m +� +r=1 +r +� +v=1 +v +r P (Av,r) = +m +� +r=1 +r +� +v=1 +1 +r +m +� +j=1 +P (Av,r, NPIj) = +m +� +r=1 +m +� +j=1 +1 +r P +� +| ˆSu| = r, NPIj +� +. +(B.1) +For each j ∈ ˆSu, we define the following events +M(j) +k +:= +� +M j +min = k +� +for +k = 1, ..., m. +28 + +Recalling the definition of M j +min = miny +� +| ˆSTj←y +u +| : j ∈ ˆSTj←y +u +� +, we have M j +min ≤ | ˆSu| if j ∈ ˆSu. Following +the proof of Theorem 1 in Benjamini and Yekutieli (2005) and using the decomposition (B.1), we have +FCR = +m +� +r=1 +m +� +j=1 +1 +r +m +� +l=1 +P +� +| ˆSu| = r, j ∈ ˆSu, M(j) +l , Yj ̸∈ PIAD +j +(Xj) +� +(i) += +m +� +r=1 +m +� +j=1 +r +� +l=1 +1 +r P +� +| ˆSu| = r, j ∈ ˆSu, M(j) +l , Yi ̸∈ PIAD +j +(Xj) +� +≤ +m +� +j=1 +m +� +r=1 +r +� +l=1 +1 +l P +� +| ˆSu| = r, j ∈ ˆSu, M(j) +l , Yi ̸∈ PIAD +j +(Xj) +� +(ii) += +m +� +j=1 +m +� +l=1 +1 +l +m +� +r=l +P +� +| ˆSu| = r, j ∈ ˆSu, M(j) +l , Yj ̸∈ PIAD +j +(Xj) +� += +m +� +j=1 +m +� +k=1 +1 +k P +� +M(j) +k , Yj ̸∈ PIAD +j +(Xj) +� +(iii) += +m +� +j=1 +m +� +k=1 +1 +k P +� +M(j) +k +� +P +� +Yj ̸∈ PIAD +j +(Xj) +� +, +(iv) +≤ +m +� +j=1 +m +� +k=1 +1 +k P +� +M(j) +k +� kα +m = α, +where (i) holds due to Mmin(T(−j)) ≤ | ˆSu|, (ii) follows from the interchange of summations over l and r, (iii) +holds since M (j) +k +is independent of the calibration set Dc and the sample j, and (iv) is true because of the +marginal coverage guarantee that for any j ∈ [m], +P +� +Yj ̸∈ PIAD +j +� +≤ α∗ +j = αMmin(TU\{j}) +m +. +(B.2) +Here we emphasize that the miscoverage probability P(Yj ̸∈ PIAD +j +(Xj)) in (iv) does not depend on the selection +condition since the summation is over [m]. Consequently, we may utilize the exchangeability between the +sample j and the calibration set to verify (B.2). +B.2 +Proof of Theorem 1 +Proof. For any given non-empty subsets Su ⊆ U and Sc ⊆ C, and for any j ∈ Su, if it holds that +α − +1 +n + 1 ≤ P +� +Yj ̸∈ PIj +��� ˆSc = Sc, ˆSu = Su +� +≤ α. +(B.3) +29 + +Following Lemma 2.1 in Lee et al. (2016), we can decompose the FCR value with non-empty ˆSu as +FCR0 = E +�� +j∈ ˆ +Su 1 {Yj ̸∈ PIj} +| ˆSu| +���| ˆSu| ̸= 0 +� += E +� +E +�� +j∈ ˆ +Su 1 {Yj ̸∈ PIj} +| ˆSu| +��� ˆSu, ˆSc +� ���| ˆSu| ̸= 0 +� += E +� +� 1 +| ˆSu| +� +j∈ ˆ +Su +P +� +j ̸∈ PIj +�� ˆSu, ˆSc +� ��| ˆSu| ̸= 0 +� +� +≤ E +� +� 1 +| ˆSu| +� +j∈ ˆ +Su +α +��| ˆSu| ̸= 0 +� +� += α, +where the last inequality holds due to the right hand side of (B.3). The FCR value can be controlled by +FCR = FCR0 ×P +� +| ˆSu| > 0 +� +≤ α. +Similarly, using the left hand side of (B.3) we can also obtain that FCR0 ≥ α− +1 +n+1. Further, if P(| ˆSu| > 0) = 1, +we can also obtain the lower bound of the FCR value by +FCR = FCR0 ×P +� +| ˆSu| > 0 +� += FCR0 ≥ α − +1 +n + 1. +Therefore, it suffices to verify (B.3) for SCOP under exchangeable assumption. +According to the construction of conformal PI and Lemma A.2, given ˆSc = Sc, we know that +{Yj ̸∈ PIj} = +� +Rj > QSc(1 − α) +� += +� +Rj > QSc∪{j}(1 − α) +� +, +where QSc∪{j}(1 − α) is the ⌈(1 − α)(|Sc| + 1)⌉-st smallest value in {Ri : i ∈ Sc ∪ {j}}. Next we will suppress +the dependency on 1 − α. Invoking Lemma A.1 to Sc ∪ {j}, we have +P +� +Rj > QSc∪{j}��� ˆSc = Sc, ˆSu = Su +� +≤ α + +1 +|Sc| + 1E +� � +k∈Sc +1 +� +Rj > QSc∪{j}� +− 1 +� +Rk > QSc∪{j}� ��� ˆSu = Su, ˆSc = Sc +� += α + +1 +|Sc| + 1 +� +k∈Sc +∆j,k, +(B.4) +where +∆j,k := P +� +Rj > QSc∪{j}��� ˆSc = Sc, ˆSu = Su +� +− P +� +Rk > QSc∪{j}��� ˆSc = Sc, ˆSu = Su +� +. +For any j ∈ Su and k ∈ Sc, we denote +ESu,j(ˆτ) = +� +� +� +� +i∈Su\{j} +{Ti ≤ ˆτ} , +� +i∈U\Su +{Ti > ˆτ} +� +� +� , +ESc,k(ˆτ) = +� +� +� +� +i∈Sc\{k} +{Ti ≤ ˆτ} , +� +i∈C\Sc +{Ti > ˆτ} +� +� +� . +30 + +Then the selection condition can be equivalently written as +� +ˆSc = Sc, ˆSu = Su +� += {Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ)} . +As a consequence, we have +∆j,k = P +� +Rj > QSc∪{j}|Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ) +� +− P +� +Rk > QSc∪{j}|Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ) +� +. +(B.5) +Let ˆτ(j,k) be the threshold obtained by swapping sample j ∈ U and k ∈ C. Under our assumption, we know +ˆτ(j,k) = ˆτ with probability 1. It follows that +{Tk ≤ ˆτ, Tj ≤ ˆτ, ESu,j(ˆτ), ESc,k(ˆτ)} = +� +Tk ≤ ˆτ(j,k), Tj ≤ ˆτ(j,k), ESu,j(ˆτ(j,k)), ESc,k(ˆτ(j,k)) +� +. +Therefore, we can guarantee ∆j,k = 0 from (B.5). +C +Proof of the results in Section 3 +In this section, we use Eu,−j[·] and Pu,−j(·) to denote the expectation and probability given data set Du,−j. +C.1 +Proof of Lemma 1 +Proof. According to the definition of FCR, we have +FCR = E +� +� 1 +| ˆSu| +� +j∈ ˆ +Su +1 {Yj ̸∈ PIj} +� +� (i) += E +� +�� +j∈U +1 +� +j ∈ ˆSu +� +ˆκ +1 {Yj ̸∈ PIj} +� +� +(ii) += E +� +�� +j∈U +1 +ˆκ(j) 1 +� +j ∈ ˆSu +� +1 {Yj ̸∈ PIj} +� +� += E +� +�� +j∈U +1 +ˆκ(j) Pu,−j +� +Yj ̸∈ PIj +��j ∈ ˆSu +� +Pu,−j +� +j ∈ ˆSu +� +� +� +≤ E +� +�� +j∈U +(α + ∆(Du,−j)) +1 +� +j ∈ ˆSu +� +ˆκ(j) +� +� +(iii) += α · E +� +�� +j∈U +1 +� +j ∈ ˆSu +� +| ˆSu| +� +� + E +� +� 1 +| ˆSu| +� +j∈ ˆ +Su +∆(Du,−j) +� +� += α + E +� +� 1 +| ˆSu| +� +j∈ ˆ +Su +∆(Du,−j) +� +� , +where the equality (i) holds due to | ˆSu| = ˆκ (c.f. the definition of ˆSu in (8)), (ii) and (iii) come from the +assumption ˆκ = ˆκ(j) under the event j ∈ ˆSu. The lower bound can be derived similarly. +31 + +C.2 +Self-driven selection +In this subsection, we provide the proofs for the results of self-driven selection procedures, where the ranking +threshold ˆκ only depends on the test set Du. +C.2.1 +Proof of Theorem 2 +Proof of Theorem 2. Recall the definitions +ˆS(j) +c += +� +i ∈ C : Ti ≤ TU\{j} +(ˆκ(j)) +� +, +Q +ˆ +S(j) +c +∪{j} = R +ˆ +S(j) +c +∪{j} +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉). +Invoking Lemma A.1, we know it holds that +α − +1 +| ˆS(j) +c | + 1 +≤ +1 +| ˆS(j) +c | + 1 +� +i∈ ˆ +S(j) +c +∪{j} +1 +� +Ri > Q +ˆ +S(j) +c +∪{j}� +≤ α. +(C.1) +In addition, we also know {j ̸∈ PIj} = {Rj > R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉)} = {Rj > R +ˆ +S(j) +c +∪{j} +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉)} by the +construction of PIj and Lemma A.2. Rearranging the inequality in the right hand side of (C.1) gives +1 {j ̸∈ PIj} = 1 +� +Rj > Q +ˆ +Sc∪{j}� +≤ α − +1 +| ˆS(j) +c | + 1 +� +i∈ ˆ +S(j) +c +∪{j} +1 +� +Ri > Q +ˆ +S(j) +c +∪{j}� ++ 1 +� +Rj > Q +ˆ +Sc∪{j}� += α − +1 +| ˆS(j) +c | + 1 +� +k∈ ˆ +S(j) +c +� +1 +� +Rk > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rj > Q +ˆ +S(j) +c +∪{j}�� ++ 1 +� +Rj > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +S(j) +c +∪{j}� +. +(C.2) +Given the dataset Du,−j, taking expectation on both sides of (C.2) conditional on the event {j ∈ ˆSu} (that is +{Tj ≤ TU +(ˆκ)}) yields +Pu,−j +� +Yj ̸∈ PIj +���Tj ≤ TU +(ˆκ) +� +≤ α − Eu,−j +� +� +1 +| ˆS(j) +c | + 1 +� +k∈ ˆ +S(j) +c +� +1 +� +Rk > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rj > Q +ˆ +S(j) +c +∪{j}�� ���Tj ≤ TU +(ˆκ) +� +� ++ Pu,−j +� +Rj > Q +ˆ +Sc∪{j}���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}���Tj ≤ TU +(ˆκ) +� +. +(C.3) +Rearranging the inequality in the left hand side of (C.1), we can also have +Pu,−j +� +Yj ̸∈ PIj +���Tj ≤ TU +(ˆκ) +� +≥ α − +1 +| ˆS(j) +c | + 1 +− Eu,−j +� +� +1 +| ˆS(j) +c | + 1 +� +k∈ ˆ +S(j) +c +� +1 +� +Rk > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rj > Q +ˆ +S(j) +c +∪{j}�� ���Tj ≤ TU +(ˆκ) +� +� ++ Pu,−j +� +Rj > Q +ˆ +Sc∪{j}���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}���Tj ≤ TU +(ˆκ) +� +. +(C.4) +Next we introduce three lemmas to control the additional terms except α in (C.3) and (C.4). And we +defer their proofs to Section C.3.1, C.3.2 and C.3.3. +32 + +Lemma C.1. Under the conditions of Theorem 2, we have +������ +Eu,−j +� +� +1 +| ˆS(j) +c | + 1 +� +k∈ ˆ +S(j) +c +� +1 +� +Rj > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rk > Q +ˆ +S(j) +c +∪{j}�� ���Tj ≤ TU +(ˆκ) +� +� +������ +≤ 2 +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +. +Lemma C.2. Under the conditions of Theorem 2, we have +���Pu,−j +� +Rj > Q +ˆ +Sc∪{j}���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}���Tj ≤ TU +(ˆκ) +���� +≤ 4C log(n ∨ m) +ρTU\{j} +(ˆκ(j)−Iu) +� +�12C log(n ∨ m) +nTU\{j} +(ˆκ(j)) ++ +2 +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +� +TU\{j} +(ˆκ(j)) +� +� + 3(n ∨ m)−C +TU\{j} +(ˆκ(j)−Iu) +. +Lemma C.3. Under the conditions of Theorem 2, we have +Eu,−j +� +1 +| ˆS(j) +c | + 1 +���Tj ≤ TU +(ˆκ) +� +≤ +TU\{j} +(ˆκ(j)) +TU\{j} +(ˆκ(j)−Iu) +� +� +2 +nTU\{j} +(ˆκ(j)) + 2 ++ (n ∨ m)−C +� +� . +Armed with Lemmas C.1, C.2 and C.3, we can obtain the result of Theorem 2 after some simplifications. +A remark on the construction of virtual calibration set. +To decouple the dependence on the candidate +j, we introduce the virtual calibration set ˆS(j) +c +by using the threshold TU\{j} +(ˆκ(j)) , where ˆκ(j) is independent of +test sample j. Another good property of ˆS(j) +c +is that ˆSc ⊆ ˆS(j) +c . In Assumption 3, we assume ˆκ ≥ γm holds +almost surely. Let us consider the following naive virtual calibration set, +ˆSnaive +c +:= +� +i ∈ C : Ti ≤ TU\{j} +(⌈γm⌉) +� +. +Even though ˆSnaive +c +possesses two good properties of ˆS(j) +c , but we cannot use ˆSnaive +c +in our proof. Notice that +| ˆSnaive +c +| − | ˆSc| = +� +i∈ ˆ +Snaive +c +1 +� +TU +(ˆκ) < Ti ≤ TU\{j} +(⌈γm⌉) +� +. +Given Du,−j, the conditional expectation of the indicator function is +Eu,−j +� +1 +� +TU +(ˆκ) < Ti ≤ TU\{j} +(⌈γm⌉) +� +|Ti ≤ TU\{j} +(⌈γm⌉) +� += +TU\{j} +(⌈γm⌉) − TU +(ˆκ) +TU\{j} +(⌈γm⌉) +. +Since the ranking threshold ˆκ can be very close to m, the conditional expectation above may scale as Op(1) +according to the spacing representation (c.f. Lemma D.1). As a consequence, we cannot have an op(1) gap for +FCR control. In Section C.3.2, we show the corresponding conditional expectation of our careful chosen ˆS(j) +c +scales as Op( log(n∨m) +m +). This explains why we cannot use ˆSnaive +c +as the virtual calibration set in the proof, and +ˆS(j) +c +is indeed a nontrivial construction. +33 + +C.2.2 +Proof of Theorem 3 +Proof. Recall that, +∆(Du,−j) = 8C log(n ∨ m) +ρTU\{j} +(ˆκ(j)−Iu) +� +�12C log(n ∨ m) +nTU\{j} +(ˆκ(j)) ++ +2 +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +� +TU\{j} +(ˆκ(j)) +� +� + 2 +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +. +From Lemma A.4, we can guarantee that +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) ≤ Iu × +max +1≤ℓ≤m−2 +� +TU\{j} +(ℓ+1) − TU\{j} +(ℓ) +� +≤ 2CIu log(n ∨ m) +m +, +holds with probability at least 1 − (n ∨ m)−C. In addition, by Lemma A.8, we have +TU\{j} +(ˆκ(j)−Iu) ≥ TU\{j} +(⌈γm⌉−Iu) ≥ TU +(⌈γm⌉−Iu) ≥ TU +(⌈(γ−Iu/m)m⌉) ≥ γ − Iu/m +2 +holds with probability at least 1 − (n ∨ m)−C. Then we have +E +� +max +1≤j≤m ∆(Du,−j) +� +≲ +log2(n ∨ m) +ργ(γ − Iu/m) +� 1 +m + 1 +n +� ++ Iu log(n ∨ m) +m(γ − Iu/m). +Then the conclusion follows from Lemma 1 immediately. +C.3 +Deferred Proofs of Section C.2 +C.3.1 +Proof of Lemma C.1 +Proof. We first notice that +Eu,−j +� +� +1 +| ˆS(j) +c | + 1 +� +k∈ ˆ +S(j) +c +� +1 +� +Rj > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rk > Q +ˆ +S(j) +c +∪{j}�� ���j ∈ ˆSu +� +� += +� +Sc⊆[m] +Pu,−j +� +ˆS(j) +c += Sc +� +|Sc| + 1 +× +Eu,−j +� � +k∈Sc +1 +� +Rj > QSc∪{j}� +− 1 +� +Rk > QSc∪{j}� ���Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +. +(C.5) +Next we will bound the conditional expectation term in (C.5). From the definition of ˆS(j) +c , we know +{ ˆS(j) +c += Sc} = +� � +k∈Sc +{Tk ≤ TU\{j} +(ˆκ(j)) } +� � +� +� +� +k∈C\Sc +{Tk > TU\{j} +(ˆκ(j)) } +� +� . +Since both sample j ∈ ˆSu and k ∈ Sc are independent of TU\{j} +(ˆκ(j)) , we can guarantee that +Pu,−j +� +Rk > Q +ˆ +S(j) +c +∪{j}�� ˆS(j) +c += Sc, Tj ≤ TU\{j} +(ˆκ(j)) +� += +Pu,−j +� +Rk > QSc∪{j}, � +i∈Sc∪{j} +� +Ti ≤ TU\{j} +(ˆκ(j)) +�� +Pu,−j +�� +i∈Sc∪{j} +� +Ti ≤ TU\{j} +(ˆκ(j)) +�� += +Pu,−j +� +Rj > QSc∪{j}, � +i∈Sc∪{j} +� +Ti ≤ TU\{j} +(ˆκ(j)) +�� +Pu,−j +�� +i∈Sc∪{j} +� +Ti ≤ TU\{j} +(ˆκ(j)) +�� += Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}�� ˆS(j) +c += Sc, Tj ≤ TU\{j} +(ˆκ(j)) +� +, +(C.6) +34 + +where the penultimate equality holds since the distribution of (Xj, Yj) and (Xk, Yk) are same. It follows that +Eu,−j +� � +k∈Sc +1 +� +Rj > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rk > QSc∪{j}� ���Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� += +� +k∈Sc +� +Pu,−j +� +Rk > QSc∪{j}���Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +− Pu,−j +� +Rk > QSc∪{j}���Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� � +− +|Sc| +|Sc| + 1 +� +Pu,−j +� +Rj > QSc∪{j}���Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +− Pu,−j +� +Rj > QSc∪{j}���Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� � ++ +� +k∈Sc +� +Pu,−j +� +Rk > QSc∪{j}���Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +− Pu,−j +� +Rj > QSc∪{j}���Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� � +=: +� +k∈Sc +∆k + +|Sc| +|Sc| + 1∆j + 0. +(C.7) +For ∆k, we have the following upper bound +∆k = +Pu,−j +� +Rk > QSc∪{j}, Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +− +Pu,−j +� +Rk > QSc∪{j}, Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +(i) +≤ +Pu,−j +� +Rk > QSc∪{j}, TU\{j} +(ˆκ(j)−Iu) < Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +(ii) +≤ +Pu,−j +� +TU\{j} +(ˆκ(j)−Iu) < Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ) +, ˆS(j) +c += Sc +� +(iii) +≤ +Pu,−j +� +TU\{j} +(ˆκ(j)−Iu) < Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)−Iu), ˆS(j) +c += Sc +� +(iv) += +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +, +(C.8) +where (i) and (iii) hold since TU\{j} +(ˆκ(j)−Iu) ≤ TU +(ˆκ) ≤ TU\{j} +(ˆκ(j)) (c.f. Lemma A.3), (ii) follows from the conclusion +(2) of Lemma A.2, and (iv) follows from the with firstly conditioning on Du,−j ∪ Dc. Similarly, we can obtain +35 + +the following lower bound +∆k = +Pu,−j +� +Rk > QSc∪{j}, Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +− +Pu,−j +� +Rk > QSc∪{j}, Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +(v) +≥ +Pu,−j +� +Rk > QSc∪{j}, Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +� +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� − 1 +� +� +(vi) +≥ − +Pu,−j +� +TU +(ˆκ) < Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +≥ − +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)) +, +(C.9) +where (v) holds because TU +(ˆκ) ≤ TU\{j} +(ˆκ(j)) , and (vi) is true since the term in the bracket is negative. Combining +(C.8) and (C.9), we have +|∆k| ≤ +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +. +(C.10) +For ∆j, we have the following upper bound +∆j = +Pu,−j +� +Rj > QSc∪{j}, Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +− +Pu,−j +� +Rj > QSc∪{j}, Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +≤ +Pu,−j +� +Rj > QSc∪{j}, Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +� +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +− 1 +� +� +≤ +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +− 1 +≤ +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +. +The lower bound of ∆j can be derived as +∆j ≥ +Pu,−j +� +Rj > QSc∪{j}, Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +− +Pu,−j +� +Rj > QSc∪{j}, Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +≥ − +Pu,−j +� +TU\{j} +(ˆκ(j)−Iu) < Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)) , ˆS(j) +c += Sc +� +≥ − +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)) +. +36 + +Hence we have +|∆j| ≤ +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +. +(C.11) +Plugging (C.10) and (C.11) into (C.7), we have +�����Eu,−j +� � +k∈Sc +1 +� +Rj > Q +ˆ +S(j) +c +∪{j}� +− 1 +� +Rk > QSc∪{j}� ���Tj ≤ TU +(ˆκ), ˆS(j) +c += Sc +������ +≤ (|Sc| + 1) · +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−Iu) +TU\{j} +(ˆκ(j)−Iu) +. +Taking consideration of (C.5), we can complete the proof of Lemma C.1. +C.3.2 +Proof of Lemma C.2 +Proof. From TU +(ˆκ) ≤ TU\{j} +(ˆκ(j)) in Lemma A.3, we know ˆSc ⊆ ˆS(j) +c . Then for any j ∈ ˆSu, we have +| ˆS(j) +c | − | ˆSc| = +� +i∈ ˆ +S(j) +c +1 +� +TU +(ˆκ) < Ti ≤ TU\{j} +(ˆκ(j)) +� +≤ +� +i∈ ˆ +S(j) +c +1 +� +TU\{j} +(ˆκ(j)−1) < Ti ≤ TU\{j} +(ˆκ(j)) +� +=: d(j), +(C.12) +where the inequality holds since TU\{j} +(ˆκ(j)−1) ≤ TU +(ˆκ) holds for j ∈ ˆSu (c.f. Lemma A.3). Recall that +Q +ˆ +Sc∪{j} = R +ˆ +Sc∪{j} +(⌈(1−α)(| ˆ +Sc|+1)⌉), +Q +ˆ +S(j) +c +∪{j} = R +ˆ +S(j) +c +∪{j} +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉). +Next we bound the rank of value R +ˆ +Sc +(⌈(1−α)(| ˆ +Sc|+1)⌉) in the set ˆS(j) +c . It follows from ˆSc ⊆ ˆS(j) +c +and (C.12) that +� +i∈ ˆ +S(j) +c +1 +� +Ri ≤ R +ˆ +Sc +(⌈(1−α)(| ˆ +Sc|+1)⌉) +� +≥ +� +i∈ ˆ +Sc +1 +� +Ri ≤ R +ˆ +Sc +(⌈(1−α)(| ˆ +Sc|+1)⌉) +� += ⌈(1 − α)(| ˆSc| + 1)⌉ +≥ ⌈(1 − α)(| ˆS(j) +c | − d(j) + 1)⌉, +and +� +i∈ ˆ +S(j) +c +1 +� +Ri ≤ R +ˆ +Sc +(⌈(1−α)(| ˆ +Sc|+1)⌉) +� +≤ d(j) + +� +i∈ ˆ +Sc +1 +� +Ri ≤ R +ˆ +Sc +(⌈(1−α)(| ˆ +Sc|+1)⌉) +� += d(j) + ⌈(1 − α)(| ˆSc| + 1)⌉ +≤ d(j) + ⌈(1 − α)(| ˆS(j) +c | + 1)⌉. +Two bounds above indicate that +R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|−d(j)+1)) ≤ R +ˆ +Sc +(⌈(1−α)(| ˆ +Sc|+1)⌉) ≤ R +ˆ +S(j) +c +(d(j)+⌈(1−α)(| ˆ +S(j) +c +|+1)). +(C.13) +37 + +By the right hand side of (C.13), we have +Pu,−j +� +Rj > Q +ˆ +Sc∪{j}���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}���Tj ≤ TU +(ˆκ) +� +(i) += Pu,−j +� +Rj > Q +ˆ +Sc +���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +���Tj ≤ TU +(ˆκ) +� +≥ −Eu,−j +� +1 +� +Rj > R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉) +� +− 1 +� +Rj > R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉+d(j)) +� ���Tj ≤ TU +(ˆκ) +� +(ii) +≥ − +Pu,−j +� +R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉) < Rj ≤ R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉+d(j)), Tj ≤ TU\{j} +(ˆκ(j)) +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)−Iu) +� +(iii) +≥ − +Eu,−j +� +R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉+d(j)) − R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉) +� +TU\{j} +(ˆκ(j)−Iu) +, +(C.14) +where (i) comes from the conclusion (2) of Lemma A.2 by dropping j, (ii) holds due to TU\{j} +(ˆκ(j)−Iu) ≤ TU +(ˆκ) ≤ +TU\{j} +(ˆκ(j)) for any j ∈ U (c.f. Lemma A.3), and (iii) holds by dropping event {Tj ≤ TU\{j} +(ˆκ(j)) }. By the left hand +side of (C.13), we similarly have +Pu,−j +� +Rj > Q +ˆ +Sc∪{j}���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}���Tj ≤ TU +(ˆκ) +� +≤ +Eu,−j +� +R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉) − R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1−d(j))⌉) +� +TU\{j} +(ˆκ(j)−Iu) +. +(C.15) +Applying Lemma A.7, with probability at least 1 − (n ∨ m)−C, it holds +| ˆS(j) +c | ≥ n · TU\{j} +(ˆκ(j)) − C +� +n log(n ∨ m) +� +TU\{j} +(ˆκ(j)) +� +1 − TU\{j} +(ˆκ(j)) +� +≥ n · TU\{j} +(ˆκ(j)) +� +�1 − C +� +� +� +�log(n ∨ m) +nTU\{j} +(ˆκ(j)) +� +� ≥ 1 +2n · TU\{j} +(ˆκ(j)) , +(C.16) +where we used the assumption 8C log(n ∨ m)/(nTU\{j} +(ˆκ(j)) ) ≤ 1 almost surely. Given Du,−j, applying Lemma +A.6 with t1 = TU\{j} +(ˆκ(j)−1) and t2 = TU\{j} +(ˆκ(j)) , we can guarantee that +d(j) +| ˆS(j) +c | += +1 +| ˆS(j) +c | +� +i∈ ˆ +S(j) +c +� +�1 +� +TU\{j} +(ˆκ(j)−1) < Ti ≤ TU\{j} +(ˆκ(j)) +� +− +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) +� +� + +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) +≤ 2 +� +eC log(n ∨ m) +| ˆS(j) +c | +� +� +� +� +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) ++ 2eC log(n ∨ m) +| ˆS(j) +c | ++ +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) +≤ 6 +� +� +� +�C log(n ∨ m) +nTU\{j} +(ˆκ(j)) +� +� +� +� +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) ++ 6C log(n ∨ m) +nTU\{j} +(ˆκ(j)) ++ +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +TU\{j} +(ˆκ(j)) +≤ 12C log(n ∨ m) +nTU\{j} +(ˆκ(j)) ++ +2 +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +� +TU\{j} +(ˆκ(j)) +=: ∆1(Du,−j), +(C.17) +38 + +holds with probability at least 1 − 2(n ∨ m)−C. Given Du,−j, with probability at least 1 − 3(n ∨ m)−C, we +have +R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉) − R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|−d(j)+1)⌉) = +⌈(1−α)(| ˆ +S(j) +c +|+1)⌉+d(j)−1 +� +ℓ=⌈(1−α)(| ˆ +S(j) +c +|−d(j)+1)⌉ +R +ˆ +S(j) +c +(ℓ+1) − R +ˆ +S(j) +c +(ℓ) +≤ d(j) +max +0≤ℓ≤| ˆ +S(j) +c +| +� +R +ˆ +S(j) +c +(ℓ+1) − R +ˆ +S(j) +c +(ℓ) +� +(i) +≤ +d(j) +| ˆS(j) +c | +· 1 +ρ +| ˆS(j) +c | +1 − 2 +� +C log(n∨m) +| ˆ +S(j) +c +|+1 +2C log(n ∨ m) +| ˆSc(t)| + 1 +(ii) +≤ ∆1(Du,−j) · 1 +ρ +2C log(n ∨ m) +1 − 2 +� +C log(n∨m) +| ˆ +S(j) +c +|+1 +(iii) +≤ 4C log(n ∨ m) +ρ +∆1(Du,−j), +(C.18) +where (i) follows from applying Lemma A.5 with t = TU\{j} +(ˆκ(j)) , (ii) comes from (C.17), and (iii) comes from +(C.16) and the assumption 8C log(n ∨ m)/(nTU\{j} +(ˆκ(j)) ) ≤ 1 almost surely. Similarly, we also have +R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉+d(j)) − R +ˆ +S(j) +c +(⌈(1−α)(| ˆ +S(j) +c +|+1)⌉) ≤ d(j) +max +0≤ℓ≤| ˆ +S(j) +c +| +� +R +ˆ +S(j) +c +(ℓ+1) − R +ˆ +S(j) +c +(ℓ) +� +≤ 4C log(n ∨ m) +ρ +∆1(Du,−j). +(C.19) +Plugging the upper bound (C.18) and (C.19) into (C.15) and (C.14) respectively, we can guarantee +Pu,−j +� +Rj > Q +ˆ +Sc∪{j}���Tj ≤ TU +(ˆκ) +� +− Pu,−j +� +Rj > Q +ˆ +S(j) +c +∪{j}���Tj ≤ TU +(ˆκ) +� +≤ 4C log(n ∨ m) +ρTU\{j} +(ˆκ(j)−Iu) +� +�12C log(n ∨ m) +nTU\{j} +(ˆκ(j)−Iu) ++ +2 +� +TU\{j} +(ˆκ(j)) − TU\{j} +(ˆκ(j)−1) +� +TU\{j} +(ˆκ(j)) +� +� + 3(n ∨ m)−C +TU\{j} +(ˆκ(j)) +. +C.3.3 +Proof of Lemma C.3 +Proof. Notice that +Eu,−j +� +1 +| ˆS(j) +c | + 1 +���Tj ≤ TU +(ˆκ) +� += +n +� +s=0 +1 +s + 1 +Pu,−j +� +| ˆS(j) +c | = s, Tj ≤ TU +(ˆκ) +� +Pu,−j +� +Tj ≤ TU +(ˆκ) +� +(i) +≤ +n +� +s=0 +1 +s + 1 +Pu,−j +� +| ˆS(j) +c | = s, Tj ≤ TU\{j} +(ˆκ(j)) +� +Pu,−j +� +Tj ≤ TU\{j} +(ˆκ(j)−Iu) +� +(ii) += +n +� +s=0 +1 +s + 1 +TU\{j} +(ˆκ(j)) Eu,−j +� +1 +� +| ˆS(j) +c | = s +�� +TU\{j} +(ˆκ(j)−Iu) += +TU\{j} +(ˆκ(j)) +TU\{j} +(ˆκ(j)−Iu) +Eu,−j +� +1 +| ˆS(j) +c | + 1 +� +, +(C.20) +39 + +where (i) holds due to TU +(ˆκ) ≥ TU\{j} +(ˆκ(j)−Iu) (c.f. Lemma A.3), and (ii) comes from the independence between Tj +and ˆS(j) +c +such that +Pu,−j +� +| ˆS(j) +c | = s, Tj ≤ TU\{j} +(ˆκ(j)) +� += Eu,−j +� +E +� +1 +� +Tj ≤ TU\{j} +(ˆκ(j)) +� +1 +� +| ˆS(j) +c | = s +� ��Du,−j, Dc +�� += TU\{j} +(ˆκ(j)) Eu,−j +� +1 +� +| ˆS(j) +c | = s +�� +. +From the relation (C.16), we know +Pu,−j +� +�| ˆS(j) +c | ≤ +nTU\{j} +(ˆκ(j)) +2 +� +� ≤ (n ∨ m)−C. +Together with (C.20), we have +Eu,−j +� +1 +| ˆS(j) +c | + 1 +���Tj ≤ TU +(ˆκ) +� +≤ +TU\{j} +(ˆκ(j)) +TU\{j} +(ˆκ(j)−Iu) +� +� +2 +nTU\{j} +(ˆκ(j)) + 2 ++ (n ∨ m)−C +� +� . +C.4 +Calibration-assisted selection +In this subsection, we provide the proofs for the results of calibration-assisted selection procedures, where the +ranking threshold ˆκ depends on the test set Du and the calibration set Dc. From now on, we use E−(j,k)[·] +and P−(j,k)[·] to denote the expectation and probability given Dc,−k and Du,−j. +C.4.1 +Proof of Theorem 4 +Proof. From the definition of Q +ˆ +Sc∪{j} and Lemma A.1, we know that +α − +1 +| ˆSc| + 1 +≤ +1 +| ˆSc| + 1 +� +i∈ ˆ +Sc∪{j} +1 +� +Ri > Q +ˆ +Sc∪{j}� +≤ α. +(C.21) +In addition, we also know {j ̸∈ PIj} = {Rj > Q +ˆ +Sc∪{j}}. Rearranging the right hand side of (C.21) gives +1 {j ̸∈ PIj} = 1 +� +Rj > Q +ˆ +Sc∪{j}� +≤ α − +1 +| ˆSc| + 1 +� +i∈ ˆ +Sc∪{j} +1 +� +Ri > Q +ˆ +Sc∪{j}� ++ 1 +� +Rj > Q +ˆ +Sc∪{j}� += α − +1 +| ˆSc| + 1 +� +k∈ ˆ +Sc +� +1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +Sc∪{j}�� +. +(C.22) +Given the dataset Du,−j, taking expectation on both sides of (C.22) conditional on the event Tj ≤ TU +(ˆκ) yields +Pu,−j +� +Yj ̸∈ PIj +���Tj ≤ TU +(ˆκ) +� +≤ α − Eu,−j +� +� +1 +| ˆSc| + 1 +� +k∈ ˆ +Sc +� +1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +Sc∪{j}�� ���Tj ≤ TU +(ˆκ) +� +� . +(C.23) +40 + +Similarly, from the left hand side of (C.21), we can have +Pu,−j +� +Yj ̸∈ PIj +���Tj ≤ TU +(ˆκ) +� +≥ α − Eu,−j +� +� +1 +| ˆSc| + 1 +� +k∈ ˆ +Sc +� +1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +Sc∪{j}�� ���Tj ≤ TU +(ˆκ) +� +� +− Eu,−j +� +1 +| ˆSc| + 1 +���Tj ≤ TU +(ˆκ) +� +. +(C.24) +For any j ∈ ˆSu and k ∈ C, we introduce a new selected virtual calibration set w.r.t. to (j, k), +ˆS(j,k) +c += +� +i ∈ C \ {k} : Ti ≤ TU\{j} +ˆκ(j,k) +� +. +According to Assumptions 2 and 4, we know +ˆκ(j,k) = ˆκj←tu,k←tc ≥ ˆκj←tu = ˆκ(j). +Together with Lemmas A.3 and A.2, we also know +TU +(ˆκ) ≤ TU\{j} +(ˆκ(j)) ≤ TU\{j} +(ˆκ(j,k)). +(C.25) +Hence we can guarantee ˆSc ⊆ ˆS(j,k) +c +∪ {k} if k ∈ ˆSc. Denote +∆(j,k) = P−(j,k) +� +Rk > Q +ˆ +Sc∪{j}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +− P−(j,k) +� +Rj > Q +ˆ +Sc∪{j}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +=: ∆(j,k) +k +− ∆(j,k) +j +. +(C.26) +Notice that +������ +Eu,−j +� +� +1 +| ˆSc| + 1 +� +k∈ ˆ +Sc +� +1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +Sc∪{j}�� ��Tj ≤ TU +(ˆκ) +� +� +������ += +������ +Eu,−j +� +�� +k∈C +1 +� +k ∈ ˆSc +� +| ˆS(j,k) +c +| + 1 +� +1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +Sc∪{j}�� ��Tj ≤ TU +(ˆκ) +� +� +������ ++ +������ +Eu,−j +� +�� +k∈C +| ˆSc| − | ˆS(j,k) +c +| +| ˆS(j,k) +c +| + 1 +1 +� +k ∈ ˆSc +� +| ˆSc| + 1 +� +1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +Sc∪{j}�� ��Tj ≤ TU +(ˆκ) +� +� +������ +(i) +≤ Eu,−j +�� +k∈C +P−(j,k) +� +k ∈ ˆSc|Tj ≤ TU +(ˆκ) +� +��∆(j,k)�� +| ˆS(j,k) +c +| + 1 +� ++ Eu,−j +� +�max +k∈C +���| ˆSc| − | ˆS(j,k) +c +| +��� +| ˆS(j,k) +c +| + 1 +� +� += Eu,−j +� +� +���∆(j,k)��� E−(j,k) +� +� +� +k∈C 1 +� +k ∈ ˆSc +� +| ˆS(j,k) +c +| + 1 +���Tj ≤ TU +(ˆκ) +� +� +� +� + Eu,−j +� +�max +k∈C +���| ˆSc| − | ˆS(j,k) +c +| +��� +| ˆS(j,k) +c +| + 1 +� +� +(ii) +≤ Eu,−j +� +max +k∈C +���∆(j,k)��� +� ++ Eu,−j +� +�max +k∈C +���| ˆSc| − | ˆS(j,k) +c +| +��� +| ˆS(j,k) +c +| + 1 +� +� , +(C.27) +41 + +where (i) follows from the tower rule and | ˆS(j,k) +c +| is independent of samples j and k, and (ii) holds since +| ˆSc| ≤ | ˆS(j,k) +c +| + 1. Further, we can decompose ∆(j,k) +j +in (C.26) as +∆(j,k) +j += +� +P−(j,k) +� +Rj > Q +ˆ +Sc∪{j}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +− P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� � ++ +� +P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +− P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� � ++ P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +=: ∆(j,k) +j,1 ++ ∆(j,k) +j,2 ++ P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +. +(C.28) +Similarly, for ∆(j,k) +k +in (C.26), we have +∆(j,k) +k += +� +P−(j,k) +� +Rk > Q +ˆ +Sc∪{j}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +− P−(j,k) +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� � ++ +� +P−(j,k) +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +− P−(j,k) +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� � ++ P−(j,k) +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +=: ∆(j,k) +k,1 ++ ∆(j,k) +k,2 ++ P−(j,k) +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +. +(C.29) +Using the identical distribution of sample j and k and the fact ˆS(j,k) +c +, TU\{j} +(ˆκ(j,k)) are independent of sample j +and k, we have +P−(j,k) +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +=P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}��Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +. +Taking account of (C.26), (C.28) and (C.29), the equality above results in +∆(j,k) = ∆(j,k) +k,1 ++ ∆(j,k) +k,2 +− +� +∆(j,k) +j,1 ++ ∆(j,k) +j,2 +� +. +(C.30) +Since ˆSc \ {k} ⊆ ˆS(j,k) +c +, under the event j ∈ ˆSu, we know that +| ˆS(j,k) +c +| − | ˆSc| + 1 ≤ +� +i∈ ˆ +S(j,k) +c +1 +� +TU +(ˆκ) < Ti ≤ TU\{j} +(ˆκ(j,k)) +� += +� +i∈ ˆ +S(j,k) +c +1 +� +TU +(ˆκ(j)) < Ti ≤ TU\{j} +(ˆκ(j,k)) +� +≤ +� +i∈ ˆ +S(j,k) +c +1 +� +TU\{j} +(ˆκ(j,k)−Ic−1) < Ti ≤ TU\{j} +(ˆκ(j,k)) +� +=: d(j,k), +(C.31) +where the last inequality holds since ˆκ(j,k) ≤ ˆκ(j) + Ic. +42 + +Bound ∆(j,k) +j,1 +and ∆(j,k) +k,1 . +We first bound the rank of value Q +ˆ +Sc∪{j} in the set {Ri : i ∈ ˆS(j,k) +c +}. For k ∈ ˆSc +(that is, Tk ≤ TU +(ˆκ)), we have +� +i∈ ˆ +S(j,k) +c +1 +� +Ri ≤ Q +ˆ +Sc∪{j}� +≥ +� +i∈ ˆ +S(j,k) +c +∪{j,k} +1 +� +Ri ≤ Q +ˆ +Sc∪{j}� +− 2 +(i) +≥ +� +i∈ ˆ +Sc∪{j} +1 +� +Ri ≤ Q +ˆ +Sc∪{j}� +− 2 += ⌈(1 − α)(1 + | ˆSc|)⌉ − 2 +(ii) +≥ ⌈(1 − α)(| ˆS(j,k) +c +| − d(j,k) + 2)⌉ − 2, +(C.32) +and +� +i∈ ˆ +S(j,k) +c +1 +� +Ri ≤ Q +ˆ +Sc∪{j}� +≤ +� +i∈ ˆ +S(j,k) +c +∪{j,k} +1 +� +Ri ≤ Q +ˆ +Sc∪{j}� +(iii) +≤ +� +i∈ ˆ +Sc∪{j} +1 +� +Ri ≤ Q +ˆ +Sc∪{j}� ++ d(j,k) − 1 +≤ ⌈(1 − α)(2 + | ˆS(j,k) +c +|)⌉ + d(j,k) − 1, +(C.33) +where (i), (ii) holds due to ˆSc ∪ {j} ⊆ ˆS(j,k) +c +∪ {j, k}, and (ii), (iii) comes from (C.31). Similarly, we have +� +i∈ ˆ +S(j,k) +c +1 +� +Ri ≤ Q +ˆ +S(j,k) +c +∪{j,k}� +≥ +� +i∈ ˆ +S(j,k) +c +∪{j,k} +1 +� +Ri ≤ Q +ˆ +S(j,k) +c +∪{j,k}� +− 2 += ⌈(1 − α)(2 + | ˆS(j,k) +c +|)⌉ − 2, +(C.34) +and +� +i∈ ˆ +S(j,k) +c +1 +� +Ri ≤ Q +ˆ +S(j,k) +c +∪{j,k}� +≤ +� +i∈ ˆ +S(j,k) +c +∪{j,k} +1 +� +Ri ≤ Q +ˆ +S(j,k) +c +∪{j,k}� += ⌈(1 − α)(2 + | ˆS(j,k) +c +|)⌉. +(C.35) +Combining (C.32)-(C.35), we can guarantee +���Q +ˆ +Sc∪{j} − Q +ˆ +S(j,k) +c +∪{j,k}��� ≤ max +� +R +ˆ +S(j,k) +c +(⌈(1−α)(2+| ˆ +S(j,k) +c +|)⌉+d(j,k)−1) − R +ˆ +S(j,k) +c +(⌈(1−α)(2+| ˆ +S(j,k) +c +|)⌉−2), +R +ˆ +S(j,k) +c +(⌈(1−α)(2+| ˆ +S(j,k) +c +|)⌉) − R +ˆ +S(j,k) +c +(⌈(1−α)(| ˆ +S(j,k) +c +|−d(j,k)+2)⌉−2) +� +≤ R +ˆ +S(j,k) +c +(⌈(1−α)(| ˆ +S(j,k) +c +|+2)⌉+d(j,k)) − R +ˆ +S(j,k) +c +(⌈(1−α)(| ˆ +S(j,k) +c +|+2−d(j,k))⌉−2) +=: R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)). +(C.36) +In addition, using Assumptions 2 and 4, we know for any j ∈ U, it holds that +TU +(ˆκ) ≥ TU\{j} +(ˆκ(j)−Iu) ≥ TU\{j} +(ˆκ(j,k)−Iu−Ic). +(C.37) +43 + +Since the samples j and k are independent of R +ˆ +S(j,k) +c +(U (j,k)) and R +ˆ +S(j,k) +c +(L(j,k)), we have +|∆(j,k) +k,1 | ≤ E−(j,k) +����1 +� +Rk > Q +ˆ +Sc∪{j}� +− 1 +� +Rk > Q +ˆ +S(j,k) +c +∪{j,k}���� +��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +(i) +≤ P−(j,k) +� +R +ˆ +S(j,k) +c +(L(j,k)) < Rk ≤ R +ˆ +S(j,k) +c +(U (j,k)) +��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +(ii) +≤ +P−(j,k) +� +R +ˆ +S(j,k) +c +(L(j,k)) < Rk ≤ R +ˆ +S(j,k) +c +(U (j,k)), Tj ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)−Iu−Ic), Tk ≤ TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +≤ +TU\{j} +(ˆκ(j,k)) +� +R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) +� +� +TU\{j} +(ˆκ(j,k)−Iu−Ic) +�2 +, +(C.38) +where (i) comes from (C.36) and (ii) holds due to (C.37). Similarly, we can also have +|∆(j,k) +j,1 | ≤ E−(j,k) +����1 +� +Rj > Q +ˆ +Sc∪{j}� +− 1 +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}���� +��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +≤ P−(j,k) +� +R +ˆ +S(j,k) +c +(L(j,k)) < Rj ≤ R +ˆ +S(j,k) +c +(U (j,k)) +��Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +≤ +P−(j,k) +� +R +ˆ +S(j,k) +c +(L(j,k)) < Rj ≤ R +ˆ +S(j,k) +c +(U (j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)−Iu−Ic), Tk ≤ TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +≤ +TU\{j} +(ˆκ(j,k)) +� +R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) +� +� +TU\{j} +(ˆκ(j,k)−Iu−Ic) +�2 +. +(C.39) +Bound ∆(j,k) +j,2 +and ∆(j,k) +k,2 . +For ∆(j,k) +j,2 , we have the following upper bound +∆(j,k) +j,2 +(i) +≤ +P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}, Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +− +P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}, Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +(ii) +≤ 1 − +P−(j,k) +� +Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� += +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), TU +(ˆκ) < Tk ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� ++ +P−(j,k) +� +TU +(ˆκ) < Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU +(ˆκ) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +(iii) +≤ +2 +� +TU\{j} +(ˆκ(j,k)) − TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +TU\{j} +(ˆκ(j)) +, +(C.40) +44 + +where (i) and (ii) holds since TU\{j} +(ˆκ(j,k)) ≥ TU +(ˆκ) in (C.25), and (iii) comes from (C.37) and the towerrule. +Similarly, we can get the lower bound of ∆(j,k) +j,2 +as +∆(j,k) +j,2 +≥ +P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}, Tj ≤ TU +(ˆκ), Tk ≤ TU +(ˆκ) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +− +P−(j,k) +� +Rj > Q +ˆ +S(j,k) +c +∪{j,k}, Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +≥ − +P−(j,k) +� +TU +(ˆκ) < Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU +(ˆκ) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +− +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), TU +(ˆκ) < Tk ≤ TU\{j} +(ˆκ(j,k)) +� +P−(j,k) +� +Tj ≤ TU\{j} +(ˆκ(j,k)), Tk ≤ TU\{j} +(ˆκ(j,k)) +� +≥ − +2 +� +TU\{j} +(ˆκ(j,k)) − TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +TU\{j} +(ˆκ(j)) +. +(C.41) +Applying the same calculations in (C.40) and (C.41) to ∆(j,k) +k,2 , we can get +|∆(j,k) +j,2 |, |∆(j,k) +k,2 | ≤ +2 +� +TU\{j} +(ˆκ(j,k)) − TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +TU\{j} +(ˆκ(j)) +. +(C.42) +Conclusion. +Substituting (C.39), (C.38) and (C.42) into (C.30), we have +|∆(j,k)| ≤ +2TU\{j} +(ˆκ(j,k)) +� +R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) +� +� +TU\{j} +(ˆκ(j,k)−Iu−Ic) +�2 ++ +4 +� +TU\{j} +(ˆκ(j,k)) − TU\{j} +(ˆκ(j,k)−Iu−Ic) +� +TU\{j} +(ˆκ(j)) +, +where U (j,k) = ⌈(1 − α)(| ˆS(j,k) +c +| + 2)⌉ + d(j,k) and L(j,k) = ⌈(1 − α)(| ˆS(j,k) +c +| + 2 − d(j,k))⌉ − 2. In addition, from +(C.31), we know +���| ˆSc| − | ˆS(j,k) +c +| +��� +| ˆS(j,k) +c +| + 1 +≤ +d(j,k) +| ˆS(j,k) +c +| + 1 +. +Plugging two upper bounds above into (C.27) can finish the proof. +C.4.2 +Proof of Theorem 5 +Proof of Theorem 5. Denote ˆSc(κ) = +� +i ∈ C \ {k} : Ti ≤ TU\{j} +(κ) +� +. From the definition of ˆS(j,k) +c +, we can write +ˆS(j,k) +c += ˆSc(ˆκ(j,k)). In addition, we write +d(j,k) = +� +i∈ ˆ +S(j,k) +c +1 +� +TU\{j} +(ˆκ(j,k)−Ic−1) < Ti ≤ TU\{j} +(ˆκ(j,k)) +� +=: d(j,k)(ˆκ(j,k)). +45 + +Then we have +R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) = +U (j,k)−1 +� +ℓ=L(j,k) +R +ˆ +S(j,k) +c +(ℓ+1) − R +ˆ +S(j,k) +c +(ℓ) +≤ +� +d(j,k) + 2 +� +max +1≤ℓ≤| ˆ +S(j,k) +c +|−1 +� +R +ˆ +S(j,k) +c +(ℓ+1) − R +ˆ +S(j,k) +c +(ℓ) +� += +� +d(j,k)(ˆκ(j,k)) + 2 +� +max +1≤ℓ≤| ˆ +Sc(ˆκ(j,k))|−1 +� +R +ˆ +Sc(ˆκ(j,k)) +(ℓ+1) +− R +ˆ +Sc(ˆκ(j,k)) +(ℓ) +� +≤ +max +⌈γm⌉≤κ≤m +�� +d(j,k)(κ) + 2 +� +max +1≤ℓ≤| ˆ +Sc(κ)|−1 +� +R +ˆ +Sc(κ) +(ℓ+1) − R +ˆ +Sc(κ) +(ℓ) +�� +. +(C.43) +Given Du,−j, applying Lemma A.6 with t1 = TU\{j} +(κ−Ic−1) and t2 = TU\{j} +(κ) +, we can have +d(j,k)(κ) ≤ | ˆSc(κ)| +� +�TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +TU\{j} +(κ) ++ 6C log(n ∨ m) +| ˆSc(κ)| +� +� , +(C.44) +holds with probability at least 1 − (n ∨ m)−C. In addition, given Du,−j, applying Lemma A.5, we get +max +1≤ℓ≤| ˆ +Sc(κ)|−1 +� +R +ˆ +Sc(κ) +(ℓ+1) − R +ˆ +Sc(κ) +(ℓ) +� +≤ 1 +ρ +1 +1 − 2 +� +C log(n∨m) +| ˆ +Sc(κ)|+1 +2C log(n ∨ m) +| ˆSc(κ)| + 1 +, +(C.45) +holds with probability at least 1 − (n ∨ m)−C. Substituting (C.44) and (C.45) into (C.43) and using union +bound, with probability at least 1 − (n ∨ m)−C+2, we have +R +ˆ +S(j,k) +c +(U (j,k)) − R +ˆ +S(j,k) +c +(L(j,k)) ≤ +max +⌈γm⌉≤κ≤m +� +� +� +1 +ρ +2C log(n ∨ m) +1 − 2 +� +C log(n∨m) +| ˆ +Sc(κ)|+1 +� +�TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +TU\{j} +(κ) ++ 6C log(n ∨ m) +| ˆSc(κ)| +� +� +� +� +� +(i) +≤ +max +⌈γm⌉≤κ≤m +� +� +� +� +� +� +� +1 +ρ +2C log(n ∨ m) +1 − 2 +� +C log(n∨m) +nTU\{j} +(κ) +/2+1 +� +�TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +TU\{j} +(κ) ++ 6C log(n ∨ m) +nTU\{j} +(κ) +/2 +� +� +� +� +� +� +� +� +� +(ii) +≤ 4C log(n ∨ m) +ρ +� +� +max⌈γm⌉≤κ≤m +� +TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +� +TU\{j} +(κ) ++ 6C log(n ∨ m) +nTU\{j} +(κ) +/2 +� +� +(iii) +≤ 4C log(n ∨ m) +ρTU\{j} +(⌈γm⌉) +�4CIc log(n ∨ m) +m + 1 ++ 12C log(n ∨ m) +n +� +, +(C.46) +where (i) follows from Lemma A.7, (ii) comes from Lemma A.8, and (iii) follows from Lemma A.4. Using +Lemma A.4 again, we have +TU\{j} +(ˆκ(j,k)) − TU\{j} +(ˆκ(j,k)−Iu−Ic) ≤ (Ic + Iu) +max +1≤ℓ≤m−2 +� +TU\{j} +(ℓ+1) − TU\{j} +(ℓ) +� +≤ 4C(Ic + Iu) log(n ∨ m) +m +, +(C.47) +46 + +holds with probability at least 1 − 2(n ∨ m)−C. With probability at least 1 − (n ∨ m)−C+2, we can guarantee +that +d(j,k) +| ˆS(j,k) +c +| + 1 +(i) +≤ +max +⌈γm⌉≤κ≤m +� +d(j,k)(κ) +| ˆSc(κ)| + 1 +� +≤ +max +⌈γm⌉≤κ≤m +� +� +� +TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +TU\{j} +(κ) ++ 6C log(n ∨ m) +| ˆSc(κ)| +� +� +� +(ii) +≤ +max +⌈γm⌉≤κ≤m +� +� +� +TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +TU\{j} +(κ) ++ 6C log(n ∨ m) +nTU\{j} +(κ) +/2 +� +� +� +≤ +max⌈γm⌉≤κ≤m +� +TU\{j} +(κ) +− TU\{j} +(κ−Ic−1) +� +TU\{j} +(⌈γm⌉) ++ 12C log(n ∨ m) +nTU\{j} +(⌈γm⌉) +(iii) +≤ 8CIc log(n ∨ m) +γm ++ 24C log(n ∨ m) +nγ +, +(C.48) +where (i) holds due to (C.44), (ii) comes from Lemma A.7, and (iii) comes from Lemma A.4 and A.8. Plugging +(C.46), (C.47) and (C.48) into (11), we can get +∆(−(j,k)) ≲ +log2(n ∨ m) +ρ(κ − (Ic + Iu)/m)2 +�Ic + Iu +m + 1 + 1 +n +� +, +where we also used the fact TU\{j} +(⌈γm⌉) ≤ TU\{j} +(ˆκ(j,k)) since ˆκ(j,k) ≥ ˆκ ≥ ⌈γm⌉. Then the conclusion follows from +Lemma 1 immediately. +C.5 +Prediction-oriented selection with conformal p-values +In section 3.3, we split the calibration set according the null hypothesis and alternative hypothesis, that is +C = C0 ∪ C1. Denote n0 = |C0| and n1 = |C1|. Let us first recall the definition of conformal p-value: +pj = 1 + � +i∈C0 1 {Ti ≤ Tj} +n0 + 1 +. +Therefore, the ranking threshold ˆκ only depends on the test set Du and the null calibration set Dc0. The +selected calibration set is defined as +ˆSc1 = +� +i ∈ C1 : Ti ≤ TU +(ˆκ) +� +. +In the following proof, we will fix the null calibration set Dc0 = {(Xi, Yi) : i ∈ C, Yi ≤ b0}, and then the +randomness of ˆκ only comes from the test set. Moreover, the index set C1 is also fixed once given Dc0. +C.5.1 +Proof of Proposition 3.1 +Proof. Notice that, there may exist ties in {pj : j ∈ U}, but there is no tie in {Tj : j ∈ U}. Let Rank(pj) = +� +i∈U 1 {pi ≤ pj} and Rank(Tj) = � +i∈U 1 {Ti ≤ Tj} be the ranks of pj and Tj in the test set, respectively. +47 + +Then we have Rank(pj) ≥ Rank(Tj), which means +� +pj ≤ pU +(ˆκ) +� += {Rank(pj) ≤ ˆκ} =⇒ {Rank(Tj) ≤ ˆκ} = +� +Tj ≤ TU +(ˆκ) +� +. +By the definition of ˆκ = max +� +τ : pU +(τ) ≤ δ(τ) +� +, we know +� +Tj ≤ TU +(ˆκ) +� += +� +Tj ≤ max +� +Ti : pi = pU +(ˆκ) +�� +=⇒ +� +pj ≤ pU +(ˆκ) +� +. +Therefore, we have proved that {p(Xj) ≤ pU +(τ)} = {Tj ≤ TU +(τ)} for any j ∈ U. +C.5.2 +Proof of Proposition 3.2 +Proof. Conclusion 1. +For any j ∈ ˆSu, the Lemma 1 in Fithian and Lei (2020) has showed ˆκ = ˆκ(j) +for the step-up procedures. Next, we prove the conclusion for j ∈ U \ ˆSu. Denote ri as the rank of Ti +in the set TU. Denote r(j) +i +as the rank of Ti in the set TU but the value of Tj is replaced by 0, that is +{0, T1, ..., Tj−1, Tj+1, ..., Tm}. Then we know +r(j) +i += ri + 1 for ri < rj; and r(j) +i += ri for ri > rj. +(C.49) +Recall the definition of ˆκ in step-up procedures, ˆκ = max{r : +pU +(r) ≤ δ(r)}. Together with (C.49) and +ˆκ ≥ ⌈γm⌉ in Assumption 3, we have +ˆκ(j) − ˆκ = +max +ˆκ+1≤r≤rj +� +r : δ(r) < pU +(r) ≤ δ(r + 1) +� +− ˆκ += +rj +� +r=ˆκ+1 +1 +� +δ(r) < pU +(r) ≤ δ(r + 1) +� +≤ +max +⌈γm⌉≤r≤m−1 +m +� +i=1 +1 {δ(r) < pi ≤ δ(r + 1)} . +(C.50) +Given the calibration set, we know {pi : i ∈ U} are i.i.d. random variables with +PDc +� +pi = +k +n0 + 1 +� += PDc +� � +k∈C0 +1 {Tk ≤ Ti} = k − 1 +� += PDc +� +TC0 +(k−1) < Ti ≤ TC0 +(k) +� += TC0 +(k) − TC0 +(k−1), +where TC0 +(k) is the k-th smallest value in TC0. Now we denote Ω(r) = {k ∈ [n0] : δ(r) < +k +n0+1 < δ(r + 1)}. +Then we have +PDc (δ(r) < pi ≤ δ(r + 1)) = +� +k∈Ω(r) +PDc +� +pi = +k +n0 + 1 +� += +� +k∈Ω(r) +TC0 +(k) − TC0 +(k−1) =: ω(r). +(C.51) +Similar to Lemma A.6, given Dc, we can verify that for any C ≥ 1, +����� +1 +m +m +� +i=1 +1 {δ(r) < pi ≤ δ(r + 1)} − ω(r) +����� ≤ 2eC log m +m ++ 2 +� +eCω(r) log m +m +, +(C.52) +48 + +holds with probability at least 1 − (n ∨ m)−C. Invoking maximal spacing in Lemma A.4, we have +P +� +ω(r) ≥ min +�2C|Ω(r)| log m +n0 + 1 +, 1 +�� +≤ m−C. +(C.53) +Combining (C.50)-(C.53), we can guarantee that +ˆκ(j) − ˆκ ≤ 4eC log m + 4m +max +1≤r≤m−1 ω(r) +≤ 4eC log m + 4m +max +1≤r≤m−1 +��2C|Ω(r)| log(n ∨ m) +n0 + 1 +� +∧ 1 +� +, +holds with probability at least 1 − 2m−C+1. +Conclusion 2. If we replace Tk with 1 for any k ∈ C, the corresponding p-value is +p(k) +i += +1 + � +l∈C0\{k} 1 {Tl ≤ Ti} +n0 + 1 +, +for i ∈ U. +It indicates that 0 ≤ pi − p(k) +i +≤ 1/(n0 + 1) for any i ∈ U. Hence we have +ˆκ(k) − ˆκ = +max +ˆκ+1≤r≤m +� +r : δ(r) < pU +(r) ≤ δ(r) + +1 +n0 + 1 +� +− ˆκ +≤ +max +1≤r≤m−1 +m +� +i=1 +1 +� +δ(r) < pi ≤ δ(r) + +1 +n0 + 1 +� +. +Notice that +���� +� +k ∈ [n0] : δ(r) < k + 1 +n0 + 1 ≤ δ(r) + +1 +n0 + 1 +����� ≤ 1. +Similar arguments yield that +ˆκ(k) − ˆκ ≤ 4eC log m + 8Cm log m +n0 + 1 +, +holds with probability at least 1 − 2m−C+1. +D +Proof of Auxiliary Lemmas +D.1 +Proof of Lemma A.3 +Proof. Let Rank(Ti) for i ∈ U be the rank of Ti in the test set {Ti : i ∈ U}. Notice that, ˆκ(j) = ˆκ if +Rank(Tj) ≤ ˆκ, and ˆκ(j) ≤ ˆκ + Iu if Rank(Tj) > ˆκ. Next we discuss the value of TU\{j} +(ˆκ(j)) in different scenarios: +• If Rank(Tj) > ˆκ, then TU\{j} +(ˆκ(j)−Iu) ≤ TU\{j} +(ˆκ) += TU +(ˆκ). +• If Rank(Tj) = ˆκ, then TU\{j} +(ˆκ(j)) = TU\{j} +(ˆκ) += TU +(ˆκ+1) and TU\{j} +(ˆκ(j)−1) = TU\{j} +(ˆκ−1) = TU +(ˆκ−1). +• If Rank(Tj) < ˆκ, then TU\{j} +(ˆκ(j)) = TU\{j} +(ˆκ) += TU +(ˆκ+1) and TU\{j} +(ˆκ(j)−1) = TU\{j} +(ˆκ−1) = TU +(ˆκ). +Then the conclusion follows immediately. +49 + +D.2 +Proof of Lemma A.4 +Lemma D.1 (Representation of spacing (Arnold et al., 2008)). Let U1, · · · , Un +i.i.d. +∼ Unif([0, 1]), and U(1) ≤ +U(2) ≤ · · · ≤ U(n) be their order statistics. Then +� +U(1) − U(0), · · · , U(n+1) − U(n) +� d= +� +V1 +�n+1 +k=1 Vk +, · · · , +Vn+1 +�n+1 +k=1 Vk +� +, +where U0 = 0, U(n+1) = 1, and V1, · · · , Vn+1 +i.i.d. +∼ Exp(1). +Lemma D.2 (Quantile transformation of order statistics, Theorem 1.2.5 in Reiss (2012)). Let X1, · · · , Xn be +i.i.d. random variables with CDF F(·), and U1, · · · , Un +i.i.d. +∼ Unif([0, 1]), then +� +F −1(U(1)), · · · , F −1(U(n)) +� d= +� +X(1), · · · , X(n) +� +. +Fact 1. For the random variable X ∼ Exp(λ), it holds that P (X ≥ x) = e−λx. +Fact 2. For the random variable X ∼ χ2 +ν, it holds that P (X − ν ≥ x) ≤ e−νx2/8. +Proof of Lemma A.4. Using the spacing representation in Lemma D.1, we have +PDu +� +� max +0≤ℓ≤n +� +U(ℓ+1) − U(ℓ) +� +≥ +1 +1 − 2 +� +C log(n∨m) +n+1 +C log(n ∨ m) +n + 1 +� +� += P +� +� max +0≤ℓ≤n +Vℓ +�n+1 +i=1 Vi +≥ +1 +1 − 2 +� +C log(n∨m) +n+1 +2C log(n ∨ m) +n + 1 +� +� +≤ P +� +max +0≤ℓ≤n Vℓ ≥ 2C log(n ∨ m) +� ++ P +� +1 +n + 1 +n+1 +� +i=1 +Vi ≤ 1 − 2 +� +C log(n ∨ m) +n + 1 +� +, +(D.1) +where U(0) = 0 and U(n+1) = 1. By the tail probability of Exp(1) in Fact 1 and union bound, we know +P +� +max +0≤ℓ≤n Vℓ ≥ 2C log(n ∨ m) +� +≤ (n + 1) P (V1 ≥ 2C log(n ∨ m)) += (n + 1) (n ∨ m)−2C ≤ (n ∨ m)−C, +(D.2) +holds for any C ≥ 1. In addition, we know that 1 +2 +�n+1 +i=1 Vi ∼ Γ(n + 1, 2) +d= χ2 +2(n+1). Using the tail bound of +χ2 distribution with ν = 2(n + 1) in Fact 2, we have +P +� +1 +n + 1 +n+1 +� +i=1 +Vi ≤ 1 − 2 +� +C log(n ∨ m) +n + 1 +� +≤ (n ∨ m)−C. +(D.3) +Substituting (D.2) and (D.3) into (D.1) gives the desired result. +D.3 +Proof of Lemma A.5 +Proof. Notice that, for any Sc ⊆ C, we can write +� +ˆSc(t) = Sc +� += +� � +i∈Sc +{Ti ≤ t} +� � +� +� � +i∈C\Sc +{Ti > t} +� +� . +(D.4) +50 + +Then for any i ∈ Sc, we have +P +� +Ri ≤ r +�� ˆSc(t) = Sc +� += P +� +Ri ≤ r +��Ti ≤ t +� += F(R,T )(r, t) +t +=: G(r). +Hence, given ˆSc(t) = Sc ⊆ C, {Ri}i∈Sc are i.i.d. random variables with the common CDF G(·). Applying +Lemma D.2, we know there exist Ui +i.i.d. +∼ Unif([0, 1]) for i ∈ Sc such that +� +RSc +(1), · · · , RSc +(|Sc|)| ˆSc(t) = Sc +� d= +� +G−1(U(1), · · · , G−1(U(|Sc|) +� +. +(D.5) +Let G−1(·) be the inverse function of G(·), and use our assumption on +d +drF(R,T )(r, t) ≥ ρt, we can get +d +drG−1(r) = +� d +drG(r) +�−1 += +t +d +drF(R,T )(r, t) ≤ 1 +ρ. +(D.6) +Then for any x ≥ 0, we have +P +� +max +0≤ℓ≤| ˆ +Sc(t)|−1 +� +R +ˆ +Sc(t) +(ℓ+1) − R +ˆ +Sc(t) +(ℓ+1) +� +≥ x +ρ +��� ˆSc(t) = Sc +� +(i) += P +� +max +0≤ℓ≤|Sc|−1 +� +RSc +(ℓ+1) − RSc +(ℓ) +� +≥ x +ρ +��� +� +i∈Sc +{Ti ≤ t} +� +(ii) += P +� +max +0≤ℓ≤|Sc|−1 +� +G−1(U(ℓ+1)) − G−1(U(ℓ)) +� +≥ x +ρ +� +(iii) +≤ P +� +max +0≤ℓ≤|Sc|−1 +� +U(ℓ+1) − U(ℓ) +� +≥ x +� +, +(D.7) +where (i) holds due to (D.4) and the fact that {Ti}i∈Sc are independent of {Ti}i∈C\Sc, (ii) follows from (D.5), +and (iii) comes from (D.6). Invoking Lemma A.4, we can finish the proof. +D.4 +Proof of Lemma A.6 +Proof of Lemma A.6. For any absolute constant C ≥ 1, we divide the subsets of C into two groups: +Ξ1 = +� +Sc ⊆ C : |Sc| > C log(n ∨ m) · +t2 +t2 − t1 +� +, +Ξ2 = +� +Sc ⊆ C : |Sc| ≤ C log(n ∨ m) · +t2 +t2 − t1 +� +. +Then for any x, y ≥ 0, it holds that +P +� +� +1 +| ˆSc(t2)| +������ +� +i∈ ˆ +Sc(t2) +Zi +������ +≥ x + y +� +� ≤ P +� +� +1 +| ˆSc(t2)| +������ +� +i∈ ˆ +Sc(t2) +Zi +������ +≥ x1 +� +ˆSc(t2) ∈ Ξ1 +� ++ y1 +� +ˆSc(t2) ∈ Ξ2 +� +� +� += +� +Sc⊆C +P +� +� +1 +| ˆSc(t2)| +������ +� +i∈ ˆ +Sc(t2) +Zi +������ +≥ x1 +� +ˆSc(t2) ∈ Ξ1 +� ++ y1 +� +ˆSc(t2) ∈ Ξ2 +� �� ˆSc(t2) = Sc +� +� × P +� +ˆSc(t2) = Sc +� += +� +Sc∈Ξ1 +P +� +1 +|Sc| +����� +� +i∈Sc +1 {t1 < Ti ≤ t2} +����� ≥ x +�� ˆSc(t2) = Sc +� +P +� +ˆSc(t2) = Sc +� ++ +� +Sc∈Ξ2 +P +� +1 +|Sc| +����� +� +i∈Sc +1 {t1 < Ti ≤ t2} +����� ≥ y +�� ˆSc(t2) = Sc +� +P +� +ˆSc(t2) = Sc +� +. +(D.8) +51 + +According to the definition of ˆSc(t2), we have +� +ˆSc(t2) = Sc +� += +� +� +� +� +l∈Sc +{Tl ≤ t2} , +� +l∈C\Sc +{Tl > t2} +� +� +� . +It implies that for any i ∈ Sc, +P +� +t1 < Ti ≤ t2 +��� ˆSc(t2) = Sc +� += P +� +t1 < Ti ≤ t2 +��Ti ≤ t2 +� += P (t1 < Ti ≤ t2) +P (Ti ≤ t2) += t2 − t1 +t2 +, +where the first equality follows from the independence of the samples in C. Recall the definition Zi = +1 {t1 < Ti ≤ t2} − t2−t1 +t2 +, then for any i ∈ Sc we have +E +� +Zi +��� ˆSc(t2) = Sc +� += E +� +�Zi +��� +� +l∈Sc +{Tl ≤ t2} , +� +l∈C\Sc +{Tl > t2} +� +� += P +� +t1 < Ti ≤ t2 +��� ˆSc(t2) = Sc +� +− t2 − t1 +t2 += 0. +(D.9) +Next we will bound the conditional moment generating function of � +i∈Sc Zi. For any λ > 0, we have +E +� +eλ � +i∈Sc Zi +��� ˆSc(t2) = Sc +� += E +� +� � +i∈Sc +eλZi +��� +� +l∈Sc +{Tl ≤ t2} , +� +l∈C\Sc +{Tl > t2} +� +� += E +� +E +� � +i∈Sc +eλZi +���Ti ≤ t2, {Tl}l∈Sc\{i} +� ��� +� +l∈Sc +{Tl ≤ t2} +� += E +� +� +� +l∈Sc\{i} +eλZlE +� +eλZi +���Ti ≤ t2 +� ��� +� +l∈Sc +{Tl ≤ t2} +� +� +(i) +≤ +� +1 + λ2E +� +Z2 +i eλ|Zi|���Ti ≤ t2 +�� +E +� +� +� +l∈Sc\{i} +eλZl +��� +� +l∈Sc +{Tl ≤ t2} +� +� +(ii) += +� +1 + λ2E +� +Z2 +i eλ|Zi|���Ti ≤ t2 +�� +E +� +� +� +l∈Sc\{i} +eλZl +��� +� +l∈Sc\{i} +{Tl ≤ t2} +� +� +≤ +� +i∈Sc +� +1 + λ2E +� +Z2 +i eλ|Zi|���Ti ≤ t2 +�� +(iii) +≤ exp +� +λ2 � +i∈Sc +E +� +Z2 +i eλ|Zi|���Ti ≤ t2 +�� +, +(D.10) +where (i) holds since (D.9) and the basic inequality ey − 1 − y ≤ y2e|y|, (ii) holds due to the independence, +and (iii) comes from the basic inequality 1 + y ≤ ey. Notice that +� +i∈Sc +E +� +Z2 +i eλ|Zi|���Ti ≤ t2 +� += |Sc|E +� +Z2 +1eλ|Z1|���T1 ≥ t2 +� +≤ eλ|Sc|E +� +1 {t1 < Ti ≤ t2} +���Ti ≤ t2 +� += eλ|Sc|t2 − t1 +t2 +=: K2 +Sc(λ), +(D.11) +52 + +where the inequality holds since |Z1| ≤ 1 {t1 < Ti ≤ t2} ≤ 1. Using Markov’s inequality and (D.10), for any +z ≥ 0, we can get +P +� +�� +i∈Sc +Zi ≥ 2KSc(1)z +��� +� +l∈Sc +{Tl ≤ t2} , +� +l∈C\Sc +{Tl > t2} +� +� += P +� +�eλ � +i∈Sc Zi ≥ e2λKSc(1)z��� +� +l∈Sc +{Tl ≤ t2} , +� +l∈C\Sc +{Tl > t2} +� +� +≤ e−2λKSc(1)zE +� +�eλ � +i∈Sc Zi +��� +� +l∈Sc +{Tl ≤ t2} , +� +l∈C\Sc +{Tl > t2} +� +� +(i) +≤ e−2λKSc(1)z exp +� +λ2 � +i∈Sc +E +� +Z2 +i eλ|Zi|���Ti ≤ t2 +�� +(ii) +≤ e−2λKSc(1)z+λ2K2 +Sc(λ), +where (i) comes from (D.10), and (ii) comes from the definition of KSc(λ) in (D.11). By the definition of Ξ1, +we know that KSc(1)2 ≥ C log(n ∨ m) for any Sc ∈ Ξ1. Taking z = (C log(n ∨ m))1/2, λ = +z +KSc(1) ≤ 1, then +we have +P +������ +� +i∈Sc +Zi +����� ≥ 2KSc(1)z +��� ˆSc(t2) = Sc +� +≤ 2 exp +� +−2z2 + z2 K2 +Sc(λ) +K2 +Sc(1) +� +≤ 2e−z2 = 2(n ∨ m)−C. +(D.12) +For Sc ∈ Ξ2 such that KSc(1)2 < C log(n ∨ m), applying (D.11) with λ = 1 gives +P +������ +� +i∈Sc +Zi +����� ≥ 2eC log(n ∨ m) +��� ˆSc(t2) = Sc +� +≤ 2e−2eC log(n∨m)E +� +e +� +i∈Sc Zi +��� ˆSc(t2) = Sc +� +≤ 2e−2eC log(n∨m)+K2 +Sc(1) +≤ 2e−2eC log(n∨m)+eC log(n∨m) +≤ 2(n ∨ m)−C. +(D.13) +Taking x = 2KSc(1)(C log(n ∨ m))1/2 and y = 2eC log(n ∨ m), then plugging (D.12) and (D.13) into (D.8) +gives +P +� +� +1 +| ˆSc(t2)| +������ +� +i∈ ˆ +Sc(t2) +Zi +������ +≥ 2 +� +eC log(n ∨ m) +| ˆSc(t2)| +� +t2 − t1 +t2 ++ 2eC log(n ∨ m) +| ˆSc(t2)| +� +� +≤ 2(n ∨ m)−C +� +� � +C⊆Ξ1 +P +� +ˆSc(t2) = Sc +� ++ +� +C⊆Ξ2 +P +� +ˆSc(t2) = Sc +� +� +� += 2(n ∨ m)−C. +Thus we have complete the proof. +53 + +D.5 +Proof of Lemma A.7 +Proof. By the definition of ˆSc(t), we have +| ˆSc(t)| − nt = +� +i∈C +1 {Ti ≤ t} − nt = +� +i∈C +1 {Ti ≤ t} − P (Ti ≤ t) . +Applying Hoeffding’s inequality, we have +P +����| ˆSc(t)| − nt +��� ≥ 2C +� +n log(n ∨ m) +� +≤ (n ∨ m)−C. +Using the assumption 8C log(n ∨ m)/(nt) ≤ 1, we can finish the proof. +D.6 +Proof of Lemma A.8 +Proof. Invoking the spacing representation in Lemma D.1, we have +P +� +TU\{j} +(⌈γm⌉) ≤ γ +2 +� += P +��⌈γm⌉ +i=1 +Vi +�m +k=1 Vi +≤ γ +2 +� += P +� +� +1 +⌈γm⌉ +�⌈γm⌉ +i=1 +Vi +1 +m +�m +k=1 Vk +≤ γ +2 +m +⌈γm⌉ +� +� +≤ P +� +� +1 +⌈γm⌉ +�⌈γm⌉ +i=1 +Vi +1 +m +�m +k=1 Vk +≤ 1 +2 +� +� +≤ P +� +� +1 +⌈γm⌉ +⌈γm⌉ +� +i=1 +Vi − 1 ≤ −1 +4 +� +� + P +� +1 +m +m +� +k=1 +(Vk − 1) ≥ 1 +2 +� +≤ (n ∨ m)−C. +54 + diff --git a/MNAyT4oBgHgl3EQfs_kD/content/tmp_files/load_file.txt b/MNAyT4oBgHgl3EQfs_kD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55c2e2006ddd15be9d9cb4807526af54889deb69 --- /dev/null +++ b/MNAyT4oBgHgl3EQfs_kD/content/tmp_files/load_file.txt @@ -0,0 +1,2300 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf,len=2299 +page_content='Selective Conformal Inference with FCR Control Yajie Baoa, Yuyang Huob, Haojie Rena and Changliang Zoub∗ aSchool of Mathematical Sciences, Shanghai Jiao Tong University Shanghai, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' China bSchool of Statistics and Data Science, Nankai University Tianjin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' China January 3, 2023 Abstract Conformal inference is a popular tool for constructing prediction intervals (PI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We consider here the scenario of post-selection/selective conformal inference, that is PIs are reported only for individuals selected from an unlabeled test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To account for multiplicity, we develop a general split conformal framework to construct selective PIs with the false coverage-statement rate (FCR) control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We first investigate the Benjamini and Yekutieli (2005)’s FCR-adjusted method in the present setting, and show that it is able to achieve FCR control but yields uniformly inflated PIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We then propose a novel solution to the problem, named as Selective COnditional conformal Predictions (SCOP), which entails performing selection procedures on both calibration set and test set and construct marginal conformal PIs on the selected sets by the aid of conditional empirical distribution obtained by the calibration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' More importantly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond exchangeablity and discuss the conditions under which the SCOP is able to control the FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' As special cases, the SCOP with quantile-based selection or conformal p-values-based multiple testing procedures enjoys valid coverage guarantee under mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Numerical results confirm the effectiveness and robustness of SCOP in FCR control and show that it achieves more narrowed PIs over existing methods in many settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Keywords: Conditional empirical distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Distribution-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Non-exchangeable conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Post- selection inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Prediction intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Split conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ∗Corresponding Author: nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='chlzou@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='00584v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='ME] 2 Jan 2023 1 Introduction To improve the prediction performance in modern data, many sophisticated machine learning algorithms including various “black-box” models are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' While often witnessing empirical success, quantifying prediction uncertainty is one of the major issues for interpretable machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Conformal inference (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 1999, 2005) provides a powerful and flexible tool to quantify the uncertainty of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Consider a typical setting that we observe one labeled data set Dl = {(Xi, Yi)}2n i=1 and a set of unla- belled/test samples Du = {Xi}2n+m i=2n+1 whose outcomes {Yi}2n+m i=2n+1 are unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Generally, suppose all (Xi, Yi) ∈ X × Y are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d from some unknown distribution, and µ(x) := Y | X = x as the prediction model associated with (X, Y ), which is usually estimated by the labeled data Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For any Xj ∈ Du and a given miscoverage level α, standard conformal prediction methods (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018), yield a prediction interval (PI) with distribution-free coverage guarantee, PIα(Xj), P(Yj ∈ PIα(Xj)) ≥ 1 − α, under independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d) (or exchangeable data) assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' With the development of big data, making predictive inference on all available data (Du) is either unnecessary or inefficient in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For example, in the recruitment decisions, only some selected viable candidates can get into interview processes (Faliagka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Shehu and Saeed, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In the drug discovery trials, researchers select promising ones based on predicting candidates’ activity for further clinical trials (Carracedo-Reboredo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Dara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Related applications also appear in financial investment and scientific discovery (Jin and Candès, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In such problems, the most common way is to select a subset of interest with some rules through some statistical/machine learning algorithms at first, and then perform statistical inference only on the selected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Formally, letting ˆSu ⊆ {2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' , 2n + m} be the selected subset, our goal is to construct the PI of Yj for each j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' As pointed by Benjamini and Yekutieli (2005), ignoring the multiplicity in construction of post-selection intervals will result in distorted average coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under the context of post-selection inference in which confidence intervals for multiple selected parameters/variables are being reported, Benjamini and Yekutieli (2005) pioneered the criterion, false coverage-statement rate (FCR), to take account for multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The FCR, an analog of the false discovery rate (FDR), can readily be adapted to the present conformal inference setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It is defined as the expected ratio of number of reported PIs failing to cover their respective true outcomes to the total number of reported PIs, say FCR := E � |{j ∈ ˆSu : Yj ̸∈ PIj}| max{| ˆSu|, 1} � , (1) where PIj is the PI for the selected sample j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Benjamini and Yekutieli (2005) provided a selection- agnostic method which adjusts the confidence level through multiplying α by a quantity which is related to the proportion of selected candidates over all candidates and then constructed the marginal confidence 2 intervals at the adjusted level for each selected candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We will hereafter call it the FCR-adjusted method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Accordingly, we may expect that the FCR-adjusted PIs enjoy valid FCR control properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, due to the dependence structure among PI| ˆ Su|α/m(Xj)’s, the results in Benjamini and Yekutieli (2005) are not directly applicable in the setting of conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Please refer to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 for detailed discussions and rigorous theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We notice that Weinstein and Ramdas (2020) also discussed the selective inference problem under the framework of conformal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The authors suggested to use the FCR-adjusted method, however, they did not provide theoretical or empirical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' While the FCR-adjusted approach can reach FCR control and is widely used, it is generally known to yield uniformly inflated confidence intervals (Weinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This is because that when calculating the noncoverage probabilities of confidence intervals, the adjusted confidence intervals do not take into account the selection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Along this line, Weinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2013), Zhao and Cui (2020) and Zhao (2022) further proposed some methods to narrow the adjusted confidence intervals by incorporating more selection information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Among some others, Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2014), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2016) and Taylor and Tibshirani (2018) proposed constructing conditional confidence intervals for each selected variables and showed that the selective error rate can be controlled given that the selected set is equal to some deterministic subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, those methods either require some tractable conditional distribution assumptions or are only applicable for some given prediction algorithms, such as normality assumptions or LASSO model, which would limit their applicability in the conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fortunately, by the virtue of the availability of Dl, distribution/model-agnostic conditional prediction intervals with theoretical guarantee can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 Our contributions In this paper, we develop a novel conformal framework to construct post-selection prediction intervals while control the FCR, named as Selective COnditional conformal Predictions (SCOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our method stems from the split conformal inference (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fithian and Lei, 2020), where the labeled data Dl is split into two disjoint parts, one as the training set for obtaining a prediction model ˆµ(X), and the remaining one as the calibration set for estimating the distribution of the discrepancy between the Y and ˆµ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then, the key ingredient of our proposal entails performing a pre-specified selective procedure on both the calibration set and the test set and construct the marginal conformal PIs on the selected sets with the help of conditional empirical distribution obtained by the calibration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The proposed SCOP procedure is model- or distribution-agnostic, in the sense that it could wrap around any prediction algorithms with commonly used selection procedures to construct PIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The main contributions of the paper can be summarized as follows: Firstly, we investigate the FCR-adjusted method in the setting of conformal inference and show that it is able to achieve FCR control under mild conditions, which lays a foundation for our subsequent development of SCOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3 Secondly, under a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the FCR at the target level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Thirdly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond exchangeablity, termed as ranking-based procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This broadens the scopes of our SCOP in theoretical guarantee and practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To address the non-exchangeability between the the post-selection test set and calibration set, we introduce a virtual post-selection calibration set in our proof, and then quantify the conditional miscoverage gap between the virtual calibration and the real calibration in SCOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This new technique may be of independent interest for conformal prediction for non-exchangeable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Finally, we illustrate the easy coupling of the SCOP with commonly used prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Numerical experiments indicate that it yields more accurate FCR control than existing methods, while offers the narrowed prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 Connections to existing works Post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Post-selection inference on a large number of variables has attracted considerable research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Besides the references mentioned before, a relevant direction is the splitting-based strategy for high-dimensional inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The number of variables is firstly reduced to a manageable size using one part of data, while confidence intervals or significance tests can be constructed by computing estimates in a low-dimensional region with the other part of data and selected variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' See Wasserman and Roeder (2009), Rinaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2019), Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' One potential related work is Chen and Bien (2020), in which the authors considered to construct confidence intervals for regression coefficients after removing the potential outliers from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our paradigm differs substantially with those works as we focus on post-selection inference for sample selection rather than variable selection, and existing works on variable selection is difficult to extend to the present problem due to the requirements on model or distribution assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Conformal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The building block of our SCOP is the conformal inference framework, which has been well studied in many settings, including non-parametric regression (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013), quantile regression (Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2019), high-dimensional regression (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018) and classification (Sadinle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' More comprehensive reviews can be found in Shafer and Vovk (2008), Zeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2020) and Angelopoulos and Bates (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Conventionally, conformal PIs enjoy distribution-free marginal coverage guarantee with the assumption that the data are exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, the exchangeability may be violated in practice and would be more severe in the post-selection conformal inference because the selection procedure might be determined by either the labelled data or the test data, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In such situations, one particularly difficult issue is that the selected set ˆSu is random and has a complex dependence structure to the labelled data and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Some conformal inference beyond exchangeability has attracted attention 4 (Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Candès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In particular, Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022) proposed a general framework to implement conformal inference when the algorithms cannot treat data exchangeable and theoretically displayed the coverage deviations compared from exchangeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, how to decouple the dependence to achieve FCR control in the present framework remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Taking a different but related perspective from multiple-testing, Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) proposed a method to construct conformal p-values with data splitting and apply it to detect outliers with finite-sample FDR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022) extended that method and proposed a Jackknife implementation combined with automatic model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Jin and Candès (2022) considered a scenario that one aims to select some individuals of interest from the test sample and proposed a conformal p-value based method to control the FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Those existing works are not concerned about the construction of PIs, which differs with our focus essentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3 Organization and notations The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We introduce the FCR-adjusted prediction and SCOP for valid FCR control in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Section 3 presents the theoretical properties of SCOP for ranking-based procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Numerical results and real-data examples are presented in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Section 5 concludes the paper, and the technical proofs are relegated to the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For a positive integer n, we use [n] to denote the index set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let A = {Ai : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', n} be a set of n real numbers, and S ⊆ [n] be an index subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We use AS (ℓ) to denote the ℓ-th smallest value in {Ai : i ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We use 1 {·} to denote the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For a real random sequences Xn and an non-negative real deterministic sequence an, we write Xn = Op(an) if for any ϵ > 0, there exists some constant C > 0 such that P(|Xn| > Can) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In our paper, the notations with subscript c or u refer to depending on the calibration set or the test set respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2 Selective conditional conformal prediction Denote the index sets for the labelled data Dl and the test data Du as L = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' , 2n} and U = {2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' , 2n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The main prediction method studied in this paper is built upon the split conformal framework (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018), which is also called “inductive conformal prediction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' That is we randomly split Dl into two disjoint parts, the training set Dt and the calibration set Dc with n samples respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We can firstly train a prediction model ˆµ(X) on the Dl, and then compute the empirical quantiles of the residuals Ri = |Yi − ˆµ(Xi)| on the calibration set Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For Xj ∈ Du, the (1 − α)-marginal conformal PI is PIM j = � ˆµ(Xj) − QC(1 − α), ˆµ(Xj) + QC(1 − α) � , (2) where QC(1 − α) is the ⌈(1 − α)(n + 1)⌉-st smallest value in RC = {Ri = |Yi − ˆµ(Xi)| : i ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (or more generally, exchangeable) assumption on Dc ∪ {(Xj, Yj)}, the marginal PI in (2) enjoys the coverage guarantee, P � Yj /∈ PIM j � ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 5 Suppose g : X → R be one plausible score function, which can be user-specified or estimated by the training data Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' A particular selection procedure S can be applied to g(Xi) for i ∈ U to find the samples of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For simplicity, denote Ti = g(Xi) and those Xi’s with smaller values of Ti tend to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote the selected set as ˆSu = {i ∈ U : Ti ≤ ˆτ}, where ˆτ is the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Different selective procedures S can be chosen from different perspectives, and we summarize the selection threshold ˆτ into three types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (Fixed threshold) The ˆτ is user-specified or independent of the whole data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For example, ˆτ = t, where t is either known as a priori or could possibly be obtained from an independent process of Dc ∪ Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (Self-driven threshold) The ˆτ is only dependent on the scores {Ti : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This type includes the Top-K which choose the first K individuals, and the quantile of Ti values in the test set which a given proportion of individuals with smallest Ti values in the test set, respectively (Fithian and Lei, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (Calibration-assisted selection) The ˆτ relies on the calibration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For example, ˆτ is some quantile of true response Yi in calibration set, or the quantile based on both calibration and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In particular, one may employ some multiple testing procedures to achieve error rate control, such like FDR control based on the Benjamini–Hochberg (BH) procedure (Benjamini and Hochberg, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Consequently, the {Ti : i ∈ C} is required to approximate the distribution of {Ti : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our goal is to construct conformal PIs for the selected subset ˆSu with the FCR control at α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 Adjusted conformal prediction We firstly adapt the Benjamini and Yekutieli (2005)’s FCR-adjusted method to the present setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Define Mj min := min y � | ˆSTj←y u | : j ∈ ˆSTj←y u � , where ˆSTj←y u denotes the selected subset when replacing Tj with value y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The FCR-adjusted conformal PIs are amount to marginally constructing larger 1 − α × Mj min/m PIs instead of 1 − α level in (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', PIAD j = � ˆµ(Xj) − QC(1 − α × Mj min), ˆµ(Xj) + QC(1 − α × Mj min) � , j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (3) Notice that given ˆµ(·) and ˆSu, PIAD j ’s are not independent of each other because they all rely on the empirical quantile obtained from Dc, and therefore the proofs in Benjamini and Yekutieli (2005) are not readily extended to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following result demonstrates that the FCR-adjusted approach can successfully control the FCR for any selection threshold that is independent of the calibration set given training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose that given Dt, {Ti : i ∈ C ∪ U} are independent random variables and the selection threshold ˆτ is independent of Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then the FCR value of the FCR-adjusted method in (3) satisfies FCRAD ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For many plausible selection rules such as fixed-threshold selection, the Mj min can be replaced by the cardinality of the selected subset | �Su|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In practice, for ease of computation, one may prefer to use this 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 Ri density Marginal Conditional Test Figure 1: The densities of Ri for i ∈ Dc (in blue), Ri for i ∈ ˆSc (in green) and the density of Rj for j ∈ ˆSu (in red), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There are 2n = 400 labeled data and m = 200 test data generated from a linear model with heterogeneous noise, where the details of the model are in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The selection rule is ˆS = {k : ˆµ(Xk) ≤ −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' simplification, even though it does not have a theoretical guarantee for many data-dependent selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The FCR-adjusted method is known to be quite conservative (Weinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013), because it does not incorporate the selection event into the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Take the Top-K selection as an intuitive example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The selected set ˆSu is fixed with | ˆSu| = K and the FCR can be written as FCR = 1 K � j∈U P � j ∈ ˆSu, Yj ̸∈ PIj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (4) Since the marginal PIAD j reaches the 1 − αK/m confidence level for any fixed K, the FCR-adjusted method achieves the FCR control via FCRAD = 1 K � j∈U P � j ∈ ˆSu, Yj ̸∈ PIAD j � ≤ 1 K � j∈U P � Yj ̸∈ PIAD j � ≤ α, where the first inequality might be rather loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' A simple yet effective remedy is to use conditional calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 Selective conditional conformal prediction (SCOP) We start by making a decomposition of the FCR according to the contribution of each sample in the selected set ˆSu, given as P � Yj ̸∈ PIj |j ∈ ˆSu � P � j ∈ ˆSu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notice that the FCR can naturally be controlled at level α if the conditional control satisfies P(Yj ̸∈ PIj |j ∈ ˆSu) ≤ α, which sheds light on the construction of conditional conformal PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In the regime of conformal inference, the conditional uncertainty of |Yj − ˆµ(Xj)| given j ∈ ˆSu can be reliably approximated using the calibration set Dc, enabling us to construct model/distribution-agnostic conditional 7 PIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To be specific, we conduct the selective algorithm S on the fitted score values {Ti = g(Xi) : i ∈ C} and obtain the post-selection calibration set ˆSc = {i ∈ C : Ti ≤ ˆτ} with the same threshold ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notice that ˆSc is formed via the same selection criterion with ˆSu, and thus we utilize the residuals Ri for i ∈ ˆSc to approximately characterize the conditional uncertainty of Rj for j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To visualize the effect, we consider a linear model with heterogeneous noise, where we use ordinary least-squares for predictions and select ˆSu = {j ∈ U : ˆµ(Xj) ≤ −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In Figure 1, we display the densities of Ri for i ∈ Dc, Ri for i ∈ ˆSc and the density of Rj for j ∈ ˆSu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The selection procedure significantly distorts the distribution of residuals, but the conditional uncertainty on ˆSu can be well approximated by that on ˆSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The conditional conformal PI for j ∈ ˆSu can accordingly be constructed as PISCOP j = � ˆµ(Xj) − Q ˆ Sc(1 − α), ˆµ(Xj) + Q ˆ Sc(1 − α) � , (5) where Q ˆ Sc(1 − α) is the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest value in R ˆ Sc = {Ri : i ∈ ˆSc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We refer this procedure as Selective COnditional conformal Prediction (SCOP) and summarize it in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following theorem shows that SCOP can control the FCR at α for exchangeable selective procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Further, if the selection scores Ti are continuous (or almost surely distinct), we can obtain a lower bound for the FCR value, guaranteeing that the SCOP is nearly exact in O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose {Ti : i ∈ C ∪ U} are exchangeable random variables, and the threshold ˆτ is also exchangeable with respective to the {Ti : i ∈ C ∪ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For each j ∈ U, the conditional miscoverage probability is bounded by P � Yj ̸∈ PISCOP j |j ∈ ˆSu � ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (6) Further, the FCR value of the SCOP algorithm is controlled at FCRSCOP ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In addition, if Ti follows a continuous distribution for i ∈ C ∪ U and P(| ˆSu| > 0) = 1, we also have P � Yj ̸∈ PISCOP j |j ∈ ˆSu � ≥ α − 1 n + 1 and FCRSCOP ≥ α − 1 n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under the exchangeable assumption, the FCR results actually match the marginal miscoverage results of original conformal PIs (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This theorem relies on exchangeability in two ways, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', the fitted selection score {Ti : i ∈ C ∪ U} are exchangeable and the selection threshold ˆτ is assumed to keep the same value by swapping Tj and Tk for any j, k ∈ C ∪ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The former one is commonly used in conformal inference and holds easily when the data are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d given Dt (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The later one imposes restrictions on the selection procedures and can be fulfilled with some practical thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The simplest case is the fixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Another popular example is that ˆτ is some quantile of {Ti : i ∈ C ∪ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, many selection procedures may be excluded, such as the Top-K selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In such cases, the threshold ˆτ is only determined by the test data U, which does not treat the data points from calibration and test sets symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We will next explore the effectiveness of the proposed SCOP for more general selection procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 8 Algorithm 1 Selective COnditional conformal Prediction (SCOP) Input: Labeled set Dl, test set Du, selection procedure S, target FCR level α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 1 (Splitting and training) Split Dl into training set Dt and calibration set Dc with equal size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fit prediction model ˆµ(·) and score function g (if needed) on the training set Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 2 (Selection) Compute the scores: TC = {Ti = g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Apply the selective procedure S to TC ∪ TU and obtain the threshold value ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Obtain the post-selection subsets: ˆSu = {i ∈ U : Ti ≤ ˆτ} and ˆSc = {i ∈ C : Ti ≤ ˆτ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 3 (Calibration) Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Find the ⌈(1−α)(| ˆSc|+1)⌉-st smallest value of RSc, Q ˆ Sc(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 4 (Construction) Construct PI for each j ∈ ˆSu as PISCOP j = [ˆµ(Xj)−Q ˆ Sc(1−α), ˆµ(Xj) +Q ˆ Sc(1−α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Output: Prediction intervals {PISCOP j : j ∈ ˆSu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In predictive inference, several works considered to approximately construct the conditional PI (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', P (Yj /∈ PI(Xj)|Xj = x) ≤ α, for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (7) However, it is well known that achieving “fully" conditional validity in (7) is impossible in distribution-free regime (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Foygel Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our conditional miscoverage control in (6) is a weaker guarantee compared with (7), since we only consider the selection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For more discussion about these two conditional guarantees, we refer to Appendix B in Weinstein and Ramdas (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The SCOP can leverage the post-selection calibration set to approximate the selective conditional distribution of residuals, which contributes to achieve a better conditional coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In addition, the conditional calibration of SCOP provides an anti-conservative lower bound for FCR value in the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3 Ranking-based selection In this section, we consider a general class of selection procedures named ranking-based selection and discuss the conditions under which the proposed SCOP is able to control the FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2, we discuss the FCR control for the self-driven selection procedures and calibration-assisted ones, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3, we demonstrate the effectiveness of the SCOP procedure when the selection procedures based on conformal p-values are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We begin with some general assumptions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For simplicity, we suppose Ti ∈ [0, 1] and the selection algorithm S conducted on {Ti : i ∈ C ∪ U} outputs a ranking threshold ˆκ ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Say, we have the selection threshold ˆτ = TU (ˆκ) as the ˆκ-th smallest value in TU = {Tj : j ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then the selected subset of the 9 test set can be rewritten as ˆSu = � j ∈ U : Tj ≤ TU (ˆκ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (8) The ranking-based procedure in (8) incorporates many practical examples, such as Top-K selection, quantile- based selection, step-up procedures (Fithian and Lei, 2020) and the well-known BH procedure1 (Benjamini and Hochberg, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' With the ranking based selection, we have | ˆSu| = ˆκ, which is usually random and coupled to each test sample Xj ∈ Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To decouple the dependence, we introduce Lemma 1 to control the FCR through conditioning on the leave-one-out data set Du,−j, which is the test set Du without the sample j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote EDu,−j[·] and PDu,−j(·) as the conditional expectation and probability given Du,−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let ˆκj←tu be the ranking threshold obtained from the selection algorithm S by replacing Tj with some deterministic value tu ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose | ˆSu| > 0 almost surely and ˆκ = ˆκj←tu holds for any j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If the conditional false coverage probability satisfies ���PDu,−j � Yj ̸∈ PIj ��j ∈ ˆSu � − α ��� ≤ ∆(Du,−j), (9) where ∆(Du,−j) only depends on the data set Du,−j, then we have | FCR −α| ≤ E � � 1 | ˆSu| � j∈ ˆ Su ∆(Du,−j) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The leave-one-out technique often appears in the literature about FDR control under dependence (Heesen and Janssen, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fithian and Lei, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The key is to decompose the FCR into the summation of conditional miscoverage probability of each candidate given other test samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e, FCR = E � �� j∈U 1 ˆκj←tu PDu,−j � Yj ̸∈ PIj |j ∈ ˆSu � PDu,−j � j ∈ ˆSu � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We may regard the term ∆(Du,−j) in (9) as the individual FCR contribution of the j-th candidate in ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The detailed proof of Lemma 1 is deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Next, we introduce two universal assumptions to find how the conditional false coverage probability in (9) holds and further control the FCR of SCOP with the ranking-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote the cumulative distribution functions (CDF) of Ri and Ti as FT (·) and FR(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let F(R,T )(·, ·) be the joint CDF of (Ri, Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The score function g depends only on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose {Ti : i ∈ C ∪ U} and {Ri : i ∈ C ∪ U} are both i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There exists some ρ ∈ (0, 1) such that d drF(R,T ) � F −1 R (r), F −1 T (t) � ≥ ρt, holds for any t ∈ (0, 1) and r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1The BH procedure is also an example of step-up procedures, see Fithian and Lei (2020) 10 Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There exists some deterministic value tu ∈ [0, 1] such that ˆκj←tu = ˆκ holds for any j ∈ ˆSu, and ˆκj←tu ≤ ˆκ + Iu holds for any j ∈ U \\ ˆSu and some positive integer Iu ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To facilitate our technical development, we impose mild distributional assumption on the joint CDF of (Ri, Ti) in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It is worth noticing that the same selected set ˆSu and ˆSc can be obtained if one applies the ranking-based selection procedure to the transformed scores {FT (Ti) : i ∈ U} instead of the scores {Ti : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Also, the transformed residuals {FR(Ri) : i ∈ C ∪ U} keep the original order as the residuals {Ri : i ∈ C ∪ U} in the conformal coverage control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Therefore, without loss of generality, we can assume that Ti i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ∼ Unif([0, 1]) and Ri i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ∼ Unif([0, 1]) for i ∈ C ∪ U in the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then the condition on CDF in Assumption 1 will reduce to d drF(R,T )(r, t) ≥ ρt which appears quite weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For Assumption 2, we can verify that ˆκ = ˆκj←tu under the event {j ∈ ˆSu} in many cases, such as the quantile-based selection procedure and BH procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For the selection procedures with fixed ranking threshold, such as quantile-based selection and Top-K selection, Assumption 2 is clearly satisfied with tu = 0 and Iu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For the BH procedure based on the conformal p-values, taking tu as 0 for each j ∈ ˆSu leads a smaller p-value pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' By the property of the BH procedure, for j ∈ ˆSu, assigning pj to a smaller value will not change the rejection set (Fithian and Lei, 2020), and hence we have κj←0 = ˆκ for j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3, we also show that Iu = Op(log m) for the BH procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' From now on, we write ˆκ(j) = ˆκj←tu for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 FCR control for self-driven selection When the self-driven selection procedures are used, the samples in the selected calibration set ˆSc and the selected test set ˆSu are not exchangeable, but the original samples from the calibration set and the test set are exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following theorem provides delicate bounds for the conditional miscoverage gap of the SCOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1 and 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' for any absolute constant C ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' if 8C log n/(nTU\\{j} (ˆκ(j)) ) ≤ 1 holds almost surely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' the conditional miscoverage probability can be bounded by PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ���j ∈ ˆSu � ≤ α + ∆(Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' and PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ���j ∈ ˆSu � ≥ α − 2∆(Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' where ∆(Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j) = 8C log n ρTU\\{j} (ˆκ(j)−Iu) � �6C log n nTU\\{j} (ˆκ(j)) + TU\\{j} (ˆκ(j)) − TU\\{j} (ˆκ(j)−1) TU\\{j} (ˆκ(j)) � � + 2 � TU\\{j} (ˆκ(j)) − TU\\{j} (ˆκ(j)−Iu) � TU\\{j} (ˆκ(j)−Iu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (10) Our theorem is closely connected to Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022)’s Theorem 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Both theorems involve assessing how the deviations from the “idealized” exchangability would affect the actual miscoverage level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, 11 the interpretations are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022) showed that under assumption that the test and calibration samples are non-exchangeable, the miscoverage gap can be bounded by an error term regarding the total variation between the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Whereas in SCOP the deviation comes from the possible violation of the similarity between the distributions of {Ri : i ∈ ˆSc} and {Rj : j ∈ ˆSu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The technical difficulty in proving Theorem 2 lies in coping with the dependence of ˆSc and ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To address this problem, we introduce virtual post-selection test set and calibration set, ˆS(j) u = � i ∈ U : Ti ≤ TU\\{j} (ˆκ(j)) � and ˆS(j) c = � i ∈ C : Ti ≤ TU\\{j} (ˆκ(j)) � respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We denote the corresponding virtual conformal PI constructed by ˆS(j) c as PIj( ˆS(j) c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For clarity, we rewrite the real conformal PI constructed by ˆSc in Algorithm 1 as PIj( ˆSc) ≡ PISCOP j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notice that, the threshold TU\\{j} (ˆκ(j)) and the virtual selected calibration set ˆS(j) c are independent of the test candidate j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Therefore, the test candidate j and the calibration candidate k are ex- changeable in the set ˆS(j) c ∪{j} under the selection conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It remains to control two conditional miscoverage gaps: PDu,−j � j ̸∈ PIj( ˆS(j) c )|j ∈ ˆS(j) u � − α and PDu,−j � j ̸∈ PIj( ˆSc)|j ∈ ˆSu � − PDu,−j � j ̸∈ PIj( ˆS(j) c )|j ∈ ˆS(j) u � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' the former can be bounded as in conventional conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our theorem shows that a tight control of the deviation term ∆(Du,−j) in (10) leads to effective FCR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Next we carefully interpret the bound and present more explicit settings in which the FCR achieves or is very close to the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Observe that controlling ∆(Du,−j) actually boils down to establishing the upper bound of the difference TU\\{j} (ˆκ(j)) − TU\\{j} (ˆκ(j)−Iu) and the lower bound of the denominator TU\\{j} (ˆκ(j)−Iu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To guarantee that the denominator will stay away from 0, we impose the following assumption on ˆκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The ranking threshold satisfies ˆκ ≥ γm for some γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The lower bound on ˆκ in Assumption 3 is mild and reasonable, since the FCR control will be extremely difficult when | ˆSu|/n = ˆκ/n = o(1) for a small level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Applying the well-known representation of spacing between consecutive order statistics (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1) to {Ti}i∈U\\{j}, together with Assumption 3, we can obtain the following FCR control result for self-driven selection procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If γ > Iu/m, the FCR value of SCOP with self-driven selection procedures can be controlled at FCRSCOP = α + O � log2(n ∨ m) ργ(γ − Iu/m) �Iu m + 1 n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In the asymptotic regime, FCRSCOP is exact if Iu = o(m), that is lim(n,m)→∞ FCRSCOP = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Recalling that for quantile-based selection and Top-K selection, we have Iu = 0, and thus Theorem 3 guarantees the FCR of SCOP with such selection procedures can attain the target level in a nearly optimal rate (up to a logarithmic factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 FCR control for calibration-assisted selection For calibration-assisted selective procedures, the analysis is more complex because the ranking threshold ˆκ depends also on the calibration set Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It implies that a more tractable ranking threshold is needed to decouple the dependence on the selected samples and the calibration samples simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' That is for any j ∈ ˆSu and k ∈ C, let ˆκ(j,k) ≡ ˆκj←tu,k←tc be the ranking threshold by replacing Tj with tu and Tk with tc simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The virtual post-selection calibration set is further defined as ˆS(j,k) c = � i ∈ C \\ {k} : Ti ≤ TU\\{j} (ˆκ(j,k)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following assumption, an analog of Assumption 2, is imposed to restrict the change in the ranking threshold after replacing one calibration score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There exists some tc ∈ R and some positive integer Ic ≤ m such that ˆκ ≤ ˆκk←tc ≤ ˆκ + Ic holds for any k ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following theorem is parallel with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' for the calibration-assisted selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' the conditional miscoverage probability of SCOP satisfies PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ��j ∈ ˆSu � ≤ α + EDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � max k ∆(D(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' and PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ��j ∈ ˆSu � ≥ α − 2EDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � max k ∆(D(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' where ∆(D(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) := 2TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � R ˆ S(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c (U (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) − R ˆ S(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c (L(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � � TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)−Iu−Ic) �2 + 4 � TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) − TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)−Iu−Ic) � TU\\{j} (ˆκ(j)) + d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) | ˆS(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c | + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (11) with d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) = � i∈ ˆ S(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c 1 � TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)−Ic−1) < Ti ≤ TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' U (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) = ⌈(1 − α)(| ˆS(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c | + 2)⌉ + d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) and L(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) = ⌈(1 − α)(| ˆS(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c | + 2 − d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k))⌉ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We can see that all the terms in (11) are independent of the samples j ∈ ˆSu and k ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The quantity d(j,k) measures the size difference between the real calibration set ˆSc and the virtual calibration set ˆS(j,k) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The term R ˆ S(j,k) c (U (j,k)) − R ˆ S(j,k) c (L(j,k)) represents the largest possible distance of the corresponding quantiles in ˆSc and ˆS(j,k) c , which can be bounded in ˆS(j,k) c conditional on the data set Du,−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The remaining parts in maxk ∆(D(j,k)) rely on the difference between thresholds from TU\\{j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Equipped with the conditional miscoverage gap in Theorem 4, we can obtain the FCR control results of SCOP with calibration-assisted selection in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If γ > (Ic + Iu)/m, the FCR value of SCOP for calibration-assisted selection can be controlled at FCRSCOP = α + O � log2(n ∨ m) ρ(γ − (Ic + Iu)/m)2 �Ic + Iu m + 1 n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 13 Similar to the results with self-driven selection, the SCOP can control the FCR around the target value with a small gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To decouple the dependence between the ranking threshold and calibration set, an addition term Ic/m appears in Theorem 5, regarding to the effect of replacing one calibration sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If Ic ∨Iu = o(m), then we can take m = exp{o(n 1 2 )} and have lim(n,m)→∞ FCRSCOP = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3 Prediction-oriented selection with conformal p-values We discuss the implementation of the SCOP with a special calibration-assisted selection procedure, the selection via multiple testing based on conformal p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The concept of the conformal p-value was proposed by Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Similar to the conformal PI, the conformal p-values enjoy model/distribution-free properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Recently, there exists some works to apply conformal p-values to implement sample selection from a multiple-testing perspective, such as Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) and Jin and Candès (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In particular, Jin and Candès (2022) investigated the prediction-oriented selection problem, aiming to select samples whose unobserved outcomes exceed some specified values while control the proportion of falsely selected units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This problem can be viewed as the following multiple hypothesis tests: for i ∈ U and some b0 ∈ R, H0,i : Yi ≥ b0 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' H1,i : Yi < b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' By choosing a monotone function g0 : R+ → [0, 1], one could take the score function as g(x) = g0(ˆµ(x) − b0) and compute the conformity scores as {Ti = g(Xi) : i ∈ C ∪ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote the null set of calibration samples as C0 = {i ∈ C : Yi ≥ b0} and its size as n0 = |C0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Given the conformity scores {Ti : i ∈ C0} in the calibration set, the conformal p-value for each test data point can be calculated by2 pj := p(Xj) = 1 + |{i ∈ C0 : Ti ≤ Tj}| n0 + 1 , for j ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (12) To control FDR at the target level β ∈ (0, 1), we may deploy BH procedure to {pj : j ∈ U} and obtain the rejection set ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let pU (1) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ≤ pU (m) be order statistics of conformal p-values in the test set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For any i ∈ U, it holds that {pi ≤ pU (ˆκ)} = {Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 indicates that using the conformal p-values in (12) to obtain ˆSu is equivalent to using the conformity scores in TU with the same ranking threshold ˆκ, that is ˆSu = {i ∈ U : pi ≤ pU (ˆκ)} ≡ {i ∈ U : Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Further, we can also obtain the post-selection calibration set by ˆSc = {i ∈ C : Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Therefore, we can frame the BH procedures with conformal p-values as a calibration-assisted selection in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2Under our continuous assumption, we present the form of conformal p-value without ties in the conformity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For the tie-breaking form, please refer to Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 14 Algorithm 2 SCOP under selection with conformal p-values Input: Training data Dt, calibration data Dc, test data Du, threshold sequence {δ(r) : r ∈ [m]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 1 Fit prediction model ˆµ(·) and score function g(·) on Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Compute the score values TC = {Ti = g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 2 Compute the conformal p-values {pi : i ∈ U} according to (12) based on DC0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Apply the BH procedure with target level β to TU and obtain (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Obtain the post-selection subsets: ˆSu = {i ∈ U : Ti ≤ TU (ˆκ)} and ˆSc = {i ∈ C : Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 3 Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Find the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest value of RSc, denoted by Q ˆ Sc(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 4 Construct PI for each j ∈ ˆSu as PIj = [ˆµ(Xj) − Q ˆ Sc(1 − α), ˆµ(Xj) + Q ˆ Sc(1 − α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Output: {PIj : j ∈ ˆSu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To study the FCR control with selection procedures based on conformal p-values, we consider a more general class of step-up procedures introduced by Fithian and Lei (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let 0 ≤ δ(1) ≤ · · · ≤ δ(m) ≤ 1 denote an increasing sequence of thresholds, we choose the ranking threshold for step-up procedures as ˆκ = max � r : pU (r) ≤ δ(r) � , (13) where pU (r) is the rth-smallest conformal p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Specially, the BH procedure takes δ(r) = rβ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We summarize the SCOP with the step-up selection procedures in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To adapt Assumptions 2 and 4, we can simply take ˆκ(j) = ˆκj←0 and ˆκ(k) = ˆκk←1 by replacing Tj with 0 for j ∈ U and Tk with 1 for k ∈ C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' From Lemma 1 in Fithian and Lei (2020), we have ˆκ(j) = ˆκ for any j ∈ ˆSu in the step-up procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The next proposition characterizes the magnitudes of ˆκ(j) − ˆκ for any j ̸∈ ˆSu and ˆκ(k) − ˆκ for any k ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose {Xi : i ∈ C ∪ U} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let Ω(r) = {ℓ ∈ [n0] : δ(r) < ℓ+1 n0+1 ≤ δ(r + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For step-up procedures (13) using conformal p-values defined in (12) and any absolute constant C > 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For any j ∈ ˆSu, we have ˆκ(j) = ˆκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In addition, for any j ∈ U \\ ˆSu, ˆκ(j) − ˆκ ≤ 12C log m + 8Cm log m n0 + 1 max ⌈γm⌉ K. Thus, it can be shown that the degree-of-freedom (DoF) of +the proposed scheme is limited by K instead of NBS [48], which in fact determines the maximum +sum-rate growth rate (the pre-log factor) as shown in Theorem 2. Next, we study the sum-rate +performance with respect to the number of antennas at BS NBS. +Theorem 3 (Sum-Rate vs. NBS): For sufficiently large L (perfect pre-cancellation of interference) +and NBS ≫ K, the sum-rate of the proposed scheme for a given K increases logarithmically with +the number of antennas at BS under favorable propagation conditions. +Proof : Based on (38), we have +lim +SNR→∞ +Rsum +log2 (NBS) = +lim +SNR→∞ +K +� +i=1 +log2 +� +6 +πe +���h(i) +1 +��� +2 +K +SNR +� ++ Klog2 (NBS) +log2 (NBS) += K, +(42) +which indicates that the sum-rate growth increases logarithmically with NBS. +■ +The conclusion in Theorem 3 aligns with Theorem 2. As the DoF is determined by the number +of users K, a larger number of NBS can only provide the SNR gain, which is consistent with the +general conclusions for MU-MIMO [48]. The correctness of the above theorems will be verified +in the coming section. + +25 +V. NUMERICAL RESULTS +In this section, we will use numerical results to verify the effectiveness of the proposed schemes. +We consider MU-MIMO-OTFS transmissions with M = 32 and N = 16, where we set the +maximum delay and Doppler indices to lmax = 5 and kmax = 7, respectively. The delay and +Doppler indices are assumed to be integer values unless otherwise specified. The fading coefficients +are generated based on the exponential power delay profile with a path loss exponent of 2.76. +The signal constellation is the quadrature phase shift keying (QPSK) constellation. Furthermore, +we present the results under both favorable propagation and practical channel conditions. For the +favorable propagation case, the received signals are generated based on (27), where both the MPSI +and IBI are ignored. For the practical case, the received signals are generated based on (25), and +a user grouping strategy is applied such that the maximum spatial correlation between different +users is no larger than 0.1, i.e., g(i) +p [j] ≤ 0.1, for i ̸= j. Meanwhile, we assume that the different +resolvable paths have AoDs that are at least 5 degrees away from each other. +A. Numerical Results under Favorable Propagation Conditions +We first present the sum-rate performance of the proposed scheme with respect to different +numbers of antennas NBS in Fig. 8(a), where we set K = 2, P = 2, and L = 1. As shown in +the figure, the sum-rate increases by K bits/s/Hz when doubling the number of antennas, which +indicates a logarithmical increase of the sum-rate with with the number of antennas NBS as indicated +by Theorem 3. The sum-rate performance for different numbers of users is presented in Fig. 8(b), +where we set P = 3 and L = 1. In particular, we apply a fixed ratio ρ = 2 between the number +of antennas NBS and number of users K. It can be seen that the sum-rate appears to increase first +with SNR and then slightly saturate in the very high SNR regime. This is because L = 1 is not +sufficient to perfectly cancel out the CTI for the considered case. But we still observe that the +sum-rate exhibits a strong increasing trend at practical SNRs, e.g., SNR from 10 dB to 30 dB. +Furthermore, we also notice that with a fixed ratio ρ, the sum-rate is doubled if the number of +users is doubled. This observation suggests a linear increase of the sum-rate with respect to the +number of users K, and it is consistent with our findings in Theorem 2. +In Fig. 8(c), the sum-rate performance with different values of L is considered, where we set +NBS = 8, K = 4, P = 3. The performance bounds given in both (38) and (39) are also drawn in +the figure. As can be observed from the figure, the proposed scheme outperforms the no precoding + +26 +0 +10 +20 +30 +40 +50 +SNR (dB) +0 +5 +10 +15 +20 +25 +30 +35 +40 +Sum-rate (bits/s/Hz) +NBS = 4 +NBS = 8 +NBS = 16 +NBS = 32 +NBS = 64 +Logarithmically increasing +(a) Sum-rate performance for K = 2 and different NBS. +0 +10 +20 +30 +40 +50 +SNR (dB) +0 +10 +20 +30 +40 +50 +60 +Sum-rate (bits/s/Hz) +K = 2, NBS = 4 +K = 3, NBS = 6 +K = 4, NBS = 8 +K = 5, NBS = 10 +Linearly increasing +(b) Sum-rate performance for different K and NBS. +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +SNR (dB) +-10 +0 +10 +20 +30 +40 +50 +60 +70 +Sum-rate (bits/s/Hz) +No precoding +L = 1 +L = 2 +L = 3 +Bound in (38) +Bound in (39) +(c) Sum-rate performance for different values of L. +0 +5 +10 +15 +20 +25 +30 +SNR (dB) +10-4 +10-3 +10-2 +10-1 +100 +BER +NBS = 8, K = 2, P = 2 +NBS = 16, K = 2, P = 2 +NBS = 8, K = 4, P = 4 +NBS = 16, K = 4, P = 4 +(d) BER performance for different K, NBS, and L. +Fig. 8. The sum-rate and BER performances of the proposed scheme with respect to different numbers of users K and antennas +NBS and different values of L. +benchmark in terms of the sum-rate. Furthermore, we also observe that the sum-rate increases with +a larger L, but the rate saturation appears at very high SNRs. This is not unexpected because +the number of CTI terms is large with a small antenna-to-user ratio and many resolvable paths. +Consequently, a large L is required to fully cancel the interference. On the other hand, it should be +noticed that the sum-rate of the proposed scheme still shows a good increasing rate with imperfect +cancellation at practical SNRs, e.g., SNR from 10 dB to 30 dB, as evidenced by the bounds. The +choice of L is important for the system designs, and more discussions on how to choose L will be +given later in Remark 1. +The bit error rate (BER) performance with various numbers of users, antennas, and resolvable +paths is presented in Fig. 8(d), where we set L = 1. As indicated by the figure, the BER + +27 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +SNR (dB) +0 +5 +10 +15 +20 +25 +30 +35 +40 +Sum-rate (bits/s/Hz) +NBS = 8, K = 2, P = 3 +NBS = 12, K = 3, P = 3 +NBS = 16, K = 4, P = 3 +NBS = 20, K = 5, P = 3 +(a) Sum-rate performance for different K and NBS. +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +SNR (dB) +0 +5 +10 +15 +20 +25 +30 +Sum-rate (bits/s/Hz) +Fractional delay Doppler, with MPSI and IBI +Integer delay Doppler, with MPSI and IBI +Integer delay Doppler, without MPSI and IBI +(b) Sum-rate comparison between various channel conditions. +0 +5 +10 +15 +20 +25 +30 +35 +SNR (dB) +10-4 +10-3 +10-2 +10-1 +BER +NBS = 20, K = 4, P = 2, OTFS + THP +NBS = 20, K = 4, P = 2, OTFS + MRT +NBS = 20, K = 4, P = 2, OFDM + ZF +(c) BER of THP, MRT [29], and OFDM with ZF. +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +SNR (dB) +0 +5 +10 +15 +20 +25 +30 +Sum-rate (bits/s/Hz) +OTFS + THP, without overhead +OTFS + THP, with overhead +OTFS + MRT +OFDM + ZF +(d) Sum-rates of THP, MRT [29], and OFDM with ZF. +Fig. 9. The sum-rate performance of the proposed scheme with different parameters and benchmark technologies. +performance with various channel conditions does not show a noticeable error floor at practical +SNRs. Furthermore, we notice that increasing P and K could degrade the BER performance. This +observation is consistent with the fact that more interference terms are introduced with an increasing +number of resolvable paths and users. On the other hand, we also observe that the BER performance +improves with an increasing number of BS antennas NBS. This observation is also consistent with +our conclusions from Fig. 8(a). +B. Numerical Results under Practical Channel Conditions +In this subsection, we present the numerical results of the proposed scheme under more realistic +channel conditions, where both the MPSI and IBI are considered. We compare the sum-rate +performance for different K and NBS in Fig. 9(a), where P = 3 and L = 1. As can be observed + +28 +from the figure, the sum-rate improves roughly linearly with the increase of K at mid-to-high +SNRs, but saturates when the SNR is larger than 30 dB. This rate saturation is mainly caused by +the MPSI and IBI. +We examine the proposed scheme with more complex channel conditions in Fig. 9(b), where we +consider NBS = 8, K = 3, P = 4, and L = 1. In particular, we present the sum-rate performance +with favorable propagation (no MPSI and IBI), practical channel (with MPSI and IBI), and practical +channel having fractional delay and Doppler. It can be observed that the proposed scheme enjoys +a sum-rate increase with the growth of SNR even in the presence of fractional delay and Doppler. +However, it suffers from a noticeable rate degradation, because the inter-Doppler and inter-delay +interferences are treated as noise in the case of fractional delay and Doppler. It should be noted +that the fractional delay and Doppler can be and should be dealt with by baseband filtering, such +as windowing [37], and pulse shaping [8], [9], [38]–[40]. On the other hand, we observe that the +influence of MPSI and IBI becomes more severe at high SNRs, which aligns with the rate saturation +observed from Fig. 9(a). +A performance comparison between the proposed scheme, the MRT precoding in [29], and +OFDM with zero-forcing (ZF) precoding is presented in Fig. 9(c) and Fig. 9(d). To have a fair +comparison, the OFDM also applies a reduced-CP structure, where no CP is appended between +the adjacent OFDM symbols. But we apply a large ZF precoder of size KN × KN on each +subcarrier to mitigate the intersymbol interference and multiuser interference. In Fig. 9(c), the BER +performance of those schemes are presented, where we consider NBS = 20, K = 4, P = 2, and +L = 1. It can be observed from the figure that the proposed scheme outperforms the MRT scheme +and the OFDM with ZF at mid-to-high SNRs. This observation validates the advantage of the +proposed THP over existing schemes. This advantage can also be demonstrated by the achieved +sum-rate gain shown in Fig. 9(d), where we consider NBS = 8, K = 4, P = 3, and L = 1. In +particular, we also include the sum-rate results of the proposed THP with and without considering +the required overhead in Fig. 9(d). It can be noticed that even though the overhead reduces the sum- +rate, the proposed scheme is still advantageous in terms of the sum-rate over the existing schemes. +However, it should be noted that the required overhead can be reduced as discussed in Section +III-D, which is a topic for future research. More importantly, the proposed THP only requires a +linear complexity of O (LKMN), while the MRT in [29] requires matrix/vector superposition and +multiplication, thus having a complexity of O (KM2N2). Furthermore, the ZF precoded OFDM + +29 +TABLE II +OVERHEAD VS. DIFFERENT NUMBERS OF USERS AND RESOLVABLE PATHS. +K = 2, L = 1 +K = 3, L = 1 +K = 3, L = 2 +P = 2 +2.9% +24.1% +34.9% +P = 3 +9.6% +25.2% +37.8% +P = 4 +12.9% +25.7% +39.0% +requires the matrix inversion and has a complexity of O (MK3N3). The superior performance and +the low implementation complexity make our proposed THP a promising candidate for downlink +MU-MIMO transmissions. +Remark 1: The pre-cancellation term L is a key parameter for our proposed THP, which +determines how many CTI interference terms are pre-cancelled in the precoding. Note that the value +of L should be selected considering the channel condition, operating SNR, and the cancellation +strategy discussed in Section III-D. In our simulations, we intentionally use small values of L, such +as L = 1, because this is the most straightforward application of the proposed THP and it also +requires the least overhead. As extensively discussed in our numerical results, L = 1 performs quite +well under various channel conditions. We argue that this is not a coincidence. Instead, this is an +expected result due to the careful user grouping strategy. The important insight here is that the CTI +interference is only severe when the BF path of one user has a direction that is sufficiently close +to the non-BF path of a different user, as depicted in Fig. 3. Therefore, it is almost impossible +that the BF paths of different users have similar AoDs overlapping with the same non-BF path +of a specific user after a reasonable user grouping. Furthermore, the possibility of multiple users’ +BF paths overlapping with different non-BF paths of the same user is generally low, and this case +can also be avoided by smart grouping strategy. Therefore, we can safely choose a relatively small +value of L in practical systems facilitated by a carefully grouping of users. +Remark 2: It is important to evaluate the required overhead of the proposed scheme. In Table II, +we compute the overhead of the proposed scheme with NBS = 16 and different K and L. The +overhead is calculated as the ratio between the number of assigned known symbols in the DD +domain and the number of DD grids in total, i.e., KMN, which is represented in the form of a +percentage. We observe that the overhead generally increases with more resolvable paths and users, +due to the increase of interference terms. On the other hand, we also notice that a larger value of L +also increases the overhead. However, we have discussed in Remark 1 that a relatively small value + +30 +of L is sufficient in practical systems, which is also consistent with our numerical results in this +section. Furthermore, it should be noted that the overhead performance can be further improved by +considering the scheduling of pre-cancellation as discussed in Section III-D. +VI. CONCLUSIONS +In this paper, we investigated the DD domain THP for MU-MIMO-OTFS. In particular, the +proposed THP implementation exploits the DD domain channel characteristics and does not require +any matrix decomposition or inversion. Furthermore, we analyzed performance for the proposed +scheme in terms of the achievable rates and investigated the scaling factors for the number of BS +antennas and users. Our derivations implied that the sum-rate increases logarithmically with the +number of antennas and linearly with the number of users (under the same antenna-to-user ratio). +Our derivations were verified by numerical results. 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Viterbo, Delay Doppler Communications: Principles and Applications. +Elsevier, 2022. +[42] J. Nam, A. Adhikary, J.-Y. Ahn, and G. Caire, “Joint spatial division and multiplexing: Opportunistic beamforming, user +grouping and simplified downlink scheduling,” IEEE J. Sel. Top. Signal Process., vol. 8, no. 5, pp. 876–890, Oct. 2014. +[43] M. Kollengode Ramachandran and A. Chockalingam, “MIMO-OTFS in high-Doppler fading channels: Signal detection and +channel estimation,” in IEEE Global Commun. Conf. (GLOBECOM), 2018, pp. 206–212. +[44] I. Bocharova, M. Handlery, R. Johannesson, and B. Kudryashov, “A BEAST for prowling in trees,” IEEE Trans. Inf. Theory, +vol. 50, no. 6, pp. 1295–1302, Jun. 2004. +[45] S. Li, B. Bai, J. Zhou, P. Chen, and Z. Yu, “Reduced-complexity equalization for faster-than-Nyquist signaling: New methods +based on Ungerboeck observation model,” IEEE Trans. Commun., vol. 66, no. 3, pp. 1190–1204, Mar. 2017. +[46] T. M. Cover and J. A. Thomas, Elements of Information Theory. +John Wiley & Sons, 2012. +[47] G. D. Forney, “Trellis shaping,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 281–300, Mar. 1992. +[48] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities +and challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60, Jan. 2012. + diff --git a/MNE0T4oBgHgl3EQfjAF4/content/tmp_files/load_file.txt b/MNE0T4oBgHgl3EQfjAF4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb1d9acf9ab48dc67c48933ef210e8efe380d45b --- /dev/null +++ b/MNE0T4oBgHgl3EQfjAF4/content/tmp_files/load_file.txt @@ -0,0 +1,1157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf,len=1156 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='02453v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='IT] 6 Jan 2023 1 Delay-Doppler Domain Tomlinson-Harashima Precoding for OTFS-based Downlink MU-MIMO Transmissions: Linear Complexity Implementation and Scaling Law Analysis Shuangyang Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Jinhong Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Paul Fitzpatrick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Taka Sakurai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' and Giuseppe Caire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IEEE Abstract Orthogonal time frequency space (OTFS) modulation is a recently proposed delay-Doppler (DD) do- main communication scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' which has shown promising performance in general wireless communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' especially over high-mobility channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In this paper, we investigate DD domain Tomlinson-Harashima precoding (THP) for downlink multiuser multiple-input and multiple-output OTFS (MU-MIMO-OTFS) transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Instead of directly applying THP based on the huge equivalent channel matrix, we propose a simple implementation of THP that does not require any matrix decomposition or inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Such a simple implementation is enabled by the DD domain channel property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', different resolvable paths do not share the same delay and Doppler shifts, which makes it possible to pre-cancel all the DD domain interference in a symbol-by-symbol manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We also study the achievable rate performance for the proposed scheme by leveraging the information-theoretical equivalent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we show that the proposed scheme can achieve a near optimal performance in the high signal-to-noise ratio (SNR) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' More importantly, scaling laws for achievable rates with respect to number of antennas and users are derived, which indicate that the achievable rate increases logarithmically with the number of antennas and linearly with the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Our numerical results align well with our findings and also demonstrate a significant improvement compared to existing MU-MIMO schemes on OTFS and orthogonal frequency- division multiplexing (OFDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Part of the paper was presented at IEEE Global Communications Conference 2022 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2 Index Terms OTFS, MU-MIMO, THP, delay-Doppler domain communication, scaling law I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' INTRODUCTION Orthogonal time frequency space (OTFS) modulation has received much attention in the past few years since its invention in [2], thanks to its capability of providing highly reliable communications over complex transmission scenarios, such as high-mobility channels [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Compared to the cur- rently deployed orthogonal frequency-division multiplexing (OFDM) modulation, OTFS modulation has demonstrated high-Doppler resilience and robust communication performance against various channel conditions [3], [5]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, OTFS modulation has been recognized as a potential solution to supporting the heterogeneous requirements of beyond fifth-generation (B5G) wireless systems, especially in high-mobility scenarios [3], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The success of OTFS originates from the delay-Doppler (DD) domain signal processing [8], [9], guided by the elegant mathematical theory of the Zak transform [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The Zak transform gives rise to the DD domain symbol placement, which potentially enables pulse localization without violating Heisenberg’s uncertainty principle [2], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, the DD domain symbol placement allows the information symbols to directly interact with the DD domain channel response, resulting in a much simpler input-output relationship compared to that of OFDM modulation over complex channels such as the high-mobility channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' More importantly, it can be shown that with DD domain modulation, each information symbol principally experiences the whole fluctuations of the time- frequency (TF) channel over an OTFS frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, the OTFS modulation offers the potential of achieving full TF diversity [12]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The DD domain channel response has several appealing properties including compactness, quasi- stationarity, separability, and sparsity [17], [18], which enables simple channel estimation and reduced-complexity detection approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For example, an embedded pilot scheme for OTFS chan- nel estimation was proposed in [19], where a sufficiently large guard interval is applied around the pilot to improve the acquisition of delay and Doppler responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Such a scheme can permit a direct channel estimation by simply checking the received signal’s value around the DD grid of the embedded pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In [20], a sparse Bayesian-learning-assisted channel estimation approach was presented, where both on-grid and off-grid (due to the virtual sampling) delay and Doppler components are used to perform sparse signal recovery in order to estimate the delay and Doppler 3 responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A message passing algorithm (MPA) was proposed in [21], where the Gaussian ap- proximation is applied to model the characteristic of DD domain interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This algorithm and its variants, such as [22], [23], and [24], take advantage of the DD domain sparsity, such that fewer iterations over the graphical model are sufficient to obtain a good error performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The aforementioned algorithms and many other excellent works [25], [26] have laid a strong foundation for single-input and single-output (SISO)-OTFS transceiver designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, related investigations on multiple-input and multiple-output (MIMO)-OTFS systems are only in the their infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' MIMO technology is an important candidate to meet the stringent requirements of the achievable rate for B5G wireless systems [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Research on MIMO-OTFS, especially multiuser MIMO-OTFS (MU-MIMO-OTFS), is important to determine whether OTFS modulation can be applied in practical multiple-antenna systems [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Unfortunately, the design of MU-MIMO-OTFS is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is because OTFS modulation does not guarantee interference-free transmission like OFDM modula- tion in static channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In fact, the DD domain received symbols generally contain interference [21] in the multi-path transmission, as the result of the “twisted convolution” between the transmitted symbols and the DD domain channel responses1 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Consequently, most of the designs of MU- MIMO-OTFS will face an equivalent channel matrix with a huge size, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', number of delay bins times number of Doppler bins times number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' With such an enormous matrix size, conventional precoding/equalization techniques, such as zero forcing and minimum mean square error (MMSE), cannot be directly applied due to the extremely high computational complexity introduced by the channel inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As a result, most of the existing works for downlink MU- MIMO-OTFS rely on simple precoding approaches, such as maximum ratio transmission (MRT) precoding [29], or approximation of channel inversion, such as [30], with an aim to reduce the computational complexity by trading off performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In this paper, we consider the precoding design for downlink MU-MIMO-OTFS from a different perspective by using the Tomlinson-Harashima precoding (THP) [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' THP is a classic non-linear precoding scheme that has been widely applied in practice, whose core idea is to pre-cancel/pre-subtract the known interference before transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' THP has shown promising performance in terms of the achievable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, it has been shown in [33] that the constant “shaping loss” is the only loss of the achievable rate for THP at high signal-to-noise ratios 1The term “twisted convolution” comes from the first OTFS paper [2], which is similar to the circular convolution but with an additional phase term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 4 (SNRs) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, we postulate that the application of THP in MU-MIMO-OTFS would result in a promising rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that the conventional implementation of THP requires QR decomposition [31], [32], such that the decomposed channel matrix has a triangular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' How- ever, with a huge matrix size in the MU-MIMO-OTFS transmission, such a decomposition could be computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In contrast to the existing works, we do not aim to design precoding directly based on the huge equivalent channel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Instead, we propose to perform interference pre-cancellation directly in the DD domain without any channel decomposition or inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is possible by exploiting the fact that different resolvable paths must be distinguishable in at least one dimension of delay and Doppler, and consequently cannot share both the same delay and Doppler shifts at the same time2 [17], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The major contributions of this paper can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We derive a concise input-output relation for downlink MU-MIMO-OTFS with beamforming (BF) in the matrix form, which lays the foundations for our digital precoder designs and later performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Using the derived system model, we conduct a detailed analysis on the DD domain interference pattern and compare it to the TF domain interference pattern for the OFDM counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we show that the DD domain received symbols suffer from three types of interference, namely multi-path self-interference (MPSI), inter-beam interference (IBI), and crosstalk interference (CTI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We unveil the physical meanings of those interference terms, and show that IBI can be ignored by considering user grouping or user scheduling, while MPSI can be mitigated by BF in practical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We propose a DD domain THP design that only entails linear complexity without any matrix decomposition or inversion based on the characteristics of DD domain channel responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we show that the DD domain interference pattern contains several cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The existence of the cycles suggests that the interference pre-cancellation can start from any DD grid in the cycle and all the interference can be cancelled out in a symbol-by-symbol manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We study the sum-rate of the proposed scheme by deriving the representative information- theoretical equivalent models according to the property of the modulo operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Based on the 2Physical channels can have multiple paths sharing the same or very similar delay and Doppler responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, due to the limited capability of distinguishing delay and Doppler for practical receivers, those paths cannot be fully resolved or separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Consequently, the receiver only sees one multi-path component (DD response) due to the combining of these paths [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5 TABLE I LIST OF MAIN SYSTEM PARAMETERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Definitions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Number of users ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Number of delay bins/subcarriers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Number of Doppler bins/time slots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Number of resolvable paths ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='NBS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Number of antennas at BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='∆f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Subcarrier spacing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Time slot duration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Number of interference terms considered for cancellation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='h(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Channel coefficient for the p-th path of the k-th user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='and k(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Delay and Doppler indices for the p-th path of the k-th user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='p [j] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='Spatial interference power of the j-th beam on i-th user’s p-th path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='X(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='DD [l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' k] and Y (i) DD [l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' k] (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' k)-th DD domain transmitted and received symbol of the i-th user derived sum-rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' we show that the proposed scheme can achieve a near-optimal performance that only has a constant rate loss (the shaping loss) compared to the optimal interference- free transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we investigate the sum-rate performance with respect to the number of antennas at the base station (BS) NBS and the number of users K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we show that the sum-rate of the proposed scheme increases linearly with K and logarithmically with NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Notations: The blackboard bold letters A, E, and C denote the constellation set, the expectation operator, and the complex number field, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' the notations (·)T and (·)H denote the transpose and the Hermitian transpose for a matrix, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' vec(·) denotes the vectorization operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' diag{·} denotes the diagonal matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' “⊗” denotes the Kronecker product operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' min (·) returns the minimum value of a function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' I (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ·) and h (·) denote the mutual information and the differential entropy, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' [·]x denotes the modulo operation with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' FN and IM denote the discrete Fourier transform (DFT) matrix of size N × N and the identity matrix of size M × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' the big-O notation O (·) describes the asymptotic growth rate of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For the sake of clarity, the main system parameters are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 6 ISFFT IFFT ( ) tx g t [ ] DD , X l k DD X [ ] TF , X m n TF X ( ) s t TD x Heisenberg Transform OTFS Modulation [ ] TD x m nM + FFT SFFT Wigner Transform OTFS Demodulation ( ) rx g t ( ) r t TD y [ ] TD y m nM + [ ] TF , y m n TF Y DD Y [ ] DD , y l k Channel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The transmitter structure of SISO-OTFS transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' SYSTEM MODEL In this section, we will derive a concise system model for MU-MIMO-OTFS transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Before going into the details of MU-MIMO-OTFS transmissions, we will briefly review some preliminaries on SISO-OTFS transmissions, which will then be used for the related discussions on MU-MIMO-OTFS transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Preliminaries on SISO-OTFS Transmissions Without loss of generality, let us consider the OTFS transmitter shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Let M be the number of delay bins/subcarriers and N be the number of Doppler bins/time slots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The corresponding subcarrier spacing and time slot duration are given by ∆f and T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Let xDD ∈ AMN be the DD domain information symbol vector of length MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the information symbol vector xDD can be arranged as a two-dimensional (2D) information symbol matrix XDD ∈ AM×N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', xDD ∆= vec (XDD), and the (l, k)-th element of XDD, XDD [l, k], is the information symbol at the l-th delay grid and the k-th Doppler grid [2], for 0 ≤ k ≤ N − 1, 0 ≤ l ≤ M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As indicated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 1, the TF domain transmitted symbol XTF [m, n] , 0 ≤ m ≤ M − 1, 0 ≤ n ≤ N − 1 can be obtained from XDD via the inverse symplectic finite Fourier transform (ISFFT) [35], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', XTF ∆= FMXDDFH N, (1) where XTF [m, n] is the (m, n)-th element in XTF, and FM and FN are the normalized DFT matrices of size M × M and N × N defined in the Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It is also convenient to write the corresponding vector form of (1), which is given by [36] xTF ∆= vec (XTF) = � FH N ⊗ FM � xDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (2) The transmitted OTFS signal s (t) can be obtained by performing the Heisenberg transform [2] to XTF with the transmitter shaping pulse gtx(t), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the Heisenberg 7 transform can be interpreted as a multicarrier modulator and a popular choice for implementing the Heisenberg transform is to apply the OFDM modulator [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' According to the OFDM modulation, the Heisenberg transform can be implemented by an inverse fast Fourier transform (IFFT) module and transmit pulse shaping, in which case the resultant transmitted OTFS signal s (t) is given by s (t) = N−1 � n=0 M−1 � m=0 XTF [m, n] gtx (t − nT) ej2πm∆f(t−nT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (3) Based on (3), it is useful to define the time-delay (TD) domain transmitted symbol vector xTD of length MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Considering the energy-normalized rectangular shaping pulse gtx (t), xTD is defined by [35] xTD ∆= vec � FH MXTF � = � FH N ⊗ IM � xDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (4) Let hDD (τ, ν) be the DD domain channel response given by hDD (τ, ν) = P � p=1 hpδ (τ − τp) δ (ν − νp) , (5) where hp, τp, and νp are the fading coefficient, the delay shift, and the Doppler shift associated with the p-th path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' According to [35], the corresponding TD domain channel response of (5) can be equivalently represented in a matrix form in the case of rectangular filtering pulse grx (t), reduced CP structure, and non-fractional delay and Doppler shifts, such that HTD = P � p=1 hpΠlp∆kp, (6) where Π is the permutation matrix (forward cyclic shift), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', Π = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 · · 0 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 · · 1 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb MN×MN , (7) and ∆ = diag{γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', γMN−1} is a diagonal matrix with γ ∆= e j2π MN [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In (6), the terms lp and kp are the indices of delay and Doppler, respectively, associated with the p-th path, respectively, where τp = lp M∆f , and νp = kp NT , (8) and we have lp ≤ lmax and −kmax ≤ kp ≤ kmax, for 1 ≤ p ≤ P, with lmax and kmax denoting the largest delay index and Doppler index, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It should be noted that the system model in (6) 8 only considers the integer delay and Doppler case, which is only valid with a sufficiently large signal bandwidth and a sufficiently long frame duration [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, it is reported in [37] that the effects of fractional Doppler could be mitigated by adding TF domain windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, some recent developments of OTFS have shown that the pulse shaping could improve the DD domain sparsity [8], [9], [38]–[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As the main focus of this paper is on the application of THP to MU-MIMO-OTFS transmissions, we restrict ourselves to the case of integer delay and Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Following on from (6), the received time-delay (TD) domain symbol vector yTD is given by yTD = HTDxTD + w, (9) where w is the corresponding additive white Gaussian noise (AWGN) sample vector in the TD domain with one-sided power spectral density (PSD) N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The OTFS demodulation can be interpreted as the concatenation of the Wigner transform and the SFFT [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Based on (9), the DD domain received symbol vector is given by3 [35], yDD = (FN ⊗ IM) yTD = HDDxDD + w, (10) where HDD is the corresponding equivalent DD domain channel matrix of the form [14] HDD ∆= P � p=1 hp (FN ⊗ IM)Πlp∆kp � FH N ⊗ IM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (11) For ease of derivation, it is useful to derive a DD domain symbol-wise input-output relation based on (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In fact, (11) has a direct connection to the inverse discrete Zak transform (IDZT), which gives rise to the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Lemma 1 (DD Domain Input-Output Relation via IDZT): Let YDD be the corresponding matrix representation of yDD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', yDD ∆= vec (YDD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Then, in the case of integer Doppler indices and rectangular shaping pulses, the input-output relation for OTFS transmissions with the reduced-CP structure can be characterized by YDD [l, k] = P � p=1 hpej2π kp(l−lp) MN αl,lp,k,kpXDD � [l − lp]M, [k − kp]N � , (12) where αl,lp,k,kp is a phase offset as the result of the quasi-periodicity property of the IDZT, and it is given by αl,lp,k,kp = \uf8f1 \uf8f2 \uf8f3 1, l − lp ≥ 0, e−j2π k−kp N , l − lp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (13) 3In (10), we use the same notation for the AWGN samples in both TD and DD domains, because they follow the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9 THP OTFS Modulation OTFS Modulation BF ( ) 1 DD s ( ) DD K sM ( ) DD K x ( ) 1 DD x M ( ) 1 TD x ( ) TD K x BS N z 1z M OTFS Demodulation OTFS Demodulation ( ) 1 DD y ( ) DD K y ( ) 1 TD y ( ) TD K y M Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The block diagram of considered THP-based downlink MU-MIMO-OTFS transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Proof: The proof is straightforward by invoking the IDZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, derivations without applying IDZT can also be found in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='2 of [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Despite the fact that Lemma 1 has already appeared in the literature [41], we still want to emphasize the importance of those results here because of the following two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Firstly, the symbol-wise DD domain input-output relation for OTFS has not been widely considered and understood in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Secondly, the results of Lemma 1 will be frequently used in the later part of this paper as the building block for our derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Based on the above descriptions of SISO-OTFS transmissions, we will der ive the system model of MU-MIMO-OTFS transmissions in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Derivations of the System Model for MU-MIMO-OTFS Transmissions Without loss of generality, let us consider the downlink MU-MIMO-OTFS transmission for K users, where the BS is equipped with K radio-frequency (RF) chains and NBS antennas with NBS ≥ K, while each user is equipped with only one antenna, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For notational consistency, we will extend the related notations from the above subsection by adding superscripts or subscripts to specify the underlying users or antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Denote by s(k) DD ∈ AMN×1 the DD domain information symbol vector of length MN for the k-th user, where 1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the DD domain information symbol vectors for the K users can be arranged into a 2D matrix SDD of size MN × K, whose k-th column is s(k) DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As indicated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2, we apply THP to SDD and the resultant symbol matrix after precoding is XDD of size MN × K, whose k-th column is the DD domain symbol vector for the k-th user after precoding, denoted by x(k) DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' After passing x(k) DD through the OTFS modulator, the TD domain symbol vector for the k-th user can be obtained by 10 x(k) TD = � FH N ⊗ IM � x(k) DD according to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, we can write XTD = � FH N ⊗ IM � XDD, (14) where XTD of size MN ×K is the TD domain symbol matrix after OTFS modulation, and its k-th column is x(k) TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For ease of derivation, let us consider the vectorized version of XTD by stacking each column of XTD into a vector, such as xTD ∆= �� x(1) TD �H , � x(2) TD �H , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', � x(K) TD �H�H = vec (XTD) = � IK ⊗ FH N ⊗ IM � xDD, (15) where xDD ∆= vec (XDD) is the DD domain symbol vector of size KMN × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We consider conventional BF for the downlink transmission as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Let VBF of size K × NBS be the BF matrix adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Then, the transmitted symbol matrix Z after BF is given by Z = XTDVBF, (16) where the n-th column of Z, zn, is the transmitted symbol vector on the n-th antenna at the BS, for 1 ≤ n ≤ NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Similar to (15), we can write the corresponding vector form of (16), which is given by z ∆= � zH 1 , zH 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', zH NBS �H = vec (Z) = � VT BF ⊗ IMN � xTD = � VT BF ⊗ FH N ⊗ IM � xDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (17) Now let us turn our attention to the wireless channel for MU-MIMO transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Without loss of generality, we assume that the antenna array at the BS is in the form of a uniform linear array (ULA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We further assume that the underlying channel between the BS and each user has P independent resolvable paths, where the angle-of-departure (AoD) for the p-th path of the k-th user, for 1 ≤ p ≤ P and 1 ≤ k ≤ K, is given by ϕ(k) p , and ϕ(k) p ̸= ϕ(k′) p′ , for p ̸= p′ or k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Then, according to the far field assumption [27] and the DD domain channel characteristics in (5), the DD domain channel for the n-th antenna and the k-th user can be modeled by h (n, k, τ, ν) = P � p=1 h(k) p exp � jπ (n − 1) sin � ϕ(k) p �� δ � τ − τ (k) p � δ � ν − ν(k) p � , (18) where we assume that the distance between adjacent antennas is equal to half of the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In (18), h(k) p ∈ C, τ (k) p , and ν(k) p are the fading coefficient, the delay shift, and the Doppler shift corresponding to the p-th path of the k-th user, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' According to (18), let us denote by l(k) p and k(k) p the delay and Doppler indices corresponding to the p-th path of the k-th user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', τ (k) p = l(k) p M∆f , ν(k) p = k(k) p NT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (19) 11 Let us further define the effective TD domain channel matrix for the p-th path of the k-th user based on (6) by ˜Hk,p TD = h(k) p Πl(k) p ∆k(k) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Similarly, based on (11), the effective DD domain channel matrix for the p-th path of the k-th user is defined by ˜Hk,p DD ∆= h(k) p (FN ⊗ IM)Πl(k) p ∆k(k) p � FH N ⊗ IM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' After some derivations, we can write the TD domain received symbol vector y(k) TD for the k-th user by y(k) TD = � NBS P � p=1 � aT � ϕ(k) p � ⊗ ˜Hk,p TD � z + w(k), (20) where a � ϕ(k) p � ∆= 1 √NBS � 1, exp � jπ sin ϕ(k) p � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', exp � jπ (NBS − 1) sin ϕ(k) p ��T, (21) is the normalized steering vector for the p-th path of the k-th user, and w(k) is the AWGN sample vector with one-sided PSD N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Next, by considering (17), (20) can be further expanded as y(k) TD = � NBS P � p=1 �� aT � ϕ(k) p � VT BF � ⊗ � ˜Hk,p TD � FH N ⊗ IM ��� xDD + w(k) = � NBS P � p=1 � ˜Hk,p TD � FH N ⊗ IM �� XDD � VBFa � ϕ(k) p �� + w(k), (22) where the second equation is due to the properties of the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Considering (22), it is convenient to define the effective spatial domain channel vector g(k) p ∆= VBFa � ϕ(k) p � to characterize the interference from different data streams to the received symbols of the k-th user from the p-th path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Finally, by performing OTFS demodulation to y(k) TD, the DD domain received symbol vector y(k) DD for the k-th user can be written by y(k) DD = � NBS P � p=1 � (FN ⊗ IM) ˜Hk,p TD � FH N ⊗ IM �� XDDg(k) p + w(k) = � NBS P � p=1 ˜Hk,p DDXDDg(k) p + w(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (23) So far, we have derived the system model of the MU-MIMO-OTFS transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In the following section, we will develop our digital THP scheme based on (23) by adopting a simple BF matrix according to the steering vectors, where the k-th row of VBF is the Hermitian transpose of the steering vector associated with the strongest path of the k-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 12 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' DD DOMAIN THP FOR DOWNLINK MU-MIMO-OTFS TRANSMISSIONS In this section, we will discuss the proposed DD domain THP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It should be noted that the direct application of THP by employing QR decomposition may require high complexity since the size of the equivalent channel matrix is KMN × KMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, we propose a DD domain THP scheme that does not require the decomposition of channel matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we assume that the channel state information (CSI) is available at the transmitter, which can be achieved by exploiting the DD domain reciprocity [17] based on uplink channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' DD Domain Interference Pattern Analysis Let us first have a close look at the interference pattern in the DD domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' To provide some insights, let us rewrite (23) as y(i) DD = � NBS P � p=1 K � j=1 g(i) p [j] ˜Hi,p DDx(j) DD + w(i), (24) where g(i) p [j] denotes the j-th element of g(i) p implying the contribution from the j-th beam to the i-th user via the i-th user’s p-th path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As implied by (24), the DD domain received symbol vector of the i-th user is related to the DD domain transmitted symbols of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, by considering (12), (24) can be expanded as Y (i) DD [l, k] = P � p=1 K � j=1 ˜g(i,j) l,l(i) p ,k,k(i) p ,pX(j) DD �� l − l(i) p � M, � k − k(i) p � N � + w(i) [l, k], (25) where Y (i) DD [l, k] denotes the (l, k)-th symbol of the received symbol matrix Y(i) DD of the i-th user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' y(i) DD ∆= vec � Y(i) DD � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' and ˜g(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='p characterizes the symbol-wise effective channel coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' including the angular domain interference from the j-th user/beam to the i-th user/beam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' the fading coefficient from the p-th path of the i-th user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' and the phase rotation due to the twisted convolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' and is given by4 ˜g(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='p= \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 √NBSg(i) p [j] h(i) p exp � j2π k(i) p � l−l(i) p � MN � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' l − l(i) p ≥0 √NBSg(i) p [j] h(i) p exp � j2π k(i) p � l−l(i) p � MN � exp � −j2π � k−k(i) p � N � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' l − l(i) p <0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (26) To further characterize the interference pattern, let us assume that the channel strengths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', absolute values of fading coefficients, associated to each user are sorted in descending order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', 4The additional phase term in the second line of (26) is the consequence of the quasi-periodicity of the Zak transform [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 13 ���h(i) 1 ��� ≥ ���h(i) 2 ��� ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ≥ ���h(i) P ���, for 1 ≤ i ≤ K, without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In this case, the BS forms multi-beams towards the directions of the first paths of all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We henceforth refer to the first path of each user as the BF path, while the other paths are called non-BF paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' With these in mind,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' we can expand (25) to yield Y (i) DD [l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' k] = ˜g(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='i) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(i) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k(i) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='1X(i) DD �� l − l(i) 1 � M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' � k − k(i) 1 � N � � �� � Desired signal + P � p=2 ˜g(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='i) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='pX(i) DD �� l − l(i) p � M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' � k − k(i) p � N � � �� � MPSI + K � j=1 j̸=i ˜g(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(i) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k(i) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='1X(j) DD �� l − l(i) 1 � M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' � k − k(i) 1 � N � � �� � IBI + P � p=2 K � j=1 j̸=i ˜g(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='l(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='k(i) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='pX(j) DD �� l − l(i) p � M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' � k − k(i) p � N � � �� � CTI +w(i) [l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (27) From (27), we notice that the value of Y (i) DD [l, k] is composed of several terms with different physical meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We can characterize those signals based on their physical meanings as follows: Desired signal: The first term in (27) is the desired signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The desired signal contains the information of the desired user and it is transmitted from the BF path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' MPSI: The second term in (27) is the MPSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The MPSI contains the interference from the desired user caused by the multi-path transmissions from the non-BF paths of the desired user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IBI: The third term in (27) is the IBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The IBI contains the interference from other users caused by the superposition among different beams, as each user has a distinctive beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' CTI: The fourth term in (27) is the CTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The CTI contains the interference from other users caused by the unintended alignment between the other users’ BF directions and the desired user’s non-BF paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A brief diagram characterizing the interference pattern is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 3, where both the IBI and CTI are clearly indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As implied by the interference descriptions above, we notice that the interference terms have different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, it should be noted that not all those interference terms make a significant contribution to the received symbol Y (i) DD [l, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, 14 CTI IBI CTI BF path User 1 User 2 BS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The brief diagram of the interference pattern for downlink MU-MIMO-OTFS transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' user scheduling is usually performed at the BS before transmitting the downlink signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' One of the objectives of performing user scheduling is to avoid severe interference among different beams, which is enabled by grouping users with diverse spatial characteristics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', AoDs [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, thanks to the nature of BF, the impact of MPSI is generally small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is because the BS only forms narrow beams towards the BF paths of each user, and consequently the residual power on the non-BF paths is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, it can be shown that the CTI could have a high impact if the BF path of one user overlaps with one of the non-BF paths from a different user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is because the transmitted signal after BF usually has a large power towards the BF direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, even though the non-BF path may not have a large channel gain, the overall received power is still non-negligible as the transmitted power towards this direction is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Approximations with User Grouping As indicated by the discussions in the previous subsection, the interference terms have different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In the following subsection, we will develop a DD domain THP scheme by exploiting the nature of those interference terms with the aid of user grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Let us consider the following assumption for user grouping: Assumption 1: We assume that the beams formulated for different users in the group are sufficiently separated (orthogonal) in the angular domain by having NBS ≫ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' With this assumption, it is reasonable to ignore the IBI between different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 15 Furthermore, it should be noted that the AoDs of different paths associated to the same user are usually separated, especially for a sufficiently large number of transmit antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' On top of that, the non-BF paths usually have much lower channel gain compared to the BF paths in practical settings thanks to the BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Those two observations give rise to the following assumption: Assumption 2: We assume that the non-BF paths associated to the same user are relatively separated in the angular domain, where the channel gains are much lower compared to that of the BF path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' With this assumption, it is reasonable to ignore the MPSI of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We henceforth refer to the transmission where both assumptions 1 and 2 hold as the favorable propagation conditions, which is realizable with NBS ≫ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Under the favorable propagation conditions, (27) becomes Y (i) DD [l, k] ≈˜g(i,i) l,l(i) 1 ,k,k(i) 1 ,1X(i) DD �� l − l(i) 1 � M, � k − k(i) 1 � N � + L � p=1 ˜g(i,Bi[p]) l,l(i) Pi[p],k,k(i) Pi[p],Pi[p]X(Bi[p]) DD �� l − l(i) Pi[p] � M, � k − k(i) Pi[p] � N � + w(i) [l, k] , (28) where the MPSI, IBI are ignored and only L CTI terms are considered with 1 ≤ L ≤ (P − 1) (K − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Here, the term L is the number of CTI terms with significant power that will be considered in the precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The introduction of L aims to strike a balance between the error performance and the computational complexity of the precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In (28), we define Bi of length L as the CTI beam vector for the i-th user and Pi of length L as the CTI path vector for the i-th user, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The CTI beam vector contains the beam indices that correspond to the L CTI terms with the most significant power for the i-th user, while the CTI path vector contains the indices of paths for the i-th user that spatially overlap with the beams with indices given in the CTI beam vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In other words, with a descending power order of the CTI terms, the p-th CTI term, for 1 ≤ p ≤ L, is caused by Bi [p]-th beam overlaping with the Pi [p]-th path of the i-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, by examining (26), the elements of Bi and Pi can be determined based on the absolute values of hi [p] g(i) p [j], for 2 ≤ p ≤ P and 1 ≤ j ≤ K, j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The approximated input-output relation in (28) has an important property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For each DD domain received symbol, all the related DD domain transmitted symbols that contribute to the interference of this received symbol are from different DD grids of other users, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is quite different from the OFDM counterpart, where all the related TF domain transmitted symbols that contribute to a specific received TF domain symbol are from the same TF grid of different users, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The rationale behind this observation is that the TF domain channel 16 operation can be characterized by an element-wise product [27], while the DD domain channel operation is characterized by the twisted convolution [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In fact, this property is the key enabler for a reduced-complexity THP for downlink MU-MIMO transmissions, which will be introduced in detail in the coming subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Rx User 2 Rx User 1 Tx User 2 Tx User 1 t n Rx User 2 Rx User 1 Tx User 2 Tx User 1 (a) MU-MIMO-OTFS transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Rx User 2 Rx User 1 Tx User 2 Tx User 1 Rx User 2 Rx User 1 Tx User 2 Tx User 1 t f (b) MU-MIMO-OFDM transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A diagram characterizing the difference of interference patterns between MU-MIMO-OTFS and MU-MIMO-OFDM, where two users are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the red arrow denotes the BF path, while the blue dashed line implies the CTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' DD Domain THP The core idea of THP is to pre-cancel the interference before transmission, where a modulo operation is applied to control the transmitted signal power [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Before introducing the considered DD domain THP, let us consider the following example as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5, where M = N = 3, P = 2, and K = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' There are in total 9 DD grids for each user and we use the capital letters A to I with different colors to refer to the DD domain transmitted symbols associated to each DD grid, where the subscripts for the capital letters denote the corresponding user indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we use the solid and dashed arrows at the “transmitter” part indicating the DD shift corresponding to each resolvable path, where we assume that l(1) 1 = 0, k(1) 1 = 0, and l(1) 2 = 0, k(1) 2 = −1 for user 1, while l(2) 1 = 1, k(2) 1 = 0, and l(2) 2 = 0, k(2) 2 = 1 for user 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Here, we assume that the positive delay and Doppler indices shift the symbol up and to the left, while the negative delay and Doppler indices shift the symbol down and to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The interference pattern corresponding to (28) is shown in the “receiver” part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5, where the symbols on the left hand side in each DD grid is the desired signal (marked in red), while the symbols on the right hand side are the interference (marked in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For a better illustration, we also use dashed circles 17 BF path Non-BF path A1 C2 B1 A2 C1 B2 D1 F2 E1 D2 F1 E2 G1 I2 H1 G2 I1 H2 D2 B1 E2 C1 F2 A1 G2 E1 H2 F1 I2 D1 A2 H1 B2 I1 C2 G1 User 1 User 2 A2 D2 E2 F2 G2 H2 I2 B2 C2 User 2 User 1 A1 D1 E1 F1 G1 H1 I1 B1 C1 Non-BF path BF path BF path Transmitter: Receiver: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' An example of the interference pattern for MU-MIMO-OTFS, where M = N = 3, P = 2, and K = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' highlighting the iteration between the users, where the color of each dashed circle corresponds to the DD grid from which the interference comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It is interesting to note from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5 that there is a possibility that we can directly pre-cancel all the interference in the DD domain by exploiting the different delay and Doppler responses associated to different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For example, the received value of the first DD grid for user 1 only consists of the desired signal A1 and the interference from C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, the interference for A1 can be perfectly canceled if we know the exact value of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Similarly, the interference for C2 can be canceled if we know the exact value of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' So on and so forth, it can be shown that there are DD domain cycles that contain several DD domain symbols for the interference cancellation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', A1 → C2 → G1 → I2 → D1 → F2 → A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, it should be noted that the pre-cancellation could change the value of the corresponding DD domain transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Consequently, due to the DD domain cycles, the pre-cancellation of interference cannot be directly applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For instance, in the considered example, to pre-cancel the interference for A1, it is required to know the value of A1 after interference cancellation as suggested by the cycle, which is a non-causal operation and 18 BF path Non-BF path 2 2 1 2 2 2 1 1 0 C2 B1 A2 0 B2 D1 F2 E1 0 F1 E2 G1 I2 H1 G2 I1 H2 0 B1 E2 0 F2 0 G2 E1 H2 F1 I2 D1 A2 H1 B2 I1 C2 G1 � � � � � � � � � � � � � � � � � � User 1 User 2 User 1 User 2 User 1 User 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The application of DD domain THP for the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' cannot be implemented in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' To solve this problem, we propose to assign known symbols to specific DD grids in order to break the DD domain cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For example, if we assign a zero to the symbol A1, then the pre-cancellation for F2 can be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Following the DD domain cycle, the interference can be pre-cancelled step by step, such as A1 → F2 → D1 → I2 → G1 → C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The corresponding pre-cancelation is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 6, where there are in total 3 DD domain cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We use numbers with different colors to represent the schedule of interference cancellation for each DD domain cycle, where we set A1, D2, and C1 as zeros and use zeros to represent the start of the pre-cancelation for each DD domain cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It is not hard to see that the considered pre-cancellation can indeed cancel all the interference without any matrix decomposition or inversion via intentionally assigning known symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Based on the above example, we are ready to present the implementation of DD domain THP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that the proposed THP follows a symbol-by-symbol pre-cancelation, and for each DD domain symbol, it is required to know where the interference comes from and which symbol should be pre-canceled next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Let us denote by ˆB of length K the interfered beam vector for all the users and ˆP of length K the interfered path vector for all the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the i-th element of ˆB is the index of the user, to whom the i-th beam (the transmitted signal of the i-th user) causes the most significant CTI, and the i-th element of ˆP is the corresponding path index, from which the ˆB[i]-th user receives the CTI due to the i-th beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Those terms indicate the precoding schedule for the considered THP scheme, as the most significant CTI from the i-th beam is likely to be included in the CTI beam vector of the ˆB [i]-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In this case, the symbols in the i-th beam after pre- cancellation are likely to be used for the pre-cancellation for the ˆB [i]-th user, thereby reducing the 19 overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, by observing (26), we have ˆB [i] ∆= arg max i ���hj [p] g(j) p [i] ���, for 2 ≤ p ≤ P and 1 ≤ j ≤ K, j ≤ i, and ˆP [i] ∆= arg max p ����h ˆB[i] [p] g( ˆB[i]) p [i] ����, for 2 ≤ p ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Corresponding to the above discussions, the details of DD domain THP are summarized in Algorithm 1, where mod [·] denotes the modulo operation in the conventional THP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Some discussions on the modulo threshold will be presented in the coming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As implied by Algorithm 1, the L most significant CTI will be pre-cancelled via THP for each DD domain symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, according to (28) and the principle of THP, the receiver side applies a single-tap equalization together with a modulo operation to recover the DD domain transmitted symbols [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we have ˆY (i) DD [l, k] = mod \uf8ee \uf8f0 1 ˜g(i,i) l,l(i) 1 ,k,k(i) 1 ,1 Y (i) DD [l, k] \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (29) Based on ˆY (i) DD [l, k], a straightforward demodulation could be applied to recover the transmitted information for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Complexity and Signaling Overhead We will discuss the computational complexity and the required signaling overhead for the considered THP in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As indicated by Algorithm 1, there are at most L times of pre- cancellation for each DD domain transmitted symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, the overall computational complexity is linear to the number of transmitted symbols with a linearity coefficient L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', O (LKMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It should be noted that such a linear complexity is lower than most of the existing precoding schemes for MU-MIMO-OTFS, including the ones in [29], [43], because the proposed THP does not rely on the complex channel decomposition or inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' On the other hand, it can be observed that the signaling overhead for the proposed THP depends on the value of L, and the channel conditions, such as the number of paths, number of users, and delay and Doppler responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, the pre-cancellation order is also of great importance for the signaling overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that Algorithm 1 is a performance-centric implementation of DD domain THP, where the algorithm aims to pre-cancel all the interference terms without considering the required overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Consequently, the total number of assigned known symbols increases if the corresponding interference symbols have not yet been pre-cancelled, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', line 9 to 13 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In contrast, there could also be an overhead-centric implementation, where the pre- cancellation is performed with the priority to the symbols, to whom the corresponding interference 20 Algorithm 1 DD Domain THP for Downlink MU-MIMO-OTFS Transmissions Input: ˜g(i,j) l,l(i) p ,k,k(i) p ,p, S(i) DD, l(i) p , k(i) p , Pi, Bi, ˆB, and ˆP, for 0 ≤ l ≤ M − 1, 0 ≤ k ≤ N − 1, 1 ≤ p ≤ P, 1 ≤ i, j ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Initialization: Set Indicator mtx[l, k, i] = 0, for 0 ≤ l ≤ M − 1, 0 ≤ k ≤ N − 1, 1 ≤ i ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Set Overhead mtx[l, k, i] = 0, for 0 ≤ l ≤ M − 1, 0 ≤ k ≤ N − 1, 1 ≤ i ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Steps: 1: for l′ from 0 to M − 1 do 2: for k′ from 0 to N − 1 do 3: for i′ from 1 to K do 4: Set l = l′, k = k′, and i = i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 5: while Indicator mtx[l, k, i] = 0 do 6: X(i) DD [l, k] = S(i) DD [l, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 7: for p from 1 to L do 8: Set delay idx = �� l − l(i) 1 � M + l(i) Pi[p] � M and Doppler idx = �� k − k(i) 1 � N + k(i) Pi[p] � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9: if Indicator mtx[delay idx, Doppler idx, Bi[p]] = 0 do 10: Set X(Bi[p]) DD [delay idx, Doppler idx] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 11: Set Indicator mtx[delay idx, Doppler idx, Bi[p]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 12: Set Overhead mtx[delay idx, Doppler idx, Bi[p]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 13: end if 14: X(i) DD [l, k] = X(i) DD [l, k] − ˜g(i,Bi[p]) l,l(i) Pi[p],k,k(i) Pi[p],Pi[p] ˜g(i,i) l,l(i) 1 ,k,k(i) 1 ,1 X(Bi[p]) DD [delay idx, Doppler idx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 15: end for 16: X(i) DD [l, k] = mod � X(i) DD [l, k] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 17: Set Indicator mtx[l, k, i] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 18: Set l = �� l − l( ˆ B[i]) ˆ P [i] � M + l( ˆ B[i]) 1 � M , k = �� k − k( ˆ B[i]) ˆ P [i] � N + k( ˆ B[i]) 1 � N , and i = ˆB [i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 19: end while 20: end for 21: end for 22: end for 23: Return ˆX(i) DD, for 1 ≤ i ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' symbols have already been pre-cancelled, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', line 14 in Algorithm 1, in order to minimized the required overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, the reduced overhead implementation is currently still an open problem and we are unable to discuss this issue in detail due to the space limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' But it should be pointed out that the searching algorithms for tree- and trellis-based graphical models may shed light on 21 MU-MIMO- OTFS S a w ˆY 1 g -% S ˆY a w% Mod-d Mod-d Mod-d (a) Equivalent diagram of the system model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' MU-MIMO- OTFS S ˆY w% Mod- Mod- Mod-d (b) Simplified diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Equivalent and simplified system models corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' this issue [44], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ACHIEVABLE RATE ANALYSIS We discuss the achievable rates of the proposed THP scheme in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Without loss of generality, we consider the quadrature amplitude modulation (QAM) constellation set5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we focus on the average achievable rate for each DD domain symbol under favorable propagation conditions by assuming that NBS ≫ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For ease of derivation, we provide an equivalent diagram of the proposed THP-based MU-MIMO-OTFS characterizing the corresponding processing between S(i) DD [l, k] and ˆY (i) DD �� l + l(i) 1 � M, � k + k(i) 1 � N � in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 7(a), where we neglect the symbol indices for notational brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Specifically, we use the term α in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 7(a) to describe the pre- cancellation of THP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As indicated by this diagram, an arbitrary DD domain symbol S after pre- cancellation with term α and modulo operation with threshold d is transmitted over the MU-MIMO- OTFS channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The received channel observation contains the corruption from the AWGN sample w, which is used for symbol detection after an single tap equalization with ˜g−1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', � ˜g(i,i) l,l(i) 1 ,k,k(i) 1 ,1 �−1 , and applying the modulo operation with threshold d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Those descriptions are consistent with our system model in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the above processing can be described by the following equation ˆY = mod �1 ˜g (˜g (mod [S + α]) + η + w) � = mod � mod [S + α] + 1 ˜g (η + w) � , (30) where η denotes the interference term due to the MU-MIMO-OTFS transmission as suggested in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that mod [mod [a] + b] = mod [a + b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, (30) can be further simplified to ˆY = mod � S + α + 1 ˜gη + 1 ˜gw � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (31) 5Although we only focus on QAM constellation here, the related discussions can be straightforwardly extended to the case of general constellations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', pulse amplitude modulation (PAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 22 Furthermore, as implied by Line 14 of Algorithm 1, the interference term η/˜g will be cancelled by pre-cancellation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', term α, with a sufficiently large number of L, in the case of user grouping and BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, we can further approximate (31) by ˆY ≈ mod � S + 1 ˜gw � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (32) The corresponding diagram to (32) is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 7(b), where ˜w = 1 ˜gw denotes the equivalent AWGN sample with one-sided PSD N0 � |˜g|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Now we focus on the achievable rate for the considered scheme based on (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the mutual information between S and ˆY is given by [33], [46] I � S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ˆY � ∆= h � ˆY � − h � ˆY |S � ≈ h � mod � S + 1 ˜gw �� − h � mod �1 ˜gw �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (33) Notice that the modulo operation strictly limits the signal value from � −d 2, d 2 � for both the real and imaginary dimensions, and the maximum entropy probability distribution for a random variable with support constrained to an interval is the independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=') uniform distribution [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, (33) can be approximately upper-bounded by I � S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ˆY � ≲ 2 log2 (d) − h � mod �1 ˜gw �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (34) Note that the values of AWGN samples are generally small in the high SNR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, in the high SNR regime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', the real/imaginary part of the noise sample is within the range of � −d 2, d 2 � ), (34) can be shown to converge to [33] I � S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ˆY � ≲ 2 log2 (d) − h �1 ˜gw � = 2log2 (d) − log2 � πe N0 |˜g|2 � = log2 � d2|˜g|2 πeN0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (35) Based on (35), we are ready to investigate the sum-rate performance for the considered THP scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Notice that there is no joint decoding among different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, with favorable propagation conditions, the sum-rate for the considered downlink MU-MIMO-OTFS can be formulated by Rsum ∆= K � i=1 I � S(i) DD [l, k] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ˆY (i) DD [l, k] � = K � i=1 log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed d2 ����˜g(i,i) l,l(i) 1 ,k,k(i) 1 ,1 ���� 2 πeN0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (36) Furthermore, by substituting (26) into (36), we have Rsum = K � i=1 log2 \uf8eb \uf8ec \uf8ed d2NBS ���h(i) 1 ��� 2 πeN0 \uf8f6 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (37) 23 As implied by (37), the sum-rate is related to the choice of modulo threshold d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' According to [33], the average power for transmitted symbol X(i) DD converges to d2/12 and d2/6 for PAM and QAM constellations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, with QAM constellations, the total transmit power for a given time slot is Kd2/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Based on the total transmit power, we can define the SNR for the THP transmission by SNR ∆= Kd2 6N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Finally, we obtain the sum-rate at high SNRs by Rsum = K � i=1 log2 � 6 πe NBS K ���h(i) 1 ��� 2 SNR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (38) Next, we discuss some important insights based on the previous analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we restrict ourselves to the high SNR regime, where the sum-rate is characterized by (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Let us first characterize the sum-rate gap of the proposed scheme to the optimal transmission scenario, where there is only one resolvable path between the BS and each user with sufficiently separated (orthogonal) angular features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The latter transmission scenario is optimal in the sense that it does not have neither MPSI, IBI, nor CTI, and therefore maximizes the throughput of the downlink transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The following lemma shows the sum-rate in the optimal transmission scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Lemma 2 (Optimal Sum-rate): In the optimal transmission scenario, where there is only one resolvable path between the BS and each user without IBI, the sum-rate is given by Ropt sum = K � i=1 log2 � 1 + NBS K ���h(i) 1 ��� 2 SNR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' (39) Proof : By considering the uniform power allocation among different users, (39) can be derived by following the capacity calculation for parallel Gaussian channels with independent noise [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The detail derivations are omitted here due to the space limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ■ Based on Lemma 2, the following theorem characterizes the sum-rate gap between the proposed scheme and the optimal case in the high SNR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Theorem 1 (Shaping Loss): For sufficiently large L (perfect pre-cancellation of interference) and NBS ≫ K, the proposed scheme only has a constant rate loss for each user compared to the optimal transmission scenario in the high SNR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Proof : Ropt sum − Rsum = K � i=1 log2 \uf8eb \uf8ec \uf8ed 1 + NBS K ���h(i) 1 ��� 2 SNR 6 πe NBS K ���h(i) 1 ��� 2 SNR \uf8f6 \uf8f7 \uf8f8 ≈ K � i=1 log2 �πe 6 � , (40) where the approximation holds in the high SNR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that 1 2log2 � πe 6 � ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='255, which is the well-known “shaping loss” for general PAM constellations in the THP literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ■ 24 As implied by Theorem 1, the proposed scheme can obtain a promising rate performance that only has a constant gap to the optimal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As pointed out by [47], this performance loss is the “shaping loss”, which is caused by the peak limitation introduced by precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Next, we will discuss the growth rate of the sum-rate with respect to different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The following theorem shows the scaling law of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Theorem 2 (Scaling Law for Sum-rate): For sufficiently large L (perfect pre-cancellation of interference) and NBS ≫ K, the sum-rate of the proposed scheme scales linearly with the number of users K under favorable propagation conditions at the asymptotically high SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Proof : Based on (38), we have lim SNR→∞ Rsum log2 (SNR) = lim SNR→∞ K� i=1 log2 � 6 πe NBS K ���h(i) 1 ��� 2� + Klog2 (SNR) log2 (SNR) = K, (41) which indicates that the sum-rate growth is linear in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ■ The conclusion in Theorem 2 is not unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that the proposed scheme contains NBS antennas and K RF, where NBS > K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Thus, it can be shown that the degree-of-freedom (DoF) of the proposed scheme is limited by K instead of NBS [48], which in fact determines the maximum sum-rate growth rate (the pre-log factor) as shown in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Next, we study the sum-rate performance with respect to the number of antennas at BS NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Theorem 3 (Sum-Rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' NBS): For sufficiently large L (perfect pre-cancellation of interference) and NBS ≫ K, the sum-rate of the proposed scheme for a given K increases logarithmically with the number of antennas at BS under favorable propagation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Proof : Based on (38), we have lim SNR→∞ Rsum log2 (NBS) = lim SNR→∞ K � i=1 log2 � 6 πe ���h(i) 1 ��� 2 K SNR � + Klog2 (NBS) log2 (NBS) = K, (42) which indicates that the sum-rate growth increases logarithmically with NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ■ The conclusion in Theorem 3 aligns with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As the DoF is determined by the number of users K, a larger number of NBS can only provide the SNR gain, which is consistent with the general conclusions for MU-MIMO [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The correctness of the above theorems will be verified in the coming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 25 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we will use numerical results to verify the effectiveness of the proposed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We consider MU-MIMO-OTFS transmissions with M = 32 and N = 16, where we set the maximum delay and Doppler indices to lmax = 5 and kmax = 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The delay and Doppler indices are assumed to be integer values unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The fading coefficients are generated based on the exponential power delay profile with a path loss exponent of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The signal constellation is the quadrature phase shift keying (QPSK) constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we present the results under both favorable propagation and practical channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For the favorable propagation case, the received signals are generated based on (27), where both the MPSI and IBI are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' For the practical case, the received signals are generated based on (25), and a user grouping strategy is applied such that the maximum spatial correlation between different users is no larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', g(i) p [j] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='1, for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Meanwhile, we assume that the different resolvable paths have AoDs that are at least 5 degrees away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Numerical Results under Favorable Propagation Conditions We first present the sum-rate performance of the proposed scheme with respect to different numbers of antennas NBS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 8(a), where we set K = 2, P = 2, and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As shown in the figure, the sum-rate increases by K bits/s/Hz when doubling the number of antennas, which indicates a logarithmical increase of the sum-rate with with the number of antennas NBS as indicated by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The sum-rate performance for different numbers of users is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 8(b), where we set P = 3 and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we apply a fixed ratio ρ = 2 between the number of antennas NBS and number of users K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It can be seen that the sum-rate appears to increase first with SNR and then slightly saturate in the very high SNR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is because L = 1 is not sufficient to perfectly cancel out the CTI for the considered case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' But we still observe that the sum-rate exhibits a strong increasing trend at practical SNRs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', SNR from 10 dB to 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we also notice that with a fixed ratio ρ, the sum-rate is doubled if the number of users is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This observation suggests a linear increase of the sum-rate with respect to the number of users K, and it is consistent with our findings in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 8(c), the sum-rate performance with different values of L is considered, where we set NBS = 8, K = 4, P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The performance bounds given in both (38) and (39) are also drawn in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As can be observed from the figure, the proposed scheme outperforms the no precoding 26 0 10 20 30 40 50 SNR (dB) 0 5 10 15 20 25 30 35 40 Sum-rate (bits/s/Hz) NBS = 4 NBS = 8 NBS = 16 NBS = 32 NBS = 64 Logarithmically increasing (a) Sum-rate performance for K = 2 and different NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 10 20 30 40 50 SNR (dB) 0 10 20 30 40 50 60 Sum-rate (bits/s/Hz) K = 2, NBS = 4 K = 3, NBS = 6 K = 4, NBS = 8 K = 5, NBS = 10 Linearly increasing (b) Sum-rate performance for different K and NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 45 50 SNR (dB) 10 0 10 20 30 40 50 60 70 Sum-rate (bits/s/Hz) No precoding L = 1 L = 2 L = 3 Bound in (38) Bound in (39) (c) Sum-rate performance for different values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 5 10 15 20 25 30 SNR (dB) 10-4 10-3 10-2 10-1 100 BER NBS = 8, K = 2, P = 2 NBS = 16, K = 2, P = 2 NBS = 8, K = 4, P = 4 NBS = 16, K = 4, P = 4 (d) BER performance for different K, NBS, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The sum-rate and BER performances of the proposed scheme with respect to different numbers of users K and antennas NBS and different values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' benchmark in terms of the sum-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we also observe that the sum-rate increases with a larger L, but the rate saturation appears at very high SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This is not unexpected because the number of CTI terms is large with a small antenna-to-user ratio and many resolvable paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Consequently, a large L is required to fully cancel the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' On the other hand, it should be noticed that the sum-rate of the proposed scheme still shows a good increasing rate with imperfect cancellation at practical SNRs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', SNR from 10 dB to 30 dB, as evidenced by the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The choice of L is important for the system designs, and more discussions on how to choose L will be given later in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The bit error rate (BER) performance with various numbers of users, antennas, and resolvable paths is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 8(d), where we set L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As indicated by the figure, the BER 27 0 5 10 15 20 25 30 35 40 45 50 SNR (dB) 0 5 10 15 20 25 30 35 40 Sum-rate (bits/s/Hz) NBS = 8, K = 2, P = 3 NBS = 12, K = 3, P = 3 NBS = 16, K = 4, P = 3 NBS = 20, K = 5, P = 3 (a) Sum-rate performance for different K and NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 45 50 SNR (dB) 0 5 10 15 20 25 30 Sum-rate (bits/s/Hz) Fractional delay Doppler, with MPSI and IBI Integer delay Doppler, with MPSI and IBI Integer delay Doppler, without MPSI and IBI (b) Sum-rate comparison between various channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 SNR (dB) 10-4 10-3 10-2 10-1 BER NBS = 20, K = 4, P = 2, OTFS + THP NBS = 20, K = 4, P = 2, OTFS + MRT NBS = 20, K = 4, P = 2, OFDM + ZF (c) BER of THP, MRT [29], and OFDM with ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 45 50 SNR (dB) 0 5 10 15 20 25 30 Sum-rate (bits/s/Hz) OTFS + THP, without overhead OTFS + THP, with overhead OTFS + MRT OFDM + ZF (d) Sum-rates of THP, MRT [29], and OFDM with ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The sum-rate performance of the proposed scheme with different parameters and benchmark technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' performance with various channel conditions does not show a noticeable error floor at practical SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we notice that increasing P and K could degrade the BER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This observation is consistent with the fact that more interference terms are introduced with an increasing number of resolvable paths and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' On the other hand, we also observe that the BER performance improves with an increasing number of BS antennas NBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This observation is also consistent with our conclusions from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Numerical Results under Practical Channel Conditions In this subsection, we present the numerical results of the proposed scheme under more realistic channel conditions, where both the MPSI and IBI are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We compare the sum-rate performance for different K and NBS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(a), where P = 3 and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As can be observed 28 from the figure, the sum-rate improves roughly linearly with the increase of K at mid-to-high SNRs, but saturates when the SNR is larger than 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This rate saturation is mainly caused by the MPSI and IBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We examine the proposed scheme with more complex channel conditions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(b), where we consider NBS = 8, K = 3, P = 4, and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we present the sum-rate performance with favorable propagation (no MPSI and IBI), practical channel (with MPSI and IBI), and practical channel having fractional delay and Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It can be observed that the proposed scheme enjoys a sum-rate increase with the growth of SNR even in the presence of fractional delay and Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, it suffers from a noticeable rate degradation, because the inter-Doppler and inter-delay interferences are treated as noise in the case of fractional delay and Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It should be noted that the fractional delay and Doppler can be and should be dealt with by baseband filtering, such as windowing [37], and pulse shaping [8], [9], [38]–[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' On the other hand, we observe that the influence of MPSI and IBI becomes more severe at high SNRs, which aligns with the rate saturation observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' A performance comparison between the proposed scheme, the MRT precoding in [29], and OFDM with zero-forcing (ZF) precoding is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' To have a fair comparison, the OFDM also applies a reduced-CP structure, where no CP is appended between the adjacent OFDM symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' But we apply a large ZF precoder of size KN × KN on each subcarrier to mitigate the intersymbol interference and multiuser interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(c), the BER performance of those schemes are presented, where we consider NBS = 20, K = 4, P = 2, and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It can be observed from the figure that the proposed scheme outperforms the MRT scheme and the OFDM with ZF at mid-to-high SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This observation validates the advantage of the proposed THP over existing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' This advantage can also be demonstrated by the achieved sum-rate gain shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(d), where we consider NBS = 8, K = 4, P = 3, and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, we also include the sum-rate results of the proposed THP with and without considering the required overhead in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 9(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' It can be noticed that even though the overhead reduces the sum- rate, the proposed scheme is still advantageous in terms of the sum-rate over the existing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, it should be noted that the required overhead can be reduced as discussed in Section III-D, which is a topic for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' More importantly, the proposed THP only requires a linear complexity of O (LKMN), while the MRT in [29] requires matrix/vector superposition and multiplication, thus having a complexity of O (KM2N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, the ZF precoded OFDM 29 TABLE II OVERHEAD VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' DIFFERENT NUMBERS OF USERS AND RESOLVABLE PATHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' K = 2, L = 1 K = 3, L = 1 K = 3, L = 2 P = 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='9% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='1% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='9% P = 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='6% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='2% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='8% P = 4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='9% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='7% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='0% requires the matrix inversion and has a complexity of O (MK3N3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The superior performance and the low implementation complexity make our proposed THP a promising candidate for downlink MU-MIMO transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Remark 1: The pre-cancellation term L is a key parameter for our proposed THP, which determines how many CTI interference terms are pre-cancelled in the precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Note that the value of L should be selected considering the channel condition, operating SNR, and the cancellation strategy discussed in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In our simulations, we intentionally use small values of L, such as L = 1, because this is the most straightforward application of the proposed THP and it also requires the least overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' As extensively discussed in our numerical results, L = 1 performs quite well under various channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We argue that this is not a coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Instead, this is an expected result due to the careful user grouping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The important insight here is that the CTI interference is only severe when the BF path of one user has a direction that is sufficiently close to the non-BF path of a different user, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, it is almost impossible that the BF paths of different users have similar AoDs overlapping with the same non-BF path of a specific user after a reasonable user grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, the possibility of multiple users’ BF paths overlapping with different non-BF paths of the same user is generally low, and this case can also be avoided by smart grouping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Therefore, we can safely choose a relatively small value of L in practical systems facilitated by a carefully grouping of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Remark 2: It is important to evaluate the required overhead of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In Table II, we compute the overhead of the proposed scheme with NBS = 16 and different K and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' The overhead is calculated as the ratio between the number of assigned known symbols in the DD domain and the number of DD grids in total, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=', KMN, which is represented in the form of a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' We observe that the overhead generally increases with more resolvable paths and users, due to the increase of interference terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' On the other hand, we also notice that a larger value of L also increases the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' However, we have discussed in Remark 1 that a relatively small value 30 of L is sufficient in practical systems, which is also consistent with our numerical results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, it should be noted that the overhead performance can be further improved by considering the scheduling of pre-cancellation as discussed in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' CONCLUSIONS In this paper, we investigated the DD domain THP for MU-MIMO-OTFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' In particular, the proposed THP implementation exploits the DD domain channel characteristics and does not require any matrix decomposition or inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Furthermore, we analyzed performance for the proposed scheme in terms of the achievable rates and investigated the scaling factors for the number of BS antennas and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Our derivations implied that the sum-rate increases logarithmically with the number of antennas and linearly with the number of users (under the same antenna-to-user ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Our derivations were verified by numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Our future work may investigate overhead reduction approaches for DD domain THP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors would like to express their thanks to the inventor of OTFS modulation, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Ronny Hadani, for his enlightening speech on MU-MIMO-OTFS, which motivates this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQfjAF4/content/2301.02453v1.pdf'} +page_content=' Yuan, P.' metadata={'source': 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new file mode 100644 index 0000000000000000000000000000000000000000..97ac01d488f6468aa41d0bb2b995c58de87aa42a --- /dev/null +++ b/NNFRT4oBgHgl3EQf3jjV/content/tmp_files/2301.13665v1.pdf.txt @@ -0,0 +1,2555 @@ +Variational Amplitude Amplification for Solving QUBO Problems +Daniel Koch1∗, Massimiliano Cutugno1, Saahil Patel1, Laura Wessing1, Paul M. Alsing1 +1Air Force Research Lab, Information Directorate, Rome, NY +and +∗Corresponding Author: daniel.koch.13@us.af.mil +We investigate the use of amplitude amplification on the gate-based model of quantum comput- +ing as a means for solving combinatorial optimization problems. This study focuses primarily on +QUBO (quadratic unconstrained binary optimization) problems, which are well-suited for qubit su- +perposition states. Specifically, we demonstrate circuit designs which encode QUBOs as ‘cost oracle’ +operations UC, which when combined with the standard Grover diffusion operator Us lead to high +probabilities of measurement for states corresponding to the optimal and near optimal solutions. In +order to achieve these probabilities, a single scalar parameter ps is required, which we show can be +found through a variational quantum-classical hybrid approach. +I. +INTRODUCTION +Amplitude amplification is a quantum algorithm strat- +egy that is capable of circumventing one of quantum com- +puting’s most difficult challenges: probabilistic measure- +ments. Originally proposed by Grover in 1996 [1], and +later shown to be optimal [2, 3], the combination of his +oracle UG and ‘diffusion’ Us operators is able to drive +a quantum system to a superposition state where one +(or multiple) basis state(s) has nearly 100% probability +of being measured. Since then, many researchers have +contributed to the study of UG and Us [4–9], seeking to +better understand how the fundamental nature of am- +plitude amplification is dependent on these two opera- +tors. Similarly, the aim of this study is to further extend +the capabilities of amplitude amplification as a means for +solving combinatorial optimization problems using gate- +based quantum computers. +The results of this paper are a continuation of our +previous work [10], in which we demonstrated an ora- +cle design which was capable of encoding and solving a +weighted directed graph problem. +The motivation for +this oracle was to address a common criticism of UG [11– +15], namely that the circuit construction of oracles too +often hardcodes the solution it aims to find, negating the +use of quantum entirely. Similar to other recent stud- +ies [16–21], we showed that this problem can be solved at +the circuit depth level by avoiding gates such as control-Z +for constructing the oracle, and instead using phase and +control-phase gates (P(θ) and CP(θ)). However, simply +changing the phase produced from UG to something other +than π is not enough [22–27]. Our oracle construction ap- +plies phases to not only a desired marked state(s), but +all states in the full 2N Hilbert Space. The phase each +basis state receives is proportional to the solutions of a +weighted combinatorial optimization problem, for which +the diffusion operator Us can be used to boost the prob- +ability of measuring states that correspond to optimal +solutions. +The consequence of using an oracle operation that ap- +plies phases to every basis state is an interesting double- +edged sword. As we show in sections II. - IV., and later in +section VII., the use of phase gates allows for amplitude +amplification to encode a broad scope of combinatorial +optimization problems into oracles, which we call ‘cost +oracles’ Uc. In particular, we demonstrate the robust- +ness of amplitude amplification for solving these kinds +of optimization problems with asymmetry and random- +ness [28–30]. However, the tradeoff for solving more com- +plex problems is twofold. Firstly, in contrast to Grover’s +oracle, using Uc is only able to achieve peak measure- +ment probabilities up to 70-90%. In section VI. we show +that these probabilities are still high enough for quan- +tum to reliably find optimal solutions, which notably are +achieved using the same O( π +4 +� +N/M ) iterations as stan- +dard Grover’s [1–3]. +The second, more challenging tradeoff when using Uc +is that the success of amplitude amplification is largely +dependant on the correct choice of a single free parame- +ter ps [10]. This scalar parameter is multiplied into ev- +ery phase gate for the construction of Uc (P(θ · ps) and +CP(θ · ps)), and is responsible for transforming the nu- +meric scale of a given optimization problem to values +which form a range of approximately 2π. This in turn is +what allows for reflections about the average amplitude +via Us to iteratively drive the probability of desired solu- +tion states up to 70-90%. The significance of ps, and the +challenges in determining it experimentally, are a major +motivation for this study. In particular, the results of +section V. demonstrate that there is a range of ps values +for which many optimal solutions can be made to become +highly probable. Additionally, our simulations show that +there is an observed correlation between the numerical +cost function value of these solutions and the ps values +where they achieve peak probabilities. This underlying +correlation supports the idea of using amplitude amplifi- +cation for a variational model of hybrid quantum-classical +computing, which is the core finding of this study. +A. +Layout +The layout of this study is as follows. Section II. be- +gins with the mathematical formalism for the optimiza- +tion problem we will seek to solve using amplitude am- +plification. Sections III. & IV. discuss the construction +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 +arXiv:2301.13665v1 [quant-ph] 31 Jan 2023 + +A +Layout +2 +of the problem as a quantum circuit, the varying degrees +of success one can expect from optimization problems +generated using random numbers, and the conditions for +which these successes can be experimentally realized. In +section V. we explore the role of ps from a heuristic per- +spective, whereby we demonstrate that many near opti- +mal solutions are capable of reaching significant proba- +bilities of measurement. Section VI. is a primarily spec- +ulative discussion, theorizing how the collective results +of section V. can be coalesced into a hybrid quantum- +classical variational algorithm. And finally, section VII. +completes the study with additional optimization prob- +lems that can be constructed as oracles and solved using +amplitude amplification. +II. +QUBO DEFINITIONS +We begin by outlining the optimization problem which +will serve as the focus for this study: QUBO (quadratic +unconstrained binary optimization). The QUBO prob- +lem has many connections to important fields of com- +puter science [31–35], making it relevant for demonstrat- +ing quantum’s potential for obtaining solutions. To date, +the two most successful quantum approaches to solving +QUBOs are annealing [36–39] and QAOA [40–43], with +a lot of interest in comparing the two [44–46]. Shown be- +low in equation 1 is the QUBO cost function C(X) which +we shall seek to solve using our quantum algorithm. +C(X) = +N +� +i +Wixi + +� +{i,j}∈S +wijxixj +(1) +The function C(X) evaluates a given binary string X +of length N, composed of individual binary variables xi. +Together, the total number of unique solutions to each +QUBO is 2N, which is also the number of quantum states +producible from N qubits. Throughout this study we will +use subscripts Xi and C(Xi) when referring to individual +solutions, and C(X) when discussing a cost function more +generally. +As shown in equation 1, a QUBO is defined by two sep- +arate summations of weighted values. The first summa- +tion evaluates weights Wi associated with each individual +binary variable, while the second summation accounts for +pairs of variables which share a weighted connection wij. +In this study we adopt the typical interpretation of QU- +BOs as graph problems, whereby each binary variable xi +represents a node. We can then define the connectivity of +a QUBO graph using the set S, which itself is a collection +of sets that describe each pair of nodes xi and xj that +share a connection. See figure 1 below for an example. +The interest of this study is to use a quantum algo- +rithm to find either Xmin or Xmax, which are the solu- +tions which minimize / maximize the cost function C(X) +respectively. For all QUBOs analyzed in the coming sec- +tions, the weight values Wi and wij are restricted to in- +FIG. 1. +(top) An example 3-qubit linear QUBO with +weighted nodes and edges. +(bottom) The set S containing +the complete connectivity of the QUBO. +tegers, randomly selected from a uniform distribution as +shown below in equations 2 and 3. +Wi, wij ∈ Z +(2) +Wi, wij ∈ [−100, 100] +(3) +In section V. we discuss the consequences of choosing +weight values in this manner and its advantage for quan- +tum. However, nearly all of the results shown throughout +this study are applicable to the continuous cases for Wi +and wij as well, with the one exception being the results +of section V.D. +A. +Linear QUBO +The cost function given in equation 1 is applicable to +any graph structure S, so long as every node and edge is +assigned a weight. For this study we will focus on one +specific S, which we refer to as a ‘linear QUBO.’ The +connectivity of these graphs is as follows: +S = {{n, n + 1} | 1 ≤ n ≤ N − 1} +(4) +As the name suggests, linear QUBOs are graphs for +which every node has connectivity with exactly two +neighboring nodes, except for the first and final nodes. +The motivation for studying QUBOs of this nature is +their efficient realizability as quantum circuits, given in +the next section. +III. +AMPLITUDE AMPLIFICATION +The quantum strategy for finding optimal solutions to +C(X) investigated in this study is amplitude amplifica- +tion [4–9], which is the generalization of Grover’s algo- +rithm [1]. The full algorithm is shown below in Alg. 1, +which notably is almost identical to Grover’s algorithm +except for the replacement of Grover’s oracle UG with +our cost oracle Uc . +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +Wi +W +3 +2 +3 +W12 +W23 +S = {{1,2],[2,3]3 +Algorithm 1 Amplitude Amplification Algorithm +Initialize Qubits: |Ψ⟩ = |0⟩⊗N +Prepare Equal Superposition: H⊗N|Ψ⟩ = |s⟩ +for k ≈ π +4 +√ +2N +do +Apply Uc|Ψ⟩ (Cost Oracle) +Apply Us|Ψ⟩ (Diffusion) +end for +Measure +By interchanging different oracle operations into the +Alg. 1, various problem types can be solved using amplti- +tude amplification. For example, Grover’s original oracle +solves an unstructured search, whereas here we are in- +terested in optimal solutions to a cost function. Later in +section VII. we discuss further oracle adaptations and the +problems they solve. For all oracles, we use the standard +diffusion operator Us, given below in equation 5. +Us = 2|s⟩⟨s| − I +(5) +This operation achieves a reflection about the average +amplitude, whereby every basis state in |Ψ⟩ is reflected +around their collective mean in the complex plane. This +operation causes states’ distance from the origin to in- +crease or decrease based on their location relative to the +mean, which in turn determines their probability of mea- +surement. Therefore, a successful amplitude amplifica- +tion is able to drive the desired basis state(s) as far from +the origin as possible, up to a maximum distance of 1 +(measurement probability of 100%). +A. +Solution Space Distribution +A prerequisite for the success of amplitude amplifica- +tion as demonstrated in this study is an optimization +problem’s underlying solution space distribution; that is, +the manner in which all possible solutions to the prob- +lem are distributed with respect to one another. +For +QUBOs, these are the 2N possible C(Xi) cost function +values. Shown below in figure 2 is a histogram of one +such solution space distribution, for the case of a length +20 linear QUBO according to equations 1 - 4. The x-axis +represents all possible cost function evaluations, and the +y-axis is the corresponding number of unique Xi solutions +that result in the same C(Xi) value. +Depicted in figure 2 are all 220 possible solutions to +an example linear QUBO. Because this QUBO was gen- +erated from randomized weights, the combination of the +Law of Large Numbers [47] and Central Limit Theorem +[48] predicts that its underlying solution space should be +approximately gaussian [49] in shape, given by equation +6. +G(x) = αe +(x−µ)2 +2σ2 +(6) +FIG. 2. Example of a solution space distribution for a 20 node +linear QUBO, with weights according to equations 2 and 3. +Indeed, the histogram shown is approximately gaus- +sian, but importantly it has imperfections resulting from +the randomized weights. At large enough problem sizes +(around N ≥ 20), these imperfections have minimal im- +pact on a problem’s aptitude for amplitude amplifica- +tion, which was a result from our previous study [10]. +Similarly, another recent study [18] demonstrated that +in addition to symmetric gaussians, solution space distri- +butions for both skewed gaussians and exponential pro- +files also lead to successful amplitude amplifications. The +commonality between these three distribution shapes is +that they all possess large clusters of solutions that are +sufficiently distanced from the optimal solutions we seek +to boost. This can be seen in figure 2 as the location of +Xmin and Xmax as compared to the central peak of the +gaussian. When appropriately encoded as an oracle Uc, +these clusters serve to create a mean point in the complex +plane which the optimal solution(s) use to reflect about +and increase in probability. +B. +Cost Oracle Uc +In order to use algorithm 1 for finding the optimal +solution to a given cost function, we must construct a +cost oracle Uc which encodes the weighted information +and connectivity of the problem. In our previous study +we referred to this operation as a ‘phase oracle’ UP [10], +and similarly it has also been called a ‘subdivided phase +oracle’ SPO [17, 18] or ‘non-boolean oracle’ [19]. How +one constructs Uc is problem specific, but the general +strategy is to primarily use two quantum gates, shown +below in equations 7 and 8. +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +3000 +2500- +2000- +pop.(C(X)) 1500 +1000 - +500. +max +0. +-600 +-400 +-200 +0 +200 +400 +C(X)B +Cost Oracle Uc +4 +P(θ) = +�1 +1 +1 eiθ +� +(7) +CP(θ) = +� +�� +1 1 1 +1 +1 1 1 +1 +1 1 1 +1 +1 1 1 eiθ +� +�� +(8) +The single and two-qubit gates P(θ) and CP(θ) are re- +ferred to as phase gates, also known as Rz(θ) and CRz(θ) +for their effect of rotating a qubit’s state around the z- +axis of the Bloch sphere. Mathematically they are capa- +ble of applying complex phases as shown below. +P(θ)|1⟩ = eiθ|1⟩ +(9) +CP(θ)|11⟩ = eiθ|11⟩ +(10) +Applying P(θ) to a qubit only affects the |1⟩ state, +leaving |0⟩ unchanged, and similarly only |11⟩ for CP(θ). +However, this is exactly what we need in order to con- +struct C(X) from equation 1. When evaluating a partic- +ular binary string Xi classically, only instances where the +binary values xi are equal to 1 yield non-zero terms in +the summations. For quantum, each binary string Xi is +represented by one of the 2N basis states |Xi⟩. Thus, our +quantum cost oracle Uc can replicate C(X) by using P(θ) +and CP(θ) to only effect basis states with qubits in the +|1⟩ and |11⟩ states. +FIG. 3. +(top) Example of a 4-qubit linear QUBO with +weighted nodes and edges. (bottom) The same QUBO en- +coded into a cost oracle Uc without scaling. Each unitary in +the circuit is P(θ) (single qubit gate) or CP(θ) (2-qubit gate). +Shown above in figure 3 is an example of a 4-qubit +QUBO cost oracle, where the weighted values Wi and wij +are used as the θ parameters for the various phase gates. +Although incomplete, we will use this oracle circuit to +demonstrate quantum’s ability to encode a cost function +C(X). For example, consider the binary solution Xi = +1101 and the corresponding quantum basis state |1101⟩. +The classical evaluation of this solution is as follows: +C(1101) = −8 + 18 − 22 − 12 += −24 +(11) +Now let us compare this to the phase of |1101⟩ after +applying Uc: +Uc|1101⟩ = ei(−8+18−22−12)|1101⟩ += e−24i|1101⟩ +(12) +The phase acquired in equation 12 is equivalent to the +classical evaluation shown in 11, which means that Uc is +an accurate encoding of C(X). If we were to now apply Uc +to the equal superposition state |s⟩ (step 2 in Alg. 1), all +2N basis states would receive phases equal to their cost +function value. This is the advantage that quantum has +to offer: simultaneously evaluating all possible solutions +of a cost function through superposition. +C. +Scaling Parameter ps +While the cost oracle shown in figure 3 is capable of re- +producing C(X), its use in algorithm 1 will not yield the +optimal solution Xmin or Xmax. This is because quantum +phases are 2π modulo, which is problematic if the numer- +ical scale of C(X) exceeds a range of 2π. Consequently if +two quantum states receive phases that differ by a mul- +tiple of 2π, then they will both undergo the amplitude +amplification process identically. If this happens unin- +tentionally via Uc, then our cost oracle cannot be used +to minimize or maximize C(X). +In order to construct Uc such that it is usable for am- +plitude amplification, a scalar parameter ps must be in- +cluded in all of the phase gates. The value of ps is prob- +lem specific, but its role is always the same: scaling the +cumulative phases applied by Uc down (or up) to a range +where [C(Xmin) , C(Xmax)] is approximately [x , x+2π]. +This range does not have to be [0 , 2π] exactly, so long +as the phases acquired by |Xmin⟩ and |Xmax⟩ are roughly +2π different. See figure 4 below for an example of ps in +Uc’s construction. +FIG. 4. +The 4-qubit linear QUBO cost oracle Uc from figure +3, now scaled by ps. +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +Wi: +-8 +18 +26 +-22 +(1) +2 +3 +4 +Wii: +-12 +33 +6 +Q +8- +Q +18 +-12 +Uc: +2 +Q +26 +33 +3 +Q +-22 +6 +4Q +-8·Ps +Q +18·Ps +-12·Ps +2 +Q +26·Ps +33·Ps +3 +Q +-22·Ps +6·Ps +4C +Scaling Parameter ps +5 +Using the scaled oracle shown in figure 4 above, let us +now show how this new Uc acts on the basis state |1101⟩ +from before: +Uc|1101⟩ = ei(−8·ps+18·ps−22·ps−12·ps)|1101⟩ += ei(−8+18−22−12)·ps|1101⟩ += e−24i·ps|1101⟩ +(13) +As shown in equation 13 above, multiplying ps into ev- +ery phase gate has the net effect of scaling the cumulative +phase applied by Uc: e−24i → e−24i·ps. Note that this is +not a global phase, which would have an additive effect +on all states rather than a multiplicative one like shown +above. +Finding the optimal ps value for boosting Xmin or Xmax +is non-trivial, and was a major focus of our previous +study [10], as well as this one. In general, the scale of +ps needed for finding the optimal solution can be ob- +tained using equation 14 below, which scales the numer- +ical range of a problem [C(Xmin) , C(Xmax)] to exactly +[x , x+2π]. +ps = +2π +C(Xmax) − C(Xmin) +(14) +Although equation 14 above is guaranteed to solve the +2π modulo phase problem mentioned previously, it is al- +most never the ps value which can be used to find Xmin or +Xmax. Only in the case of a perfectly symmetric solution +space distribution is equation 14 the optimal ps value, +in which case the states |Xmin⟩ and |Xmax⟩ undergo the +amplitude amplification process together. However, re- +alistic optimization problems can be assumed to have a +certain degree of randomness or asymmetry to their so- +lution space, producing distributions more akin to figure +7. For this reason, equation 14 is better thought of as +the starting point for finding the true optimal ps, which +we discuss later in section IV.B. For now, equation 14 is +sufficient for demonstrating ps’s role in creating an av- +erage amplitude suitable for boosting |Xmin⟩ or |Xmax⟩, +shown in figure 5. +The bottom plot in figure 5 shows |Ψ⟩ after the first +application of Uc in algorithm 1. Note the location of the +average amplitude (red ‘x’), which is only made possible +by the majority of quantum states which recieve phases +near the center of the gaussian in the top plot. Optimal +amplitude amplification occurs when the desired state +for boosting is exactly π phase different with the mean +[2, 3], which is very close to the situation seen in figure +5. However, since this Uc is derived from a QUBO with +randomized weights, the ps value provided from equa- +tion 14 does not exactly produce a π phase difference +between the optimal states (black star) and the mean +amplitude (red ‘x’). +Consequently, the state(s) which +does become highly probable from amplitude amplifica- +tion for this particular ps is not |Xmin⟩ and |Xmax⟩, which +will be the subject of the coming two sections. +FIG. 5. +(top) The 20-qubit linear QUBO histogram from +figure 2, scaled by ps according to equation 14. (bottom) All +220 quantum states after applying Uc|s⟩, plotted in amplitude +space (the complex plane). The red-blue color scale shows the +density of quantum states in the bottom plot, corresponding +to the y-axis of the top histogram. +The states |Xmin⟩ and +|Xmax⟩ are marked with a black star, the origin a black ‘+’, +and average amplitude with a red ‘x’. +IV. +GAUSSIAN AMPLITUDE AMPLIFICATION +The amplitude space plot depicted at the bottom of +figure 5 is useful for visualizing how a gaussian solution +space distribution can be used for boosting, but the full +amplitude amplification process is far more complicated. +This is especially true for the QUBOs of this study, which +are generated with randomized weights. Consequently, +all of the results which follow throughout the remainder +of this study are produced from classical simulations of +amplitude amplification using cost oracles derived from +linear QUBOs according to equations 1 - 4. For a deeper +mathematical insight into these processes, please see [16– +18]. +A. +Achievable Probabilities +Amplitude amplifiation is an appealing quantum al- +gorithm because it solves one of the most fundamental +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +3000 +2500 +2000 +pop.(C(X) 1500 +1000· +500 +0 +¥-2 +3 +0 +2 +C(X)·Ps +Ucls)A +Achievable Probabilities +6 +problems of quantum computing: measurement proba- +bility. For example, a single marked state using Grover’s +oracle with 30 qubits is capable of achieving a final prob- +ability that is only less than 100% by one billionth of a +percent [1]. Thus, a natural question to ask when using +Uc is what kinds of probabilities can it produce for |Xmin⟩ +or |Xmax⟩? To answer this we conducted a statistically +study of linear QUBOs ranging from length N = 17 to 27. +For each N we generated numerous QUBOs according to +equations 1 - 4, totals given in appendix A. We then let a +classical simulator find the ps value which maximized the +probability of measuring |Xmin⟩ for each QUBO (and for +certain cases the optimal ps for |Xmax⟩ aswell). Results +for each problem size are shown below in figure 6. +FIG. 6. +Results from studying randomly generated linear +QUBOs of various sizes N. +The number of QUBOs stud- +ied per N is provided in appendix A. For each QUBO, the +optimal ps value for producing the highest probability of mea- +surement for |Xmin⟩ was used to record three trends: average +probability of |Xmin⟩ (black triangle), highest recorded proba- +bility (red star), and average scaled standard deviation (blue +circle). Error bars showing one standard deviation of each σ’ +are provided aswell. +µ = 1 +2N +2N +� +i +C(Xi) +(15) +σ = +��2N +i +(C(Xi) − µ)2 +2N +(16) +σ′ = σ · ps +(17) +Figure 6 tracks three noteworthy trends found across +the various QUBO sizes: +the average peak probabil- +ity achievable for |Xmin⟩ (black triangle), the highest +recorded probability for |Xmin⟩ (red star), and the aver- +age scaled standard deviation σ′ (blue circle). For clarity, +the derivation of σ′ is given by equations 15 - 17. This +quantity is the standard deviation of a QUBO’s solution +space distribution after being scaled by ps, making it a +comparable metric for all QUBO sizes. In our previous +study we demonstrated a result in agreement with fig- +ure 6, which is the correlation between higher achievable +probabilities for |Xmin⟩ (red star) and smaller scaled stan- +dard deviations σ′ (blue circle) [10]. The latter is what +is responsible for increasing the distance between |Xmin⟩ +and the average amplitude like shown in figure 5. +B. +Solution Space Skewness +The relation between N, σ′, and highest prob.(|Xmin⟩) +from figure 6 can be summarized as follows: larger prob- +lem sizes tend to produce smaller standard deviations, +which in turn lead to better probabilities produced from +amplitude amplification. However, there is a very appar- +ent disconnect between the probabilities capable of each +problem size (red stars) versus the average (black trian- +gle). To explain this, we must first introduce the quantity +X∆ given in equation 18 below. +X∆ = 2µ − (C(Xmax) + C(Xmin)) +(18) +The quantity X∆ from equation 18 is the difference +between C(Xmin) and µ (the mean) minus the difference +between µ and C(Xmax). A positive value for X∆ indi- +cates that the mean is closer to C(Xmax), and vice versa +for a negative valued X∆. In essence, it is a measure of +skewness that describes the assymetry of a solution space +distribution. Figure 7 shows example QUBO distribu- +tions for three cases of X∆, for N = 25, demonstrating +the impact X∆ has on the ability to boost |Xmin⟩ versus +|Xmax⟩. While σ′ is a strong indicator of a problem’s over- +all aptitude for amplitude amplification, X∆ determines +whether the optimal minimum or maximum solution is +boostable, and which is not. Further evidence of this can +be seen in figure 8, which shows 1000 randomly generated +linear QUBOs of length N = 23, and the peak probabil- +ities achievable for |Xmin⟩ and |Xmax⟩ as a function of +X∆. +If we compare the average peak probabilites for |Xmin⟩ +from figure 6 with the full data of QUBOs shown in fig- +ure 8, we can see why the average peak probability is +significantly lower than the highest recorded. +Across +the 1000 QUBOs studied, it is clear that X∆ = 0 is +a dividing point for whether |Xmin⟩ or |Xmax⟩ is capa- +ble of reaching a significant probability of measurement +through amplitude amplification. For N = 23, the aver- +age prob.(|Xmin⟩) reported in figure 6 is approximately +64%. However, if instead we only consider QUBOs with +X∆ > 0 from figure 8, then the average peak probability +for |Xmin⟩ is around 86%, and likewise for |Xmax⟩ when +X∆ < 0. +Together, figures 7 and 8 demonstrate the significance +of knowing X∆ from an experimenter’s perspective. De- +pending on the optimization problem of interest, it is +reasonable to assume that an experimenter may be in- +terested in finding only Xmin or Xmax. But without any +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +1.0 +0.9 +0.8: +0.7. +0.6. +0.5. +Avg. Prob(IXmin >) +0.4: +AV +★★ +Best Prob(Xmin +0.3 +17 +19 +21 +23 +25 +27 +NB +Solution Space Skewness +7 +FIG. 7. +Three randomly generated QUBO distributions +for N = 25, illustrating X∆ cases for largely positive (top), +largely negative (middle), and near zero (bottom). In all three +plots the exact X∆ value is reported, as well as the highest +achievable probability for |Xmin⟩ and |Xmax⟩ (each from a dif- +ferent ps value). Also shown in each plot are the values for +C(Xmin) and C(Xmax), and their numerical distance to the +mean µ (red-dashed line). +a priori knowledge of a problem’s underlying solution +space, specifically X∆, the experimenter may unknow- +ingly be searching for a solution which is probabilistically +near impossible to find through amplitude amplification. +For example, consider the QUBO distribution illustrated +in the top plot of figure 7, and the peak probability for +boosting |Xmax⟩: 0.16%. +Although it is ideal to have +insight into a particular problem’s X∆ before using am- +plitude amplification, as we demonstrate in section V., +information about X∆ can be inferred through measure- +ment results. +FIG. 8. +A total of 1000 randomly generated linear QU- +BOs of size N = 23. For each QUBO, the highest achievable +probability for |Xmin⟩ (black circle) and |Xmax⟩ (red triangle) +are plotted as a function of X∆. The top plot includes both +data points per QUBO, while the bottom plot only shows the +higher of the two values. +C. +Sampling for ps +If a particular optimization problem is suitable for am- +plitude amplification, then the speed of the quantum +algorithm outlined in this study is determined by how +quickly the optimal ps value can be found. Here we shall +show that sampling a cost function C(X) can provide reli- +able information for approximating ps from equation 14, +which can then be used to begin the variational approach +outlined in sections V. and VI. Importantly, the number +of cost function evaluations needed is significantly less +than either a classical or quantum solving speed. The +strategy outlined in equations 19 - 29 below can be used +for approximating ps when the experimenter is expecting +an underlying solution space describable by a gaussian +function (equation 6). If another type of distribution is +expected, then the function used in equation 22 could +in principle be modified accordingly (for example, sinu- +soidal, polynomial, exponential [18]). +Suppose we sample a particular cost function C(X) M +times, where M << 2N. We will define the set M as the +collection of values C(Xi) obtained from these samples. +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +55000-Prob.(IXmin >): 88.2% +Prob.(IXmax >): 0.16% +K:331.5 +1180.75 +849.25 +-516 +925 +0 +60000 +Prob.(IXmin>): 0.01% +Prob.(IXmax ): 88.1% +X: -330.0 +721 +1054 +-589 +1186 +0 +Prob.(IXmin )): 45.7 % +Prob.(IXmax ): 46.1 % +80000-1 +:0.5 +pop.(C(X) +720.75 +720.25 +-1591 +439 +-0 +C(X)1.0 +Both +0.8 +0.6 +0.4 +0.2 +0 +1.0 +Highest +0.8 +Probability +0.6 +[Xmin +[Xmax, +0.4 +0.2 +0 +-400 +-200 +0 +200 +400 +XAC +Sampling for ps +8 +M = {C(X1), C(X2), ..., C(XM)} +(19) +Using these M values, we can compute an approximate +mean and standard deviation. +˜µ = 1 +M +� +c∈M +c +(20) +˜σ = +�� +c∈M (c − ˜µ)2 +M +(21) +In order to use equation 14 for obtaining ps, we need +approximations for C(Xmin) and C(Xmax). If we assume +an underlying gaussian structure to the problem’s solu- +tion space, then we can write down the following equation +to describe it: +2N = +� ∞ +−∞ +˜αe +(x−˜µ)2 +2˜σ2 dx +(22) += −˜α ˜σ +�π +2 erf +� ˜µ − x +√ +2˜σ +�∞ +−∞ +(23) += −˜α ˜σ +�π +2 · [−1 − 1] +(24) +where erf() is the gaussian error function. Using equa- +tion 24, we can rearrange terms and solve for an approx- +imation to the height of the gaussian. +˜α = 2N−1 +˜σ� π +2 +(25) +With the values ˜µ, ˜σ, and ˜α obtained from sampling, +we can now approximate C(Xmin) and C(Xmax) using +equation 26 below. +˜G(x) = ˜αe +(x−˜µ)2 +2˜σ2 += 1 +(26) +Solving for x yields the following two values: +x± = ˜µ ± ˜σ +� +−2ln +� 1 +˜α +� +(27) +which can be expressed in terms of the two quantities +originally derived from sampling: +x± = ˜µ ± ˜σ +� +� +� +�−2ln +� +˜σ +� +π/2 +2N−1 +� +(28) +And finally, the solutions x± can be used to obtain ps. +˜ps = +2π +x+ − x− +(29) +The reason we set equation 26 equal to 1, and the +integral in equation 22 equal to 2N, is because ˜G(x) is +modeling the histogram of a QUBO’s solution space, like +shown in figure 2. This means that the total number of +solutions to C(X) is 2N, and similarly the minimum num- +ber of distinct C(Xi) solutions for a given cost function +is 1. Therefore, after setting the integral in equation 22 +equal to 2N, solving ˜G(x)= 1 yields approximations for +C(Xmin) and C(Xmax) on the tails of the gaussian. +To demonstrate how well sampling is able to approxi- +mate equation 14, we tested the strategy outlined above +against the 1000 QUBOs from figure 8 (N = 23). For +four values of M: 100, 500, 1000, and 2000, each QUBO +was used for 50 trials of random sampling to produce ap- +proximate ˜ps values. These values were then compared to +the true value of ps from equation 14, as given by equa- +tion 30 below, and finally averaged together to produce +table I. +˜ps Error = |˜ps − ps| +ps +(30) +M +100 +500 +1000 +2000 +Average ˜ps Error +7.28% +6.37% +6.31% +6.29% +TABLE I. Average error in approximating ps using equations +19 - 29. Each value comes from 50,000 independent sampling +trials on linear QUBOs of size N = 23. +The significant result from table I is that sampling 100 +- 500 times, on a cost function of 223 solutions, is accu- +rate enough to produce an approximate ˜ps value with an +expected error of only 7%. And as we show in the next +section, this is enough accuracy to use for either a heuris- +tic or variational approach for finding optimal solutions. +V. +VARIATIONAL AMPLITUDE +AMPLIFICATION +The results of sections II - IV. demonstrate quantum’s +aptitude for encoding and solving a QUBO problem using +amplitude amplification. In this section we discuss how +this potential can be realized from an experimental per- +spective. In particular, we focus on amplitude amplifica- +tion’s ability to find optimal solutions under realistic cir- +cumstances with limited information. The results of this +section are then used to motivate section VI., in which +we discuss how amplitude amplification can be used in a +hybrid classical-quantum model of computing, similar to +other successful variational approaches [40, 41, 50]. +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +9 +A. +Boosting Near-Optimal Solutions +The results shown in figures 6 - 8 focus on quantum’s +potential for finding |Xmin⟩ and |Xmax⟩, the optimal so- +lutions which minimize/maximize a given cost function +C(X). However, in order to understand how amplitude +amplification can be used in a variational model, it is +equally as important that non-optimal |Xi⟩ states are +also capable of boosting. +As discussed in the conclusion of our previous study +[10], as well as sections III.C and IV.C, the most difficult +aspect of using algorithm Alg.1 from an experimental +standpoint is finding ps. +More specifically, finding an +optimal ps for boosting |Xmin⟩ or |Xmax⟩ is a challenge +due to the limited amount of information that one can +learn through measurements alone. An example of this +can be seen in figure 9, which shows the peak achievable +probabilities of the three lowest |Xi⟩ states as a function +of ps (|Xmin⟩ and the next two minimum solutions), for +the QUBO corresponding to X∆ = 331.5 from figure 7. +FIG. 9. +Plots of |Xi⟩ state probability from amplitude ampli- +fication as a function of ps, for |Xmin⟩ (blue-solid) and the next +two minimal solutions (black and red-dashed). Cost function +values C(Xi) are reported next to each state’s plot, corre- +sponding to the QUBO from the top plot in figure 7. The +bottom plot is a zoomed in scale of the top plot, depicting +the same data points. +The challenge presented in figure 9 is the narrow +range of ps values for which each |Xi⟩ state is able to +achieve meaningful probabilities of measurement. From +an experimental perspective, the ps regions outside these +bands are only capable of producing quantum superposi- +tion states which are slightly better than |s⟩, the equal su- +perposition starting state. Thus, an experimenter could +use a ps value that is incredibly close to optimal, but +only see seemingly random measurement results through +repeat implementations of Alg.1. +Our proposed solution to the ps problem as described +above is twofold: 1) We must widen our view of useful +ps values and see where other |Xi⟩ states become highly +probable, and 2) put less burden on quantum to find +optimal solutions alone when an assisting classical ap- +proach may be more suitable. In this section we focus on +addressing (1), which will then motive (2) in section VI. +Suppose we aren’t solely interested in using quantum +to find the exact optimal solution C(Xmin), but instead +are content with any Xi within the best 50 answers (50 +lowest C(X) values). In order for amplitude amplification +to be viable in this heuristic context, it requires signifi- +cant probabilities of measurement for these non-optimal +solution states, similar to figure 9. Additionally, an ex- +perimenter must be able to quickly and reliably find the +ps values which produce them. Shown below in figure +10 is a plot which provides insight into the feasibility of +both of these questions, for the QUBO corresponding to +figure 9. +Figure 10 shows the full ps range for which an exper- +imenter could find the 50 best solutions for minimizing +C(X) via quantum measurements. The black circles in- +dicate on the x-axis the ps values where each |Xi⟩ state +(or multiple states) is maximally probable, aligning with +its corresponding C(Xi) value along the y-axis. Numeric +values for peak probabilities of the best 20 solutions are +provided in the table below the plot, as well as a lin- +ear regression best fit (red-dotted line) for the overall 50 +data points. The reported R correlation value is given by +equation B5 in appendix B. +There are several significant results displayed in figure +10, the first of which requires returning to equation 2. By +limiting the allowed weighted values for Wi and wij to +integers, all solutions to C(X) are consequently integers +as well. This means that the linear correlation shown in +the figure can in principle be used to predict ps values +where integer C(Xi) solutions must exist. If Wi and wij +are instead allowed to take on float values, which is more +general of realistic optimization problems, the linearity +of solutions like shown still persists but cannot be used +for predictions of allowed C(X) values as reliably. +The linear best fit shown in figure 10 is accurate for +the top 50 solutions, but extending the ps scale further +reveals that it is only an approximation applicable to a +small percentage of states. +This is shown in figure 11 +below, which once again is a ps vs. C(X) plot for the +same QUBO, but now for the best 400 minimizing so- +lutions. It is clear from the data in this figure that the +top 400 solutions are in no way linearly aligned, which +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +0.8 +-1553→ +0.6 +0.4- +0.2 +0 +0.00264 +0.00268 +0.00272 +0.00276 +0.81 +C(X): -1591- +-1590 +Probability +0.6 +0.4- +0.2. +0.00266 +0.002665 +0.00267 +0.002675 +PsA +Boosting Near-Optimal Solutions +10 +FIG. 10. +(top) A plot of the 50 lowest C(Xi) values as a +function of ps, for the X∆ = 331.5 QUBO from figure 7. Each +data point represents the ps value where the |Xi⟩ state(s) +is most probable. +A linear regression best-fit is shown by +the red-dotted line, with its R correlation value reported at +the top (equation B5 from appendix B). (bottom) A table +of values for the 20 best solutions. Each entry reports: the +cost function value C(Xi), the peak probability for the |Xi⟩ +state(s), and the number of unique Xi solutions that result in +the same C(Xi) value. +is a more expected result given the complicated nature +of these imperfect gaussian distributions undergoing am- +plitude amplification. However, although the data is not +linear, there is very clearly a curved structure that could +be utilized to predict ps values in the same manner de- +scribed above. +It is important to note that in both figures 10 and 11, +the manner in which the solution states |Xi⟩ are found +to be most probable is sequential. This means that if a +particular state |Xi⟩ is most probable at a certain value +ps = x, all solutions C(Xj) < C(Xi) will have peak prob- +abilities at values ps < x. However, the bottom plot in +figure 11 shows that the further a solution state is from +|Xmin⟩ the lower its achievable peak probability. +This +means that there is a limit to how many solutions are +viable for amplitude amplification to find. As we discuss +in the coming subsections, these are the key underlying +features that we must consider when constructing a vari- +ational amplitude amplification algorithm. +FIG. 11. +(top) A plot of the 400 lowest C(Xi) values as a +function of ps, for the X∆ = 331.5 QUBO from figure 7. Each +data point represents the ps value where the |Xi⟩ state(s) is +most probable. The red box in the lower left corner represents +the ps region depicted in figure 10. (bottom) The probabilities +achieved for these 400 lowest |Xi⟩ states using the ps values +shown in the top plot. Each state is plotted in order of it’s +rank from 1 (Xmin) to 400 (400th lowest C(Xi) solution). +B. +Constant Iterations +In order to construct an algorithm which capitalizes +on the structure and probabilities shown in figure 11, we +must consider an additional piece of information not il- +lustrated in the figure: step 3 of Alg. 1, iterations k. The +data points in the figure are indeed the ps values which +produce the highest probabilities of measurements, but +unfortunately they are achieved using different iteration +counts. +In principle this means that an experimenter +must decide both ps and k before each amplitude am- +plification attempt, further complicating the information +learned from measurement results. +Unlike ps, which is difficult to learn because it depends +on the collective 2N solutions to C(X), approximating a +good iteration count k is easier. It turns out that the +standard number of Grover iterations kG = +π +4 +� +N/M, +where N is the total number of quantum states and M is +the number of marked states, is equally applicable when +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +-1460 +R: 0.9994 +-1480 +-1500 : +C(X) -1520 +-1540: +-1560: +-1580: +0.0027 +0.0028 +0.0029 +0.003 +Ps +C(X) +Probability +# of States +-1528 +82.9% +2 +1 +-1591 +88.2% +1 +-1527 +83.1% +1 +-1590 +88.2% +1 +-1526 +83.7% +2 +-1553 +86.3% +1 +-1525 +83.8% +1 +-1552 +86.5% +2 +-1524 +84.1% +1 +-1550 +85.6% +1 +-1523 +85.5% +2 +-1549 +85.3% +2 +-1517 +85.4% +1 +-1548 +86.0% +1 +-1516 +85.6% +2 +-1529 +84.3% +2 +-1514 +86.4% +1-1100 - +-1200 : +-1300 - +C(X) +-1400 +-1500 +-1600 +0.0027 +0.0033 +0.0039 +0.0045 +Ps +1.0- +0.8- +Probability +0.6. +0.4. +0.2 +上0 +1 +100 +200 +300 +400 +Lowest C(Xi) Solution StatesB +Constant Iterations +11 +FIG. 12. +Plots of |Xi⟩ state probabilities as a function of ps, for the N = 25 QUBO shown in figures 10 and 11. The top +three panels show individual state probabilities as solid-colored lines, for three different constant k iterations (1000, 2000, and +3000) across the ps region depicted on the x-axis. An additional black-dashed line is also shown, which records the cumulative +probability of the five most probable solutions |Xi⟩ at any given ps value. These cumulative probabilities are also replotted in +the bottom most panel for comparison. +using Uc as well. +If an experimentor can use k ≈ kG +iterations for a cost oracle Uc and find significant proba- +bilities of measurment, then a variational amplitude am- +plification strategy can be reduced to a single parameter +problem: ps. Figure 12 demonstrates that this is indeed +viable, showcasing |Xi⟩ solution state probabilities as a +function of ps for three different choices of k. +The QUBO corresponding to figure 12 is the same +N = 25 example for figures 10 and 11. +For instances +where multiple states correspond to the same numeri- +cal solution (C(Xi) = C(Xj)), the solid-color line shown +represents their joint probability: Prob.( |Xi⟩ ) + Prob.( +|Xj⟩ ) (note that these individual probabilities are always +equal). Examples of this can also be seen in the table in- +cluded in figure 10. Additionally, a black-dashed line is +shown in the top three plots, tracking the collective prob- +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +3000 +0.8 +0.6 +0.4- +0.2 +2000 +0.8 +0.6 +0.4 +0.2 +1000 +0.8: +0.6. +0.4 +0.2 - +0 +3000 +0.8 +2000 +Probability +1000 +0.6 +0.4 +0.2 +0.0029 +0.002902 +0.002904 +0.002906 +0.002908 +0.003 +0.00302 +0.00304 +0.00306 +0.00308 +0.0031 +pC +Information Through Measurements +12 +ability of the five most probable solutions at any given +ps. These three lines are then replotted in the bottom +panel for comparison. +The ps region shown in figure 12 was chosen to il- +lustrate a scenario where variational amplitude ampli- +fication is most viable. +For ps > 0.00291, nearly ev- +ery possible integer solution C(Xi) ≥ −1497 exists via +some binary combination for this particular QUBO prob- +lem. The exceptions where certain integer solutions do +not exist can be seen clearly in the ps regions with very +low probability, for example 0.0029065 ≤ ps ≤ 0.002907. +Contrast to the region shown in this figure, once ps be- +comes closer to where |Xmin⟩ is maximally probable, then +measurment probabilities become more akin to figure 9. +Thus, it is more strategic for a hybrid algorithm to start +in a ps region like figure 12, where measurement results +can consistently yield useful information. +C. +Information Through Measurements +From an experimental perspective, a significant result +from figure 12 are the black-dashed lines shown in the +top three plots. At k = 3000 (kG ≈ 4500 for 25 qubits, +M = 1), the black-dashed line is almost entirely com- +posed of the single most probable solution state(s). With +probabilities around 70 − 80% for many of the states +shown, it is realistic that the same |Xi⟩ state could be +measured twice in only 2 − 4 amplitude amplification +attempts. Two measurements yielding the same C(Xi) +value (possibly from different Xi) is a strong experimen- +tal indicator that the ps value used is very close to op- +timal for that solution, corresponding to the data points +from figures 10 and 11. Confirming 3 − 4 different data +points in this manner can then be used to approximate +the underlying curved structure of these figures, which +in turn could be used to predict ps values where |Xmin⟩ +may exist. +While using k closer to kG is good for getting the max- +imal probability out of solution states, the k = 1000 +and 2000 plots in figure 12 support a different strat- +egy for quantum. At k = 2000, the black-dashed line +is still primarily composed of the single most probable +|Xi⟩ state(s), but critically it does not have the same +dips in probability between neighboring solutions. +In- +stead, the cumulative probability stays just as high for +these in-between ps regions, sometimes even higher! If we +now look at the k = 1000 plot, this trend becomes even +more prevelant, whereby the cumulative probability plot +is on average 20 − 30% higher than any individual |Xi⟩ +state. Interestingly, the bottom panel of figure 12 shows +that cumulative probability plot for k = 1000 is higher +than the k = 3000 line in many regions. Thus, if the +role of quantum is to simply provide a heuristic answer +[51], not necessarily |Xmin⟩, then using lower k values is +favorable for a few reasons. Firstly, we can anticipate +solutions in a ps region where multiple states share the +same cost function value, so one can expect M > 1 more +frequently when using kG = π +4 +� +N/M. Secondly, the am- +plitude amplification process itself is faster due to smaller +k, which makes it more achievable on noisy qubits due to +shallower circuit depths. +The optimal use of k is a non-trivial challenge to an +experimentor. However, as illustrated in figure 12, ampli- +tude amplification can still be effective with a wide range +of different k values. To further demonstrate this, figure +13 shows three plots of simulated measurements over the +ps range depicted in figure 12. Using the k values 1000, +2000, and 3000, each plot shows data points representing +probabilistic measurements at regular intervals of ps. In +order to compare the k value’s effectiveness more equally, +the number of measurements taken per ps value, t, was +chosen such that t·k = 12000 is consistent across all three +experiments. Thus, each of the three plots in figure 13 +represents the same total number of amplitude amplifi- +cation iterations divided among t experimental runs. +The data points shown in figure 13 are separated into +two categories, which are easily recognizable from an +experimental perspective. Measurements which yielded +C(Xi) < −1350 are plotted as red circles, while all other +measurements are plotted as black triangles. +As illus- +trated for all three values of k, the red data points can be +seen as producing near linear slopes, all of which would +signal to the experimenter that these measurement re- +sults are leading to Xmin. The motivation for figure 13 +is to demonstrate that the same underlying information +can be experimentally realized using different k values. +Thus, when to use k = 3000 versus k = 1000 is a matter +of optimization, which we discuss in section VI. as the +role of a classical optimizer for a hybrid model. +D. +Quantum Verification +The results of the previous subsections demonstrate +the capacity for amplitude amplification as a means for +finding a range of optimal Xi solutions. However, regard- +less of whether these solutions are found via quantum or +classical, a separate problem lies in verifying if a given +solution is truly the global minimum Xi = Xmin. If it is +not, then Xi is refered to as a local minimum. Classically, +evolutionary (or genetic) algorithms [52–55] are one ex- +ample strategy for overcoming local minima. Similarly, +quantum algorithms have also demonstrated success in +this area for both annealing [56, 57] and gate-based [58– +60]. +The strategy for verifying a local versus global mini- +mum using amplitude amplification can be seen by com- +paring the region 0.0029 ≤ ps ≤ 0.00291 in figures 12 and +13. For the linear QUBO corresponding to these figures, +there exists a solution C(Xi) = −1497 which becomes +maximally probable at ps ≈ 0.002914, followed by the +next lowest solution C(Xi) = −1491 at ps ≈ 0.002892. +Because there are no binary combinations Xi that can +produce values −1492 ≥ C(Xi) ≥ −1496, the ps region +that would correspond to their solutions instead produces +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +D +Quantum Verification +13 +FIG. 13. +Simulated measurement results corresponding to +the probabilities shown in figure 12, produced by amplitude +amplification for various values of ps (x-axis) and k (1000, +2000, and 3000). +At each of the ps values simulated, the +number of measurements per experiment t was chosen based +on k as follows (t,k): (4,3000) , (6,2000) , (12,1000), such that +t · k = 12000. +Measurement results which yielded C(Xi)< +−1350 are plotted as red circles, otherwise as black triangles. +Blue lines for C(Xmin) and C(Xmax) are plotted as well. +nothing measurably significant. This can be seen by the +low cumulative probabilities in figure 12, as well as ex- +perimentally in figure 13 as a gap in red data points for +this ps region across all three simulations. +The ability for quantum to determine if an Xi solution +is locally or globally minimum is achieved by searching +past the ps value corresponding to the solution. Doing so +will result in one of two outcomes: either a lower C(Xj) +value will be probabilistically found (confirming Xi was +a local minimum), or the experimenter will only find ran- +dom measurement results (Xi was the global minimum). +Examples of this can be seen in figure 14, showcasing sim- +ulated measurement results as an experimenter searches +past the optimal ps value for |Xmin⟩. +The simulated experiments shown in figure 14 were +chosen to highlight both favorable (bottom) and unfavor- +FIG. 14. +Simulated measurement results for ps regions above +and below the optimal point for finding |Xmin⟩. Each plot cor- +responds to a different linear QUBO of size N = 25, k = 4000, +with X∆ values reported for each (top plot corresponds to the +QUBO from figures 9 - 13). The point where Xmin is mea- +sured is indicated in both plots by the intersection of the blue +(horizontal) and grey (vertical) lines. Red-circle data points +represent measurement results within the best 30 minimizing +solutions to C(X), otherwise as black triangles. +able (top) cases for quantum. The commonality between +both experiments is that there is a clear point in ps (grey +line) in which decreasing ps further results in only noisy +random measurements. However, determining this cut- +off point using measurement results alone is challenging. +The top plot corresponds to the QUBO from figures 10 +- 12, which is the non-ideal situation in which there are +significant gaps in solutions between the best 20 minimiz- +ing C(Xi). Experimentally this manifests as numerous +ps regions that could be wrongly interpreted as the Xmin +cutoff point. Conversely, the bottom plot represents the +ideal case where the best minimizing C(Xi) solutions are +all closely clustered together. This leads to a much more +consistent correlation of measurement results leading to +Xmin, followed by an evident switch to randomness. +The significance here is that amplitude amplification +has an experimentally verifiable means for identifying the +global minimum Xmin of a cost function. Similarly, the +same methodology can be in principle used to check for +the existence of an Xi solution corresponding to any given +cost function value, which we discuss further in section +VII.C. However, the obvious drawback is that this verifi- +cation technique relies on numerous amplitude amplifica- +tion measurements finding nothing, which costs further +runtime as well as being probabilistic. As we discuss in +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +3000 +250. +0 +-250· +-500 +-750 +-1000 +-1250 +-1500 +2000 +250 +0 +-250 +-500 +-750 +-1000 - +-1250 +-1500 +1000 +250 +0 +-250 +500 +C(X) +-750 +-1000· +-1250 +-1500 +0.0029 +0.00295 +0.003 +0.00305 +0.0031Xa= 331.5 +-500 +-1000 +-1500 +X +min +0.0025 +0.0026 +0.0027 +0.0028 +0.0029 +Xa= 212.5 +400 +C(X) +-800 +-1200 +0.003 +0.0031 +0.0032 +0.0033 +0.0034 +psD +Quantum Verification +14 +the next section, a more realistic application of this quan- +tum feature is to help steer a classical algorithm past lo- +cal minima, leaving the veification of Xmin as joint effort +between both quantum and classical. +VI. +HYBRID SOLVING +The results of section V. were all features of amplitude +amplification using Uc that were found through classical +simulations of quantum systems. They represent the pri- +mary motivation of this study, which is to demonstrate +amplitude amplifaction’s potential and the conditions for +which it can be experimentally realized. +By contrast, +the discussions of section VI. here are more speculative. +Given all of the results from sections III. - V., we now +discuss how the strengths and weaknesses of amplitude +amplification synergize with a parallel classical computer. +The plots shown in figures 13 and 14 represent a very +non-optimal approach to finding Xmin, functionally a +quantum version of an exhaustive search. If the ultimate +goal is to solve a cost function problem as quickly as pos- +sible, then it is in our best interest to use any and all +tools available. This means using a quantum computer +when it is advantageous, and similarly also recognizing +when the use of a classical computer is more appropri- +ate. +In this section we discuss this interplay between +quantum and classical, and the situations in which an +experimenter may favor one or the other. Shown below +in figure 15 is the general outline of a variational ampli- +tude amplification model which relies solely on quantum +to produce Xmin. +FIG. 15. +The general outline of a variational amplitude am- +plification workflow. Information from amplitude amplifica- +tion in the form of measurements is fed to a classical optimizer +between runs. The optimizer then processes this information +to supply the quantum computer with the next set of values +ps and k, repeating this process until Xmin or another suitable +solution is found. +Given the current state of qubit technologies [61–63], +performing one complete amplitude amplification circuit +should be considered a scarce resource. +As such, it is +the role of a classical optimizer to determine the most +effective use of this resource, choosing ps and k values +which will probabilistically get the most value out of each +attempt. Determining optimal values to adjust a quan- +tum circuit is the typical hybrid strategy found among +other popular variational models of quantum computing +[40, 41, 50]. The majority of the computational workload +is placed on the QPU (quantum processing unit), while +a classical optimizer is used in between runs to adjust +quantum circuit parameters accordingly. As evidenced +by figures 13 and 14, this model is possible for amplitude +amplification as well. However, there is a different model +of hybrid computing which better utilizes both quantum +and classical’s strengths, shown below in figures 16 and +17. +FIG. 16. +Workflow of a hybrid model of computing, utilizing +both a quantum and classical computer. Both the QPU and +CPU are run in parallel, and the information obtained from +both are fed into the same classical optimizer, which in turn +determines the most effective use for each processor. +The advantage to hybrid computing using the model +shown in figure 16 is that both processors are working +in tandem to solve the same problem, utilizing infor- +mation gained from one another. Information obtained +through amplitude amplification measurements can be +used to speedup a classical algorithm, and vice versa. As +we discuss further in the next subsection, this pairing of +quantum and classical is maximally advantageous when +the strengths of both computers compliment each other’s +weaknesses. +A. +Supporting Greedy Algorithms +One notable strength of classical computing is ‘greedy’ +algorithms, which oftentimes provide heuristic solutions +for use cases ranging from biology and chemistry [51, 64] +to finance [65]. Greedy algorithms are particularly vi- +able for problems that possess certain structures which +can be exploited [66]. +The key feature to these algo- +rithms is that they focus on making locally optimal de- +cisions which yield the maximal gain towards being op- +timal. Consequently, they are very good at finding near +optimal solutions quickly, but are also prone to getting +bottlenecked into local minima [67]. +The motivation for pairing amplitude amplification +with a classical greedy algorithm is best exemplified by +figures 12 and 13. The quantum states illustrated in these +figures represent |Xi⟩ states which rank as the 30th−80th +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +Quantum +Classical +Perform Amplitude +Evaluate C(X) with +1) +Amplification: Ps & k +all previous results +2) +Measure | X: ) +2) +Determine new ps & kOptimizer +Evaluate C(X) values +1) +from both QPU & CPU +Supply both processors +2) +with new parameters +Quantum +Classical +Perform Amplitude +Amplification: ps & k +Perform best +optimization algorithm +2) +Measure X; >A +Supporting Greedy Algorithms +15 +best minimizing solutions to C(X). Under the right condi- +tions it is reasonable to expect that a quantum computer +could yield a solution in this range within 1 − 5 ampli- +tude amplification attempts. The question then becomes +how quickly a classical greedy algorithm could achieve +the same feat? Without problem specific structures to +exploit, and as problem sizes scale like 2N, it becomes +increasingly unlikely that classical can compete heuristi- +cally with quantum, which we argue is quantum’s first +advantage over classical in a hybrid model. +Now, supposing that amplitude amplification does +yield a low C(Xi) solution faster than classical, the prob- +lem then flips back to being classically advantageous. +This is because the Xi solution provided by quantum +is now new information available to the classical greedy +algorithm. As such, beginning the greedy approach from +this new binary string is likely to yield even lower C(Xi) +solutions in a time frame faster than amplitude amplifica- +tion. For example, this is the exact scenario in which ge- +netic algorithms shine [52–55, 65], where a near-optimal +solution is provided from which they can manipulate and +produce more solutions. And if a fast heuristic solution +is all that is needed, then quantum’s job is done, and +the best minimal solution found by the classical greedy +algorithm completes the hybrid computation. +But if a heuristic solution is not enough, then we can +continue to use a hybrid quantum-classical strategy for +finding Xmin. Referring back now to figures 13 and 14, +the strategy for quantum is to use multiple amplitude +amplification trials and measurements to approximate +the underlying correlation from figures 10 and 11. The +fastest means for achieving this is to work in a ps re- +gion analogous to figure 12, where experimentally one +has the highest probabilities of measuring useful infor- +mation. Simultaneously, the classical greedy algorithm +can also find Xi solutions in this area as it searches for +Xmin. Knowledge of these Xi solutions can be directly +fed back to quantum, which can be used to predict ps +values where solutions are known to exist, speeding up +the process of determining a ps vs. +C(X) correlation. +Thus, after quantum initially aided classical, subsequent +information obtained from classical is then used to speed +up quantum. +In the time it takes for quantum to experimentally ver- +ify enough ps and C(Xi) values to create a predictive cor- +relation, we expect the classical algorithm to find a new +lowest C(Xi) solution, labeled X’i in figure 17. After in- +vesting additional trials into the amplitude amplification +side of the computation, it is now time for quantum’s +second advantage: verifying local versus global minima. +Using an approximate ps vs C(X) best-fit, the quantum +computer can skip directly to the ps value corresponding +to best currently known X’i solution. As discussed in sec- +tion V.D, searching past this ps value will result in one of +two outcomes. Either the quantum computer will find a +new best solution C(Xj) < C(X’i), or confirm that X’i is +indeed the global minimum Xmin. In the former case, the +greedy algorithm now starts again from the new lowest +solution Xj, repeating this cycle between quantum and +classical until Xmin is found. Figure 17 below shows a +workflow outline of this hyrbid strategy. +FIG. 17. +Workflow for a hybrid model of computing between +quantum amplitude amplification and a classical greedy algo- +rithm. The full strategy is broken up into three phases: 1) +Amplitude amplification provides the first heuristic solution +Xi. 2) A classical greedy algorithm uses Xi to find a more +optimal solution X’i. Simultaneously, other near optimal so- +lutions Xj are used to assist amplitude amplification in de- +termining a ps vs. C(X) correlation (see figures 10 - 13). 3) +The correlation best-fit is used to predict ps values where so- +lutions C(Xj) < C(X’i) must exist (or C(Xj) > C(X’i) for +maximization problems). Amplitude amplification attempts +for these ps values will either produce a new best Xj for the +greedy classical algorithm to use, or confirm X’i = Xmin. +The biggest advantage to using a hybrid model like +shown in figure 17 is that it can be adapted to each prob- +lem’s uniqueness. +Problems with known fast heuristic +techniques can lean on the classical side of the computa- +tion more heavily [68, 69], while classically difficult prob- +lems can put more reliance on quantum [70, 71]. +But +above all else, this model of computation incorporates +and synergizes the best known classical algorithms with +quantum, rather than competing against them. +VII. +MORE ORACLE PROBLEMS +All of the results from sections III. - V. were derived +from linear QUBOs according to equations 1 - 4. How- +ever, these results can be applied to more challenging +and realistic optimization problems provided that 1) all +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +Quantum +Classical +Amplitude Amplification +Fast Sample: compute ps +in estimated ps region for a fast +heuristic solution X, +Greedy algorithm starting +1) +from solution X, +Continue amplitude amplification +using known X, solutions +Record all new solutions +to create p, vs. C(X) correlation +2) X, better than X;, and the +new best solution X; +X +Search p, values for +Greedy algorithm starting +solutions C(X,) < C(X'): +1) +from new solution X, +2A) +Find new solution X +Record new +or +2) +2B) +best solution X +Confirm X', is X. +min16 +possible solutions can be encoded via phases by an ap- +propriate oracle operation Uc, and 2) the distribution of +all possible answers is suitable for boosting the solution +we seek (gaussian, polynomial, exponential, etc. [18]). +Here we will briefly note some additional optimization +problems which meet both of these criteria. +A. +Weighted & Unweighted Max-Cut +The Maximum Cut problem (‘Max-Cut’) is famously +NP-Hard [70], where the objective is to partition every +vertex in a graph S into two subsets P1 and P2 such +that the number of edges between them is maximized. +In the weighted Max-Cut version of the problem, each +edge is given a weight wij, and the goal is to maximize +the sum of weights contained on edges between P1 and +P2. The unweighted Max-Cut problem has already been +demonstrated as a viable use for amplitude amplifica- +tion [17], which we will build upon further here via the +weighted version. Equation 31 below is the cost function +C(X) for the weighted Max-Cut problem, which can be +converted to the unweighted case by setting every edge +weight wij = 1. The binary variables xi here represent +being partitioned into P1 or P2. +C(X) = +� +{i,j}∈S +wij|xi − xj| +(31) +Shown in figure 18 is an example graph S and one +of its solutions. This example graph is composed of 10 +vertices, labeled 1 - 10, and a total of 15 connecting edges. +Encoding this graph requires one qubit per vertex, where +the basis states |1⟩ and |0⟩ represent belonging to the +subsets P1 and P2 respectively. See the bottom graph in +figure 18 for an example solution state. +The cost oracle Uc for solving Max-Cut must correctly +evaluate all 2N solution states |Xi⟩ based on the edges +of S according to equation 31. For example, if vertices 1 +and 2 are partitioned into different sets, then Uc needs +to affect their combined states |Q1Q2⟩ = |01⟩ and |10⟩ +with the correct phase, weighted or unweighted. Just like +figure 3 from earlier, we can achieve this with a control- +phase gate CP(θ),with the intent of scaling by ps later +(see figure 4). The caveat here is that we need this phase +on |01⟩ and |10⟩, not |11⟩, which means that additional X +gates are required for the contruction of Uc, shown below +in equation 32. +X = +� +0 1 +1 0 +� +(32) +For the complete Uc quantum circuit which encodes +the graph S in figure 18, please see appendix C. Once +properly scaled by ps, the solutions which are capable of +boosting are determined by the underlying solution space +distribution of the problem, which can be seen in figure +FIG. 18. +(top) A graph S composed of 10 nodes and 15 +connections. Each node is labeled 1 - 10, corresponding to the +qubits Q1 - Q10 shown below. (bottom) An example Max-Cut +solution Xi, along with its quantum state representation |Xi⟩. +Nodes colored red correspond to the partition P1, quantum +state |1⟩, while nodes colored white correspond to partition +P2, quantum state |0⟩. ‘Cuts’ are represented in the graph as +dashed lines, totaling 8 for this example. +19 below. +The histogram in this figure shows all 210 +C(Xi) solutions to the graph S from figure 18. Even for +a 10 qubit problem size such as this, it is clear that the +underlying solution space distribution shows gaussian- +like structure. +FIG. 19. +A histogram of all 210 solutions for an unweighted +Max-Cut on graph S from figure 18. +One interesting feature of Max-Cut is that all solutions +come in equal and opposite pairs. For example, the op- +timal solutions to S from figure 19 are |0100101110⟩ and +|1011010001⟩, which both yield 13 ‘cuts’. Mathematically +there is no difference between swapping all vertices in P1 +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +3 +S: +10 +[Xi> = IQQ2::·Q10) +8 +[X;>=|1001001101) +P, = 1,4,7,8,10] +P2 = {2,3,5,6,9]200- +150 +pop.(C(X) +100- +50- +00 +2 +8 +6 +10 +4 +12 +C(X)A +Weighted & Unweighted Max-Cut +17 +and P2, but physically it means that there are always +two optimal solution states. Consequently, these states +will always share the effect of amplitude amplification +together, which is something an experimenter must be +aware of when choosing iterations k. +Finally, moving from the unweighted to weighted ver- +sion of Max-Cut increases the problem’s difficulty, but +notably does not change the circuit depth of Uc. Rather +than using θ = 1 for all of the control-phase gates, each +θ now corresponds to a weighted edge wij of the graph. +Similar to the QUBO distributions shown in figure 7, this +increase in complexity allows for more distinct C(Xi) so- +lutions, and consequently more variance in features such +as σ′ and X∆. +B. +Graph Coloring +A similar optimization problem to Max-Cut is Graph +Coloring, also known as Vertex Coloring [70], which ex- +tends the number of allowed partition sets Pi up to any +integer number k (k = 2 is equivalent to Max-Cut). +Given a graph of vertices and edges S, the goal is to +assign every vertex to a set Pi such that the number of +edges between vertices within the same sets is minimized. +Shown below in equation 33 is the cost function C(X) for +a k-coloring problem, where the values of each vertex xi +are no longer binary, but can take on k different integer +values. The quantity inside the parentheses is equal to 1 +if xi = xj, and 0 for all other combinations xi ̸= xj. Just +like with Max-Cut, setting all wij = 1 is the unweighted +version of the problem. +C(X) = +� +{i,j}∈S +wij +� +1 − +�|xi − xj| +k +�� +(33) +The name ‘coloring’ is in reference to the problem’s +origins, whereby the sets Pi all represent different colors +to be applied to a diagram, such as a map. Shown below +in figure 20 is an example picture composed of overlap- +ping shapes, where each section must be assigned one of +k colors such that the number of adjacent sections with +the same color is minimized. Example solutions for k = 3 +and k = 4 are shown, along with their vertex and quan- +tum state representations of the problem. +In order to encode graph coloring as an oracle Uc, the +choice of k determines whether qubits or another form +of quantum computational unit is appropriate. +While +qubits are capable of producing superposition between +two quantum states, qudits are the generalized unit of +quantum information capable of achieving superposition +between d states [72–75]. To see why this is necessary, +let us compare the k = 3 and 4 examples from figure 20, +and the quantum states needed to represent partitioning +each vertex. +For k = 4, we need four distinct quantum states to +represent a vertex belonging to one of the Pi partitions. +FIG. 20. +(top) On the left, a two dimensional bounded +picture with overlapping geometric shapes. On the right, a +graph S representing the 12 distinct regions of the picture +as nodes. Connections between nodes in S represent regions +in the picture which share a border, not counting adjacent +corners. (middle) A k = 3 coloring example, with a corre- +sponding d = 3 qudit state representation. (bottom) A k = 4 +coloring example, with a corresponding d = 4 qudit state rep- +resentation. +While a single qubit can’t do this, a pair of qubits can. +Thus, every vertex in S can be encoded as a pair of qubits, +letting the basis states |00⟩, |01⟩, |10⟩, and |11⟩ each rep- +resent a different color. +Alternatively, we could use a +d = 4 qudit to represent each vertex, assigning each par- +tition a unique basis state: |0⟩, |1⟩, |2⟩, or |3⟩, such as the +state shown in figure 20. Mathematically the two encod- +ings are identical, so the choice between whether to use +qubits or qudits is a matter of experimental realization +(i.e. which technology is easier to implement). +For k = 3 however, two qubits is too many states, and +a single qubit is not enough. So in order to represent +three colors exactly in quantum, the appropriate unit is +a ‘qutrit’ (the common name for a d = 3 qudit). Simi- +larly, all prime numbers d can only be encoded as their +respective d-qudit, while all composite values can be built +up from combinations of smaller qudits. Once an appro- +priate mixed-qudit quantum system is determined, con- +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +S: +4 +11 +2 +8 +3 +5 +10 +6 +[Xi>=IQiQ2···Q12) +k +二 +=<0 +11>= +X;>=1010212010120) +K +10>= += |101232010321)B +Graph Coloring +18 +structing Uc is the same as the Max-Cut problem from +earlier, but now with k state-state interactions. For an +example of qudit quantum circuits and their use for am- +plitude amplification, please see Wang et al. [72] and our +previous work on the Traveling Salesman problem[10]. +C. +Subset Sum +For all of the oracles discussed thus far, the circuit +depth and total gate count for Uc is determined by the +size and connection complexity of S, the graphical repre- +sentation of the problem. By contrast, the simplest pos- +sible quantum circuit that can be used as Uc corresponds +to the Subset Sum problem [70]. The cost function for +this problem is given in equation 34. +C(X) = +N +� +i +Wixi +(34) +Rather than optimizing equation 34, which is trivial, +the Subset Sum problem is to determine if there exists +a particular combination such that C(Xi) = T, where +T is some target sum value. The boolean variables xi +represent which Wi values to use as contributors to the +sum. Figure 21 below shows an N = 10 example. Note +that this problem is equally applicable to any of the other +oracles discussed thus far, whereby we can ask if a target +value T exists for some graph S. +FIG. 21. +(top) A set of 10 integer values, shown in ascending +order, from which we are intereted in solving the Subset Sum +problem for T = 22. (bottom) An example solution state |Xi⟩ +corresponding to the cost function value C(X) = 22. +The reason why equation 34 is the simplest Uc oracle +one can construct is because the cost function doesn’t +contain any weights wij that depend on two variables. +Consequently, the construction of Uc doesn’t use any 2- +qubit phase gates CP(θ), instead only requiring a single +qubit phase gate P(θ) for every qubit. In principle, all of +these single qubit operations can be applied in parallel, +such as in figure 3, which means that the circuit depth +of Uc is exactly one. +Although this is the most gate efficient Uc, using it +to solve the Subset Sum problem comes with some lim- +itations. Firstly, it can only solve for T values within a +limited range. This is illustrated by the results of figure +11, which demonstrate that amplitude amplification can +only produce meaningful probabilities of measurement up +to a certain threshold away from Xmin or Xmax. Conse- +quently, one can only use Uc here if the target sum value +T is within this threshold distance from the extrema. +The second limitation to consider is the discussion +from section V.D, whereby the information of whether +a state C(Xi) = T exists or not may rely on measure- +ments finding nothing. Previously we discussed how an +experimenter might iteratively decrease ps and eventually +expect to find regions where cost function values do not +exist (see figure 14) as one approaches Xmin. Here things +are easier, since an experimenter can test for ps values +above and below where C(Xi) = T (except for the case +where T is the global extrema). Using a ps vs. C(X) cor- +relation in this manner can confirm exactly where the ps +value for C(Xi) = T must be. Testing this ps window will +then either confirm the existence of a solution for T via +a measurement, or conversely confirm no solution exists +through multiple trials of random measurement results. +VIII. +CONCLUSION +The results of this study demonstrate that the gate- +based model of amplitude amplification is a viable means +for solving combinatorial optimization problems, partic- +ularly QUBOs. +The ability to encode information via +phases and let the 2N superposition of qubits naturally +produce all possible combinations is a feature entirely +unique to quantum. Harnessing this ability into a use- +ful algorithmic form was the primary motivation for this +study, and as we’ve shown, is not without its own set of +challenges. In particular, the discussions of sections IV.A +& IV.B highlight that this algorithm is not a ‘one size fits +all’ strategy that can be blindly applied to any QUBO. +Depending on how the numerical values of a given prob- +lem form a solution space distribution, it may simply +be impossible for amplitude amplification to find one ex- +trema or the other. Figure 8 shows that at least one of +the extrema solutions is always viable for quantum to +find, it just may not happen to be the one that is of +interest to the experimenter. +For cases where the desired solution is well-suited for +quantum to find, that is |Xmin⟩ or |Xmax⟩ is capable of +achieving a high probability of measurement, a different +challenge lies in finding the correct ps value to use in or- +der to boost these states. However, the results of section +V. illustrate that this challenge is solvable via quantum +measurement results. If the best an experimenter could +do is simply guess at ps and hope for success, then am- +plitude amplification would not be a practical algorithm. +But the correlations shown in figures 10 and 11 illustrate +that that is not the case, and that information about ps +can be experimentally learned and used to find extrema +solutions. How quickly this information can be exper- +imentally produced, analyzed, and used is exactly how +quickly quantum can find the optimal solution, which is +an open question for further research. +While the free parameter ps can be considered the bot- +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +{-18，-13，-6，-1，2，3，5，11，16，21 +0011011001> = -6－ 1 ＋3 +5＋21 += 2219 +tleneck of our algorithm for finding optimal solutions, +there is a second important metric by which we can judge +the usefulness of amplitude amplification: as a heuristic +algorithm. A major finding of this study is depicted in +figure 12, which shows that there is a wide range of ps +values for which quantum can find an answer within the +best 1 − 5% of all solutions. And as we demonstrated +in section IV.C with sampling, it is not unrealistic that +a classical computation can estimate this ps region very +quickly. The question then becomes how does this com- +pare to classical greedy algorithms, and how quickly can +they achieve the same feat in a timescale compared to +quantum’s O( π +4 +� +N/M) for problem sizes of 2N. +The +answer to this question will vary from problem to prob- +lem, but certainly in some cases such as highly intercon- +nected QUBOs we view this as the first practical use for +amplitude amplification. +And finally, there is one important sentiment from sec- +tion VI that we would like to reiterate again here, namely +that amplitude amplification is a technique that bene- +fits tremendously from working in parallel with a classi- +cal computer. +The information learned through quan- +tum measurements can equally be of use to speeding +up quantum as well as a classical algorithm. And vice +versa, information learned through a classical greedy al- +gorithm can be used to speed up quantum. The goal of +this hyrbid computing model is to utilize the advantages +both computers have to offer, and ultimately to find op- +timal solutions faster than either computer can achieve +alone. Understanding which optimization problems this +scenario may be applicable to is the future direction of +our research. +ACKNOWLEDGMENTS +Any opinions, findings, conclusions or recommenda- +tions expressed in this material are those of the author(s) +and do not necessarily reflect the views of AFRL. +DATA & CODE AVAILABILITY +The data and code files that support the findings of +this study are available from the corresponding author +upon reasonable request. +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +20 +[1] L. K. Grover, A fast quantum mechanical algorithm for +database search. arXiv: 9605043 (1996) +[2] M. Boyer, G. Brassard, P. Hoyer, A. Tapp, Tight bounds +on quantum searching. Fortschritte der Physik 46, p.493- +506 (1998) +[3] C. H. Bennett, E. 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Every +QUBO was simulated through amplitude amplification, +and the ps value which yielded the highest probability of +measurement for |Xmin⟩ was recorded. +N +# of QUBOs studied +17 +5000 +18 +3000 +19 +2000 +20 +1500 +21 +1200 +22 +1000 +23 +1000 +24 +600 +25 +500 +26 +400 +27 +100 +TABLE II. +Table of values showing the number of linear +QUBOs generated and studied per size N. +Appendix B: Linear Regression +In order to determine how linearly correlated the data +points in figure 10 were, a regression best-fit was per- +formed according to equations B1 - B5 below. The col- +lection of (x,y) data points D in equation B1 corresponds +to the (ps,C(Xi)) points in the figure. The resulting lin- +ear correlation factor R is reported at the top of figure +10. +D = ((x1, y1), (x2, y2), ..., (xN, yN)) +(B1) +¯X = +N +� +i +xi +¯ +X2 = +N +� +i +(xi)2 +(B2) +¯Y = +N +� +i +yi +¯ +Y 2 = +N +� +i +(yi)2 +(B3) +¯ +XY = +N +� +i +xi · yi +(B4) +R = +N ¯ +XY − ¯X ¯Y +� +(N ¯ +X2 − ( ¯X)2)(N ¯ +Y 2 − ( ¯Y )2) +(B5) +Appendix C: Max-Cut Circuit +To illustrate how any graph structure S can be encoded +as an oracle Uc, figure 23 below is the quantum circuit +corresponding to S from figure 18. Because this oracle +needs to represent a Max-Cut problem (weighted or un- +weighted), the states which must acquire phases are |01⟩ +and |10⟩. To make the circuit less cluttered, let us define +the custom gate given in figure 22. +FIG. 22. +Quantum circuit which achieves the 2-qubit unitary +from equation C1. +The quantum circuit shown in figure 22, drawn similar +to a CP(θ) gate but with an extra box around it, is an +operation which achieves the following unitary: +U(α|00⟩ + β|01⟩ + γ|10⟩ + ρ|11⟩ +) +(C1) +=α|00⟩ + eiθβ|01⟩ + eiθγ|10⟩ + ρ|11⟩ +The unitary U from equation C1 is the required oper- +ation for representing the cost oracle given in equation +31. If two nodes (qubits) share a connection in S, then a +‘cut’ corresponds to them being partitioned into different +sets, which is represented by the qubit states |0⟩ and |1⟩. +Figure 23 uses the operation in figure 22 to create the +complete Uc circuit for encoding all 15 connections in S. +FIG. 23. +Quantum circuit which achieves the oracle Uc +corresponding to S from figure 18, for the Max-Cut problem. +Each gate shown represents one of the 15 connections in S, +corresponding to the custom gate defined in figure 22. The +placement of gates shown here are spread out for clarity, while +a real implementation could be more parallelized. +Approved for Public Release; Distribution Unlimited: PA#: AFRL 2023-0204 + +Q +X +X +三 +Q +0 +0 +X +0 +XQ +Q +Wi2 Ps +Q +W2s Ps +Q +Wi4· Ps +Q +Wys· Ps +Wis· Ps +W3s- Ps +5 +Q +Ws P. +6 +Q +WoPs +W3 Ps +Q +Wes P, +W4s· Ps +8 +Q +Wo P, +9 +Q +Wr1i Ps +Wo1 Ps +Wa: Ps +-10 \ No newline at end of file diff --git a/NNFRT4oBgHgl3EQf3jjV/content/tmp_files/load_file.txt b/NNFRT4oBgHgl3EQf3jjV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1a4d9e1ebeb6a33232b700a33f66bdeda903c85 --- /dev/null +++ b/NNFRT4oBgHgl3EQf3jjV/content/tmp_files/load_file.txt @@ -0,0 +1,1307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf,len=1306 +page_content='Variational Amplitude Amplification for Solving QUBO Problems Daniel Koch1∗, Massimiliano Cutugno1, Saahil Patel1, Laura Wessing1, Paul M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Alsing1 1Air Force Research Lab, Information Directorate, Rome, NY and ∗Corresponding Author: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='koch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='13@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='mil We investigate the use of amplitude amplification on the gate-based model of quantum comput- ing as a means for solving combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This study focuses primarily on QUBO (quadratic unconstrained binary optimization) problems, which are well-suited for qubit su- perposition states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Specifically, we demonstrate circuit designs which encode QUBOs as ‘cost oracle’ operations UC, which when combined with the standard Grover diffusion operator Us lead to high probabilities of measurement for states corresponding to the optimal and near optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In order to achieve these probabilities, a single scalar parameter ps is required, which we show can be found through a variational quantum-classical hybrid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' INTRODUCTION Amplitude amplification is a quantum algorithm strat- egy that is capable of circumventing one of quantum com- puting’s most difficult challenges: probabilistic measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Originally proposed by Grover in 1996 [1], and later shown to be optimal [2, 3], the combination of his oracle UG and ‘diffusion’ Us operators is able to drive a quantum system to a superposition state where one (or multiple) basis state(s) has nearly 100% probability of being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Since then, many researchers have contributed to the study of UG and Us [4–9], seeking to better understand how the fundamental nature of am- plitude amplification is dependent on these two opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Similarly, the aim of this study is to further extend the capabilities of amplitude amplification as a means for solving combinatorial optimization problems using gate- based quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The results of this paper are a continuation of our previous work [10], in which we demonstrated an ora- cle design which was capable of encoding and solving a weighted directed graph problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The motivation for this oracle was to address a common criticism of UG [11– 15], namely that the circuit construction of oracles too often hardcodes the solution it aims to find, negating the use of quantum entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Similar to other recent stud- ies [16–21], we showed that this problem can be solved at the circuit depth level by avoiding gates such as control-Z for constructing the oracle, and instead using phase and control-phase gates (P(θ) and CP(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, simply changing the phase produced from UG to something other than π is not enough [22–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Our oracle construction ap- plies phases to not only a desired marked state(s), but all states in the full 2N Hilbert Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The phase each basis state receives is proportional to the solutions of a weighted combinatorial optimization problem, for which the diffusion operator Us can be used to boost the prob- ability of measuring states that correspond to optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The consequence of using an oracle operation that ap- plies phases to every basis state is an interesting double- edged sword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As we show in sections II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' - IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', and later in section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', the use of phase gates allows for amplitude amplification to encode a broad scope of combinatorial optimization problems into oracles, which we call ‘cost oracles’ Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In particular, we demonstrate the robust- ness of amplitude amplification for solving these kinds of optimization problems with asymmetry and random- ness [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, the tradeoff for solving more com- plex problems is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Firstly, in contrast to Grover’s oracle, using Uc is only able to achieve peak measure- ment probabilities up to 70-90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' we show that these probabilities are still high enough for quan- tum to reliably find optimal solutions, which notably are achieved using the same O( π 4 � N/M ) iterations as stan- dard Grover’s [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The second, more challenging tradeoff when using Uc is that the success of amplitude amplification is largely dependant on the correct choice of a single free parame- ter ps [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This scalar parameter is multiplied into ev- ery phase gate for the construction of Uc (P(θ · ps) and CP(θ · ps)), and is responsible for transforming the nu- meric scale of a given optimization problem to values which form a range of approximately 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This in turn is what allows for reflections about the average amplitude via Us to iteratively drive the probability of desired solu- tion states up to 70-90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The significance of ps, and the challenges in determining it experimentally, are a major motivation for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In particular, the results of section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' demonstrate that there is a range of ps values for which many optimal solutions can be made to become highly probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Additionally, our simulations show that there is an observed correlation between the numerical cost function value of these solutions and the ps values where they achieve peak probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This underlying correlation supports the idea of using amplitude amplifi- cation for a variational model of hybrid quantum-classical computing, which is the core finding of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Layout The layout of this study is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' be- gins with the mathematical formalism for the optimiza- tion problem we will seek to solve using amplitude am- plification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Sections III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' & IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' discuss the construction Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='13665v1 [quant-ph] 31 Jan 2023 A Layout 2 of the problem as a quantum circuit, the varying degrees of success one can expect from optimization problems generated using random numbers, and the conditions for which these successes can be experimentally realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' we explore the role of ps from a heuristic per- spective, whereby we demonstrate that many near opti- mal solutions are capable of reaching significant proba- bilities of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' is a primarily spec- ulative discussion, theorizing how the collective results of section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' can be coalesced into a hybrid quantum- classical variational algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' And finally, section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' completes the study with additional optimization prob- lems that can be constructed as oracles and solved using amplitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' QUBO DEFINITIONS We begin by outlining the optimization problem which will serve as the focus for this study: QUBO (quadratic unconstrained binary optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The QUBO prob- lem has many connections to important fields of com- puter science [31–35], making it relevant for demonstrat- ing quantum’s potential for obtaining solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' To date, the two most successful quantum approaches to solving QUBOs are annealing [36–39] and QAOA [40–43], with a lot of interest in comparing the two [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown be- low in equation 1 is the QUBO cost function C(X) which we shall seek to solve using our quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) = N � i Wixi + � {i,j}∈S wijxixj (1) The function C(X) evaluates a given binary string X of length N, composed of individual binary variables xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Together, the total number of unique solutions to each QUBO is 2N, which is also the number of quantum states producible from N qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Throughout this study we will use subscripts Xi and C(Xi) when referring to individual solutions, and C(X) when discussing a cost function more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As shown in equation 1, a QUBO is defined by two sep- arate summations of weighted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The first summa- tion evaluates weights Wi associated with each individual binary variable, while the second summation accounts for pairs of variables which share a weighted connection wij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In this study we adopt the typical interpretation of QU- BOs as graph problems, whereby each binary variable xi represents a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' We can then define the connectivity of a QUBO graph using the set S, which itself is a collection of sets that describe each pair of nodes xi and xj that share a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' See figure 1 below for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The interest of this study is to use a quantum algo- rithm to find either Xmin or Xmax, which are the solu- tions which minimize / maximize the cost function C(X) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For all QUBOs analyzed in the coming sec- tions, the weight values Wi and wij are restricted to in- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) An example 3-qubit linear QUBO with weighted nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) The set S containing the complete connectivity of the QUBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' tegers, randomly selected from a uniform distribution as shown below in equations 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Wi, wij ∈ Z (2) Wi, wij ∈ [−100, 100] (3) In section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' we discuss the consequences of choosing weight values in this manner and its advantage for quan- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, nearly all of the results shown throughout this study are applicable to the continuous cases for Wi and wij as well, with the one exception being the results of section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Linear QUBO The cost function given in equation 1 is applicable to any graph structure S, so long as every node and edge is assigned a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For this study we will focus on one specific S, which we refer to as a ‘linear QUBO.’ The connectivity of these graphs is as follows: S = {{n, n + 1} | 1 ≤ n ≤ N − 1} (4) As the name suggests, linear QUBOs are graphs for which every node has connectivity with exactly two neighboring nodes, except for the first and final nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The motivation for studying QUBOs of this nature is their efficient realizability as quantum circuits, given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' AMPLITUDE AMPLIFICATION The quantum strategy for finding optimal solutions to C(X) investigated in this study is amplitude amplifica- tion [4–9], which is the generalization of Grover’s algo- rithm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The full algorithm is shown below in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 1, which notably is almost identical to Grover’s algorithm except for the replacement of Grover’s oracle UG with our cost oracle Uc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 Wi W 3 2 3 W12 W23 S = {{1,2],[2,3]3 Algorithm 1 Amplitude Amplification Algorithm Initialize Qubits: |Ψ⟩ = |0⟩⊗N Prepare Equal Superposition: H⊗N|Ψ⟩ = |s⟩ for k ≈ π 4 √ 2N do Apply Uc|Ψ⟩ (Cost Oracle) Apply Us|Ψ⟩ (Diffusion) end for Measure By interchanging different oracle operations into the Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 1, various problem types can be solved using amplti- tude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, Grover’s original oracle solves an unstructured search, whereas here we are in- terested in optimal solutions to a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Later in section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' we discuss further oracle adaptations and the problems they solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For all oracles, we use the standard diffusion operator Us, given below in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Us = 2|s⟩⟨s| − I (5) This operation achieves a reflection about the average amplitude, whereby every basis state in |Ψ⟩ is reflected around their collective mean in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This operation causes states’ distance from the origin to in- crease or decrease based on their location relative to the mean, which in turn determines their probability of mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Therefore, a successful amplitude amplifica- tion is able to drive the desired basis state(s) as far from the origin as possible, up to a maximum distance of 1 (measurement probability of 100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Solution Space Distribution A prerequisite for the success of amplitude amplifica- tion as demonstrated in this study is an optimization problem’s underlying solution space distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' that is, the manner in which all possible solutions to the prob- lem are distributed with respect to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For QUBOs, these are the 2N possible C(Xi) cost function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown below in figure 2 is a histogram of one such solution space distribution, for the case of a length 20 linear QUBO according to equations 1 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The x-axis represents all possible cost function evaluations, and the y-axis is the corresponding number of unique Xi solutions that result in the same C(Xi) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Depicted in figure 2 are all 220 possible solutions to an example linear QUBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Because this QUBO was gen- erated from randomized weights, the combination of the Law of Large Numbers [47] and Central Limit Theorem [48] predicts that its underlying solution space should be approximately gaussian [49] in shape, given by equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' G(x) = αe (x−µ)2 2σ2 (6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Example of a solution space distribution for a 20 node linear QUBO, with weights according to equations 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Indeed, the histogram shown is approximately gaus- sian, but importantly it has imperfections resulting from the randomized weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' At large enough problem sizes (around N ≥ 20), these imperfections have minimal im- pact on a problem’s aptitude for amplitude amplifica- tion, which was a result from our previous study [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Similarly, another recent study [18] demonstrated that in addition to symmetric gaussians, solution space distri- butions for both skewed gaussians and exponential pro- files also lead to successful amplitude amplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The commonality between these three distribution shapes is that they all possess large clusters of solutions that are sufficiently distanced from the optimal solutions we seek to boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This can be seen in figure 2 as the location of Xmin and Xmax as compared to the central peak of the gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' When appropriately encoded as an oracle Uc, these clusters serve to create a mean point in the complex plane which the optimal solution(s) use to reflect about and increase in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Cost Oracle Uc In order to use algorithm 1 for finding the optimal solution to a given cost function, we must construct a cost oracle Uc which encodes the weighted information and connectivity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In our previous study we referred to this operation as a ‘phase oracle’ UP [10], and similarly it has also been called a ‘subdivided phase oracle’ SPO [17, 18] or ‘non-boolean oracle’ [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' How one constructs Uc is problem specific, but the general strategy is to primarily use two quantum gates, shown below in equations 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 3000 2500- 2000- pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (C(X)) 1500 1000 - 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' max 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 600 400 200 0 200 400 C(X)B Cost Oracle Uc 4 P(θ) = �1 1 1 eiθ � (7) CP(θ) = � �� 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 eiθ � �� (8) The single and two-qubit gates P(θ) and CP(θ) are re- ferred to as phase gates, also known as Rz(θ) and CRz(θ) for their effect of rotating a qubit’s state around the z- axis of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Mathematically they are capa- ble of applying complex phases as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' P(θ)|1⟩ = eiθ|1⟩ (9) CP(θ)|11⟩ = eiθ|11⟩ (10) Applying P(θ) to a qubit only affects the |1⟩ state, leaving |0⟩ unchanged, and similarly only |11⟩ for CP(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, this is exactly what we need in order to con- struct C(X) from equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' When evaluating a partic- ular binary string Xi classically, only instances where the binary values xi are equal to 1 yield non-zero terms in the summations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For quantum, each binary string Xi is represented by one of the 2N basis states |Xi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, our quantum cost oracle Uc can replicate C(X) by using P(θ) and CP(θ) to only effect basis states with qubits in the |1⟩ and |11⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) Example of a 4-qubit linear QUBO with weighted nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) The same QUBO en- coded into a cost oracle Uc without scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each unitary in the circuit is P(θ) (single qubit gate) or CP(θ) (2-qubit gate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown above in figure 3 is an example of a 4-qubit QUBO cost oracle, where the weighted values Wi and wij are used as the θ parameters for the various phase gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Although incomplete, we will use this oracle circuit to demonstrate quantum’s ability to encode a cost function C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, consider the binary solution Xi = 1101 and the corresponding quantum basis state |1101⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The classical evaluation of this solution is as follows: C(1101) = −8 + 18 − 22 − 12 = −24 (11) Now let us compare this to the phase of |1101⟩ after applying Uc: Uc|1101⟩ = ei(−8+18−22−12)|1101⟩ = e−24i|1101⟩ (12) The phase acquired in equation 12 is equivalent to the classical evaluation shown in 11, which means that Uc is an accurate encoding of C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If we were to now apply Uc to the equal superposition state |s⟩ (step 2 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 1), all 2N basis states would receive phases equal to their cost function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This is the advantage that quantum has to offer: simultaneously evaluating all possible solutions of a cost function through superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Scaling Parameter ps While the cost oracle shown in figure 3 is capable of re- producing C(X), its use in algorithm 1 will not yield the optimal solution Xmin or Xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This is because quantum phases are 2π modulo, which is problematic if the numer- ical scale of C(X) exceeds a range of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Consequently if two quantum states receive phases that differ by a mul- tiple of 2π, then they will both undergo the amplitude amplification process identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If this happens unin- tentionally via Uc, then our cost oracle cannot be used to minimize or maximize C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In order to construct Uc such that it is usable for am- plitude amplification, a scalar parameter ps must be in- cluded in all of the phase gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The value of ps is prob- lem specific, but its role is always the same: scaling the cumulative phases applied by Uc down (or up) to a range where [C(Xmin) , C(Xmax)] is approximately [x , x+2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This range does not have to be [0 , 2π] exactly, so long as the phases acquired by |Xmin⟩ and |Xmax⟩ are roughly 2π different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' See figure 4 below for an example of ps in Uc’s construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The 4-qubit linear QUBO cost oracle Uc from figure 3, now scaled by ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 Wi: 8 18 26 22 (1) 2 3 4 Wii: 12 33 6 Q 8- Q 18 12 Uc: 2 Q 26 33 3 Q 22 6 4Q 8·Ps Q 18·Ps 12·Ps 2 Q 26·Ps 33·Ps 3 Q 22·Ps 6·Ps 4C Scaling Parameter ps 5 Using the scaled oracle shown in figure 4 above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' let us now show how this new Uc acts on the basis state |1101⟩ from before: Uc|1101⟩ = ei(−8·ps+18·ps−22·ps−12·ps)|1101⟩ = ei(−8+18−22−12)·ps|1101⟩ = e−24i·ps|1101⟩ (13) As shown in equation 13 above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' multiplying ps into ev- ery phase gate has the net effect of scaling the cumulative phase applied by Uc: e−24i → e−24i·ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Note that this is not a global phase, which would have an additive effect on all states rather than a multiplicative one like shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Finding the optimal ps value for boosting Xmin or Xmax is non-trivial, and was a major focus of our previous study [10], as well as this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In general, the scale of ps needed for finding the optimal solution can be ob- tained using equation 14 below, which scales the numer- ical range of a problem [C(Xmin) , C(Xmax)] to exactly [x , x+2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ps = 2π C(Xmax) − C(Xmin) (14) Although equation 14 above is guaranteed to solve the 2π modulo phase problem mentioned previously, it is al- most never the ps value which can be used to find Xmin or Xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Only in the case of a perfectly symmetric solution space distribution is equation 14 the optimal ps value, in which case the states |Xmin⟩ and |Xmax⟩ undergo the amplitude amplification process together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, re- alistic optimization problems can be assumed to have a certain degree of randomness or asymmetry to their so- lution space, producing distributions more akin to figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For this reason, equation 14 is better thought of as the starting point for finding the true optimal ps, which we discuss later in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For now, equation 14 is sufficient for demonstrating ps’s role in creating an av- erage amplitude suitable for boosting |Xmin⟩ or |Xmax⟩, shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The bottom plot in figure 5 shows |Ψ⟩ after the first application of Uc in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Note the location of the average amplitude (red ‘x’), which is only made possible by the majority of quantum states which recieve phases near the center of the gaussian in the top plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Optimal amplitude amplification occurs when the desired state for boosting is exactly π phase different with the mean [2, 3], which is very close to the situation seen in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, since this Uc is derived from a QUBO with randomized weights, the ps value provided from equa- tion 14 does not exactly produce a π phase difference between the optimal states (black star) and the mean amplitude (red ‘x’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Consequently, the state(s) which does become highly probable from amplitude amplifica- tion for this particular ps is not |Xmin⟩ and |Xmax⟩, which will be the subject of the coming two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) The 20-qubit linear QUBO histogram from figure 2, scaled by ps according to equation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) All 220 quantum states after applying Uc|s⟩, plotted in amplitude space (the complex plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The red-blue color scale shows the density of quantum states in the bottom plot, corresponding to the y-axis of the top histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The states |Xmin⟩ and |Xmax⟩ are marked with a black star, the origin a black ‘+’, and average amplitude with a red ‘x’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' GAUSSIAN AMPLITUDE AMPLIFICATION The amplitude space plot depicted at the bottom of figure 5 is useful for visualizing how a gaussian solution space distribution can be used for boosting, but the full amplitude amplification process is far more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This is especially true for the QUBOs of this study, which are generated with randomized weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Consequently, all of the results which follow throughout the remainder of this study are produced from classical simulations of amplitude amplification using cost oracles derived from linear QUBOs according to equations 1 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For a deeper mathematical insight into these processes, please see [16– 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Achievable Probabilities Amplitude amplifiation is an appealing quantum al- gorithm because it solves one of the most fundamental Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 3000 2500 2000 pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (C(X) 1500 1000· 500 0 ¥-2 3 0 2 C(X)·Ps Ucls)A Achievable Probabilities 6 problems of quantum computing: measurement proba- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, a single marked state using Grover’s oracle with 30 qubits is capable of achieving a final prob- ability that is only less than 100% by one billionth of a percent [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, a natural question to ask when using Uc is what kinds of probabilities can it produce for |Xmin⟩ or |Xmax⟩?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' To answer this we conducted a statistically study of linear QUBOs ranging from length N = 17 to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For each N we generated numerous QUBOs according to equations 1 - 4, totals given in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' We then let a classical simulator find the ps value which maximized the probability of measuring |Xmin⟩ for each QUBO (and for certain cases the optimal ps for |Xmax⟩ aswell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Results for each problem size are shown below in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Results from studying randomly generated linear QUBOs of various sizes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The number of QUBOs stud- ied per N is provided in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For each QUBO, the optimal ps value for producing the highest probability of mea- surement for |Xmin⟩ was used to record three trends: average probability of |Xmin⟩ (black triangle), highest recorded proba- bility (red star), and average scaled standard deviation (blue circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Error bars showing one standard deviation of each σ’ are provided aswell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' µ = 1 2N 2N � i C(Xi) (15) σ = ��2N i (C(Xi) − µ)2 2N (16) σ′ = σ · ps (17) Figure 6 tracks three noteworthy trends found across the various QUBO sizes: the average peak probabil- ity achievable for |Xmin⟩ (black triangle), the highest recorded probability for |Xmin⟩ (red star), and the aver- age scaled standard deviation σ′ (blue circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For clarity, the derivation of σ′ is given by equations 15 - 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This quantity is the standard deviation of a QUBO’s solution space distribution after being scaled by ps, making it a comparable metric for all QUBO sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In our previous study we demonstrated a result in agreement with fig- ure 6, which is the correlation between higher achievable probabilities for |Xmin⟩ (red star) and smaller scaled stan- dard deviations σ′ (blue circle) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The latter is what is responsible for increasing the distance between |Xmin⟩ and the average amplitude like shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Solution Space Skewness The relation between N, σ′, and highest prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (|Xmin⟩) from figure 6 can be summarized as follows: larger prob- lem sizes tend to produce smaller standard deviations, which in turn lead to better probabilities produced from amplitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, there is a very appar- ent disconnect between the probabilities capable of each problem size (red stars) versus the average (black trian- gle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' To explain this, we must first introduce the quantity X∆ given in equation 18 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' X∆ = 2µ − (C(Xmax) + C(Xmin)) (18) The quantity X∆ from equation 18 is the difference between C(Xmin) and µ (the mean) minus the difference between µ and C(Xmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A positive value for X∆ indi- cates that the mean is closer to C(Xmax), and vice versa for a negative valued X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In essence, it is a measure of skewness that describes the assymetry of a solution space distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Figure 7 shows example QUBO distribu- tions for three cases of X∆, for N = 25, demonstrating the impact X∆ has on the ability to boost |Xmin⟩ versus |Xmax⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' While σ′ is a strong indicator of a problem’s over- all aptitude for amplitude amplification, X∆ determines whether the optimal minimum or maximum solution is boostable, and which is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Further evidence of this can be seen in figure 8, which shows 1000 randomly generated linear QUBOs of length N = 23, and the peak probabil- ities achievable for |Xmin⟩ and |Xmax⟩ as a function of X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If we compare the average peak probabilites for |Xmin⟩ from figure 6 with the full data of QUBOs shown in fig- ure 8, we can see why the average peak probability is significantly lower than the highest recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Across the 1000 QUBOs studied, it is clear that X∆ = 0 is a dividing point for whether |Xmin⟩ or |Xmax⟩ is capa- ble of reaching a significant probability of measurement through amplitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For N = 23, the aver- age prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (|Xmin⟩) reported in figure 6 is approximately 64%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, if instead we only consider QUBOs with X∆ > 0 from figure 8, then the average peak probability for |Xmin⟩ is around 86%, and likewise for |Xmax⟩ when X∆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Together, figures 7 and 8 demonstrate the significance of knowing X∆ from an experimenter’s perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' De- pending on the optimization problem of interest, it is reasonable to assume that an experimenter may be in- terested in finding only Xmin or Xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' But without any Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Prob(IXmin >) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4: AV ★★ Best Prob(Xmin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='3 17 19 21 23 25 27 NB Solution Space Skewness 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Three randomly generated QUBO distributions for N = 25, illustrating X∆ cases for largely positive (top), largely negative (middle), and near zero (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In all three plots the exact X∆ value is reported, as well as the highest achievable probability for |Xmin⟩ and |Xmax⟩ (each from a dif- ferent ps value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Also shown in each plot are the values for C(Xmin) and C(Xmax), and their numerical distance to the mean µ (red-dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' a priori knowledge of a problem’s underlying solution space, specifically X∆, the experimenter may unknow- ingly be searching for a solution which is probabilistically near impossible to find through amplitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, consider the QUBO distribution illustrated in the top plot of figure 7, and the peak probability for boosting |Xmax⟩: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Although it is ideal to have insight into a particular problem’s X∆ before using am- plitude amplification, as we demonstrate in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', information about X∆ can be inferred through measure- ment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A total of 1000 randomly generated linear QU- BOs of size N = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For each QUBO, the highest achievable probability for |Xmin⟩ (black circle) and |Xmax⟩ (red triangle) are plotted as a function of X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The top plot includes both data points per QUBO, while the bottom plot only shows the higher of the two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Sampling for ps If a particular optimization problem is suitable for am- plitude amplification, then the speed of the quantum algorithm outlined in this study is determined by how quickly the optimal ps value can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Here we shall show that sampling a cost function C(X) can provide reli- able information for approximating ps from equation 14, which can then be used to begin the variational approach outlined in sections V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' and VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Importantly, the number of cost function evaluations needed is significantly less than either a classical or quantum solving speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The strategy outlined in equations 19 - 29 below can be used for approximating ps when the experimenter is expecting an underlying solution space describable by a gaussian function (equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If another type of distribution is expected, then the function used in equation 22 could in principle be modified accordingly (for example, sinu- soidal, polynomial, exponential [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Suppose we sample a particular cost function C(X) M times, where M << 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' We will define the set M as the collection of values C(Xi) obtained from these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 55000-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (IXmin >): 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2% Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (IXmax >): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='16% K:331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 1180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='75 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='25 516 925 0 60000 Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (IXmin>): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='01% Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (IXmax ): 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='1% X: -330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0 721 1054 589 1186 0 Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (IXmin )): 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='7 % Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (IXmax ): 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='1 % 80000-1 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (C(X) 720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='75 720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='25 1591 439 0 C(X)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0 Both 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0 Highest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 [Xmin [Xmax, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 0 400 200 0 200 400 XAC Sampling for ps 8 M = {C(X1), C(X2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', C(XM)} (19) Using these M values, we can compute an approximate mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ˜µ = 1 M � c∈M c (20) ˜σ = �� c∈M (c − ˜µ)2 M (21) In order to use equation 14 for obtaining ps, we need approximations for C(Xmin) and C(Xmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If we assume an underlying gaussian structure to the problem’s solu- tion space, then we can write down the following equation to describe it: 2N = � ∞ −∞ ˜αe (x−˜µ)2 2˜σ2 dx (22) = −˜α ˜σ �π 2 erf � ˜µ − x √ 2˜σ �∞ −∞ (23) = −˜α ˜σ �π 2 · [−1 − 1] (24) where erf() is the gaussian error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Using equa- tion 24, we can rearrange terms and solve for an approx- imation to the height of the gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ˜α = 2N−1 ˜σ� π 2 (25) With the values ˜µ, ˜σ, and ˜α obtained from sampling, we can now approximate C(Xmin) and C(Xmax) using equation 26 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ˜G(x) = ˜αe (x−˜µ)2 2˜σ2 = 1 (26) Solving for x yields the following two values: x± = ˜µ ± ˜σ � −2ln � 1 ˜α � (27) which can be expressed in terms of the two quantities originally derived from sampling: x± = ˜µ ± ˜σ � � � �−2ln � ˜σ � π/2 2N−1 � (28) And finally, the solutions x± can be used to obtain ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ˜ps = 2π x+ − x− (29) The reason we set equation 26 equal to 1, and the integral in equation 22 equal to 2N, is because ˜G(x) is modeling the histogram of a QUBO’s solution space, like shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This means that the total number of solutions to C(X) is 2N, and similarly the minimum num- ber of distinct C(Xi) solutions for a given cost function is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Therefore, after setting the integral in equation 22 equal to 2N, solving ˜G(x)= 1 yields approximations for C(Xmin) and C(Xmax) on the tails of the gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' To demonstrate how well sampling is able to approxi- mate equation 14, we tested the strategy outlined above against the 1000 QUBOs from figure 8 (N = 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For four values of M: 100, 500, 1000, and 2000, each QUBO was used for 50 trials of random sampling to produce ap- proximate ˜ps values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' These values were then compared to the true value of ps from equation 14, as given by equa- tion 30 below, and finally averaged together to produce table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ˜ps Error = |˜ps − ps| ps (30) M 100 500 1000 2000 Average ˜ps Error 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='28% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='37% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='31% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='29% TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Average error in approximating ps using equations 19 - 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each value comes from 50,000 independent sampling trials on linear QUBOs of size N = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The significant result from table I is that sampling 100 500 times, on a cost function of 223 solutions, is accu- rate enough to produce an approximate ˜ps value with an expected error of only 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' And as we show in the next section, this is enough accuracy to use for either a heuris- tic or variational approach for finding optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' VARIATIONAL AMPLITUDE AMPLIFICATION The results of sections II - IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' demonstrate quantum’s aptitude for encoding and solving a QUBO problem using amplitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In this section we discuss how this potential can be realized from an experimental per- spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In particular, we focus on amplitude amplifica- tion’s ability to find optimal solutions under realistic cir- cumstances with limited information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The results of this section are then used to motivate section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', in which we discuss how amplitude amplification can be used in a hybrid classical-quantum model of computing, similar to other successful variational approaches [40, 41, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Boosting Near-Optimal Solutions The results shown in figures 6 - 8 focus on quantum’s potential for finding |Xmin⟩ and |Xmax⟩, the optimal so- lutions which minimize/maximize a given cost function C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, in order to understand how amplitude amplification can be used in a variational model, it is equally as important that non-optimal |Xi⟩ states are also capable of boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As discussed in the conclusion of our previous study [10], as well as sections III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='C and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='C, the most difficult aspect of using algorithm Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='1 from an experimental standpoint is finding ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' More specifically, finding an optimal ps for boosting |Xmin⟩ or |Xmax⟩ is a challenge due to the limited amount of information that one can learn through measurements alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' An example of this can be seen in figure 9, which shows the peak achievable probabilities of the three lowest |Xi⟩ states as a function of ps (|Xmin⟩ and the next two minimum solutions), for the QUBO corresponding to X∆ = 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 from figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Plots of |Xi⟩ state probability from amplitude ampli- fication as a function of ps, for |Xmin⟩ (blue-solid) and the next two minimal solutions (black and red-dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Cost function values C(Xi) are reported next to each state’s plot, corre- sponding to the QUBO from the top plot in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The bottom plot is a zoomed in scale of the top plot, depicting the same data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The challenge presented in figure 9 is the narrow range of ps values for which each |Xi⟩ state is able to achieve meaningful probabilities of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' From an experimental perspective, the ps regions outside these bands are only capable of producing quantum superposi- tion states which are slightly better than |s⟩, the equal su- perposition starting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, an experimenter could use a ps value that is incredibly close to optimal, but only see seemingly random measurement results through repeat implementations of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Our proposed solution to the ps problem as described above is twofold: 1) We must widen our view of useful ps values and see where other |Xi⟩ states become highly probable, and 2) put less burden on quantum to find optimal solutions alone when an assisting classical ap- proach may be more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In this section we focus on addressing (1), which will then motive (2) in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Suppose we aren’t solely interested in using quantum to find the exact optimal solution C(Xmin), but instead are content with any Xi within the best 50 answers (50 lowest C(X) values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In order for amplitude amplification to be viable in this heuristic context, it requires signifi- cant probabilities of measurement for these non-optimal solution states, similar to figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Additionally, an ex- perimenter must be able to quickly and reliably find the ps values which produce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown below in figure 10 is a plot which provides insight into the feasibility of both of these questions, for the QUBO corresponding to figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Figure 10 shows the full ps range for which an exper- imenter could find the 50 best solutions for minimizing C(X) via quantum measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The black circles in- dicate on the x-axis the ps values where each |Xi⟩ state (or multiple states) is maximally probable, aligning with its corresponding C(Xi) value along the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Numeric values for peak probabilities of the best 20 solutions are provided in the table below the plot, as well as a lin- ear regression best fit (red-dotted line) for the overall 50 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The reported R correlation value is given by equation B5 in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' There are several significant results displayed in figure 10, the first of which requires returning to equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' By limiting the allowed weighted values for Wi and wij to integers, all solutions to C(X) are consequently integers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This means that the linear correlation shown in the figure can in principle be used to predict ps values where integer C(Xi) solutions must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If Wi and wij are instead allowed to take on float values, which is more general of realistic optimization problems, the linearity of solutions like shown still persists but cannot be used for predictions of allowed C(X) values as reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The linear best fit shown in figure 10 is accurate for the top 50 solutions, but extending the ps scale further reveals that it is only an approximation applicable to a small percentage of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This is shown in figure 11 below, which once again is a ps vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) plot for the same QUBO, but now for the best 400 minimizing so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' It is clear from the data in this figure that the top 400 solutions are in no way linearly aligned, which Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8 1553→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='81 C(X): -1591- 1590 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002675 PsA Boosting Near-Optimal Solutions 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) A plot of the 50 lowest C(Xi) values as a function of ps, for the X∆ = 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 QUBO from figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each data point represents the ps value where the |Xi⟩ state(s) is most probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A linear regression best-fit is shown by the red-dotted line, with its R correlation value reported at the top (equation B5 from appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) A table of values for the 20 best solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each entry reports: the cost function value C(Xi), the peak probability for the |Xi⟩ state(s), and the number of unique Xi solutions that result in the same C(Xi) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' is a more expected result given the complicated nature of these imperfect gaussian distributions undergoing am- plitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, although the data is not linear, there is very clearly a curved structure that could be utilized to predict ps values in the same manner de- scribed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' It is important to note that in both figures 10 and 11, the manner in which the solution states |Xi⟩ are found to be most probable is sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This means that if a particular state |Xi⟩ is most probable at a certain value ps = x, all solutions C(Xj) < C(Xi) will have peak prob- abilities at values ps < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, the bottom plot in figure 11 shows that the further a solution state is from |Xmin⟩ the lower its achievable peak probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This means that there is a limit to how many solutions are viable for amplitude amplification to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As we discuss in the coming subsections, these are the key underlying features that we must consider when constructing a vari- ational amplitude amplification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) A plot of the 400 lowest C(Xi) values as a function of ps, for the X∆ = 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 QUBO from figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each data point represents the ps value where the |Xi⟩ state(s) is most probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The red box in the lower left corner represents the ps region depicted in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) The probabilities achieved for these 400 lowest |Xi⟩ states using the ps values shown in the top plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each state is plotted in order of it’s rank from 1 (Xmin) to 400 (400th lowest C(Xi) solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Constant Iterations In order to construct an algorithm which capitalizes on the structure and probabilities shown in figure 11, we must consider an additional piece of information not il- lustrated in the figure: step 3 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 1, iterations k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The data points in the figure are indeed the ps values which produce the highest probabilities of measurements, but unfortunately they are achieved using different iteration counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In principle this means that an experimenter must decide both ps and k before each amplitude am- plification attempt, further complicating the information learned from measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Unlike ps, which is difficult to learn because it depends on the collective 2N solutions to C(X), approximating a good iteration count k is easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' It turns out that the standard number of Grover iterations kG = π 4 � N/M, where N is the total number of quantum states and M is the number of marked states, is equally applicable when Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 1460 R: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='9994 1480 1500 : C(X) -1520 1540: 1560: 1580: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='003 Ps C(X) Probability # of States 1528 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='9% 2 1 1591 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2% 1 1527 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='1% 1 1590 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2% 1 1526 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='7% 2 1553 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='3% 1 1525 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8% 1 1552 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5% 2 1524 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='1% 1 1550 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6% 1 1523 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5% 2 1549 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='3% 2 1517 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4% 1 1548 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0% 1 1516 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6% 2 1529 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='3% 2 1514 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4% 1-1100 - 1200 : 1300 - C(X) 1400 1500 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0045 Ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8- Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 上0 1 100 200 300 400 Lowest C(Xi) Solution StatesB Constant Iterations 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Plots of |Xi⟩ state probabilities as a function of ps, for the N = 25 QUBO shown in figures 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The top three panels show individual state probabilities as solid-colored lines, for three different constant k iterations (1000, 2000, and 3000) across the ps region depicted on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' An additional black-dashed line is also shown, which records the cumulative probability of the five most probable solutions |Xi⟩ at any given ps value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' These cumulative probabilities are also replotted in the bottom most panel for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' using Uc as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If an experimentor can use k ≈ kG iterations for a cost oracle Uc and find significant proba- bilities of measurment, then a variational amplitude am- plification strategy can be reduced to a single parameter problem: ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Figure 12 demonstrates that this is indeed viable, showcasing |Xi⟩ solution state probabilities as a function of ps for three different choices of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The QUBO corresponding to figure 12 is the same N = 25 example for figures 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For instances where multiple states correspond to the same numeri- cal solution (C(Xi) = C(Xj)), the solid-color line shown represents their joint probability: Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ( |Xi⟩ ) + Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ( |Xj⟩ ) (note that these individual probabilities are always equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Examples of this can also be seen in the table in- cluded in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Additionally, a black-dashed line is shown in the top three plots, tracking the collective prob- Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 - 0 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='8 2000 Probability 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0031 pC Information Through Measurements 12 ability of the five most probable solutions at any given ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' These three lines are then replotted in the bottom panel for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The ps region shown in figure 12 was chosen to il- lustrate a scenario where variational amplitude ampli- fication is most viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For ps > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00291, nearly ev- ery possible integer solution C(Xi) ≥ −1497 exists via some binary combination for this particular QUBO prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The exceptions where certain integer solutions do not exist can be seen clearly in the ps regions with very low probability, for example 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0029065 ≤ ps ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Contrast to the region shown in this figure, once ps be- comes closer to where |Xmin⟩ is maximally probable, then measurment probabilities become more akin to figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, it is more strategic for a hybrid algorithm to start in a ps region like figure 12, where measurement results can consistently yield useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Information Through Measurements From an experimental perspective, a significant result from figure 12 are the black-dashed lines shown in the top three plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' At k = 3000 (kG ≈ 4500 for 25 qubits, M = 1), the black-dashed line is almost entirely com- posed of the single most probable solution state(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' With probabilities around 70 − 80% for many of the states shown, it is realistic that the same |Xi⟩ state could be measured twice in only 2 − 4 amplitude amplification attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Two measurements yielding the same C(Xi) value (possibly from different Xi) is a strong experimen- tal indicator that the ps value used is very close to op- timal for that solution, corresponding to the data points from figures 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Confirming 3 − 4 different data points in this manner can then be used to approximate the underlying curved structure of these figures, which in turn could be used to predict ps values where |Xmin⟩ may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' While using k closer to kG is good for getting the max- imal probability out of solution states, the k = 1000 and 2000 plots in figure 12 support a different strat- egy for quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' At k = 2000, the black-dashed line is still primarily composed of the single most probable |Xi⟩ state(s), but critically it does not have the same dips in probability between neighboring solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In- stead, the cumulative probability stays just as high for these in-between ps regions, sometimes even higher!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If we now look at the k = 1000 plot, this trend becomes even more prevelant, whereby the cumulative probability plot is on average 20 − 30% higher than any individual |Xi⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Interestingly, the bottom panel of figure 12 shows that cumulative probability plot for k = 1000 is higher than the k = 3000 line in many regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, if the role of quantum is to simply provide a heuristic answer [51], not necessarily |Xmin⟩, then using lower k values is favorable for a few reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Firstly, we can anticipate solutions in a ps region where multiple states share the same cost function value, so one can expect M > 1 more frequently when using kG = π 4 � N/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Secondly, the am- plitude amplification process itself is faster due to smaller k, which makes it more achievable on noisy qubits due to shallower circuit depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The optimal use of k is a non-trivial challenge to an experimentor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, as illustrated in figure 12, ampli- tude amplification can still be effective with a wide range of different k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' To further demonstrate this, figure 13 shows three plots of simulated measurements over the ps range depicted in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Using the k values 1000, 2000, and 3000, each plot shows data points representing probabilistic measurements at regular intervals of ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In order to compare the k value’s effectiveness more equally, the number of measurements taken per ps value, t, was chosen such that t·k = 12000 is consistent across all three experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, each of the three plots in figure 13 represents the same total number of amplitude amplifi- cation iterations divided among t experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The data points shown in figure 13 are separated into two categories, which are easily recognizable from an experimental perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Measurements which yielded C(Xi) < −1350 are plotted as red circles, while all other measurements are plotted as black triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As illus- trated for all three values of k, the red data points can be seen as producing near linear slopes, all of which would signal to the experimenter that these measurement re- sults are leading to Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The motivation for figure 13 is to demonstrate that the same underlying information can be experimentally realized using different k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, when to use k = 3000 versus k = 1000 is a matter of optimization, which we discuss in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' as the role of a classical optimizer for a hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Quantum Verification The results of the previous subsections demonstrate the capacity for amplitude amplification as a means for finding a range of optimal Xi solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, regard- less of whether these solutions are found via quantum or classical, a separate problem lies in verifying if a given solution is truly the global minimum Xi = Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If it is not, then Xi is refered to as a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Classically, evolutionary (or genetic) algorithms [52–55] are one ex- ample strategy for overcoming local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Similarly, quantum algorithms have also demonstrated success in this area for both annealing [56, 57] and gate-based [58– 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The strategy for verifying a local versus global mini- mum using amplitude amplification can be seen by com- paring the region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0029 ≤ ps ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00291 in figures 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For the linear QUBO corresponding to these figures, there exists a solution C(Xi) = −1497 which becomes maximally probable at ps ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002914, followed by the next lowest solution C(Xi) = −1491 at ps ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='002892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Because there are no binary combinations Xi that can produce values −1492 ≥ C(Xi) ≥ −1496, the ps region that would correspond to their solutions instead produces Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 D Quantum Verification 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Simulated measurement results corresponding to the probabilities shown in figure 12, produced by amplitude amplification for various values of ps (x-axis) and k (1000, 2000, and 3000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' At each of the ps values simulated, the number of measurements per experiment t was chosen based on k as follows (t,k): (4,3000) , (6,2000) , (12,1000), such that t · k = 12000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Measurement results which yielded C(Xi)< −1350 are plotted as red circles, otherwise as black triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Blue lines for C(Xmin) and C(Xmax) are plotted as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' nothing measurably significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This can be seen by the low cumulative probabilities in figure 12, as well as ex- perimentally in figure 13 as a gap in red data points for this ps region across all three simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The ability for quantum to determine if an Xi solution is locally or globally minimum is achieved by searching past the ps value corresponding to the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Doing so will result in one of two outcomes: either a lower C(Xj) value will be probabilistically found (confirming Xi was a local minimum), or the experimenter will only find ran- dom measurement results (Xi was the global minimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Examples of this can be seen in figure 14, showcasing sim- ulated measurement results as an experimenter searches past the optimal ps value for |Xmin⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The simulated experiments shown in figure 14 were chosen to highlight both favorable (bottom) and unfavor- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Simulated measurement results for ps regions above and below the optimal point for finding |Xmin⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each plot cor- responds to a different linear QUBO of size N = 25, k = 4000, with X∆ values reported for each (top plot corresponds to the QUBO from figures 9 - 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The point where Xmin is mea- sured is indicated in both plots by the intersection of the blue (horizontal) and grey (vertical) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Red-circle data points represent measurement results within the best 30 minimizing solutions to C(X), otherwise as black triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' able (top) cases for quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The commonality between both experiments is that there is a clear point in ps (grey line) in which decreasing ps further results in only noisy random measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, determining this cut- off point using measurement results alone is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The top plot corresponds to the QUBO from figures 10 12, which is the non-ideal situation in which there are significant gaps in solutions between the best 20 minimiz- ing C(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Experimentally this manifests as numerous ps regions that could be wrongly interpreted as the Xmin cutoff point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Conversely, the bottom plot represents the ideal case where the best minimizing C(Xi) solutions are all closely clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This leads to a much more consistent correlation of measurement results leading to Xmin, followed by an evident switch to randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The significance here is that amplitude amplification has an experimentally verifiable means for identifying the global minimum Xmin of a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Similarly, the same methodology can be in principle used to check for the existence of an Xi solution corresponding to any given cost function value, which we discuss further in section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, the obvious drawback is that this verifi- cation technique relies on numerous amplitude amplifica- tion measurements finding nothing, which costs further runtime as well as being probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As we discuss in Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 3000 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 0 250· 500 750 1000 1250 1500 2000 250 0 250 500 750 1000 - 1250 1500 1000 250 0 250 500 C(X) 750 1000· 1250 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='00305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0031Xa= 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 500 1000 1500 X min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0029 Xa= 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='5 400 C(X) 800 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='0034 psD Quantum Verification 14 the next section, a more realistic application of this quan- tum feature is to help steer a classical algorithm past lo- cal minima, leaving the veification of Xmin as joint effort between both quantum and classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' HYBRID SOLVING The results of section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' were all features of amplitude amplification using Uc that were found through classical simulations of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' They represent the pri- mary motivation of this study, which is to demonstrate amplitude amplifaction’s potential and the conditions for which it can be experimentally realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' By contrast, the discussions of section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' here are more speculative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Given all of the results from sections III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' - V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', we now discuss how the strengths and weaknesses of amplitude amplification synergize with a parallel classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The plots shown in figures 13 and 14 represent a very non-optimal approach to finding Xmin, functionally a quantum version of an exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' If the ultimate goal is to solve a cost function problem as quickly as pos- sible, then it is in our best interest to use any and all tools available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This means using a quantum computer when it is advantageous, and similarly also recognizing when the use of a classical computer is more appropri- ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In this section we discuss this interplay between quantum and classical, and the situations in which an experimenter may favor one or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown below in figure 15 is the general outline of a variational ampli- tude amplification model which relies solely on quantum to produce Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The general outline of a variational amplitude am- plification workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Information from amplitude amplifica- tion in the form of measurements is fed to a classical optimizer between runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The optimizer then processes this information to supply the quantum computer with the next set of values ps and k, repeating this process until Xmin or another suitable solution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Given the current state of qubit technologies [61–63], performing one complete amplitude amplification circuit should be considered a scarce resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As such, it is the role of a classical optimizer to determine the most effective use of this resource, choosing ps and k values which will probabilistically get the most value out of each attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Determining optimal values to adjust a quan- tum circuit is the typical hybrid strategy found among other popular variational models of quantum computing [40, 41, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The majority of the computational workload is placed on the QPU (quantum processing unit), while a classical optimizer is used in between runs to adjust quantum circuit parameters accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As evidenced by figures 13 and 14, this model is possible for amplitude amplification as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' However, there is a different model of hybrid computing which better utilizes both quantum and classical’s strengths, shown below in figures 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Workflow of a hybrid model of computing, utilizing both a quantum and classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Both the QPU and CPU are run in parallel, and the information obtained from both are fed into the same classical optimizer, which in turn determines the most effective use for each processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The advantage to hybrid computing using the model shown in figure 16 is that both processors are working in tandem to solve the same problem, utilizing infor- mation gained from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Information obtained through amplitude amplification measurements can be used to speedup a classical algorithm, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As we discuss further in the next subsection, this pairing of quantum and classical is maximally advantageous when the strengths of both computers compliment each other’s weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Supporting Greedy Algorithms One notable strength of classical computing is ‘greedy’ algorithms, which oftentimes provide heuristic solutions for use cases ranging from biology and chemistry [51, 64] to finance [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Greedy algorithms are particularly vi- able for problems that possess certain structures which can be exploited [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The key feature to these algo- rithms is that they focus on making locally optimal de- cisions which yield the maximal gain towards being op- timal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Consequently, they are very good at finding near optimal solutions quickly, but are also prone to getting bottlenecked into local minima [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The motivation for pairing amplitude amplification with a classical greedy algorithm is best exemplified by figures 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The quantum states illustrated in these figures represent |Xi⟩ states which rank as the 30th−80th Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 Quantum Classical Perform Amplitude Evaluate C(X) with 1) Amplification: Ps & k all previous results 2) Measure | X: ) 2) Determine new ps & kOptimizer Evaluate C(X) values 1) from both QPU & CPU Supply both processors 2) with new parameters Quantum Classical Perform Amplitude Amplification: ps & k Perform best optimization algorithm 2) Measure X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' >A Supporting Greedy Algorithms 15 best minimizing solutions to C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Under the right condi- tions it is reasonable to expect that a quantum computer could yield a solution in this range within 1 − 5 ampli- tude amplification attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The question then becomes how quickly a classical greedy algorithm could achieve the same feat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Without problem specific structures to exploit, and as problem sizes scale like 2N, it becomes increasingly unlikely that classical can compete heuristi- cally with quantum, which we argue is quantum’s first advantage over classical in a hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Now, supposing that amplitude amplification does yield a low C(Xi) solution faster than classical, the prob- lem then flips back to being classically advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This is because the Xi solution provided by quantum is now new information available to the classical greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As such, beginning the greedy approach from this new binary string is likely to yield even lower C(Xi) solutions in a time frame faster than amplitude amplifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, this is the exact scenario in which ge- netic algorithms shine [52–55, 65], where a near-optimal solution is provided from which they can manipulate and produce more solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' And if a fast heuristic solution is all that is needed, then quantum’s job is done, and the best minimal solution found by the classical greedy algorithm completes the hybrid computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' But if a heuristic solution is not enough, then we can continue to use a hybrid quantum-classical strategy for finding Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Referring back now to figures 13 and 14, the strategy for quantum is to use multiple amplitude amplification trials and measurements to approximate the underlying correlation from figures 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The fastest means for achieving this is to work in a ps re- gion analogous to figure 12, where experimentally one has the highest probabilities of measuring useful infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Simultaneously, the classical greedy algorithm can also find Xi solutions in this area as it searches for Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Knowledge of these Xi solutions can be directly fed back to quantum, which can be used to predict ps values where solutions are known to exist, speeding up the process of determining a ps vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, after quantum initially aided classical, subsequent information obtained from classical is then used to speed up quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In the time it takes for quantum to experimentally ver- ify enough ps and C(Xi) values to create a predictive cor- relation, we expect the classical algorithm to find a new lowest C(Xi) solution, labeled X’i in figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' After in- vesting additional trials into the amplitude amplification side of the computation, it is now time for quantum’s second advantage: verifying local versus global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Using an approximate ps vs C(X) best-fit, the quantum computer can skip directly to the ps value corresponding to best currently known X’i solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' As discussed in sec- tion V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='D, searching past this ps value will result in one of two outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Either the quantum computer will find a new best solution C(Xj) < C(X’i), or confirm that X’i is indeed the global minimum Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In the former case, the greedy algorithm now starts again from the new lowest solution Xj, repeating this cycle between quantum and classical until Xmin is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Figure 17 below shows a workflow outline of this hyrbid strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Workflow for a hybrid model of computing between quantum amplitude amplification and a classical greedy algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The full strategy is broken up into three phases: 1) Amplitude amplification provides the first heuristic solution Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 2) A classical greedy algorithm uses Xi to find a more optimal solution X’i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Simultaneously, other near optimal so- lutions Xj are used to assist amplitude amplification in de- termining a ps vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) correlation (see figures 10 - 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 3) The correlation best-fit is used to predict ps values where so- lutions C(Xj) < C(X’i) must exist (or C(Xj) > C(X’i) for maximization problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Amplitude amplification attempts for these ps values will either produce a new best Xj for the greedy classical algorithm to use, or confirm X’i = Xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The biggest advantage to using a hybrid model like shown in figure 17 is that it can be adapted to each prob- lem’s uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Problems with known fast heuristic techniques can lean on the classical side of the computa- tion more heavily [68, 69], while classically difficult prob- lems can put more reliance on quantum [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' But above all else, this model of computation incorporates and synergizes the best known classical algorithms with quantum, rather than competing against them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' MORE ORACLE PROBLEMS All of the results from sections III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' - V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' were derived from linear QUBOs according to equations 1 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' How- ever, these results can be applied to more challenging and realistic optimization problems provided that 1) all Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 Quantum Classical Amplitude Amplification Fast Sample: compute ps in estimated ps region for a fast heuristic solution X, Greedy algorithm starting 1) from solution X, Continue amplitude amplification using known X, solutions Record all new solutions to create p, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) correlation 2) X, better than X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=', and the new best solution X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=" X Search p, values for Greedy algorithm starting solutions C(X,) < C(X'): 1) from new solution X, 2A) Find new solution X Record new or 2) 2B) best solution X Confirm X', is X." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' min16 possible solutions can be encoded via phases by an ap- propriate oracle operation Uc, and 2) the distribution of all possible answers is suitable for boosting the solution we seek (gaussian, polynomial, exponential, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Here we will briefly note some additional optimization problems which meet both of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Weighted & Unweighted Max-Cut The Maximum Cut problem (‘Max-Cut’) is famously NP-Hard [70], where the objective is to partition every vertex in a graph S into two subsets P1 and P2 such that the number of edges between them is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In the weighted Max-Cut version of the problem, each edge is given a weight wij, and the goal is to maximize the sum of weights contained on edges between P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The unweighted Max-Cut problem has already been demonstrated as a viable use for amplitude amplifica- tion [17], which we will build upon further here via the weighted version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Equation 31 below is the cost function C(X) for the weighted Max-Cut problem, which can be converted to the unweighted case by setting every edge weight wij = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The binary variables xi here represent being partitioned into P1 or P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) = � {i,j}∈S wij|xi − xj| (31) Shown in figure 18 is an example graph S and one of its solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' This example graph is composed of 10 vertices, labeled 1 - 10, and a total of 15 connecting edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Encoding this graph requires one qubit per vertex, where the basis states |1⟩ and |0⟩ represent belonging to the subsets P1 and P2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' See the bottom graph in figure 18 for an example solution state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The cost oracle Uc for solving Max-Cut must correctly evaluate all 2N solution states |Xi⟩ based on the edges of S according to equation 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, if vertices 1 and 2 are partitioned into different sets, then Uc needs to affect their combined states |Q1Q2⟩ = |01⟩ and |10⟩ with the correct phase, weighted or unweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Just like figure 3 from earlier, we can achieve this with a control- phase gate CP(θ),with the intent of scaling by ps later (see figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The caveat here is that we need this phase on |01⟩ and |10⟩, not |11⟩, which means that additional X gates are required for the contruction of Uc, shown below in equation 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' X = � 0 1 1 0 � (32) For the complete Uc quantum circuit which encodes the graph S in figure 18, please see appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Once properly scaled by ps, the solutions which are capable of boosting are determined by the underlying solution space distribution of the problem, which can be seen in figure FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) A graph S composed of 10 nodes and 15 connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Each node is labeled 1 - 10, corresponding to the qubits Q1 - Q10 shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) An example Max-Cut solution Xi, along with its quantum state representation |Xi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Nodes colored red correspond to the partition P1, quantum state |1⟩, while nodes colored white correspond to partition P2, quantum state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' ‘Cuts’ are represented in the graph as dashed lines, totaling 8 for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 19 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The histogram in this figure shows all 210 C(Xi) solutions to the graph S from figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Even for a 10 qubit problem size such as this, it is clear that the underlying solution space distribution shows gaussian- like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' A histogram of all 210 solutions for an unweighted Max-Cut on graph S from figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' One interesting feature of Max-Cut is that all solutions come in equal and opposite pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For example, the op- timal solutions to S from figure 19 are |0100101110⟩ and |1011010001⟩, which both yield 13 ‘cuts’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Mathematically there is no difference between swapping all vertices in P1 Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 3 S: 10 [Xi> = IQQ2::·Q10) 8 [X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='>=|1001001101) P, = 1,4,7,8,10] P2 = {2,3,5,6,9]200- 150 pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (C(X) 100- 50- 00 2 8 6 10 4 12 C(X)A Weighted & Unweighted Max-Cut 17 and P2, but physically it means that there are always two optimal solution states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Consequently, these states will always share the effect of amplitude amplification together, which is something an experimenter must be aware of when choosing iterations k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Finally, moving from the unweighted to weighted ver- sion of Max-Cut increases the problem’s difficulty, but notably does not change the circuit depth of Uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Rather than using θ = 1 for all of the control-phase gates, each θ now corresponds to a weighted edge wij of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Similar to the QUBO distributions shown in figure 7, this increase in complexity allows for more distinct C(Xi) so- lutions, and consequently more variance in features such as σ′ and X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Graph Coloring A similar optimization problem to Max-Cut is Graph Coloring, also known as Vertex Coloring [70], which ex- tends the number of allowed partition sets Pi up to any integer number k (k = 2 is equivalent to Max-Cut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Given a graph of vertices and edges S, the goal is to assign every vertex to a set Pi such that the number of edges between vertices within the same sets is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown below in equation 33 is the cost function C(X) for a k-coloring problem, where the values of each vertex xi are no longer binary, but can take on k different integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' The quantity inside the parentheses is equal to 1 if xi = xj, and 0 for all other combinations xi ̸= xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Just like with Max-Cut, setting all wij = 1 is the unweighted version of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' C(X) = � {i,j}∈S wij � 1 − �|xi − xj| k �� (33) The name ‘coloring’ is in reference to the problem’s origins, whereby the sets Pi all represent different colors to be applied to a diagram, such as a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Shown below in figure 20 is an example picture composed of overlap- ping shapes, where each section must be assigned one of k colors such that the number of adjacent sections with the same color is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Example solutions for k = 3 and k = 4 are shown, along with their vertex and quan- tum state representations of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' In order to encode graph coloring as an oracle Uc, the choice of k determines whether qubits or another form of quantum computational unit is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' While qubits are capable of producing superposition between two quantum states, qudits are the generalized unit of quantum information capable of achieving superposition between d states [72–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' To see why this is necessary, let us compare the k = 3 and 4 examples from figure 20, and the quantum states needed to represent partitioning each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For k = 4, we need four distinct quantum states to represent a vertex belonging to one of the Pi partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (top) On the left, a two dimensional bounded picture with overlapping geometric shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' On the right, a graph S representing the 12 distinct regions of the picture as nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Connections between nodes in S represent regions in the picture which share a border, not counting adjacent corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (middle) A k = 3 coloring example, with a corre- sponding d = 3 qudit state representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' (bottom) A k = 4 coloring example, with a corresponding d = 4 qudit state rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' While a single qubit can’t do this, a pair of qubits can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Thus, every vertex in S can be encoded as a pair of qubits, letting the basis states |00⟩, |01⟩, |10⟩, and |11⟩ each rep- resent a different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Alternatively, we could use a d = 4 qudit to represent each vertex, assigning each par- tition a unique basis state: |0⟩, |1⟩, |2⟩, or |3⟩, such as the state shown in figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Mathematically the two encod- ings are identical, so the choice between whether to use qubits or qudits is a matter of experimental realization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' which technology is easier to implement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' For k = 3 however, two qubits is too many states, and a single qubit is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' So in order to represent three colors exactly in quantum, the appropriate unit is a ‘qutrit’ (the common name for a d = 3 qudit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Simi- larly, all prime numbers d can only be encoded as their respective d-qudit, while all composite values can be built up from combinations of smaller qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Once an appro- priate mixed-qudit quantum system is determined, con- Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content=' Distribution Unlimited: PA#: AFRL 2023-0204 S: 4 11 2 8 3 5 10 6 [Xi>=IQiQ2···Q12) k 二 =<0 11>= X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFRT4oBgHgl3EQf3jjV/content/2301.13665v1.pdf'} +page_content='>=1010212010120) K 10>= rkP(F) for +all a ∈ E \ F. The flats of P form a lattice, denoted LP. The loops of a polymatroid are the +elements of the minimal flat. We say that a polymatroid is loopless if the empty set is a flat, or +equivalently, if rkP(i) > 0 for all i ∈ E. Given a flat F of P, the subset π−1(F) ⊆ E is a flat of the +multisymmetric lift Mπ(P). Flats of Mπ(P) of this form are called geometric flats of Mπ(P). The +key property of geometric flats is the following. +Proposition 3.3. [CHL+, Lemma 2.8] Every flat F of Mπ(P) contains a unique maximal geomet- +ric flat F geo. We have that rkMπ(P)(F geo) = rkP(π(F)), and rkMπ(P)(F) = rkMπ(P)(F geo) + |F \ +F geo|. + +INTERSECTION THEORY OF POLYMATROIDS +11 +Remark 3.4. As in Remark 2.4, let Γa be the product � +i∈E Sπ−1(i) of permutation groups. The +terminology “multisymmetric” is justified by the fact that the obvious action of the group Γa on +E preserves the rank function of Mπ(P). In fact, this property characterizes multisymmetric lifts: +[CHL+, Theorem 2.9] states that a matroid Mπ on E such that the action of Γa preserves the rank +function is of the form Mπ(P) for a polymatroid P of type a.1 Moreover, the map F �→ π−1(F) +induces an isomorphism from the lattice LP of flats of P to the lattice of Γa-fixed flats of Mπ(P) +[CHL+, Corollary 2.7]. +We now discuss polymatroid duality, see, e.g., [McD75]. Our main conclusion is that taking +multisymmetric lift commutes with polymatroid duality. +Definition 3.5. For a polymatroid P on E of type a and rank r, its dual polymatroid P⊥ is a +polymatroid on E of type a and rank n − r whose rank function is +rkP⊥(S) = +� +i∈S +ai + rk(E \ S) − r. +Alternatively, duality can also be described via polytopes as follows. The rank function de- +scription for P⊥ above implies that +B(P⊥) = −B(P) + a, +or, equivalently, since I(P) = {x ∈ � +i∈E[0, ai]: y − x ∈ RE +≥0 for some y ∈ B(P)}, we have +−I(P⊥) + a = {x ∈ � +i∈E[0, ai]: x − y ∈ RE +≥0 for some y ∈ B(P)}. +FIGURE 1. Polytopes associated to a polymatroid and its dual. +When P is realized by L ⊆ V = � +i∈E Vi, its dual P⊥ is realized by (V/L)∨ ⊆ � +i∈E V ∨ +i +obtained by dualizing the surjection V ↠ V/L. When a = (1, . . . , 1), polymatroid duality agrees +with the usual notion of matroid duality. +Proposition 3.6. For a polymatroid P on E of type a, one has Mπ(P⊥) = Mπ(P)⊥. +Proof. This follows from Lemma 3.2 since B(P⊥) = −B(P) + a and pπ(� +j∈E ej) = a. +□ +1In the proof of this theorem, the authors of [CHL+] make the additional assumption that rkP(i) = ai, but this +assumption is never used. + +a +-I(P-)+a +B(P) +I(P)12 +CHRISTOPHER EUR AND MATT LARSON +3.2. Augmented Bergman fans of polymatroids. Let P be a polymatroid on E of type a. We +now introduce the augmented Bergman fan ΣP of a polymatroid. +Definition 3.7. The augmented Bergman fan ΣP of P is the subfan of Σa consisting of cones σS≤F, +where S is a subset of E and F = {F1 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} is a chain of proper flats of P +satisfying +(1) For all T ⊆ S, one has rkP(π(T)) ≥ |T|, and +(2) for all F ∈ F and all T ⊆ S \ π−1(F), one has rkP(F ∪ π(T)) > rkP(F) ∪ |T|. +When a = (1, . . . , 1), that is, when P is a matroid M on E, the augmented Bergman fan of P +coincides with the augmented Bergman fan ΣM introduced in [BHM+22]. Explicitly, the fan ΣM +is the subfan of the stellahedral fan ΣE consisting of cones σI≤F where I ⊆ E is an independent +set of M and F = {F1 ⊊ · · · ⊊ Fk ⊊ E} is a chain of proper flats of M such that I ⊆ F1. +Theorem 3.8. The augmented Bergman fan ΣP of P is the subfan of Σa whose support equals the +support of the augmented Bergman fan ΣMπ(P) of the multisymmetric lift of P. More precisely, +ΣP is the coarsening of the fan ΣMπ(P) such that it is a subfan of Σa. +This is the key property of ΣP that we will repeatedly use. The rest of this subsection is +dedicated to the proof of the theorem. +Let us prepare by reviewing the theory of building sets; for proofs and details we point to +[DCP95, FS05]. A building set on a loopless matroid M on ground set E is a collection G of +nonempty flats of M such that, for all nonempty flats of F of M, the natural map of lattices +� +G∈max G≤F +[∅, G] → [∅, F] +is an isomorphism. Here, max G≤F denotes the maximal elements of G contained in the interval +[∅, F] ⊆ LM. All building sets that we consider will contain the maximal flat E. A nested set +is a subset N ⊆ G that does not contain E such that, for all pairwise incomparable subsets +{F1, . . . , Fk} ⊆ N with k ≥ 2, the join �k +i=1 Fi of {F1, . . . , Fk} is not in G. Nested sets form +a simplicial complex, which is realized as a simplicial fan ΣM,G in RE/ReE whose cones are +{image in RE/ReE of cone{ei : i ∈ N} ⊂ RE : N a nested set}. We call ΣM,G the Bergman fan +of M with respect to the building set G. The support of ΣM,G does not depend on the choice of +building set [FY04, Theorem 4], and ΣM,G is always a unimodular fan [FY04, Proposition 2]. +We prove Theorem 3.8 by identifying the fan ΣP with a Bergman fan of a matroid closely +related to the multisymmetric lift Mπ(P). Let Mπ(P) × 0 denote the free coextension of the multi- +symmetric lift Mπ(P), which is a matroid on the ground set E ⊔ {0} with flats +{F ∪ 0: F ⊆ E flat of Mπ(P)} ∪ {I ⊆ E: I independent in Mπ(P)}. +Note that Mπ(P) × 0 is always loopless. We now define a building set on Mπ(P) × 0 whose +Bergman fan will be the augmented Bergman fan of P. + +INTERSECTION THEORY OF POLYMATROIDS +13 +Lemma 3.9. Let G be the set of all flats of Mπ(P) × 0 of the form F ∪ {0} for F a geometric flat +of Mπ(P), or {j} for j ∈ E not a loop of Mπ(P). Then G is a building set. +Proof. Consider a flat of Mπ(P) × 0 of the form H ∪ 0 for H a flat of Mπ(P). By Lemma 3.3, H +contains a unique maximal geometric flat Hgeo, and, for any subset S with Hgeo ⊆ S ⊆ H, we +have that rkP(S) = rkP(Hgeo) + |S \ Hgeo|. This implies the desired decomposition for H ∪ 0. If +we have a flat of Mπ(P)×0 of the form I for I ⊆ E independent, then the desired decomposition +is automatic. +□ +Before computing the nested sets of G, we need a preparatory lemma. +Lemma 3.10. Let F be a geometric flat of a multisymmetric matroid Mπ(P), and let S be a subset +of F such that |S| ≥ rkMπ(P)(F) or |S| > rkMπ(P)(F). Then there is a geometric flat G of Mπ(P) +and a subset S′ ⊆ S ∩ G such that |S′| ≥ rkMπ(P)(G) (respectively |S′| > rkMπ(P)(G)) and S′ +spans G. +Proof. We do the case when |S| > rkMπ(P)(F), the other case is identical. We induct on the rank +of F; if rkMπ(P)(F) = 0 then the claim is obvious. Let H be the closure of S. Using Lemma 3.3, +we have that +rkMπ(P)(H) = rkMπ(P)(Hgeo) + |H \ Hgeo| ≥ rkMπ(P)(Hgeo) + |S| − |S ∩ Hgeo|. +On the other hand, we have that rkMπ(P)(H) ≤ rkMπ(P)(F) < |S|, so rkMπ(P)(Hgeo) < |S ∩Hgeo|. +Either Hgeo = F and we are done, or we conclude by induction. +□ +Lemma 3.11. With G as in Lemma 3.9, the nested sets of G are given by chains of flats F = {F1 ⊊ +· · · ⊊ Fk ⊊ Fk+1 = E} of P and a subset S of the non-loops of P such that: +(1) For all T ⊆ S, rkP(π(T)) ≥ |T|, and +(2) for all F ∈ F and all T ⊆ S \ π−1(F), rkP(F ∪ π(T)) > rkP(F) ∪ |T|. +Proof. Let S and F = {F1 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} be pair satisfying the two condition of +the lemma. We check that the corresponding set is nested. The incomparable subsets are either +given by a collection T ⊆ S, or a flat F ∈ F and S ⊆ S \ π−1(F). +The closure of T ⊆ S in Mπ(P) × 0 is T if T is independent, and it is clMπ(P)(T) ∪ 0 if T is +dependent. In the first case, T is not in G if |T| > 1. If T is dependent, then (1) guarantees that +rkMπ(P)(T) < rkP(π(T)), so the closure is not in G. Similarly, if we have T ⊆ S \ π−1(F), then +the closure of F ∪ T cannot be geometric. +Now let N be a nested set, which consists of a subset S of the non-loops of Mπ(P) and flats +of the form F ∪ 0 for F a geometric flat. As the join of two geometric flats is a geometric flat, the +flats of the form F ∪ 0 must form a chain. +Suppose there is a subset T ⊆ S with rkP(π(T)) < |T|. Let F = π−1(clP(π(T))), which is a +geometric flat containing T of rank less than |T|. By Lemma 3.10, there is T ′ ⊆ T and a geometric +flat G such that T ′ spans G and |T ′| > rkMπ(P)(G). Then the closure of T ′ in Mπ(P) × 0 is G ∪ 0, +contradicting that N is nested. + +14 +CHRISTOPHER EUR AND MATT LARSON +Now suppose that there is F ∈ F and T ⊆ S \ π−1(F) with rkP(F ∪ π(T)) ≤ rkP(F) + |T|. +Let G = π−1(clP(π(F) ∪ T)). Applying Lemma 3.10 to the contraction Mπ(P)/π−1(F), we find +a geometric flat H ⊃ F and T ′ ⊆ T ∩ H such that T ′ ∪ π−1(F) spans H. This contradicts that N +is nested. +□ +Proof of Theorem 3.8. Let H be the building set on the lattice of flats of Mπ(P) × 0 given by view- +ing Mπ(P) viewed as a polymatroid of type (1, . . . , 1). By [FY04, Theorem 4] the support of +ΣMπ(P)×0,G coincides with the support of ΣMπ(P)×0,H. By [EHL, Lemma 5.14], under the iso- +morphism RE → RE∪0/R obtained by sending ej to ej, the support of ΣMπ(P)×0,H coincides +with the support of ΣMπ(P). Under this isomorphism, ΣMπ(P)×0,G is identified with ΣP. +□ +3.3. Augmented Bergman classes of polymatroids. We begin by reviewing briefly balanced +fans and their Chow homology classes; for details and proofs we point to [FS97] and [AHK18, +Section 5]. +A pure-dimensional simplicial rational fan Σ of dimension d is balanced if for any cone τ ∈ Σ +of codimension 1, one has � +σ⊋τ uσ\τ ∈ τ, where uσ\τ denotes the primitive vector of the unique +ray in σ that is not in τ. Suppose a balanced fan Σ is a subfan of a complete unimodular fan �Σ. +Let Ad(X�Σ) be the d-th graded piece of the Chow ring of the toric variety X�Σ, which is spanned +by {[Zσ]: σ a d-dimensional cone in �Σ}, where Zσ is the torus-orbit closure in X�Σ corresponding +to σ. One then obtains a linear functional wΣ ∈ Hom(Ad(X�Σ), Z) determined by wΣ([Zσ]) = 1 if +σ ∈ �Σ and wΣ([Zσ]) = 0 otherwise. By the Poincaré duality property of the Chow ring A•(X�Σ), +the functional wΣ defines an element [Σ] ∈ Ad(XΣ). +Returning to polymatroids, let P be a polymatroid on E of type a and rank r. As the support +of the augmented Bergman fan ΣP coincides with the support of a Bergman fan, [GS21, Theorem +3.8] implies that ΣP is a balanced subfan of the polystellahedral fan of type a +Definition 3.12. The augmented Bergman class of P is the Chow homology class [ΣP] ∈ Ar(Xa) +obtained by considering ΣP as a balanced subfan of the polystellahedral fan of type a. +We will repeatedly use the following relation between the classes associated to a polymatroid +and its multisymmetric lift. Recall the birational map u: XE → Xa induced by refinement of +respective fans (Proposition 2.2). +Lemma 3.13. The pullback u∗[ΣP] is equal to the augmented Bergman class [ΣMπ(P)] of the +multisymmetric lift. +Proof. The lemma follows from applying the formula [FS97, Corollary 3.7] for computing pull- +backs in terms of Minkowski weights to Proposition 2.2 and Theorem 3.8. +□ +We use the lemma to compute how augmented Bergman classes of polymatroids multiply as +elements in the Chow ring A•(Xa). We will need the following combinatorial notions. +Given two polymatroids P1 and P2 on E of type a, we define the polymatroid union P1 ∨ P2 to +be the polymatroid of type a whose independence polytope is (I(P1)+I(P2))∩� +i∈E[0, ai]. That + +INTERSECTION THEORY OF POLYMATROIDS +15 +this is indeed the independence polytope of a polymatroid follows from [Edm70, (35)]. Define +the polymatroid intersection of P1 and P2 to be P1 ∧ P2 := (P⊥ +1 ∨ P⊥ +2 )⊥. If we view Mπ(Pi) as a +polymatroid of type (1, . . . , 1), by Lemma 3.2 we have that Mπ(P1) ∨ Mπ(P2) = Mπ(P1 ∨ P2). +Therefore Mπ(P1) ∧ Mπ(P2) = Mπ(P1 ∧ P2) by Proposition 3.6. +Theorem 3.14. Let P1 and P2 be polymatroids of type a and ranks r1 and r2, respectively. Then, +we have +[ΣP1] · [ΣP2] = +� +� +� +[ΣP1∧P2] +(n − r1) + (n − r2) = n − rank(P1 ∧ P2) +0 +otherwise. +When a = (1, . . . , 1), the above theorem is [EHL, Theorem 1.6]. Our proof is a reduction to +this case. +Proof. Applying Lemma 3.13 and using that Mπ(P1)∧Mπ(P2) = Mπ(P1 ∧P2), one obtains from +[EHL, Theorem 1.6] that +u∗[ΣP1] · u∗[ΣP2] = +� +� +� +u∗[ΣP1∧P2] +(n − r1) + (n − r2) = n − rank(P1 ∧ P2) +0 +otherwise. +The result now follows from the injectivity of u∗ (Lemma 2.12). +□ +Corollary 3.15. The augmented Bergman classes of polymatroids of type a span A•(Xa) as an +abelian group. +Proof. Recall that A•(Xa) is generated as a ring by the simplicial generators {hS}, and in par- +ticular, the monomials in the {hS} span A•(Xa) as an abelian group. By Theorem 3.14, we are +done once we show that each simplicial generator hS is an augmented Bergman class. +For each nonempty subset S ⊆ E, let HS be the polymatroid on E of type a whose dual +polymatroid has the simplex ∆0 +S as its independence polytope. By Proposition 3.6, the multi- +symmetric lift Mπ(HS) is the matroid on E whose unique circuit is π−1(S). In [EHL, Section 7.2], +it is shown that the augmented Bergman class of this matroid is equal to hπ−1(S) ∈ A1(XE). We +thus conclude [ΣHS] = hS by Lemma 2.12 and Lemma 3.13. +□ +Remark 3.16. Arguing similarly as in [EHL, Section 7.2], one can show that the set of monomials +{hd1 +F1 · · · hdk +Fk : ∅ ⊊ F1 ⊊ · · · ⊊ Fk ⊆ E, d1 ≤ |π−1(F1)| and di ≤ |π−1(Fi\Fi−1)| for all 2 ≤ i ≤ k} +form a Z-basis for A•(Xa). Moreover, combining with Theorem 3.14, one can further show that +these monomials are equal to the augmented Bergman classes of polymatroids whose multi- +symmetric lifts are Γa-fixed Schubert matroids on ground set E. In particular, A•(Xa) is generated +by the augmented Bergman classes of realizable polymatroids of type a. This basis can also be +obtained from the techniques of [DF10] and Theorem 1.6. + +16 +CHRISTOPHER EUR AND MATT LARSON +3.4. Augmented Chow rings of polymatroids. This subsection records the properties of the +augmented Chow ring of a polymatroid, but is not logically necessary for subsequent sections +of this paper. +Theorem 3.17. Let ℓ ∈ A1(XΣP) be an element corresponding to a strictly convex piecewise +linear function on ΣP. Then the following hold: +(1) (Poincaré duality) There is an isomorphism degP : Ar(XΣP) → Z such that, for 0 ≤ k ≤ +r/2, the pairing +Ak(XΣP) × Ar−k(XΣP) → Z, +(x, y) �→ degP(xy) +is unimodular. +(2) (Hard Lefschetz) For every 0 ≤ k ≤ r/2, the map +Ak(XΣP) ⊗ Q → Ar−k(XΣP) ⊗ Q, +x �→ ℓr−2kx +is an isomorphism. +(3) (Hodge-Riemann) For every 0 ≤ k ≤ r/2, the bilinear form +Ak(XΣP) ⊗ Q × Ak(XΣP) ⊗ Q → Q, +(x, y) �→ (−1)k degP(ℓr−2kxy) +is positive definite on the kernel of multiplication by ℓr−2k+1. +Proof. The support of ΣP is the same at the support of the Bergman fan of Mπ(P) × 0. The result +then follows from [ADH22, Theorem 1.6] and [AHK18]. For more details, see [CHL+, Proof of +Corollary 4.7]. +□ +As XΣP is a subvariety of Xa, there is a restriction map A•(Xa) → A•(XΣP). We note that the +degree map of Theorem 3.17 satisfies the following version of the projection formula: for any +x ∈ A•(Xa), the degree of the image of x in A•(XΣP) is equal to the degree in A•(Xa) of x · [ΣP]. +Corollary 3.18. The kernel of A•(Xa) → A•(XΣP) is ann([ΣP]), so we may identify A•(P) with +A•(XΣP). +Proof. By Poincaré duality, an element x ∈ Ak(Xa) is in the kernel of the map to A•(XΣP) if and +only if, for all y ∈ An−r−k(Xa), deg(x · [ΣP] · y) = 0. By Poincaré duality on A•(Xa), we see that +x · [ΣP] = 0. Therefore the kernel of A•(Xa) → A•(XΣP) is ann([ΣP]). +□ +Corollary 3.19. We have that +A•(P) = Z[xF , yi : F flat, i ∈ E non-loop] +I1 + I2 + I3 + I4 +, where +I1 = ⟨xF1xF2 : F1, F2 incomparable flats⟩, +I2 = ⟨ +� +i∈S +yai +i : ai > 0, +� +ai > rkP(S)⟩, +I3 = ⟨ +� +i∈T +yai +i xF : T ∩ F = ∅, ai > 0, rkP(F ∪ T) ≤ rkP(F) + +� +ai⟩, and I4 = ⟨yi − +� +F ̸∋i +xF ⟩. + +INTERSECTION THEORY OF POLYMATROIDS +17 +Proof. As XΣP is a toric variety, its Chow ring is generated by classes corresponding to rays of +ΣP, with monomial relations coming from minimal non-faces of the simplicial complex given +by the faces of ΣP and a linear relation for each element of E. The rays of ΣP correspond to +non-loops of E and flats of P. For j1, j2 non-loops in E with π(j1) = π(j2), the relation ej1 − ej2 +implies that the corresponding divisor classes are equal. +Every non-face of the complex of cones in ΣP contains either {F1, F2} for F1, F2 incom- +parable, {j1, . . . , jk} with rkP(π(j1, . . . , jk)) < k, or {j1, . . . , jℓ, F} for π−1(F) disjoint from +{j1, . . . , jℓ} and rkP(F ∪ π({j1, . . . , jℓ})) ≤ rkP(F) + ℓ. Putting this all together implies the +result. +□ +3.5. Augmented wonderful varieties of polymatroids. We sketch the geometric origins of the +notions introduced in this section. Recall that, given a realization L ⊆ V = � +i∈E Vi of a poly- +matroid P, its augmented wonderful variety WL is the closure L in � +∅⊊S⊆E P(� +i∈S Vi ⊕ k). In +the proof of Proposition 2.3, we described Xa as a sequence of blow-ups from P(V ⊕ k) along +centers disjoint from V ⊂ P(V ⊕ k). Hence, we have a natural inclusion of V into Xa, and the +variety WL is equivalently the closure of L ⊆ V in Xa. +Proposition 3.20. Let L ⊆ � +i∈E Vi be a realization of a polymatroid P of type a. Then the +homology class [WL] is equal to [ΣP]. +Proof. Because GLa = � +i∈E GL(Vi) is connected, its action on A•(Xa) is trivial, so for any g ∈ +GLa, we have that [WL] = [g · WL] = [Wg·L]. If we choose a general g ∈ GLa, then since k +is infinite, g · L is general with respect to the (fixed) choice of isomorphisms Vi +∼ +→ kπ−1(i), so +g · L ⊆ kE is a realization of Mπ(P). +By [EHL, Corollary 5.11(3)], the homology class of the closure of g · L in XE is [ΣMπ(P)]. As +u: XE → Xa is an isomorphism over g · L, we have u∗[ΣMπ(P)] = [Wg·L]. By Lemma 3.13, +[ΣMπ(P)] = u∗[ΣP], so the result follows because u∗u∗ is the identity (Lemma 2.12). +□ +Remark 3.21. The closure of L in XΣP ⊂ Xa is WL, and the restriction map A•(XΣP) → A•(WL) +is an isomorphism. Indeed, the iterated blow-up description of WL implies that A•(WL) is +generated as a ring by the restriction of hE and the classes of strict transforms of exceptional di- +visors on WL, so the restriction map A•(Xa) → A•(WL) is surjective. As WL is the union of strict +transforms of exceptional divisors and L, the inclusion WL �→ Xa factors through XΣP. There- +fore the restriction map A•(Xa) → A•(WL) factors through A•(XΣP), so A•(XΣP) → A•(WL) +is surjective. By [GS21, Proposition 3.5], A•(WL) satisfies Poincaré duality. A surjective map +between Poincaré duality algebras of the same dimension is an isomorphism, so we conclude +by Theorem 3.17(1). +4. THE EXCEPTIONAL ISOMORPHISM +In this section, we deduce the isomorphism � +r≥0 Valr(a) ≃ � +r≥0 Ar(Xa) of graded abelian +groups in Theorem 1.6. An intermediary object is the Grothendieck ring K(Xa) of vector bun- +dles on Xa, which admits a polyhedral description as a polytope algebra. + +18 +CHRISTOPHER EUR AND MATT LARSON +4.1. The polytope algebra. Let us review the polytope algebra [McM89] and its relationship to +the K-ring of a smooth projective toric variety [Mor93], following [EHL, Appendix A]. +For a subset S ⊆ Rℓ, recall that 1S : Rℓ → Z denotes its indicator function. Let Σ be a +projective fan in Rℓ that is unimodular over Zℓ. It defines a projective toric variety XΣ. A +(lattice) polytope Q ⊆ Rℓ is said to be a (lattice) deformation of Σ if its normal fan ΣQ coarsens Σ. +Definition 4.1. Let I(Σ) be the subgroup of Z(Rℓ) generated by {1Q | Q a lattice deformation of Σ}, +and let transl(Σ) to be the subgroup of I(Σ) generated by {1Q − 1Q+u | u ∈ Zℓ}. We define the +polytope algebra to be the quotient +I(Σ) = I(Σ)/ transl(Σ). +For a lattice deformation Q, denote by [Q] its class in the polytope algebra I(Σ). The multipli- +cation in the polytope algebra is induced by Minkowski sum, that is, by [Q1] · [Q2] = [Q1 + Q2]. +As mentioned in Section 2.3, a correspondence between lattice deformations of Σ and nef toric +divisors on XΣ [CLS11, Chapter 6] associates to each lattice deformation Q a nef divisor DQ. +This identifies the polytope algebra with the K-ring as follows. +Theorem 4.2. [EHL, Theorem A.9] There is an isomorphism I(Σ) +∼ +→ K(XΣ) defined by [Q] �→ +[OXΣ(DQ)]. +This isomorphism implies that a refinement of fans induces an injection of polytope algebras. +Proposition 4.3. Let Σ and Σ′ be projective unimodular fans such that Σ refines Σ′, so a lattice +deformation Q of Σ′ is also a lattice deformation of Σ. Then, the map I(Σ′) → I(Σ) that sends +[Q] ∈ I(Σ′) to [Q] ∈ I(Σ) is injective. +Proof. Let f : XΣ → XΣ′ be the corresponding toric birational map of the toric varieties induced +by the map of fans Σ → Σ′. The given map I(Σ′) → I(Σ), under the isomorphism Theorem 4.2, +is the pullback map f ∗ : K(XΣ′) → K(XΣ). Its injectivity now follows from [CLS11, Theorem +9.2.5] and the projection formula. +□ +Applying Theorem 4.2 to the polystellahedral variety Xa, noting that deformations of the +polystellahedral fan Σa are exactly expansions of polymatroids on E (Proposition 2.7), we have +the following. +Corollary 4.4. The map sending an expanded polymatroid π∗(P) on E to [OXa(Dπ∗(P))] defines +an isomorphism I(Σa) ≃ K(Xa). +We will thus use these two notions, the polytope algebra and the K-ring, interchangeably +for the polystellahedral varieties. We will use Proposition 4.3 in conjunction with the follow- +ing method of “breaking-up” a K-class on a polystellahedral variety into smaller pieces when +considered as a K-class on the stellahedral variety. +Proposition 4.5. Let P be a polymatroid on E of rank r ≤ n. Then, the class [I(π∗(P))] ∈ I(ΣE) +equals a linear combination [I(Mπ(P))] + � +k ak[I(Mk)] where Mk’s are matroids on E of rank +strictly less than r. + +INTERSECTION THEORY OF POLYMATROIDS +19 +We will need the following lemma. +Lemma 4.6. [EHL, Lemma 7.3] An intersection of the independence polytope I(P) ⊂ RE with +an integral translate of the unit cube [0, 1]E, if nonempty, is an integral translate of I(M) for +some matroid M on E. +Proof of Proposition 4.5. By tiling RE by integral translates of the unit cube [0, 1]E, we obtain a +polyhedral subdivision of I(π∗(P)), with every cell of the subdivision being integral translates +of I(M) for some matroid M on E by Lemma 4.6. By Lemma 3.2, the polytope I(Mπ(P)) is one +of the maximal interior cells of this subdivision. All other interior cells of the subdivision are of +the form I(M) + v for 0 ̸= v ∈ ZE +≥0, which implies that such matroids M are of rank strictly less +than r since π∗(P) has rank r. +□ +4.2. The exceptional isomorphism. We now use the map u: XE → Xa to construct an excep- +tional ring isomorphism φa : K(Xa) +∼ +→ A•(Xa). Its “exceptional” nature is that it differs from +the Chern character map, which is an isomorphism ch: K(X)⊗Q → A•(X)⊗Q for any smooth +projective variety X. Similar exceptional isomorphisms appeared in [BEST, EHL, LLPP]. We +prepare by recalling the case of a = (1, . . . , 1) established in [EHL]. +Theorem 4.7. [EHL, Theorem 1.8] There is a unique ring isomorphism φE : K(XE) → A•(XE) +such that φE([OXE(hS)]) = 1 + hS for all nonempty S ⊆ E. Moreover, for any matroid M on E of +rank r, the map φE satisfies +φE([I(M)]) = ξ0 + ξ1 + · · · + ξr +where ξi ∈ Ai(XE) for all i and ξr = [ΣM⊥]. +The generalization to type a is as follows. Recall that we have a birational toric map u: XE → +Xa induced by the fact that the fan Σa is a coarsening of ΣE. +Theorem 4.8. There exists a (necessarily unique) isomorphism φa : K(Xa) +∼ +→ A•(Xa) such that +we have a commuting diagram +K(Xa) +A(Xa) +K(XE) +A(XE). +φa +u∗ +u∗ +φE +Moreover, for any polymatroid P on E of type a and rank r, the map φa satisfies +φa +� +[I(π∗(P))] +� += ξ0 + ξ1 + · · · + ξr +where ξi ∈ Ai(Xa) for all i and ξr = [ΣP⊥]. +Proof. That the two vertical maps are injections follows from Lemma 2.12 and Proposition 4.3. +With these injections, we now need show that the map φE restricts to give a well-defined map φa +that is surjective. Recall that the Chow ring A•(Xa) is also generated by the simplicial generators + +20 +CHRISTOPHER EUR AND MATT LARSON +hS. We claim that K(Xa) is also generated as a ring by the line bundles [OXa(hS)]. Both the well- +definedness and the surjectivity of φa would then follow from Theorem 4.7 since u∗hS = hπ−1(S) +by Lemma 2.12. +For the claim, one notes that for any deformation Q of a projective unimodular fan Σ, the +inverse [Q]−1 of the class [Q] ∈ I(Σ) is a polynomial in [Q]. See for instance [EHL, Proof of +Lemma A.11]. The claim thus follows because the simplicial generators form a basis of A1(Xa). +For the second statement about φa +� +[I(π∗(P))] +� +, consider [I(π∗(P))] as an element in K(XE) +via the injection u∗. Proposition 4.5 and Theorem 4.7 implies that φE([I(π∗(P))]) = ξ0 + · · · + ξr +where ξi ∈ Ai(XE) and ξr = [ΣMπ(P)⊥]. Lastly, Lemma 3.13 and Proposition 3.6 implies that +[ΣMπ(P)⊥] = u∗[ΣP⊥]. +□ +Remark 4.9. Let χ: K(Xa) → Z be the sheaf Euler characteristic map. We sketch how one can +show, arguing similarly as in [EHL, Section 8.1], that the isomorphism φa satisfies +χ(ξ) = degXa +� +φa(ξ) · +� +i∈E +(1 + yi)ai� +for all ξ ∈ K(Xa). +By conjugating the isomorphism φa with the map that sends the K-class of a vector bundle to +its dual and the map that is multiplication by (−1)k on Ak(Xa), one obtains an isomorphism +ζa such that ζa([OWL]) = [WL] for any realization L ⊆ V of a type a polymatroid. Combining +Proposition 3.20 with Remark 3.16, one shows that A•(Xa) is spanned as an abelian group by +{[WL]: L ⊆ V }, and hence ζa satisfies χ(ξ) = degXa +� +ζa(ξ) · (1 + hE + · · · + hn +E) +� +. One then +computes that the anti-canonical divisor of Xa is hE +� +i∈E aiyi, and by Serre duality concludes +the desired formula. +5. PROOFS OF MAIN THEOREMS +We now use Theorem 4.8 to prove Theorem 1.6 and Theorem 1.3. +5.1. The valuative group is isomorphic to the Chow homology group. +Proof of Theorem 1.6. Since B(P⊥) = −B(P) + a and I(π∗(P)) = +� +p−1 +π (B(P)) + RE +≤0 +� +∩ RE +≥0, +the assignment 1B(P) �→ 1I(π∗(P⊥)) gives a well-defined map �n +r=0 Valr(a) → I(Σa), because +all the operations — negation, translation, inverse image, Minkowski sum, and restriction — +behave well with respect to indicator functions. +Hence, we have a map of abelian groups +�n +r=0 Valr(a) → K(Xa) defined by 1B(P) �→ [I(π∗(P⊥))]. Let ψ be the composition of this +map with the map φa : K(Xa) → A•(Xa) in Theorem 4.8. Note that ψ is upper-triangular with +respect to the gradings on �n +r=0 Valr(a) and A•(Xa). +Corollary 3.15, stating that A•(Xa) is spanned by {[ΣP]: P a polymatroid of type a}, implies +surjectivity of ψ. For injectivity, suppose we have polymatroids P1, . . . , Pk of type a and integers +c1, . . . , ck such that �k +j=1 cj[ΣPj] = 0. Then by Lemma 3.13, the validity of Theorem 1.6 when +a = (1, . . . , 1), established in [EHL, Theorem 1.5], implies that � +j cj1B(Mπ(Pj)) = 0. Since +each Pj is of type a, and since the image under the projection pπ of the unit cube [0, 1]E is the +box � +i∈E[0, ai] ⊂ RE, Lemma 3.2 implies that pπ +� +B(Mπ(Pj)) +� += B(Pj). We thus conclude + +INTERSECTION THEORY OF POLYMATROIDS +21 +� +j cj1B(Pj) = 0, proving the injectivity of ψ. Therefore ψ is an isomorphism, and so the map +that sends 1B(P) to [ΣP] is an isomorphism. +□ +Let ψ be the map as constructed in the proof above. Noting that polymatroid duality induces +an involution of �n +r=0 Valr(a), by composing ψ with the inverse φ−1 +a +of the isomorphism in +Theorem 4.8, we conclude the following. +Corollary 5.1. The map of abelian groups �n +r=0 Valr(a) → K(Xa) defined by 1B(P) �→ [I(π∗(P))] +is an isomorphism. +5.2. The Hall–Rado formula. We first note a reinterpretation of the Hall–Rado condition. +Lemma 5.2. [McD75, Theorem 2] A collection of subsets S1, . . . , Sr of E satisfies the Hall–Rado +condition with respect to a polymatroid P = (E, rk) of rank r if and only if there exists a map +f : [r] → E with f(i) ∈ Si such that �r +i=1 ef(i) ∈ B(P). +Proof of Theorem 1.3. For a nonempty subset S ⊆ E, we showed in the proof of Corollary 3.15 +that if HS is the polymatroid whose dual polymatroid has the simplex ∆0 +S as its independence +polytope, then [ΣHS] = hS. Applying this to Theorem 4.8, we have φa([I(π∗(H⊥ +S ))]) = 1 + hS. +Thus, as the degree map degXa is zero on Ai(Xa) for i < n, Theorem 4.8 implies that +degXa +� +φa([I(π∗(P⊥))][I(π∗(H⊥ +S1))] · · · [I(π∗(H⊥ +Sr))]) +� += degXa +� +[ΣP] · hS1 · · · hSr +� +. +Let �P be the polymatroid of rank n on E whose independence polytope is I(P⊥)+∆0 +S1+· · ·+∆0 +Sr. +Since multiplication in the polytope algebra is Minkowski sum and expansion commutes with +Minkowski sum, we have that [I(π∗(�P))] equals the class [I(π∗(P⊥))][I(π∗(H⊥ +S1))] · · · [I(π∗(H⊥ +Sr))] +in the left-hand-side of the equation above. By Lemma 5.2 and that B(P⊥) = −B(P) + a, we +have that a ∈ I(�P) if and only if S1, . . . , Sr satisfies the Hall–Rado condition with respect to P. +The theorem now follows from the following Lemma 5.3. +□ +Lemma 5.3. For �P a polymatroid of rank n on E, not necessarily of type a, we have that +degXa(φa([I(π∗(�P)])) = +� +� +� +1 +if a ∈ I(�P) +0 +otherwise. +Proof. By Proposition 4.5 and the commuting diagram in Theorem 4.8, we have that +degXa(φa([I(π∗(�P)])) = degXE([ΣMπ(�P)⊥]), +which is zero unless Mπ(�P) has rank n. When Mπ(�P) has rank n, that is, it is the Boolean +matroid on E, we have that [ΣMπ(�P)⊥] is the class of a point in A0(XE) = An(XE), and hence +degXE([ΣMπ(�P)⊥]) = 1 in this case. Now, note that Mπ(�P) has rank n, or equivalently (1, . . . , 1) ∈ +I(Mπ(�P)), if and only if a ∈ I(�P) by Lemma 3.2. +□ +Proof of Corollary 1.4. Follows from Lemma 5.2 and Theorem 1.6. +□ + +22 +CHRISTOPHER EUR AND MATT LARSON +Remark 5.4. At least when P is realizable, Corollary 1.4 implies Theorem 1.3. One can associate +to a realization L ⊆ � +i∈E Vi of P a realization of a polymatroid with ground set {S : ∅ ⊊ S ⊆ E} +via the composition +L �→ +� +i∈E +Vi �→ +� +∅⊊S⊆E +⊕i∈SVi. +The bases of this polymatroid are the sets S1, . . . , Sr that satisfy the Hall–Rado condition, so +applying Corollary 1.4 recovers Theorem 1.3. +Remark 5.5. One can also prove Corollary 1.4 by using Theorem 1.6 to reduce to the case of +realizable polymatroids, when Corollary 1.4 is [CCRMMn, Proposition 7.15] (and can also be +deduced from [Li18]). By Remark 3.16, in order to check that two valuative functions are equal, +it suffices to check on realizable polymatroids. The valuativity of [ΣP] implies that the volume +polynomial of A•(P) is valuative, and it is clear from the definition of valuativity that the basis +generating function of a polymatroid is valuative. +6. POLYPERMUTOHEDRA +Let π: E → E be of type a. The polystellahedral fan Σπ has the distinguished ray ρ∅ = +R≥0(−eE). The star of the fan Σπ at the ray ρ∅ is the polypermutohedral fan Σπ introduced in +[CHL+] as the Bergman fan of the boolean polymatroid of type a. Explicitly, the cones of Σπ are +in bijection with pairs S ≤ F, where F = {∅ ⊊ F1 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} is a flag of proper +subset of E and S is a subset of E containing no fiber of π. Let Xa be the associated toric variety, +which we call the polypermutohedral variety of type a, with the embedding ι: Xa �→ Xa as the +toric divisor corresponding to the ray ρ∅. +Suppose P is a polymatroid of type a and rank r. We note the following computation of the +pullback ι∗[ΣP] ∈ Ar−1(Xa). The augmented Bergman fan ΣP contains the ray ρ∅ if and only if +P is loopless. Hence, if P has a loop, then ι∗[ΣP] = 0. If P is loopless, the star of ΣP at the ray +ρ∅ is the Bergman fan ΣP of P introduced in [CHL+, Definition 1.6]. It is an (r − 1)-dimensional +balanced subfan of Σπ, and the resulting the Bergman class [ΣP] ∈ Ar−1(Xa) equals the pullback +ι∗[ΣP]. +Using Bergman fans and Bergman classes of loopless polymatroids, we establish analogues +of the main theorems Theorem 1.6 and Theorem 1.3 in the polypermutohedral setting. +6.1. The valuative group of loopless polymatroids. Define a subgroup of Valr(a) by +Val◦ +r(a) = the subgroup generated by {1B(P) : P a loopless polymatroid of type a and rank r}. +Note that Val◦ +0(a) = 0. We have the following analogue of Theorem 1.6. +Theorem 6.1. For any 1 ≤ r ≤ n, the map that sends a loopless polymatroid P of type a and +rank r to the Bergman class [ΣP] induces an isomorphism Val◦ +r(a) +∼ +→ Ar−1(Xa). + +INTERSECTION THEORY OF POLYMATROIDS +23 +We will deduce Theorem 6.1 from Theorem 1.6 by identifying the kernel of the map Valr(a) +∼ +→ +Ar(Xa) +ι∗ +→ Ar−1(Xa) with the subgroup of Valr(a) generated by polymatroids with loops. An +alternate proof that does not rely on Theorem 1.6 but proceeds by developing the polypermuto- +hedral analogue of Theorem 4.8 is sketched in Remark 6.3. +Before proving Theorem 6.1, we relate the Poincaré polynomial of the polystellahedral variety +to the Poincaré polynomials of polypermutohedral varieties. For J ⊆ E, let a \ J be the vector +obtained by removing the entries corresponding to J. +Lemma 6.2. We have that +n +� +i=0 +rank Ai(Xa)ti = +� +J⊆E +t|π−1(J)| +n−|π−1(J)|−1 +� +i=0 +rank Ai(Xa\J)ti. +Proof. As the Poincaré polynomial of a smooth projective toric variety is the h-polynomial of its +fan, it is enough to show that +f(Σa)(t) = (1 + t)n + +� +∅⊊J⊆E +t(1 + t)|π−1(J)|f(Σa\J)(t), +where f(Σ) is the f-polynomial of a fan Σ. We prove this bijectively. To each cone σ of some Σa\J +corresponding to a pair S ≤ F, we obtain 2|π−1(J)| cones of Σa by adding J to every element of +the flag and then add all possible 2|π−1(J)| subsets of π−1(J) to S. When J ̸= E and we add k +elements to S, this gives a cone of dimension dim σ +k +1. When J = E and we add k elements +to S, this gives a cone of dimension k. +□ +Proof of Theorem 6.1. For any i ∈ E, a polymatroid base polytope B(P) is always contained in +the half-space {x ∈ RE : xi ≥ 0}, and it is contained in the hyperplane {x ∈ RE : xi = 0} if and +only if P has i as a loop. Thus, the claim in the proof of [BEST, Lemma 5.9] implies that we have +a decomposition +Valr(a) = +� +J⊆E +Val◦ +r(a \ J) +given by sending a loopless polymatroid P of rank r on E \ J to the polymatroid on E with +rk(S) = rkP(S ∩ J). We now induct on the size of E, where the base case |E| = 1 is straightfor- +ward. Comparing the decomposition of Valr(a) above with Lemma 6.2, we see that the induc- +tion hypothesis implies rank Ar−1(Xa) = rank Val◦ +r(a). +By construction of the permutohedral fan Σπ as the star of ray ρ∅ in Σπ, every ray of Σπ is +the image of a ray in Σπ that forms a cone with ρ∅. Hence, the pullback ι∗ : A•(Xa) → A•(Xa) is +surjective because ι∗ : A1(Xa) → A1(Xa) is. We thus have a surjection ι∗ : Ar(Xa) → Ar−1(Xa) +that satisfies ι∗[ΣP] = [ΣP] if P is loopless and ι∗[ΣP] = 0 otherwise. Therefore the composition +Val◦ +r(a) → Ar(Xa) → Ar−1(Xa) +is a surjection of finite free abelian groups of the same rank, and hence is an isomorphism. +□ + +24 +CHRISTOPHER EUR AND MATT LARSON +Remark 6.3. We sketch an alternate proof of Theorem 6.1. First, arguing as in [EFLS, Proof of +Theorem D], one shows an isomorphism �n +r=1 Val◦ +r(a) ≃ K(Xa) when a = (1, . . . , 1), and uses +it to deduce Theorem 6.1 for the a = (1, . . . , 1) case. Now, using that polypermutohedral fans +are coarsenings of the permutohedral fan ΣE, just as polystellahedral fans are coarsenings of the +stellahedral fan, one similarly deduces the polypermutohedral analogue of Theorem 4.8. Then, +one deduces Theorem 6.1 the same way that we proved Theorem 1.6 here. +6.2. The dragon Hall–Rado formula. Let Xa be the polypermutohedral variety, with the em- +bedding ι: Xa �→ Xa as the toric divisor corresponding to the ray ρ∅. The following theorem +generalizes [BES, Theorem 5.2.4]. +Theorem 6.4. For a polymatroid P = (E, rk) of rank r, a collection of subsets S1, . . . , Sr−1 is +said to satisfy the dragon Hall–Rado condition if +rk +� � +j∈J +Sj +� +≥ |J| + 1 +for all nonempty J ⊆ [r − 1]. +Then, if P is loopless, we have +degXa(hS1 · · · hSr−1[ΣP][Xa]) = +� +� +� +1 +if the dragon Hall–Rado condition is satisfied +0 +otherwise. +Proof. Note that, in A•(Xa), we have that x∅ = − � +∅⊊S⊆E(−1)|S|hS. Then, for any S1, . . . , Sr−1, +(1) +degXa(hS1 · · · hSr−1[ΣP][Xa]) = − +� +∅⊊S⊆E +(−1)|S| degP(hS1 . . . hSr−1hS). +Suppose we have sets S1, . . . , Sr−1 that satisfy the dragon Hall–Rado condition. Because P is +loopless, every term in the above sum corresponds to r sets that satisfy the Hall–Rado condition, +and so each term is (−1)|S|. Because the sum is over nonempty sets, this gives the result. +Suppose that S1, . . . , Sr−1 fails the dragon Hall–Rado condition, and let T1, T2 be nonempty +subsets of E. If S1, . . . , Sr−1, T1 and S1, . . . , Sr−1, T2 both fail the Hall–Rado condition, then so +does S1, . . . , Sr−1, T1 ∩ T2. We claim that S1, . . . , Sr−1, T1 ∪ T2 fails the Hall–Rado condition. +Indeed, if there is a function f : [r] → E as in Lemma 5.2 with f(r) ∈ T1 ∪ T2, then f(r) lies in T1 +or T2, contradicting the assumption. +This implies that the set {T : ∅ ⊊ T ⊊ E, S1, . . . , Sr−1, T fails dragon Hall–Rado} is a boolean +sublattice. Therefore the sum in (1) is zero. +□ +Remark 6.5. Theorem 6.4 can be alternatively proved along the lines of Theorem 1.3, by using +the polypermutohedral analogue of Theorem 4.8 and a reformation of the dragon Hall-Rado- +condition in terms of a matching condition as in [BES, Proposition 5.2.3]. +REFERENCES +[AA] +Marcelo Aguiar and Federico Ardila. 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IMRN, (6):1026–1106, +2009. 1, 3, 5, 9 +[Wel76] +D. J. A. Welsh. Matroid theory. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, +1976. L. M. S. Monographs, No. 8. 1 +[Yuz02] +Sergey Yuzvinsky. Small rational model of subspace complement. Trans. Amer. Math. Soc., 354(5):1921–1945, +2002. 3 +Email address: ceur@math.harvard.edu +Email address: mwlarson@stanford.edu + diff --git a/NdAyT4oBgHgl3EQf6_p-/content/tmp_files/load_file.txt b/NdAyT4oBgHgl3EQf6_p-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e37f4119418b6dcebb84367c895e72304988891 --- /dev/null +++ b/NdAyT4oBgHgl3EQf6_p-/content/tmp_files/load_file.txt @@ -0,0 +1,1238 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf,len=1237 +page_content='INTERSECTION THEORY OF POLYMATROIDS CHRISTOPHER EUR AND MATT LARSON ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By introducing augmented Chow rings of polymatroids, modeled after augmented wonderful varieties of sub- space arrangements, we generalize several algebro-geometric techniques developed in recent years to study matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We show that intersection numbers in the augmented Chow ring of a poly- matroid are determined by a matching property known as the Hall–Rado condition, which is new even in the case of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTRODUCTION Let E = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , m} be a finite set, and let a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , am) be a sequence of nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A polymatroid P on E of type a is a function rkP : 2E → Z≥0 satisfying (1) (Submodularity) rkP(S1) + rkP(S2) ≥ rkP(S1 ∩ S2) + rkP(S1 ∪ S2) for any S1, S2 ⊆ E, (2) (Monotonicity) rkP(S1) ≤ rkP(S2) for any S1 ⊆ S2 ⊆ E, (3) (Normalization) rkP(∅) = 0, and (4) (Type) rkP(i) ≤ ai for any i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We say that rkP is the rank function of the polymatroid P, and that P has rank r = rkP(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A polymatroid of type (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1) is a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For the fundamentals of matroid theory we point to [Wel76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Introduced as generalizations of matroids [Edm70], and also known as generalized permutohedra, polymatroids are the central objects in the polyhedral study of combinatorial structures related to the symmetric group [AA, Pos09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In those works, two polytopes associated a polymatroid P = (E, rkP) are the independence polytope I(P), defined by I(P) = � x ∈ RE ≥0 : � i∈S xi ≤ rkP(S) for all S ⊆ E � , and the base polytope B(P), which is the face of I(P) defined by B(P) = I(P) ∩ � x ∈ RE : � i∈E xi = rkP(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Both polytopes are re-encodings of the polymatroid P as follows [Edm70]: The polytope B(P) determines I(P) by I(P) = {x ∈ RE ≥0 : y − x ∈ RE ≥0 for some y ∈ B(P)}, and the rank function of P is recovered by rkP(S) = max{� i∈S xi : x ∈ I(P)} = max{� i∈S xi : x ∈ B(P)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We connect polyhedral properties of polymatroids to algebro-geometric properties arising from the intersection theory of varieties associated to their realizations by linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='00831v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='AG] 2 Jan 2023 2 CHRISTOPHER EUR AND MATT LARSON Let V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Vm be vector spaces of dimensions a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , am respectively over a field k, and let V = � i∈E Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A subspace L ⊆ V defines a polymatroid P on E of type a whose rank function is rkP(S) = dim � image of L under the projection V → � i∈S Vi � for any S ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We say that L ⊆ V is a realization of the polymatroid P in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A realization L ⊆ V defines a subspace arrangement on L that consists of subspaces {Li}i∈E where Li = ker(L → Vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In terms of the subspace arrangement, the rank function of P is equivalently described as rkP(S) = codimL � � i∈S Li � for any S ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The key geometric object for us is the following compactification of L ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The augmented wonderful variety WL of a subspace L ⊆ V = � i∈E Vi is WL = the closure of the image of L in � ∅⊊S⊆E P � � i∈S Vi ⊕ k � , where the map L → P(� i∈S Vi ⊕ k) is the composition of the projection L → � i∈S Vi with the projective completion � i∈S Vi �→ P(� i∈S Vi ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In the context of matroids and hyperplane arrangements, the augmented wonderful variety was introduced in [BHM+22], and it played a central role in the proof of Dowling-Wilson top- heavy conjecture [BHM+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Augmented wonderful varieties are closely related to the wonderful compactifications of subspace arrangement complements introduced in [DCP95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A special role is played by the boolean arrangement L = � i∈E Vi of type a, whose augmented wonderful variety is called the polystellahedral variety of type a, denoted Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let A•(Xa) be the Chow cohomology ring of Xa, which in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5 we show has the presentation A•(Xa) = Z[xS, yi : ∅ ⊆ S ⊊ E, i ∈ E] ⟨xS1xS2 : S1, S2 incomparable⟩ + ⟨xSyai i : i ̸∈ S⟩ + ⟨yi − � S̸∋i xS : i ∈ E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Its grading satisfies A•(Xa) = �a1+···+am k=0 Ak(Xa), and it is equipped with the degree map, which is an isomorphism degXa : Aa1+···+am(Xa) ∼ → Z determined by the property degXa(ya1 1 · · · yam m ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The Chow homology group A•(Xa) is the graded group �a1+···+am k=0 Ak(Xa) where Ak(Xa) = Aa1+···+am−k(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polymatroid P = (E, rkP) of type a and rank r, we define a homology class [ΣP ] ∈ Ar(Xa) called the augmented Bergman class of P (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We define the augmented Chow ring A•(P) of P by A•(P) = A•(Xa)/ ann([ΣP]), where ann([ΣP]) = {x ∈ A•(Xa): x · [ΣP] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' See Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='19 for an explicit presentation of A•(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Its grading satisfies A•(P) = �r k=0 Ak(P), and it is equipped with the degree map, which is an isomorphism degP : Ar(P) ∼ → Z defined by degP(ξ) = degXa(ξ′ · [ΣP]) for any lift ξ′ ∈ A•(Xa) of ξ ∈ A•(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 3 When a subspace L ⊆ V realizes P, one has an embedding WL �→ Xa by the construction of the augmented wonderful variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The resulting homology class [WL] ∈ Ar(Xa) equals [ΣP] (Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='20), and the Chow ring A•(WL) of the augmented wonderful variety WL coincides with the augmented Chow ring A•(P) (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The embedding Xa �→ � ∅⊊S⊆E P(� i∈S Vi ⊕ k) provides the following useful set of gen- erators for the Chow ring of Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For each nonempty subset S ⊆ E, let hS ∈ A1(Xa) be the pullback of the hyperplane class on P(� i∈S Vi ⊕ k) along the map induced by the embedding Xa �→ � ∅⊊S⊆E P(� i∈S Vi ⊕k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We show that {hS : ∅ ⊊ S ⊆ E} generates A•(Xa), and that the monomials in these are all of the form [ΣP] for some polymatroid P of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polymatroid P, we define hS ∈ A1(P) to be image of hS under the quotient map A•(Xa) → A•(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We call these the simplicial generators of A•(P), motivated by similar terminology in the case of matroids [BES, LLPP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' These generators were also considered in [Yuz02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We show that the intersection numbers of the simplicial generators are described by the Hall– Rado condition: A sequence S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr of nonempty subsets of E is said to satisfy the Hall–Rado condition (with respect to a polymatroid P = (E, rkP)) if rkP � � j∈J Sj � ≥ |J| for all J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' See Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 for an interpretation of this condition in terms of a matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let P be a polymatroid of rank r, and let S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr be a sequence of nonempty subsets of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then degP(hS1 · · · hSr) = � � � 1 S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr satisfies the Hall–Rado condition, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' At least when P is realizable, the fact that degP(hS1 · · · hSr) = 0 if S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr does not satisfy the Hall–Rado condition has a simple geometric explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If rkP(Si1 ∪ · · · ∪ Sik) < k, then the degree k element hSi1 · · · hSik is zero because it is pulled back from the image of WL in P(� i∈Si1 Vi ⊕ k) × · · · × P(� i∈Sik Vi ⊕ k), which has dimension rkP(Si1 ∪ · · · ∪ Sik) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 generalizes several previous results [AB16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3(c)], [BES, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4], [Li18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1], [Pos09, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3], and [CCRMMn, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='15] about poly- matroids and volume polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We highlight here the following corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let P be a polymatroid on E of rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then 1 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' degP � (� i∈E tih{i})r� , the volume polynomial of A•(P) with respect to {h{i} : i ∈ E} ⊂ A1(P), equals the basis exponential generating function of P, which is the polynomial in Q[ti : i ∈ E] given by � u∈B(P)∩ZE tu u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', where tu = tu1 1 · · · tum m and u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' = u1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' · · · um!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='. Many invariants of matroids behave well with respect to matroid polytope decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This leads to the study of the valuative group of matroids [AFR10, BEST, DF10], which gives a 4 CHRISTOPHER EUR AND MATT LARSON powerful tool to study invariants of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We consider the following notion of valuativity for polymatroids of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polytope Q ⊂ RE, let 1Q : RE → Z be its indicator function defined by 1Q(x) = 1 if x ∈ Q and 1Q(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The valuative group Valr(a) of rank r polymatroids of type a is the subgroup of Z(RE) generated by 1B(P) for P a polymatroid of rank r and type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We show that the valuative group is isomorphic to the homology groups of the polystellahe- dral variety, generalizing [EHL, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For any 0 ≤ r ≤ a1 + · · · + am, the map that sends a polymatroid P of type a and rank r to [ΣP] induces an isomorphism Valr(a) ∼ → Ar(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6, we show that a choice of isomorphism Vi ≃ kai for each i ∈ E realizes Xa as a toric variety (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This gives a description of the Grothendieck ring of vector bundles K(Xa) in terms of certain polytopes in Ra1+···+am (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We relate � r Valr(a) to this polytopal description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We then prove an exceptional Hirzebruch–Riemann–Roch-type theorem (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8) that leads to the proofs of both Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In section 2, we discuss polystellahedral varieties from the point of view of toric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In section 3, we construct the augmented Bergman fan of a polymatroid and develop its basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In section 4, with study the K-ring of the polystellahedral variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In section 5, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In section 6, we prove analogs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 for the polypermutohedral variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We thank June Huh for many invaluable conversations related to polyma- troids, including suggesting the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The first author is supported by NSF Grant DMS-2001854, and the second author is supported by an NDSEG fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' All varieties are over an algebraically closed field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a subset S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , ℓ}, let eS = � i∈S ei be the sum of standard basis vectors in Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Denote by (·, ·) the standard inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For polyhedra and toric varieties, we follow conventions of [Ful93, CLS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a rational polyhedral fan Σ, we let XΣ be the toric variety associated to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' THE TORIC GEOMETRY OF POLYSTELLAHEDRAL VARIETIES We introduce the polystellahedral fan (of type a) and study the properties of the associated toric variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This amounts to developing basic properties of the polystellahedral variety Xa, since we will show that any choice of isomorphisms Vi ≃ kai for all i ∈ E induces an isomorphism between Xa and the toric variety associated to the polystellahedral fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Polystellahedral fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Set n = a1 + · · · + am, and let E be a set of cardinality n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A map π: E → E, which defines a partition E = � i∈E π−1(i), is said to have type a if |π−1(i)| = ai for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A compatible pair with respect to a map π: E → E is a pair I ≤ F consisting of a subset I ⊆ E and a chain F = {F1 ⊊ F2 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} of proper subsets of E such that if π−1(S) ⊆ I for a subset S ⊆ E, then S ⊆ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The polystellahedral fan Σπ is the fan in RE whose cones are in bijection with compatible pairs, with a compatible pair I ≤ F corresponding to the cone σI≤F = cone(−eE\\π−1(F1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , −eE\\π−1(Fk)) + cone(ei : i ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Its rays are denoted ρi = R≥0ei for i ∈ E and ρF = R≥0(−eE\\π−1(F )) for ∅ ⊆ F ⊊ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that the fan Σπ depends only on the map E → π(E), not the codomain E of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A polystellahedral fan Σa of type a is a fan Σπ where π has type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We note two extreme cases: When π has type (n), the fan Σπ is the inner normal fan of the n-dimensional standard simplex conv({0} ∪ {ej : j ∈ E}) in RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We denote this fan by Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When π has type (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), the fan Σπ is the stellahedral fan on E in [EHL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We denote this fan by ΣE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A general polystellahedral fan in RE is both a refinement of Σn and a coarsening of ΣE in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For two maps π: E → E and π′ : E → E′, let us say π refines π′, denoted π ⪰ π′, if the corresponding partitions form a refinement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', for every i ∈ E one has π−1(i) ⊆ π′−1(i′) for some i′ ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall that for a simplicial fan Σ and a vector v in its support, the stellar subdivision of Σ by v is the new fan whose set of rays are {rays of Σ} ∪ {ρv = R≥0v} and the set of cones are {σ ∈ Σ: v /∈ σ} ∪ {σ ∪ ρv : σ ∈ Σ such that v /∈ σ and v ∈ σ′ for some σ ⊂ σ′ ∈ Σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a refinement π ⪰ π′, let (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sk) be a sequence consisting of the subsets S ⊆ E such that π−1(S) ̸= π′−1(S′) for any S′ ⊆ E′, ordered in a way that |S1| ≥ · · · ≥ |Sk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then the fan Σπ is the result of the sequence of stellar subdivisions of the fan Σπ′ by the sequence of vectors (−eE\\π−1(S1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , −eE\\π−1(Sk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Moreover, at each step of the sequence of stellar subdivisions, the resulting fan is projective and unimodular with respect to the lattice ZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We prove the proposition using building sets, which were introduced in [DCP95] and studied in [FY04, FS05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We review the special case of building sets on boolean lattice here following [Pos09, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A building set on E is a collection G ⊆ 2E of subsets of E such that G contains E and {i} for each i ∈ E, and if S and S′ are in G and S ∩ S′ ̸= ∅, then S ∪ S′ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A nested set of G is a collection {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Xk} ⊆ G such that for every subcollection {Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Xiℓ} with ℓ ≥ 2 consisting only of pairwise incomparable elements, one has �ℓ j=1 Xij /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The fan associated G is the fan ΣG in RE/ReE whose cones are {the image in RE/ReE of cone{eX1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , eXk} ⊂ RE : {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Xk} a nested set of G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let E ∪ {0} be the disjoint union of E with an extra element 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We have an isomorphism RE∪{0}/ReE∪{0} ≃ RE induced by ei �→ ei for i ∈ E and e0 �→ − � i∈E ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' It is straightforward to verify that, under this isomorphism, the fan Σπ equals the fan ΣGπ in RE∪0/R1 associated to the building set Gπ = E ∪ {π−1(S) ∪ 0: ∅ ⊆ S ⊆ E} on the boolean lattice of E ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If π ⪰ π′, 6 CHRISTOPHER EUR AND MATT LARSON then we have Gπ ⊇ Gπ′, and the desired statements in the proposition are now special cases of [FM05, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2] and [FY04, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Polystellahedral varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let us fix the following notation for the rest of a paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let E be a set of size n := a1 + · · · + am, and let π: E → E to be a map of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Xπ be the toric variety associated to the polystellahedral fan Σπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We record some proper- ties of Xπ arising from the properties of the fan Σπ, starting with the fact that Xπ is isomorphic to the polystellahedral variety Xa of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As before, let V = � i∈E Vi be the direct sum of vector spaces where dim Vi = ai = |π−1(i)| for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Denote by GLa the group � i∈E GL(Vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall that Xa is the closure of the image of the map V → � ∅⊊S⊆E P(� i∈S Vi ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Because this map is GLa-equivariant, the group GLa acts naturally on the variety Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Any choice of isomorphisms Vi ≃ kπ−1(i) for each i ∈ E, which gives a natural embedding of the torus (k∗)E �→ GLa, identifies Xa with the toric variety Xπ of the fan Σπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Thus, from this point on we will identify Xa with the toric variety Xπ, although the identifi- cation depends on the choice of isomorphisms Vi ≃ kπ−1(i) for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' With the isomorphisms Vi ≃ kπ−1(i) for all i ∈ E, the projective space PE = P(kE ⊕ k) ≃ P(V ⊕ k) with the obvious action of (k∗)E is the toric variety of the fan Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a subset S ⊆ E, let LS = kπ−1(E\\S) ⊕ 0 ⊂ kπ−1(E\\S) ⊕ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If S is a proper subset, then P(LS) is the hyperplane at infinity of the coordinate subspace P(kπ−1(E\\S) ⊕ k) ≃ P(� i∈E\\S Vi ⊕ k) of PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note the complementation, and note that P(LS) is GLa-invariant for any ∅ ⊆ S ⊊ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 with π′ being the map from E to a singleton set, which describes the fan Σπ as a sequence of stellar subdivisions of the fan Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Translated into toric geometry terms, it states that the toric variety Xπ of the fan Σπ is obtained from PE via a sequence of blow-ups as follows: Order the proper subsets of E so that their cardinalities are non-strictly decreasing, then sequentially blow-up the (strict transforms of) the loci P(LS) in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This sequential blow- up is also the description of the wonderful compactification of the complement of the subspace arrangement {P(L{i}): i ∈ E} in PE, introduced in [DCP95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' [DCP95, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 Proposition (2)] moreover states that this wonderful compactification is also the closure of the image of the rational map PE ��� � ∅⊊S⊆E P � (kE ⊕ k)/LS � , which, when restricted to V ≃ kE ⊂ PE, is exactly the map V → � ∅⊊S⊆E P(� i∈S Vi ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Γa be the product � i∈E Sπ−1(i) of permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Because Γa acts nat- urally on the fan Σπ by permuting the coordinates of RE, the group Γa acts on the variety Xπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Under the identification Xa ≃ Xπ, this action agrees with the action of Γa embedded in GLa via the isomorphism � i∈E Vi ≃ � i∈E kπ−1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We record the following presentation of the Chow ring of Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a proper subset S of E and an element j ∈ E, let xS and �yj denote the toric divisors of Xa corresponding to the rays ρS and ρj of Σa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 7 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For each i ∈ E, the divisors in the set {�yj : j ∈ π−1(i)} are all equal to each other as divisor classes in A1(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Denote this divisor class by yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The Chow ring A•(Xa) of Xπ equals A•(Xπ) = Z[xS, yi : ∅ ⊆ S ⊊ E, i ∈ E] ⟨xS1xS2 : S1, S2 incomparable⟩ + ⟨xSyai i : i ̸∈ S⟩ + ⟨yi − � S̸∋i xS : i ∈ E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a unimodular and projective fan Σ in RE with rays Σ(1) and primitive ray vectors {uρ ∈ ZE : ρ ∈ Σ(1)}, [Ful93, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 Proposition] states that the Chow ring of the smooth projective toric variety XΣ equals A•(XΣ) = Z[xρ : ρ ∈ Σ(1)] ⟨� ρ∈S xρ : {ρi}i∈S do not form a cone in Σ⟩ + ⟨� ρ∈Σ(1)(uρ, v)xρ : v ∈ ZE⟩ where (u, v) here denotes the standard inner product on RE and xρ represents the toric divisor of XΣ corresponding to the ray ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We apply this with Σ = Σπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Setting v = ej1−ej2 for any i ∈ E and j1, j2 ∈ π−1(i), the linear relations � ρ∈Σ(1)(uρ, v)xρ = 0 imply the first statement that {�yj}j∈π−1(i) are all equal as elements in A1(Xπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Setting v = ej for any i ∈ E and j ∈ π−1(i) then gives the relations {yi − � S̸∋i xS = 0: i ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The rest of the corollary follows when one notes that the minimal non-faces of Σπ are the following: the sets of the form {ρS1, ρS2} for incomparable proper subsets S1 and S2 of E, or the sets of the form {ρS} ∪ {ρj : j ∈ π−1(i)} for a proper subset S of E and i ∈ E \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Nef divisors, deformations, and expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a fan Σ in RE, a (lattice) polytope Q ⊂ RE is a (lattice) deformation of Σ if its inner normal fan ΣQ coarsens the fan Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We describe the deformations of the polystellahedral fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As before, let π: E → E be a map of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Define a linear map pπ : RE → RE by ei �→ eπ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let P = (E, rkP) be a polymatroid on E of arbitrary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The expansion (with respect to π) of P is the polymatroid π∗(P) on E whose rank function is given by rkP ◦π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Equiv- alently, the polymatroid π∗(P) is defined by setting its independence polytope to be I(π∗(P)) = p−1 π (I(P)) ∩ RE ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A lattice polytope Q in RE is a deformation of Σa if and only if Q is a translate of I(π∗(P)) for a polymatroid P on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We deduce the proposition by using a standard result in toric geometry that identifies defor- mations with nef toric divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We prepare with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that, by the linear relations for the Chow ring A•(Xa) in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5, the set of divisor classes {xS : ∅ ⊆ S ⊊ E} is a basis of A1(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A divisor class D ∈ A1(Xa) is nef if and only if, when we write D = � S⊊E aSxS, the function S �→ aE\\S is the rank function of a polymatroid on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 8 CHRISTOPHER EUR AND MATT LARSON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let ϕD be the piecewise linear function corresponding to the divisor D = � S⊊E aSxS, which satisfies ϕD(ej) = 0 for all j ∈ E and ϕD(−eE\\π−1(S)) = −aS for S ⊊ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We use a criterion for a line bundle on a smooth projective toric variety to be nef from [CLS11, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9], which states that D is nef if and only if the support function ϕD satisfies an inequality for each minimal non-face of the fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This gives the following inequalities: For S, S′ ⊊ E incomparable, the minimal non-face spanned by ρS and ρS′ gives the inequality ϕD(−eE\\π−1(S) − eE\\π−1(S′)) ≥ ϕD(−eE\\π−1(S)) + ϕD(−eE\\π−1(S′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Because −eE\\π−1(S) − eE\\π−1(S′) = −eE\\π−1(S∩S′) − eE\\π−1(S∪S′) and ϕD is linear on the cone spanned by −eE\\π−1(S∩S′) and −eE\\π−1(S∪S′), we get that aS∩S′ + aS∪S′ ≤ aS + aS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For S ⊊ E and i ̸∈ S, the minimal non-face spanned by ρS ∪ {ρj : j ∈ π−1(i)} gives the inequality ϕD(−eE\\π−1(S) + � j∈π−1(i) ej) ≥ ϕD(−eE\\π−1(S)) + � j∈π−1(i) ϕD(ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As −eE\\π−1(S) + � j∈π−1(i) ej = −eE\\π−1(S∪i) and ϕD(ej) = 0, this gives the inequality aS∪i ≤ aS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' These two inequalities are equivalent to the statement that S �→ aE\\S is a polymatroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The standard correspondence between nef toric divisors and deforma- tions [CLS11, Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7], when applied to the fan Σa, states that a nef divisor D = � S⊊E aSxS on Xa corresponds to the lattice deformation QD of Σa defined by QD = {y ∈ RE : (y, ej) ≥ 0 for all j ∈ E and (y, −eE\\π−1(S)) ≥ −aS for all ∅ ⊆ S ⊊ E}, which is exactly the independence polytope of the expansion of the polymatroid with rank function S �→ aE\\S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Moreover, the correspondence implies that every lattice deformation of Σa arise as a translate of the polytope corresponding to a nef divisor D = � S⊊E aSxS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ We distinguish the following set of nef divisors on Xa arising from the standard simplices in RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that, for each nonempty subset S ⊆ E, the simplex ∆0 S = conv({0}∪{ei : i ∈ S}) ⊂ RE is the independence polytope of the polymatroid on E whose rank function is rk(T) = � � � 1 if T ∩ S ̸= ∅ 0 otherwise for ∅ ⊆ T ⊆ E, or equivalently, rk(E \\ T) = 1 exactly when T ̸⊇ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For each nonempty subset S ⊆ E, we define hS ∈ A1(Xa) to be the nef divisor hS = � ∅⊆T ⊊E T ̸⊇S xT INTERSECTION THEORY OF POLYMATROIDS 9 corresponding to the simplex ∆0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We call the divisor classes {hS : ∅ ⊊ S ⊆ E} the simplicial generators of Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The simplicial generators of Xa form a basis of A1(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In particular, their monomials span A•(Xa) as an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Möbius inversion, every divisor class in the basis {xT : ∅ ⊆ T ⊊ E} of A1(Xa) is a linear combination of the simplicial generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The definition of hS here agrees with its definition in the introduction as the pullback of the hyperplane class of P(� i∈S Vi ⊕ k) along the map Xa �→ � ∅⊊S⊆2E P(� i∈S Vi ⊕ k) → P(� i∈S Vi ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' To see this, one notes that the independence polytope of the expansion of the polymatroid of ∆0 S is the simplex ∆0 π−1(S) = conv({0} ∪ {ej : j ∈ π−1(S)}) ⊂ RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The lattice points of ∆0 π−1(S), considered as global sections of the corresponding line bundle, induce the map Xa → P(� i∈S Vi ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We conclude by discussing the behavior of Chow rings under refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 implies that Σπ is a coarsening of the stellahedral fan ΣE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Thus, we have a toric birational map u: XE → Xa induced by the refinement of fans ΣE ⪰ Σπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We record the following properties of u for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The pullback map u∗ : A•(Xa) → A•(XE) satisfies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' (1) u∗ is a split injection, with the splitting given by the pushforward map u∗ : A•(XE) → A•(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' (2) If D ∈ A1(Xa) is a nef divisor class corresponding to a deformation Q of Σa, then the pullback u∗D ∈ A1(XE) is a nef divisor class corresponding to Q considered as a defor- mation of ΣE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' (3) For a nonempty subset S ⊆ E, the simplicial generator hS ∈ A1(Xa) pulls back to the simplicial generator u∗hS = hπ−1(S) ∈ A1(XE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The first statement is a standard consequence of the birationality of u and the projection formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The second statement follows from [CLS11, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The third statement follows from the second, since the independence polytope of the expansion of the polymatroid of ∆0 S is the simplex ∆0 π−1(S) = conv({0} ∪ {ej : j ∈ π−1(S)}) ⊂ RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let the polystellahedron of type a to be the polytope Πa in RE defined by Πa = I(π∗(P)), where P is the polymatroid on E with B(P) = conv{w · (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , m): w ∈ SE}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The face B(π∗(P)) of Πa was introduced as the polypermutohedron of type a in [CHL+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Using the results in this subsection, one can verify that the polystellahedral fan Σa is the normal fan of the polystellahedron Πa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Alternatively, using the building set associated to a polystellahedral fan given in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2, one can verify that Πa is the corresponding nestohedron [Pos09, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 10 CHRISTOPHER EUR AND MATT LARSON 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' AUGMENTED GEOMETRY OF POLYMATROIDS For a polymatroid P of type a and rank r, we define its augmented Bergman fan ΣP as a subfan of the polystellahedral fan of type a, and use its properties to define the augmented Bergman class [ΣP] ∈ Ar(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We then record some of their geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Multisymmetric lift and duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We begin with a construction of a matroid from a poly- matroid P of type a which has appeared many times in the literature [Hel72, Lov77, McD75, Ngu86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We use the terminology of [CHL+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The multisymmetric lift of a polymatroid P on E is the matroid Mπ(P) on E whose rank function is given by rkMπ(P)(S) = min{rkP(A) + |S \\ π−1(A)|: A ⊆ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Alternatively, the multisymmetric lift can be described via polytopes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let [0, 1]E be the unit cube in RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, we have I(Mπ(P)) = I(π∗(P)) ∩ [0, 1]E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We need to show that a subset S ⊆ E is independent in the matroid Mπ(P) if and only if eS ∈ I(π∗(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By the definition of I(π∗(P)), we have that eS ∈ I(π∗(P)) if and only if, for all U ⊆ E, one has |S ∩ U| ≤ rkP(π(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' It suffices to check whether this holds when U is a fiber of π, so this condition becomes |S ∩ π−1(A)| ≤ rkP(A), or, equivalently, rkP(A) + |S \\ π−1(A)| = rkP(A) + |S| − |S ∩ π−1(A)| ≥ |S|, for all A ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' That is, the condition is equivalent to rkMπ(P)(S) = |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ When P is of type a, the lemma implies that I(Mπ(P)) maps onto I(P) under the linear pro- jection pπ : RE → RE because the unit cube [0, 1]E ⊂ RE maps onto the box � i∈E[0, ai] ⊂ RE under pπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Equivalently, a type a polymatroid P is recovered from Mπ(P) via the formula rkP(S) = rkMπ(P)(π−1(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For an arbitrary type P, the multisymmetric lift Mπ(P) may not recover P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When P is realized by a subspace arrangement L ⊆ � i∈E Vi, the multisymmetric lift Mπ(P) is realized by the hyperplane arrangement L ⊆ kE obtained by a general choice of isomorphisms Vi ≃ kπ−1(i) for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In particular, the subspaces {Li : i ∈ E} in the arrangement appear as subspaces arising as intersections of the hyperplanes in the arrangement L ⊆ kE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polymatroid P = (E, rkP), a subset F ⊆ E is a flat of P if rkP(F ∪ a) > rkP(F) for all a ∈ E \\ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The flats of P form a lattice, denoted LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The loops of a polymatroid are the elements of the minimal flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We say that a polymatroid is loopless if the empty set is a flat, or equivalently, if rkP(i) > 0 for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Given a flat F of P, the subset π−1(F) ⊆ E is a flat of the multisymmetric lift Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Flats of Mπ(P) of this form are called geometric flats of Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The key property of geometric flats is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' [CHL+, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8] Every flat F of Mπ(P) contains a unique maximal geomet- ric flat F geo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We have that rkMπ(P)(F geo) = rkP(π(F)), and rkMπ(P)(F) = rkMπ(P)(F geo) + |F \\ F geo|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 11 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4, let Γa be the product � i∈E Sπ−1(i) of permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The terminology “multisymmetric” is justified by the fact that the obvious action of the group Γa on E preserves the rank function of Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In fact, this property characterizes multisymmetric lifts: [CHL+, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9] states that a matroid Mπ on E such that the action of Γa preserves the rank function is of the form Mπ(P) for a polymatroid P of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1 Moreover, the map F �→ π−1(F) induces an isomorphism from the lattice LP of flats of P to the lattice of Γa-fixed flats of Mπ(P) [CHL+, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We now discuss polymatroid duality, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', [McD75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Our main conclusion is that taking multisymmetric lift commutes with polymatroid duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polymatroid P on E of type a and rank r, its dual polymatroid P⊥ is a polymatroid on E of type a and rank n − r whose rank function is rkP⊥(S) = � i∈S ai + rk(E \\ S) − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Alternatively, duality can also be described via polytopes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The rank function de- scription for P⊥ above implies that B(P⊥) = −B(P) + a, or, equivalently, since I(P) = {x ∈ � i∈E[0, ai]: y − x ∈ RE ≥0 for some y ∈ B(P)}, we have −I(P⊥) + a = {x ∈ � i∈E[0, ai]: x − y ∈ RE ≥0 for some y ∈ B(P)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Polytopes associated to a polymatroid and its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When P is realized by L ⊆ V = � i∈E Vi, its dual P⊥ is realized by (V/L)∨ ⊆ � i∈E V ∨ i obtained by dualizing the surjection V ↠ V/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), polymatroid duality agrees with the usual notion of matroid duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polymatroid P on E of type a, one has Mπ(P⊥) = Mπ(P)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 since B(P⊥) = −B(P) + a and pπ(� j∈E ej) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 1In the proof of this theorem, the authors of [CHL+] make the additional assumption that rkP(i) = ai, but this assumption is never used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' a I(P-)+a B(P) I(P)12 CHRISTOPHER EUR AND MATT LARSON 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Augmented Bergman fans of polymatroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let P be a polymatroid on E of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We now introduce the augmented Bergman fan ΣP of a polymatroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The augmented Bergman fan ΣP of P is the subfan of Σa consisting of cones σS≤F, where S is a subset of E and F = {F1 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} is a chain of proper flats of P satisfying (1) For all T ⊆ S, one has rkP(π(T)) ≥ |T|, and (2) for all F ∈ F and all T ⊆ S \\ π−1(F), one has rkP(F ∪ π(T)) > rkP(F) ∪ |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), that is, when P is a matroid M on E, the augmented Bergman fan of P coincides with the augmented Bergman fan ΣM introduced in [BHM+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Explicitly, the fan ΣM is the subfan of the stellahedral fan ΣE consisting of cones σI≤F where I ⊆ E is an independent set of M and F = {F1 ⊊ · · · ⊊ Fk ⊊ E} is a chain of proper flats of M such that I ⊆ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The augmented Bergman fan ΣP of P is the subfan of Σa whose support equals the support of the augmented Bergman fan ΣMπ(P) of the multisymmetric lift of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' More precisely, ΣP is the coarsening of the fan ΣMπ(P) such that it is a subfan of Σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This is the key property of ΣP that we will repeatedly use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The rest of this subsection is dedicated to the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let us prepare by reviewing the theory of building sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' for proofs and details we point to [DCP95, FS05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A building set on a loopless matroid M on ground set E is a collection G of nonempty flats of M such that, for all nonempty flats of F of M, the natural map of lattices � G∈max G≤F [∅, G] → [∅, F] is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Here, max G≤F denotes the maximal elements of G contained in the interval [∅, F] ⊆ LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' All building sets that we consider will contain the maximal flat E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A nested set is a subset N ⊆ G that does not contain E such that, for all pairwise incomparable subsets {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Fk} ⊆ N with k ≥ 2, the join �k i=1 Fi of {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Fk} is not in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Nested sets form a simplicial complex, which is realized as a simplicial fan ΣM,G in RE/ReE whose cones are {image in RE/ReE of cone{ei : i ∈ N} ⊂ RE : N a nested set}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We call ΣM,G the Bergman fan of M with respect to the building set G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The support of ΣM,G does not depend on the choice of building set [FY04, Theorem 4], and ΣM,G is always a unimodular fan [FY04, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8 by identifying the fan ΣP with a Bergman fan of a matroid closely related to the multisymmetric lift Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Mπ(P) × 0 denote the free coextension of the multi- symmetric lift Mπ(P), which is a matroid on the ground set E ⊔ {0} with flats {F ∪ 0: F ⊆ E flat of Mπ(P)} ∪ {I ⊆ E: I independent in Mπ(P)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that Mπ(P) × 0 is always loopless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We now define a building set on Mπ(P) × 0 whose Bergman fan will be the augmented Bergman fan of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 13 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let G be the set of all flats of Mπ(P) × 0 of the form F ∪ {0} for F a geometric flat of Mπ(P), or {j} for j ∈ E not a loop of Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then G is a building set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Consider a flat of Mπ(P) × 0 of the form H ∪ 0 for H a flat of Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3, H contains a unique maximal geometric flat Hgeo, and, for any subset S with Hgeo ⊆ S ⊆ H, we have that rkP(S) = rkP(Hgeo) + |S \\ Hgeo|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This implies the desired decomposition for H ∪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If we have a flat of Mπ(P)×0 of the form I for I ⊆ E independent, then the desired decomposition is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Before computing the nested sets of G, we need a preparatory lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let F be a geometric flat of a multisymmetric matroid Mπ(P), and let S be a subset of F such that |S| ≥ rkMπ(P)(F) or |S| > rkMπ(P)(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then there is a geometric flat G of Mπ(P) and a subset S′ ⊆ S ∩ G such that |S′| ≥ rkMπ(P)(G) (respectively |S′| > rkMπ(P)(G)) and S′ spans G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We do the case when |S| > rkMπ(P)(F), the other case is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We induct on the rank of F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' if rkMπ(P)(F) = 0 then the claim is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let H be the closure of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3, we have that rkMπ(P)(H) = rkMπ(P)(Hgeo) + |H \\ Hgeo| ≥ rkMπ(P)(Hgeo) + |S| − |S ∩ Hgeo|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' On the other hand, we have that rkMπ(P)(H) ≤ rkMπ(P)(F) < |S|, so rkMπ(P)(Hgeo) < |S ∩Hgeo|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Either Hgeo = F and we are done, or we conclude by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' With G as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9, the nested sets of G are given by chains of flats F = {F1 ⊊ · · ⊊ Fk ⊊ Fk+1 = E} of P and a subset S of the non-loops of P such that: (1) For all T ⊆ S, rkP(π(T)) ≥ |T|, and (2) for all F ∈ F and all T ⊆ S \\ π−1(F), rkP(F ∪ π(T)) > rkP(F) ∪ |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let S and F = {F1 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} be pair satisfying the two condition of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We check that the corresponding set is nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The incomparable subsets are either given by a collection T ⊆ S, or a flat F ∈ F and S ⊆ S \\ π−1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The closure of T ⊆ S in Mπ(P) × 0 is T if T is independent, and it is clMπ(P)(T) ∪ 0 if T is dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In the first case, T is not in G if |T| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If T is dependent, then (1) guarantees that rkMπ(P)(T) < rkP(π(T)), so the closure is not in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Similarly, if we have T ⊆ S \\ π−1(F), then the closure of F ∪ T cannot be geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Now let N be a nested set, which consists of a subset S of the non-loops of Mπ(P) and flats of the form F ∪ 0 for F a geometric flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As the join of two geometric flats is a geometric flat, the flats of the form F ∪ 0 must form a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Suppose there is a subset T ⊆ S with rkP(π(T)) < |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let F = π−1(clP(π(T))), which is a geometric flat containing T of rank less than |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='10, there is T ′ ⊆ T and a geometric flat G such that T ′ spans G and |T ′| > rkMπ(P)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then the closure of T ′ in Mπ(P) × 0 is G ∪ 0, contradicting that N is nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 14 CHRISTOPHER EUR AND MATT LARSON Now suppose that there is F ∈ F and T ⊆ S \\ π−1(F) with rkP(F ∪ π(T)) ≤ rkP(F) + |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let G = π−1(clP(π(F) ∪ T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='10 to the contraction Mπ(P)/π−1(F), we find a geometric flat H ⊃ F and T ′ ⊆ T ∩ H such that T ′ ∪ π−1(F) spans H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This contradicts that N is nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let H be the building set on the lattice of flats of Mπ(P) × 0 given by view- ing Mπ(P) viewed as a polymatroid of type (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By [FY04, Theorem 4] the support of ΣMπ(P)×0,G coincides with the support of ΣMπ(P)×0,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By [EHL, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='14], under the iso- morphism RE → RE∪0/R obtained by sending ej to ej, the support of ΣMπ(P)×0,H coincides with the support of ΣMπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Under this isomorphism, ΣMπ(P)×0,G is identified with ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Augmented Bergman classes of polymatroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We begin by reviewing briefly balanced fans and their Chow homology classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' for details and proofs we point to [FS97] and [AHK18, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A pure-dimensional simplicial rational fan Σ of dimension d is balanced if for any cone τ ∈ Σ of codimension 1, one has � σ⊋τ uσ\\τ ∈ τ, where uσ\\τ denotes the primitive vector of the unique ray in σ that is not in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Suppose a balanced fan Σ is a subfan of a complete unimodular fan �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Ad(X�Σ) be the d-th graded piece of the Chow ring of the toric variety X�Σ, which is spanned by {[Zσ]: σ a d-dimensional cone in �Σ}, where Zσ is the torus-orbit closure in X�Σ corresponding to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' One then obtains a linear functional wΣ ∈ Hom(Ad(X�Σ), Z) determined by wΣ([Zσ]) = 1 if σ ∈ �Σ and wΣ([Zσ]) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By the Poincaré duality property of the Chow ring A•(X�Σ), the functional wΣ defines an element [Σ] ∈ Ad(XΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Returning to polymatroids, let P be a polymatroid on E of type a and rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As the support of the augmented Bergman fan ΣP coincides with the support of a Bergman fan, [GS21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8] implies that ΣP is a balanced subfan of the polystellahedral fan of type a Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The augmented Bergman class of P is the Chow homology class [ΣP] ∈ Ar(Xa) obtained by considering ΣP as a balanced subfan of the polystellahedral fan of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We will repeatedly use the following relation between the classes associated to a polymatroid and its multisymmetric lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall the birational map u: XE → Xa induced by refinement of respective fans (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The pullback u∗[ΣP] is equal to the augmented Bergman class [ΣMπ(P)] of the multisymmetric lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The lemma follows from applying the formula [FS97, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7] for computing pull- backs in terms of Minkowski weights to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ We use the lemma to compute how augmented Bergman classes of polymatroids multiply as elements in the Chow ring A•(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We will need the following combinatorial notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Given two polymatroids P1 and P2 on E of type a, we define the polymatroid union P1 ∨ P2 to be the polymatroid of type a whose independence polytope is (I(P1)+I(P2))∩� i∈E[0, ai].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' That INTERSECTION THEORY OF POLYMATROIDS 15 this is indeed the independence polytope of a polymatroid follows from [Edm70, (35)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Define the polymatroid intersection of P1 and P2 to be P1 ∧ P2 := (P⊥ 1 ∨ P⊥ 2 )⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If we view Mπ(Pi) as a polymatroid of type (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 we have that Mπ(P1) ∨ Mπ(P2) = Mπ(P1 ∨ P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Therefore Mπ(P1) ∧ Mπ(P2) = Mπ(P1 ∧ P2) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let P1 and P2 be polymatroids of type a and ranks r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, we have [ΣP1] · [ΣP2] = � � � [ΣP1∧P2] (n − r1) + (n − r2) = n − rank(P1 ∧ P2) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), the above theorem is [EHL, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Our proof is a reduction to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13 and using that Mπ(P1)∧Mπ(P2) = Mπ(P1 ∧P2), one obtains from [EHL, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6] that u∗[ΣP1] · u∗[ΣP2] = � � � u∗[ΣP1∧P2] (n − r1) + (n − r2) = n − rank(P1 ∧ P2) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The result now follows from the injectivity of u∗ (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The augmented Bergman classes of polymatroids of type a span A•(Xa) as an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall that A•(Xa) is generated as a ring by the simplicial generators {hS}, and in par- ticular, the monomials in the {hS} span A•(Xa) as an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='14, we are done once we show that each simplicial generator hS is an augmented Bergman class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For each nonempty subset S ⊆ E, let HS be the polymatroid on E of type a whose dual polymatroid has the simplex ∆0 S as its independence polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6, the multi- symmetric lift Mπ(HS) is the matroid on E whose unique circuit is π−1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In [EHL, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2], it is shown that the augmented Bergman class of this matroid is equal to hπ−1(S) ∈ A1(XE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We thus conclude [ΣHS] = hS by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Arguing similarly as in [EHL, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2], one can show that the set of monomials {hd1 F1 · · · hdk Fk : ∅ ⊊ F1 ⊊ · · · ⊊ Fk ⊆ E, d1 ≤ |π−1(F1)| and di ≤ |π−1(Fi\\Fi−1)| for all 2 ≤ i ≤ k} form a Z-basis for A•(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Moreover, combining with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='14, one can further show that these monomials are equal to the augmented Bergman classes of polymatroids whose multi- symmetric lifts are Γa-fixed Schubert matroids on ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In particular, A•(Xa) is generated by the augmented Bergman classes of realizable polymatroids of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This basis can also be obtained from the techniques of [DF10] and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 16 CHRISTOPHER EUR AND MATT LARSON 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Augmented Chow rings of polymatroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This subsection records the properties of the augmented Chow ring of a polymatroid, but is not logically necessary for subsequent sections of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let ℓ ∈ A1(XΣP) be an element corresponding to a strictly convex piecewise linear function on ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then the following hold: (1) (Poincaré duality) There is an isomorphism degP : Ar(XΣP) → Z such that, for 0 ≤ k ≤ r/2, the pairing Ak(XΣP) × Ar−k(XΣP) → Z, (x, y) �→ degP(xy) is unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' (2) (Hard Lefschetz) For every 0 ≤ k ≤ r/2, the map Ak(XΣP) ⊗ Q → Ar−k(XΣP) ⊗ Q, x �→ ℓr−2kx is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' (3) (Hodge-Riemann) For every 0 ≤ k ≤ r/2, the bilinear form Ak(XΣP) ⊗ Q × Ak(XΣP) ⊗ Q → Q, (x, y) �→ (−1)k degP(ℓr−2kxy) is positive definite on the kernel of multiplication by ℓr−2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The support of ΣP is the same at the support of the Bergman fan of Mπ(P) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The result then follows from [ADH22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6] and [AHK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For more details, see [CHL+, Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ As XΣP is a subvariety of Xa, there is a restriction map A•(Xa) → A•(XΣP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We note that the degree map of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='17 satisfies the following version of the projection formula: for any x ∈ A•(Xa), the degree of the image of x in A•(XΣP) is equal to the degree in A•(Xa) of x · [ΣP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The kernel of A•(Xa) → A•(XΣP) is ann([ΣP]), so we may identify A•(P) with A•(XΣP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Poincaré duality, an element x ∈ Ak(Xa) is in the kernel of the map to A•(XΣP) if and only if, for all y ∈ An−r−k(Xa), deg(x · [ΣP] · y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Poincaré duality on A•(Xa), we see that x · [ΣP] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Therefore the kernel of A•(Xa) → A•(XΣP) is ann([ΣP]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We have that A•(P) = Z[xF , yi : F flat, i ∈ E non-loop] I1 + I2 + I3 + I4 , where I1 = ⟨xF1xF2 : F1, F2 incomparable flats⟩, I2 = ⟨ � i∈S yai i : ai > 0, � ai > rkP(S)⟩, I3 = ⟨ � i∈T yai i xF : T ∩ F = ∅, ai > 0, rkP(F ∪ T) ≤ rkP(F) + � ai⟩, and I4 = ⟨yi − � F ̸∋i xF ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As XΣP is a toric variety, its Chow ring is generated by classes corresponding to rays of ΣP, with monomial relations coming from minimal non-faces of the simplicial complex given by the faces of ΣP and a linear relation for each element of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The rays of ΣP correspond to non-loops of E and flats of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For j1, j2 non-loops in E with π(j1) = π(j2), the relation ej1 − ej2 implies that the corresponding divisor classes are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Every non-face of the complex of cones in ΣP contains either {F1, F2} for F1, F2 incom- parable, {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , jk} with rkP(π(j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , jk)) < k, or {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , jℓ, F} for π−1(F) disjoint from {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , jℓ} and rkP(F ∪ π({j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , jℓ})) ≤ rkP(F) + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Putting this all together implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Augmented wonderful varieties of polymatroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We sketch the geometric origins of the notions introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall that, given a realization L ⊆ V = � i∈E Vi of a poly- matroid P, its augmented wonderful variety WL is the closure L in � ∅⊊S⊆E P(� i∈S Vi ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3, we described Xa as a sequence of blow-ups from P(V ⊕ k) along centers disjoint from V ⊂ P(V ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Hence, we have a natural inclusion of V into Xa, and the variety WL is equivalently the closure of L ⊆ V in Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let L ⊆ � i∈E Vi be a realization of a polymatroid P of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then the homology class [WL] is equal to [ΣP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Because GLa = � i∈E GL(Vi) is connected, its action on A•(Xa) is trivial, so for any g ∈ GLa, we have that [WL] = [g · WL] = [Wg·L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If we choose a general g ∈ GLa, then since k is infinite, g · L is general with respect to the (fixed) choice of isomorphisms Vi ∼ → kπ−1(i), so g · L ⊆ kE is a realization of Mπ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By [EHL, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='11(3)], the homology class of the closure of g · L in XE is [ΣMπ(P)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As u: XE → Xa is an isomorphism over g · L, we have u∗[ΣMπ(P)] = [Wg·L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13, [ΣMπ(P)] = u∗[ΣP], so the result follows because u∗u∗ is the identity (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The closure of L in XΣP ⊂ Xa is WL, and the restriction map A•(XΣP) → A•(WL) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Indeed, the iterated blow-up description of WL implies that A•(WL) is generated as a ring by the restriction of hE and the classes of strict transforms of exceptional di- visors on WL, so the restriction map A•(Xa) → A•(WL) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As WL is the union of strict transforms of exceptional divisors and L, the inclusion WL �→ Xa factors through XΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' There- fore the restriction map A•(Xa) → A•(WL) factors through A•(XΣP), so A•(XΣP) → A•(WL) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By [GS21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5], A•(WL) satisfies Poincaré duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A surjective map between Poincaré duality algebras of the same dimension is an isomorphism, so we conclude by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='17(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' THE EXCEPTIONAL ISOMORPHISM In this section, we deduce the isomorphism � r≥0 Valr(a) ≃ � r≥0 Ar(Xa) of graded abelian groups in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' An intermediary object is the Grothendieck ring K(Xa) of vector bun- dles on Xa, which admits a polyhedral description as a polytope algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 18 CHRISTOPHER EUR AND MATT LARSON 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The polytope algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let us review the polytope algebra [McM89] and its relationship to the K-ring of a smooth projective toric variety [Mor93], following [EHL, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a subset S ⊆ Rℓ, recall that 1S : Rℓ → Z denotes its indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Σ be a projective fan in Rℓ that is unimodular over Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' It defines a projective toric variety XΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' A (lattice) polytope Q ⊆ Rℓ is said to be a (lattice) deformation of Σ if its normal fan ΣQ coarsens Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let I(Σ) be the subgroup of Z(Rℓ) generated by {1Q | Q a lattice deformation of Σ}, and let transl(Σ) to be the subgroup of I(Σ) generated by {1Q − 1Q+u | u ∈ Zℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We define the polytope algebra to be the quotient I(Σ) = I(Σ)/ transl(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a lattice deformation Q, denote by [Q] its class in the polytope algebra I(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The multipli- cation in the polytope algebra is induced by Minkowski sum, that is, by [Q1] · [Q2] = [Q1 + Q2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3, a correspondence between lattice deformations of Σ and nef toric divisors on XΣ [CLS11, Chapter 6] associates to each lattice deformation Q a nef divisor DQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This identifies the polytope algebra with the K-ring as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' [EHL, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9] There is an isomorphism I(Σ) ∼ → K(XΣ) defined by [Q] �→ [OXΣ(DQ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This isomorphism implies that a refinement of fans induces an injection of polytope algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Σ and Σ′ be projective unimodular fans such that Σ refines Σ′, so a lattice deformation Q of Σ′ is also a lattice deformation of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, the map I(Σ′) → I(Σ) that sends [Q] ∈ I(Σ′) to [Q] ∈ I(Σ) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let f : XΣ → XΣ′ be the corresponding toric birational map of the toric varieties induced by the map of fans Σ → Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The given map I(Σ′) → I(Σ), under the isomorphism Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2, is the pullback map f ∗ : K(XΣ′) → K(XΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Its injectivity now follows from [CLS11, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5] and the projection formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 to the polystellahedral variety Xa, noting that deformations of the polystellahedral fan Σa are exactly expansions of polymatroids on E (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7), we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The map sending an expanded polymatroid π∗(P) on E to [OXa(Dπ∗(P))] defines an isomorphism I(Σa) ≃ K(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We will thus use these two notions, the polytope algebra and the K-ring, interchangeably for the polystellahedral varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We will use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 in conjunction with the follow- ing method of “breaking-up” a K-class on a polystellahedral variety into smaller pieces when considered as a K-class on the stellahedral variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let P be a polymatroid on E of rank r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, the class [I(π∗(P))] ∈ I(ΣE) equals a linear combination [I(Mπ(P))] + � k ak[I(Mk)] where Mk’s are matroids on E of rank strictly less than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 19 We will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' [EHL, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3] An intersection of the independence polytope I(P) ⊂ RE with an integral translate of the unit cube [0, 1]E, if nonempty, is an integral translate of I(M) for some matroid M on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By tiling RE by integral translates of the unit cube [0, 1]E, we obtain a polyhedral subdivision of I(π∗(P)), with every cell of the subdivision being integral translates of I(M) for some matroid M on E by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2, the polytope I(Mπ(P)) is one of the maximal interior cells of this subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' All other interior cells of the subdivision are of the form I(M) + v for 0 ̸= v ∈ ZE ≥0, which implies that such matroids M are of rank strictly less than r since π∗(P) has rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The exceptional isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We now use the map u: XE → Xa to construct an excep- tional ring isomorphism φa : K(Xa) ∼ → A•(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Its “exceptional” nature is that it differs from the Chern character map, which is an isomorphism ch: K(X)⊗Q → A•(X)⊗Q for any smooth projective variety X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Similar exceptional isomorphisms appeared in [BEST, EHL, LLPP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We prepare by recalling the case of a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1) established in [EHL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' [EHL, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8] There is a unique ring isomorphism φE : K(XE) → A•(XE) such that φE([OXE(hS)]) = 1 + hS for all nonempty S ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Moreover, for any matroid M on E of rank r, the map φE satisfies φE([I(M)]) = ξ0 + ξ1 + · · · + ξr where ξi ∈ Ai(XE) for all i and ξr = [ΣM⊥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The generalization to type a is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall that we have a birational toric map u: XE → Xa induced by the fact that the fan Σa is a coarsening of ΣE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' There exists a (necessarily unique) isomorphism φa : K(Xa) ∼ → A•(Xa) such that we have a commuting diagram K(Xa) A(Xa) K(XE) A(XE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' φa u∗ u∗ φE Moreover, for any polymatroid P on E of type a and rank r, the map φa satisfies φa � [I(π∗(P))] � = ξ0 + ξ1 + · · · + ξr where ξi ∈ Ai(Xa) for all i and ξr = [ΣP⊥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' That the two vertical maps are injections follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' With these injections, we now need show that the map φE restricts to give a well-defined map φa that is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Recall that the Chow ring A•(Xa) is also generated by the simplicial generators 20 CHRISTOPHER EUR AND MATT LARSON hS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We claim that K(Xa) is also generated as a ring by the line bundles [OXa(hS)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Both the well- definedness and the surjectivity of φa would then follow from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7 since u∗hS = hπ−1(S) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For the claim, one notes that for any deformation Q of a projective unimodular fan Σ, the inverse [Q]−1 of the class [Q] ∈ I(Σ) is a polynomial in [Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' See for instance [EHL, Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The claim thus follows because the simplicial generators form a basis of A1(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For the second statement about φa � [I(π∗(P))] � , consider [I(π∗(P))] as an element in K(XE) via the injection u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='7 implies that φE([I(π∗(P))]) = ξ0 + · · · + ξr where ξi ∈ Ai(XE) and ξr = [ΣMπ(P)⊥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lastly, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 implies that [ΣMπ(P)⊥] = u∗[ΣP⊥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let χ: K(Xa) → Z be the sheaf Euler characteristic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We sketch how one can show, arguing similarly as in [EHL, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1], that the isomorphism φa satisfies χ(ξ) = degXa � φa(ξ) · � i∈E (1 + yi)ai� for all ξ ∈ K(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By conjugating the isomorphism φa with the map that sends the K-class of a vector bundle to its dual and the map that is multiplication by (−1)k on Ak(Xa), one obtains an isomorphism ζa such that ζa([OWL]) = [WL] for any realization L ⊆ V of a type a polymatroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='20 with Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='16, one shows that A•(Xa) is spanned as an abelian group by {[WL]: L ⊆ V }, and hence ζa satisfies χ(ξ) = degXa � ζa(ξ) · (1 + hE + · · · + hn E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' One then computes that the anti-canonical divisor of Xa is hE +� i∈E aiyi, and by Serre duality concludes the desired formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' PROOFS OF MAIN THEOREMS We now use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8 to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The valuative group is isomorphic to the Chow homology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Since B(P⊥) = −B(P) + a and I(π∗(P)) = � p−1 π (B(P)) + RE ≤0 � ∩ RE ≥0, the assignment 1B(P) �→ 1I(π∗(P⊥)) gives a well-defined map �n r=0 Valr(a) → I(Σa), because all the operations — negation, translation, inverse image, Minkowski sum, and restriction — behave well with respect to indicator functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Hence, we have a map of abelian groups �n r=0 Valr(a) → K(Xa) defined by 1B(P) �→ [I(π∗(P⊥))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let ψ be the composition of this map with the map φa : K(Xa) → A•(Xa) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that ψ is upper-triangular with respect to the gradings on �n r=0 Valr(a) and A•(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='15, stating that A•(Xa) is spanned by {[ΣP]: P a polymatroid of type a}, implies surjectivity of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For injectivity, suppose we have polymatroids P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Pk of type a and integers c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , ck such that �k j=1 cj[ΣPj] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='13, the validity of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 when a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), established in [EHL, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5], implies that � j cj1B(Mπ(Pj)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Since each Pj is of type a, and since the image under the projection pπ of the unit cube [0, 1]E is the box � i∈E[0, ai] ⊂ RE, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 implies that pπ � B(Mπ(Pj)) � = B(Pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We thus conclude INTERSECTION THEORY OF POLYMATROIDS 21 � j cj1B(Pj) = 0, proving the injectivity of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Therefore ψ is an isomorphism, and so the map that sends 1B(P) to [ΣP] is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Let ψ be the map as constructed in the proof above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Noting that polymatroid duality induces an involution of �n r=0 Valr(a), by composing ψ with the inverse φ−1 a of the isomorphism in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8, we conclude the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The map of abelian groups �n r=0 Valr(a) → K(Xa) defined by 1B(P) �→ [I(π∗(P))] is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The Hall–Rado formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We first note a reinterpretation of the Hall–Rado condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' [McD75, Theorem 2] A collection of subsets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr of E satisfies the Hall–Rado condition with respect to a polymatroid P = (E, rk) of rank r if and only if there exists a map f : [r] → E with f(i) ∈ Si such that �r i=1 ef(i) ∈ B(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a nonempty subset S ⊆ E, we showed in the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='15 that if HS is the polymatroid whose dual polymatroid has the simplex ∆0 S as its independence polytope, then [ΣHS] = hS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Applying this to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8, we have φa([I(π∗(H⊥ S ))]) = 1 + hS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Thus, as the degree map degXa is zero on Ai(Xa) for i < n, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8 implies that degXa � φa([I(π∗(P⊥))][I(π∗(H⊥ S1))] · · · [I(π∗(H⊥ Sr))]) � = degXa � [ΣP] · hS1 · · · hSr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let �P be the polymatroid of rank n on E whose independence polytope is I(P⊥)+∆0 S1+· · ·+∆0 Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Since multiplication in the polytope algebra is Minkowski sum and expansion commutes with Minkowski sum, we have that [I(π∗(�P))] equals the class [I(π∗(P⊥))][I(π∗(H⊥ S1))] · · · [I(π∗(H⊥ Sr))] in the left-hand-side of the equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 and that B(P⊥) = −B(P) + a, we have that a ∈ I(�P) if and only if S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr satisfies the Hall–Rado condition with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The theorem now follows from the following Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For �P a polymatroid of rank n on E, not necessarily of type a, we have that degXa(φa([I(π∗(�P)])) = � � � 1 if a ∈ I(�P) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5 and the commuting diagram in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8, we have that degXa(φa([I(π∗(�P)])) = degXE([ΣMπ(�P)⊥]), which is zero unless Mπ(�P) has rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When Mπ(�P) has rank n, that is, it is the Boolean matroid on E, we have that [ΣMπ(�P)⊥] is the class of a point in A0(XE) = An(XE), and hence degXE([ΣMπ(�P)⊥]) = 1 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Now, note that Mπ(�P) has rank n, or equivalently (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1) ∈ I(Mπ(�P)), if and only if a ∈ I(�P) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 22 CHRISTOPHER EUR AND MATT LARSON Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' At least when P is realizable, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' One can associate to a realization L ⊆ � i∈E Vi of P a realization of a polymatroid with ground set {S : ∅ ⊊ S ⊆ E} via the composition L �→ � i∈E Vi �→ � ∅⊊S⊆E ⊕i∈SVi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The bases of this polymatroid are the sets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr that satisfy the Hall–Rado condition, so applying Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4 recovers Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' One can also prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4 by using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 to reduce to the case of realizable polymatroids, when Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4 is [CCRMMn, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='15] (and can also be deduced from [Li18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='16, in order to check that two valuative functions are equal, it suffices to check on realizable polymatroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The valuativity of [ΣP] implies that the volume polynomial of A•(P) is valuative, and it is clear from the definition of valuativity that the basis generating function of a polymatroid is valuative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' POLYPERMUTOHEDRA Let π: E → E be of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The polystellahedral fan Σπ has the distinguished ray ρ∅ = R≥0(−eE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The star of the fan Σπ at the ray ρ∅ is the polypermutohedral fan Σπ introduced in [CHL+] as the Bergman fan of the boolean polymatroid of type a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Explicitly, the cones of Σπ are in bijection with pairs S ≤ F, where F = {∅ ⊊ F1 ⊊ · · · ⊊ Fk ⊊ Fk+1 = E} is a flag of proper subset of E and S is a subset of E containing no fiber of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Xa be the associated toric variety, which we call the polypermutohedral variety of type a, with the embedding ι: Xa �→ Xa as the toric divisor corresponding to the ray ρ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Suppose P is a polymatroid of type a and rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We note the following computation of the pullback ι∗[ΣP] ∈ Ar−1(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The augmented Bergman fan ΣP contains the ray ρ∅ if and only if P is loopless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Hence, if P has a loop, then ι∗[ΣP] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If P is loopless, the star of ΣP at the ray ρ∅ is the Bergman fan ΣP of P introduced in [CHL+, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' It is an (r − 1)-dimensional balanced subfan of Σπ, and the resulting the Bergman class [ΣP] ∈ Ar−1(Xa) equals the pullback ι∗[ΣP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Using Bergman fans and Bergman classes of loopless polymatroids, we establish analogues of the main theorems Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3 in the polypermutohedral setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The valuative group of loopless polymatroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Define a subgroup of Valr(a) by Val◦ r(a) = the subgroup generated by {1B(P) : P a loopless polymatroid of type a and rank r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that Val◦ 0(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We have the following analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For any 1 ≤ r ≤ n, the map that sends a loopless polymatroid P of type a and rank r to the Bergman class [ΣP] induces an isomorphism Val◦ r(a) ∼ → Ar−1(Xa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' INTERSECTION THEORY OF POLYMATROIDS 23 We will deduce Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1 from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 by identifying the kernel of the map Valr(a) ∼ → Ar(Xa) ι∗ → Ar−1(Xa) with the subgroup of Valr(a) generated by polymatroids with loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' An alternate proof that does not rely on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 but proceeds by developing the polypermuto- hedral analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8 is sketched in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Before proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1, we relate the Poincaré polynomial of the polystellahedral variety to the Poincaré polynomials of polypermutohedral varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For J ⊆ E, let a \\ J be the vector obtained by removing the entries corresponding to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We have that n � i=0 rank Ai(Xa)ti = � J⊆E t|π−1(J)| n−|π−1(J)|−1 � i=0 rank Ai(Xa\\J)ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' As the Poincaré polynomial of a smooth projective toric variety is the h-polynomial of its fan, it is enough to show that f(Σa)(t) = (1 + t)n + � ∅⊊J⊆E t(1 + t)|π−1(J)|f(Σa\\J)(t), where f(Σ) is the f-polynomial of a fan Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We prove this bijectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' To each cone σ of some Σa\\J corresponding to a pair S ≤ F, we obtain 2|π−1(J)| cones of Σa by adding J to every element of the flag and then add all possible 2|π−1(J)| subsets of π−1(J) to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When J ̸= E and we add k elements to S, this gives a cone of dimension dim σ +k +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' When J = E and we add k elements to S, this gives a cone of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For any i ∈ E, a polymatroid base polytope B(P) is always contained in the half-space {x ∈ RE : xi ≥ 0}, and it is contained in the hyperplane {x ∈ RE : xi = 0} if and only if P has i as a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Thus, the claim in the proof of [BEST, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='9] implies that we have a decomposition Valr(a) = � J⊆E Val◦ r(a \\ J) given by sending a loopless polymatroid P of rank r on E \\ J to the polymatroid on E with rk(S) = rkP(S ∩ J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We now induct on the size of E, where the base case |E| = 1 is straightfor- ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Comparing the decomposition of Valr(a) above with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2, we see that the induc- tion hypothesis implies rank Ar−1(Xa) = rank Val◦ r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' By construction of the permutohedral fan Σπ as the star of ray ρ∅ in Σπ, every ray of Σπ is the image of a ray in Σπ that forms a cone with ρ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Hence, the pullback ι∗ : A•(Xa) → A•(Xa) is surjective because ι∗ : A1(Xa) → A1(Xa) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We thus have a surjection ι∗ : Ar(Xa) → Ar−1(Xa) that satisfies ι∗[ΣP] = [ΣP] if P is loopless and ι∗[ΣP] = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Therefore the composition Val◦ r(a) → Ar(Xa) → Ar−1(Xa) is a surjection of finite free abelian groups of the same rank, and hence is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ 24 CHRISTOPHER EUR AND MATT LARSON Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We sketch an alternate proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' First, arguing as in [EFLS, Proof of Theorem D], one shows an isomorphism �n r=1 Val◦ r(a) ≃ K(Xa) when a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1), and uses it to deduce Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1 for the a = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , 1) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Now, using that polypermutohedral fans are coarsenings of the permutohedral fan ΣE, just as polystellahedral fans are coarsenings of the stellahedral fan, one similarly deduces the polypermutohedral analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, one deduces Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='1 the same way that we proved Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='6 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The dragon Hall–Rado formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Let Xa be the polypermutohedral variety, with the em- bedding ι: Xa �→ Xa as the toric divisor corresponding to the ray ρ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The following theorem generalizes [BES, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' For a polymatroid P = (E, rk) of rank r, a collection of subsets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1 is said to satisfy the dragon Hall–Rado condition if rk � � j∈J Sj � ≥ |J| + 1 for all nonempty J ⊆ [r − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, if P is loopless, we have degXa(hS1 · · · hSr−1[ΣP][Xa]) = � � � 1 if the dragon Hall–Rado condition is satisfied 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Note that, in A•(Xa), we have that x∅ = − � ∅⊊S⊆E(−1)|S|hS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Then, for any S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1, (1) degXa(hS1 · · · hSr−1[ΣP][Xa]) = − � ∅⊊S⊆E (−1)|S| degP(hS1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' hSr−1hS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Suppose we have sets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1 that satisfy the dragon Hall–Rado condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Because P is loopless, every term in the above sum corresponds to r sets that satisfy the Hall–Rado condition, and so each term is (−1)|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Because the sum is over nonempty sets, this gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Suppose that S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1 fails the dragon Hall–Rado condition, and let T1, T2 be nonempty subsets of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' If S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1, T1 and S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1, T2 both fail the Hall–Rado condition, then so does S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1, T1 ∩ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' We claim that S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1, T1 ∪ T2 fails the Hall–Rado condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Indeed, if there is a function f : [r] → E as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2 with f(r) ∈ T1 ∪ T2, then f(r) lies in T1 or T2, contradicting the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' This implies that the set {T : ∅ ⊊ T ⊊ E, S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' , Sr−1, T fails dragon Hall–Rado} is a boolean sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Therefore the sum in (1) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='4 can be alternatively proved along the lines of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3, by using the polypermutohedral analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='8 and a reformation of the dragon Hall-Rado- condition in terms of a matching condition as in [BES, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' REFERENCES [AA] Marcelo Aguiar and Federico Ardila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Hopf monoids and generalized permutahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Simplicial generation of Chow rings of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' (to appear).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Springer-Verlag, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 10 [Li18] Binglin Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Images of rational maps of projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' IMRN, (13):4190–4228, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 3, 22 26 CHRISTOPHER EUR AND MATT LARSON [LLPP] Matt Larson, Shiyue Li, Sam Payne, and Nicholas Proudfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' K-rings of wonderful varieties and matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='03169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 3, 19 [Lov77] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Lovász.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Flats in matroids and geometric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' In Combinatorial surveys (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Sixth British Combinatorial Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', Royal Holloway Coll.' metadata={'source': 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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', 78(1):76–130, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 18 [Mor93] Robert Morelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' The K-theory of a toric variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', 100(2):154–182, 1993.' metadata={'source': 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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=', 354(5):1921–1945, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content=' 3 Email address: ceur@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQf6_p-/content/2301.00831v1.pdf'} +page_content='harvard.' metadata={'source': 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submission January 12, 2023) +We present numerical analysis of the lateral movement of a deformable spherical capsule in a pul- +satile channel flow, with a Newtonian fluid in almost inertialess condition and at a small confinement +ratio R/a = 2.5, where R and a are the channel and capsule radius. We find that the speed of the +axial migration of the capsule can be accelerated by the flow pulsation at a specific frequency. The +migration speed increases with the oscillatory amplitude, while the most effective frequency remains +basically unchanged and independent of the amplitude. Our numerical results form a fundamental +basis for further studies on cellular flow mechanics, since pulsatile flows are physiologically relevant +in human circulation, potentially affecting the dynamics of deformable particles and red blood cells, +and can also be potentially exploited in cell focusing techniques. +High-throughput +measurements +of +single-cell +be- +haviour under confined channel flow is of fundamental +importance and technical requirement in bioengineering +applications such as cellular-level diagnoses for blood dis- +eases. Although several attempts have addressed this is- +sue and gained insights into (soft) particle dynamics in +microchannels [1–3], cell manipulation including label- +free cell alignment, sorting, and separation still face ma- +jor challenges. +Along with the aforementioned experi- +mental studies, recent numerical simulations revealed the +mechanical background regarding the lateral movement +of particles, e.g. in [4–6]. The lateral movement of de- +formable spherical particles in almost inertialess condi- +tions was originally reported in Karnis et al. [7], and +these results have been the fundamental basis to describe +the phenomena observed in microfluidics [8] but also in +in vivo microcirculations [9]. In particular, it was found +that a deformable spherical particle tends to move to- +wards the channel axis and settles there. Hereafter we +will call this phenomenon as “axial migration”. In a more +recent work, the framework of the axial migration of a +droplet has been extended by Santra and Chakraborty +[10] by including the effect of an electric field, and find- +ing that as the strength of the electric field increases, +droplets can reach the centreline at a faster rate with +reduced axial oscillations. Furthermore, a deformation- +dependent propulsion of soft particles, including biologi- +cal cells, were confirmed experimentally by Krauss et al. +[11] and numerically by [12]. Despite these efforts, the +effect of a pulsatile flow on the axial migration of cap- +sules, where the internal fluid is enclosed by a thin elas- +tic membrane, has not been described and understood +yet. The objective of this study is thus to clarify whether +frequency-dependent axial migration of the spherical cap- +sule occurs in confined channel flows. More precisely, can +the time necessary for the axial migration be controlled +by the channel pulsations? Is there an optimal pulsation +frequency to do that? +FIG. 1. Visualization of a spherical capsule with radius a0 in +a tube with radius of R under a pulsatile flow with velocity +V ∞, which can be decomposed into the steady parabolic flow +V ∞ +0 +and the oscillatory flow V ∞ +osci in the absence of any cells. +The capsule, initially placed near the wall, exhibits axial mi- +gration. +To answer these fundamental questions, we perform a +series of fully resolved numerical simulations. We con- +sider the motion of an initially spherical capsule with di- +ameter d0 (= 2a0 = 8 µm) flowing in a circular channel +of diameter D (= 2R = 20 µm), see Fig. 1. The capsule +is made by an elastic membrane, separating two New- +tonian fluids, which satisfy the incompressible Navier– +Stokes equations, and have the same density ρ but dif- +ferent viscosity (inside) µ1 and (outside) µ0. The mem- +brane is elastic, modeled as an isotropic and hyperelastic +material based on the Skalak constitutive law [13], with +surface shear elastic modulus Gs. For instance, the sur- +face shear elastic modulus is equal to Gs = 4 µN/m when +considering the human red blood cells [14, 15]. The flow +in the channel is sustained by a uniform pressure gradient +∇p0, which can be related to the maximum fluid velocity +in the channel as ∇p0 = −4µ0V ∞ +max/R2. The pulsation +is instead given by a superimposed sinusoidal function, +such that the total pressure gradient is +∇p(t) = ∇p0 + (∇pamp) sin (2πft). +(1) + +op ++ +R +V +X +81 +osci +Z +82 +The problem is governed by six main non-dimensional +numbers: i) the Reynolds number Re = ρDV ∞ +max/µ0; +ii) the capillary number Ca = µ0 ˙γma0/Gs, where ˙γm = +V ∞ +max/4R; iii) the viscosity ratio between the two flu- +ids λ = µ1/µ0; iv) the confinement ratio d0/D; v) the +non-dimensional pulsation frequency f ∗ = f/ ˙γm; vi) the +non-dimensional pulsation amplitude ∇pamp/∇p0 . +In +this work, all simulations are performed in an almost in- +ertialess condition, keeping the Reynolds number low and +fixed to the value Re = 0.2; also, we limit our main anal- +ysis to a confinement ratio of 0.8. In the Supplemental +Materials we verify the sensitivity of the results to these +two parameters [16]. Instead here we comprehensively +vary the amplitude and frequency of the pulsation, the +viscosity ratio and the capillary number. As we will de- +scribe in the following, our investigation of the capsule +dynamic show that it indeed exists an optimal frequency +to speed-up the capsule axial migration by up to 80% in +the range of parameters investigated here. +FIG. 2. (a) Time history of the radial position of the capsule +centroid r/R for different non-dimensional frequency f ∗. The +inset images represent the capsule initial state (r0/R = 0.55 +at ˙γmt = 0) and the final stable state at the channel center +line ((r/R) ≈ 0) at ˙γmt = 50). (b-d) Side views of the capsule +during its axial migration for f ∗ = 0.5 and different oscilla- +tory amplitude: (b) ∇pamp = ∇p0 and (c) ∇pamp = 4∇p0. +The snapshots are taken at the time instants marked in (d), +showed over the time history of r/R. All the results are ob- +tained with Ca = 1.2, and λ = 1. +To prove this, first, we investigate the trajectory of +the capsule centroids for different frequencies f ∗ = f/ ˙γm. +The time history of the radial position of the capsule cen- +troid r is shown in Fig. 2(a), together with the capsule +shape at the initial (˙γmt = 0) and final states (˙γmt = 50). +The capsule, initially spherical, migrates towards the +channel centerline while deforming, finally reaching its +equilibrium position at the centerline, where it achieves +an axial-symmetric shape. While the trajectory obtained +with the highest frequency investigated (f ∗ = 5) well +collapses on that obtained with a steady flow, see the +Supplemental Materials [16], when f ∗ is small enough, +the trajectory paths depend on the pulsation frequency, +with the appearance of oscillations and with different ax- +ial migration speed. The migration speed is also affected +by the amplitude of the oscillation ∇pamp, as shown in +Figs. 2(b)–2(d). Indeed, as ∇pamp increases, the capsule +appears to migrate faster toward the channel centerline. +To properly quantify the changes in axial migration, +we define the migration time T ∗ as the time needed by +the capsule centroid to reach the centerline (within a dis- +tance of ∼6% of its radius to account for the oscillations +in the capsule trajectory). The ratio of the elapsed time +T ∗ and that in a steady flow is reported in Fig. 3(a) as a +function of f ∗, for various pulsation amplitudes. The re- +sults clearly suggest that there exist a specific frequency +to minimize the migration time. A very minor increase of +the optimal frequency with the pulsation amplitude can +be observed in the data. While the optimal frequency +is almost independent of the pulsation amplitude, the +migration time can be strongly reduced by its increase. +Indeed, while the elapsed time is reduced by 18% at the +lowest amplitude investigated (∇pamp = ∇p0/4), it is re- +duced by 80% at the highest one (∇pamp = 4∇p0). The +changes in the migration time are clearly reflected in the +migration speed V∗ = V/V ∞ +max, reported in Fig. 3(b), +which shows that when the migration time is minimum, +the axial migration speed reaches almost its maximum. +Here, the migration speed V is defined as the ratio of +the elapsed time T and the traveled distance L (i.e., +V = L/T ), defined as L = � L +0 |dr| = � L +0 dr·ˆt = � T +0 vdt·ˆt, +where ˆt = r/|dr| is the unit tangential vector along the +trajectory of the capsule centroid and v is the the cap- +sule centroid velocity. The distance traveled by the cap- +sule before completing the axial migration is reported +in Fig. 3(c) for the sake of completeness, showing that +the optimal frequency to minimize the migration time, +roughly corresponds to the minimization of the the trav- +eled distance too. +In summary, we have shown that, for a fixed Ca and λ, +there is an optimal frequency for the channel pulsation, +able to minimize the capsule migration time by maxi- +mizing the migration speed and minimizing the traveled +distance. To complete our investigation, the effects of Ca +and λ on the migration time T ∗ are shown in Fig. 4. In + +0.6 +=Vp +*=5 +Vn +0.5 +* = 0.5 + = 0.1 +Initial state (ro/R = 0.55) +0.4 +f* = 0.05 +f* = 0.01 +0.3 +*= 0.005 +0.2 +0.1 +Final state (r/R = O(10-3) +900 +10 +20 +30 +40 +50 +Ym +(b) +(c) +(p) +0.6 +(*1) +f* = 0.5 +(*1) +0.5 +de +Vpo +Vp += 4Vpo +amp +Vp +(*2) +0.4 +(*2) +Time +(*3) +R +0.3 +(*3) +0.2 +(*4) +0.1 +(*4) +0.0 +10 +20 +30 +40 +0 +50 +X +Vpamp = 4Vpo3 +FIG. 3. (a) The migration time T ∗, (b) the migration speed +V∗, and (c) the distance traveled during the migration L∗, +normalized with those obtained in a steady flow (T ∗ +steady, +V∗ +steady, and L∗ +steady) as a function of f ∗ and for different +∇pamp. The results are obtained with Ca = 1.2, and λ = +1. The filled symbols in each panels represent the case with +the optimal frequency which minimizes the migration time. +particular, the results in Fig. 4(a) shows that the migra- +tion time depends on Ca, thus suggesting that the opti- +mal frequency f ∗ is also a function of Ca. On the other +hand, as shown in Fig. 4(b), the migration time remains +almost independent of the viscosity ratio for λ <∼ 5. +In conclusion, we have proved that the axial migra- +tion speed of an elastic capsule in a pipe flow can be +substantially accelerated by making the driving pressure +FIG. 4. The migration time (a) as a function of Ca at λ = 1 +and f ∗ = 0.01 and (b) as a function of λ at Ca = 1.2 and +f ∗ = 0.01. The filled symbol in (a) represent the case with +the optimal Ca(= 0.1). +gradient oscillating in time. +We found that, the axial +migration speed increases with the amplitude of the os- +cillation, while the most effective frequency revealed to +be independent of the oscillatory amplitude. +Also, we +showed that the optimal frequency depends on Ca, but +is basically independent of the viscosity ratio λ, over- +all proving that the changes in the axial migration are +mostly due to the membrane elasticity. +Our numerical results provide a fundamental basis for +further studies on unsteady cellular flow dynamics. Also, +since the axial migration of the capsule can be controlled +by the background flow strength and frequency, our re- +sults can be easily employed for label-free cell align- +ment/sorting/separation techniques to precisely diagnose +patients with hematologic disorders, or for the analysis +of anticancer drug efficacy in cancer patients. +ACKNOWLEDGMENTS +N.T. was supported by JSPS KAKENHI Grant Num- +ber JP20H04504, and by the Keihanshin Consortium +for Fostering the Next Generation of Global Leaders in + +(a) +1.0 +f* = 0.01, 2 = 1 +0.8 + steady +0.6 +*L/* +0.4 +0.2 +0.0 +10-2 +101 +10-1 +100 +Ca +(b) +1.0 +f* = 0.01, Ca = 1.2 +0.8 + steady +0.6 +*L/*L +0.4 +D- +0.2 +0.0 +101 +102 +10-1 +100 +入(a) +1.0 +Vpamp = Vp/4 +Ca = 1.2 +白 +白 +0.8 += 2Vpo +amp +- +白 + steady += 4Vpo +0.6 +X +*L/*L +0.4 +- +0.2 +0.0 +10-4 +10-3 +10-2 +10-1 +100 +10-s +101 +(b) +4.0 +3.5 +3.0 +0- - +2.5 +2.0 F +1.5 E +口 +1.0 +10-2 +10-1 +10-s +104 +103 +101 +100 +(c) +0.2 +0.1 + steady +0.0 +*T/ +0.15 +-0.1 +L* +口 +0.10E +-0.2 +0.05 E +国 +-0.3 +0.00 +10-2 +10-4 +10-3 +10-1 +-0.4 +10-1 +10° +101 +10-4 +103 +102 +10-s4 +Research (K-CONNEX), established by the Human Re- +source Development Program for Science and Technol- +ogy. +M.E.R. was supported by the Okinawa Institute +of Science and Technology Graduate University (OIST) +with subsidy funding from the Cabinet Office, Govern- +ment of Japan. Finally, the collaborative research was +supported by the SHINKA grant provided by OIST. +∗ takeishi.naoki.es@osaka-u.ac.jp +† marco.rosti@oist.jp +[1] A. T. Ciftlik, M. Ettori, +and M. Gijs, Small 9, 2764 +(2013). +[2] B. Fregin, F. Czerwinski, D. Biedenweg, S. Girardo, +S. Gross, K. Aurich, +and O. Otto, Nat. Commun. 10, +415 (2019). +[3] H. +Ito, +R. +Murakami, +S. +Sakuma, +C.-H. +Tsai, +T. Gutsmann, K. Brandenburg, J. Po¨oschl, F. Arai, +M. Kaneko, and M. Tanaka, Sci. Rep. 7, 43134 (2017). +[4] D. Alghalibi, M. E. Rosti, +and L. Brandt, Phys. Rev. +Fluids 4, 104201 (2019). +[5] N. Takeishi, H. Yamashita, T. Omori, N. Yokoyama, and +M. Sugihara-Seki, Micromachines 12, 1162 (2021). +[6] N. Takeishi, H. Yamashita, T. Omori, N. Yokoyama, +S. Wada, +and M. Sugihara-Seki, J. Fluid Mech. 952, +A35 (2022). +[7] A. Karnis, H. L. Goldsmith, +and S. G. Mason, Nature +200, 159 (1963). +[8] B. Kim, S. S. Lee, T. H. Yoo, S. Kim, S. Y. Kim, S.-H. +Choi, and J. M. Kim, Sci. Adv. 5, eaav4819 (2019). +[9] T. W. Secomb, Annu. Rev. Fluid Mech. 49, 443 (2017). +[10] S. Santra and S. Chakraborty, J. Fluid Mech. 907, A8 +(2021). +[11] S. W. Krauss, P.-Y. Gires, +and M. Weiss, Phys. Rev. +Fluids 7, L082201 (2022). +[12] W. Schmidt, A. F¨ortsch, M. Laumann, and W. Zimmer- +mann, Phys. Rev. Fluids 7, L032201 (2022). +[13] R. Skalak, A. Tozeren, R. P. Zarda, and S. Chien, Bio- +phys. J. 13, 245 (1973). +[14] N. Takeishi, Y. Imai, K. Nakaaki, T. Yamaguchi, +and +T. Ishikawa, Physiol. Rep. 2, e12037 (2014). +[15] N. Takeishi, M. E. Rosti, Y. Imai, S. Wada, +and +L. Brandt, J. Fluid Mech. 872, 818 (2019). +[16] See the Supplemental Materials for additional details on +the numerical method, and results, which include Refs.[5, +6, 13–15, 17–21]. +[17] D. Barth´es-Biesel, A. Diaz, and E. Dheni, J. Fluid Mech. +460, 211 (2002). +[18] J. Li, M. Dao, C. T. Lim, and S. Suresh, Phys. Fluids +88, 3707 (2005). +[19] S. Chen and G. D. Doolen, Annu. Rev. Fluid. Mech. 30, +329 (1998). +[20] J. Walter, A. V. Salsac, D. Barth´es-Biesel, +and P. L. +Tallec, Int. J. Numer. Meth. Eng. 83, 829 (2010). +[21] C. S. Peskin, Acta Numer. 11, 479 (2002). + diff --git a/V9E3T4oBgHgl3EQfFAnd/content/tmp_files/load_file.txt b/V9E3T4oBgHgl3EQfFAnd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8785954a7aa15e8b5aae70721bc40d763e724aab --- /dev/null +++ b/V9E3T4oBgHgl3EQfFAnd/content/tmp_files/load_file.txt @@ -0,0 +1,345 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf,len=344 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='04302v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='flu-dyn] 11 Jan 2023 Enhanced axial migration of a deformable capsule in pulsatile channel flows Naoki Takeishi1, ∗ and Marco Edoardo Rosti2, † 1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 2Complex Fluids and Flows Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' (Dated: First submission January 12, 2023) We present numerical analysis of the lateral movement of a deformable spherical capsule in a pul- satile channel flow, with a Newtonian fluid in almost inertialess condition and at a small confinement ratio R/a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5, where R and a are the channel and capsule radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' We find that the speed of the axial migration of the capsule can be accelerated by the flow pulsation at a specific frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The migration speed increases with the oscillatory amplitude, while the most effective frequency remains basically unchanged and independent of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Our numerical results form a fundamental basis for further studies on cellular flow mechanics, since pulsatile flows are physiologically relevant in human circulation, potentially affecting the dynamics of deformable particles and red blood cells, and can also be potentially exploited in cell focusing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' High-throughput measurements of single-cell be- haviour under confined channel flow is of fundamental importance and technical requirement in bioengineering applications such as cellular-level diagnoses for blood dis- eases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Although several attempts have addressed this is- sue and gained insights into (soft) particle dynamics in microchannels [1–3], cell manipulation including label- free cell alignment, sorting, and separation still face ma- jor challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Along with the aforementioned experi- mental studies, recent numerical simulations revealed the mechanical background regarding the lateral movement of particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' in [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The lateral movement of de- formable spherical particles in almost inertialess condi- tions was originally reported in Karnis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' [7], and these results have been the fundamental basis to describe the phenomena observed in microfluidics [8] but also in in vivo microcirculations [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In particular, it was found that a deformable spherical particle tends to move to- wards the channel axis and settles there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Hereafter we will call this phenomenon as “axial migration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In a more recent work, the framework of the axial migration of a droplet has been extended by Santra and Chakraborty [10] by including the effect of an electric field, and find- ing that as the strength of the electric field increases, droplets can reach the centreline at a faster rate with reduced axial oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Furthermore, a deformation- dependent propulsion of soft particles, including biologi- cal cells, were confirmed experimentally by Krauss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' [11] and numerically by [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Despite these efforts, the effect of a pulsatile flow on the axial migration of cap- sules, where the internal fluid is enclosed by a thin elas- tic membrane, has not been described and understood yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The objective of this study is thus to clarify whether frequency-dependent axial migration of the spherical cap- sule occurs in confined channel flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' More precisely, can the time necessary for the axial migration be controlled by the channel pulsations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Is there an optimal pulsation frequency to do that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Visualization of a spherical capsule with radius a0 in a tube with radius of R under a pulsatile flow with velocity V ∞, which can be decomposed into the steady parabolic flow V ∞ 0 and the oscillatory flow V ∞ osci in the absence of any cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The capsule, initially placed near the wall, exhibits axial mi- gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' To answer these fundamental questions, we perform a series of fully resolved numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' We con- sider the motion of an initially spherical capsule with di- ameter d0 (= 2a0 = 8 µm) flowing in a circular channel of diameter D (= 2R = 20 µm), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The capsule is made by an elastic membrane, separating two New- tonian fluids, which satisfy the incompressible Navier– Stokes equations, and have the same density ρ but dif- ferent viscosity (inside) µ1 and (outside) µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The mem- brane is elastic, modeled as an isotropic and hyperelastic material based on the Skalak constitutive law [13], with surface shear elastic modulus Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' For instance, the sur- face shear elastic modulus is equal to Gs = 4 µN/m when considering the human red blood cells [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The flow in the channel is sustained by a uniform pressure gradient ∇p0, which can be related to the maximum fluid velocity in the channel as ∇p0 = −4µ0V ∞ max/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The pulsation is instead given by a superimposed sinusoidal function, such that the total pressure gradient is ∇p(t) = ∇p0 + (∇pamp) sin (2πft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' (1) op + R V X 81 osci Z 82 The problem is governed by six main non-dimensional numbers: i) the Reynolds number Re = ρDV ∞ max/µ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' ii) the capillary number Ca = µ0 ˙γma0/Gs, where ˙γm = V ∞ max/4R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' iii) the viscosity ratio between the two flu- ids λ = µ1/µ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' iv) the confinement ratio d0/D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' v) the non-dimensional pulsation frequency f ∗ = f/ ˙γm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' vi) the non-dimensional pulsation amplitude ∇pamp/∇p0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In this work, all simulations are performed in an almost in- ertialess condition, keeping the Reynolds number low and fixed to the value Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' also, we limit our main anal- ysis to a confinement ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In the Supplemental Materials we verify the sensitivity of the results to these two parameters [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Instead here we comprehensively vary the amplitude and frequency of the pulsation, the viscosity ratio and the capillary number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' As we will de- scribe in the following, our investigation of the capsule dynamic show that it indeed exists an optimal frequency to speed-up the capsule axial migration by up to 80% in the range of parameters investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' (a) Time history of the radial position of the capsule centroid r/R for different non-dimensional frequency f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The inset images represent the capsule initial state (r0/R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='55 at ˙γmt = 0) and the final stable state at the channel center line ((r/R) ≈ 0) at ˙γmt = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' (b-d) Side views of the capsule during its axial migration for f ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 and different oscilla- tory amplitude: (b) ∇pamp = ∇p0 and (c) ∇pamp = 4∇p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The snapshots are taken at the time instants marked in (d), showed over the time history of r/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' All the results are ob- tained with Ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2, and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' To prove this, first, we investigate the trajectory of the capsule centroids for different frequencies f ∗ = f/ ˙γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The time history of the radial position of the capsule cen- troid r is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 2(a), together with the capsule shape at the initial (˙γmt = 0) and final states (˙γmt = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The capsule, initially spherical, migrates towards the channel centerline while deforming, finally reaching its equilibrium position at the centerline, where it achieves an axial-symmetric shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' While the trajectory obtained with the highest frequency investigated (f ∗ = 5) well collapses on that obtained with a steady flow, see the Supplemental Materials [16], when f ∗ is small enough, the trajectory paths depend on the pulsation frequency, with the appearance of oscillations and with different ax- ial migration speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The migration speed is also affected by the amplitude of the oscillation ∇pamp, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 2(b)–2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Indeed, as ∇pamp increases, the capsule appears to migrate faster toward the channel centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' To properly quantify the changes in axial migration, we define the migration time T ∗ as the time needed by the capsule centroid to reach the centerline (within a dis- tance of ∼6% of its radius to account for the oscillations in the capsule trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The ratio of the elapsed time T ∗ and that in a steady flow is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 3(a) as a function of f ∗, for various pulsation amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The re- sults clearly suggest that there exist a specific frequency to minimize the migration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' A very minor increase of the optimal frequency with the pulsation amplitude can be observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' While the optimal frequency is almost independent of the pulsation amplitude, the migration time can be strongly reduced by its increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Indeed, while the elapsed time is reduced by 18% at the lowest amplitude investigated (∇pamp = ∇p0/4), it is re- duced by 80% at the highest one (∇pamp = 4∇p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The changes in the migration time are clearly reflected in the migration speed V∗ = V/V ∞ max, reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 3(b), which shows that when the migration time is minimum, the axial migration speed reaches almost its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Here, the migration speed V is defined as the ratio of the elapsed time T and the traveled distance L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=', V = L/T ), defined as L = � L 0 |dr| = � L 0 dr·ˆt = � T 0 vdt·ˆt, where ˆt = r/|dr| is the unit tangential vector along the trajectory of the capsule centroid and v is the the cap- sule centroid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The distance traveled by the cap- sule before completing the axial migration is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 3(c) for the sake of completeness, showing that the optimal frequency to minimize the migration time, roughly corresponds to the minimization of the the trav- eled distance too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In summary, we have shown that, for a fixed Ca and λ, there is an optimal frequency for the channel pulsation, able to minimize the capsule migration time by maxi- mizing the migration speed and minimizing the traveled distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' To complete our investigation, the effects of Ca and λ on the migration time T ∗ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='6 =Vp =5 Vn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='1 Initial state (ro/R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='4 f* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='05 f* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='1 Final state (r/R = O(10-3) 900 10 20 30 40 50 Ym (b) (c) (p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='6 (*1) f* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 (*1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 de Vpo Vp = 4Vpo amp Vp (*2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='4 (*2) Time (*3) R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='3 (*3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 (*4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='1 (*4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 10 20 30 40 0 50 X Vpamp = 4Vpo3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' (a) The migration time T ∗, (b) the migration speed V∗, and (c) the distance traveled during the migration L∗, normalized with those obtained in a steady flow (T ∗ steady, V∗ steady, and L∗ steady) as a function of f ∗ and for different ∇pamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The results are obtained with Ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2, and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The filled symbols in each panels represent the case with the optimal frequency which minimizes the migration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' particular, the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 4(a) shows that the migra- tion time depends on Ca, thus suggesting that the opti- mal frequency f ∗ is also a function of Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' On the other hand, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 4(b), the migration time remains almost independent of the viscosity ratio for λ <∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' In conclusion, we have proved that the axial migra- tion speed of an elastic capsule in a pipe flow can be substantially accelerated by making the driving pressure FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The migration time (a) as a function of Ca at λ = 1 and f ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='01 and (b) as a function of λ at Ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 and f ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' The filled symbol in (a) represent the case with the optimal Ca(= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' gradient oscillating in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' We found that, the axial migration speed increases with the amplitude of the os- cillation, while the most effective frequency revealed to be independent of the oscillatory amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Also, we showed that the optimal frequency depends on Ca, but is basically independent of the viscosity ratio λ, over- all proving that the changes in the axial migration are mostly due to the membrane elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Our numerical results provide a fundamental basis for further studies on unsteady cellular flow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Also, since the axial migration of the capsule can be controlled by the background flow strength and frequency, our re- sults can be easily employed for label-free cell align- ment/sorting/separation techniques to precisely diagnose patients with hematologic disorders, or for the analysis of anticancer drug efficacy in cancer patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' ACKNOWLEDGMENTS N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' was supported by JSPS KAKENHI Grant Num- ber JP20H04504, and by the Keihanshin Consortium for Fostering the Next Generation of Global Leaders in (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 f* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='01, 2 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='8 steady 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='6 L/* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 10-2 101 10-1 100 Ca (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 f* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='01, Ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='8 steady 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='6 L/*L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='4 D- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 101 102 10-1 100 入(a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 Vpamp = Vp/4 Ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 白 白 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='8 = 2Vpo amp 白 steady = 4Vpo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='6 X L/*L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 10-4 10-3 10-2 10-1 100 10-s 101 (b) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 0- - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='5 E 口 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 10-2 10-1 10-s 104 103 101 100 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='1 steady 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='0 T/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='1 L* 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='10E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='05 E 国 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='00 10-2 10-4 10-3 10-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='4 10-1 10° 101 10-4 103 102 10-s4 Research (K-CONNEX), established by the Human Re- source Development Program for Science and Technol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' was supported by the Okinawa Institute of Science and Technology Graduate University (OIST) with subsidy funding from the Cabinet Office, Govern- ment of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Finally, the collaborative research was supported by the SHINKA grant provided by OIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' ∗ takeishi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='naoki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='es@osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='jp † marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='rosti@oist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='jp [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Ciftlik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Ettori, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Gijs, Small 9, 2764 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Fregin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Czerwinski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Biedenweg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Girardo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Gross, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Aurich, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Otto, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 10, 415 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Ito, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Murakami, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Sakuma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Tsai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Gutsmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Brandenburg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Po¨oschl, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Arai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Kaneko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Tanaka, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} +page_content=' 7, 43134 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfFAnd/content/2301.04302v1.pdf'} 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diff --git a/XtAzT4oBgHgl3EQf1_4E/content/tmp_files/2301.01806v1.pdf.txt b/XtAzT4oBgHgl3EQf1_4E/content/tmp_files/2301.01806v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..acb0f78a32674f2d63dfe25e821f95357fba11d7 --- /dev/null +++ b/XtAzT4oBgHgl3EQf1_4E/content/tmp_files/2301.01806v1.pdf.txt @@ -0,0 +1,463 @@ +arXiv:2301.01806v1 [hep-ph] 4 Jan 2023 +Renormalon-chain contributions to two-point +correlators of nonlocal quark currents +S. V. Mikhailova,1, N. Volchanskiya,b,2 +a Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia +b Research Institute of Physics, Southern Federal University, +Prospekt Stachki 194, 344090, Rostov-na-Donu, Russia +We calculate, within massless QCD, a two-point correlator of nonlocal (compos- +ite) vector quark currents with arbitrary-length chains of the simplest fermion loops +being inserted into gluon lines. Within the large nf (or large β0) approximation, +the correlator defines a perturbative contribution to the leading-twist distribution +amplitudes for light mesons. +Our results are consistent with a number of spe- +cial cases in the literature. We consider functionals of the correlator, which are +important for the phenomenology, and their properties as function series. +PACS: 11.15.Pg; 11.25.Db; 12.38.-t; 12.38.Bx +Introduction +We consider two-point correlators Πn(x, y; L) of nonlocal vector quark +currents within large-β0 approximation to massless perturbative QCD1 in +MS scheme, +− ias +π2NcCFAnΠn(x, y; L) = +� +dDη eipη⟨0|ˆT +� +J†(η; x)J(0; y) +� +|0⟩ += +n ++ +n ++ +n ++ +n ++ . . . +(1) +Here, L = ln(−p2/µ2) with p being an external momentum and µ being +the renormalization scale and the constant A = +4 +3asTFnf can be replaced +by −asβ0 as prescribed by the naive nonabelization trick. +In Eq. (1), +the nonlocal vector quark current J(η; x) is defined as the inverse Mellin +transform ˆM−1 of a quark bilinear involving the Nth derivative of a quark +field operator2: +J(η; x) = ˆM−1J(η; N), +J(η; N) = ¯d(η)˜n/ (i˜n∇)N u(η), +(2) +1E-mail: mikhs@theor.jinr.ru +2E-mail: nikolay.volchanskiy@gmail.com +1We work in QCD with nf = 3 massless quark flavors; Nc = 3 is the number of colors; +the Casimir invariants are CA = 3 and CF = 4/3; β0 = 11 +3 CA − 4 +3TF nf = 9 is the one-loop +β function coefficient; TF = 1 +2; as = αs/(4π) is the coupling constant. +2Note that, in this paper, arguments of the Mellin transform are underlined, i.e. f(a) = +ˆMf(x) = +� 1 +0 dx f(x)xa. + +2 +where x is a Bjorken fraction, η is a space-time point, ∇µ = ∂µ − igtaAa +µ +is the QCD covariant derivative, ˜nµ is a light-like vector, ˜n2 = 0. In QCD, +the nonlocal current (2) emerges naturally in the description of hard ex- +clusive processes—its projection on a helicity-zero meson state gives twist-2 +distribution amplitude (DA) of a meson. DA accumulates information about +long-distance dynamics of partons constituting the meson and carrying a +fraction xp of the meson momentum p. +Within the approach of QCD sum rules (SR), the Borel transform ˆB of +the correlator (1) determines the perturbative contributions into meson DA, +DA(x; LB) = ˆBas +π2NcCF +� +n⩾0 +AnΠn(x, 0; L), +Πn(x, 0; L) = +� 1 +0 +Πn(x, y; L) dy, +(3) +where LB = ln(M2 +B/µ2) is the logarithm of the Borel parameter MB. +In +the approximation of large β0 (or nf), the pQCD part of SR is completely +determined by diagrams (1) of two-loop topology with gluon lines dressed by +one-loop fermion insertions—renormalon chains +n += +� +�� +� +n +. +1. The generating function for the correlator Πn(x, 0; L) +Let us now discuss the properties of Πn(x, 0; L), which is the two-point +correlator of one nonlocal and one local quark current. The general expression +for the corresponding diagram of two-loop topology (1) with nonlocal vertices +and arbitrary exponent of internal line propagator was derived in [1]. This +“kite” diagram can be represented in terms of the hypergeometric functions +3F2(x) and 3F2(¯x), ¯x = 1−x. Due to this, the sequence of Πn(x, 0; L) can be +condensed as two generating functions, an exponential Π′ +n and an ordinary +Π′′ +n: +Πn(x, 0; L) = Π′ +n(x, 0; L) + Π′′ +n(x, 0; L), +(4) +� +n⩾0 +An +n! +d +dLΠ′ +n(x, 0; L) += ˆS +� +eA(L−5/3)xA +A2(1 + A)(2 + A) +� +−¯x(A + 4x) + 2x¯x(πA)2 cot(πA) +xA sin(πA) ++ Ax(2¯x + A)B¯x(A, 1 − A) + 2x2¯xA2 +(1 + A)2 3F2 +� 1, 1, 1 + A +2 + A, 2 + A +���� x +��� +, +(5) + +3 +� +n⩾0 +An d +dLΠ′′ +n(x, 0; L) = − 1 +A +� A +0 +da F(x; a), +(6) +where +F(x; a) = 1 +2a +� 1 +0 +dy y¯y +�V (x, y; a) +h1(a) +− V (x, y; 0) +� ++(x) +, +(7) +h1(a) = +(1 − a)Γ(1 + a)Γ3(1 − a) +(1 − 2a/3)(1 − 2a)Γ(1 − 2a), +(8) +V (x, y; a) = 2 ˆS +� +Θ(y > x) +�x +y +�1−a � +1 − a + +1 +y − x +�� +. +(9) +Here, the function h1(ε) comes from ε-dependence of the simplest quark +loop in the gluon propagator (D = 4 − 2ε is the space-time dimension), +V (x, y; a) is a generalization of one-loop ERBL evolution-equation kernels, +f(x, y)+(x) = f(x, y)−δ(x−y)f(0, y) is the plus distribution, and ˆS [f(x, y)] = +f(x, y)+f(¯x, ¯y). Note that in the scope of this paper we are not interested in +the nonlogarithmic term of the correlator, Πn(x, 0; L = 0), since ˆB (const) = +0. The part of the correlator that is represented as the ordinary generating +function (6) is related to the counterterms in the nonlocal vertex. +From (4)–(6), we can derive explicit coefficients of the L-expansion of the +correlator +Πn(x, 0; L) = (−1)nn! +n+1 +� +k=0 +(−L)k +k! +Πk +n(x, 0). +(10) +The highest degree term Πn+2 +n +(x, 0) is equal to 0 because of the gauge sym- +metry and current conservation. The first nonvanishing coefficient reads +Πn+1 +n +(x, 0) = 1 +2 +ˆS +� +x ln x + δ0,n +� +−x ln x + 1 +2x¯x +�π2 +3 − 5 − ln2 x +¯x +��� +, (11) +which is in agreement with the previous calculations. The following terms +grow increasingly lengthy to be written out in proceedings. Nevertheless, +what (highest transcendency) types of functions they are expressed in terms +of can still be specified: +Πn +n>0(x, 0) ∼ ˆS Li3 x + simpler polylogarithms, +Πn−1 +n>1(x, 0) ∼ ˆS Li4 x + . . . , +Πk>0 +n +(x, 0) ∼ ˆS Hµ(x) + . . . , +µ = µ1, . . . µr : µi > 0, +� +µi = n − k + 3, +where Hµ(x) are harmonic polylogarithms [2], +Hµ(z) = +� +σ +zm1 +r� +i=1 +1 +mµi +i +, +|z| < 1, +µ = µ1, . . . µr, +σ = {∀mi ∈ N, i = 1, . . . r : m1 > m2 > · · · > mr > 0} . + +4 +0.0 0.2 0.4 0.6 0.8 1.0 +0.00 +0.01 +0.02 +0.03 +0.04 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Fig. 1. +Left panel: +LO (−), NLO (−), β0N2LO (- -), β2 +0N3LO (· · ·), and +β3 +0N4LO (- · -) contributions to DAs for pseudoscalar or longitudinally polarized +vector mesons. +Right panel: the ratios NLO/LO (−), β0N2LO/LO (- -), and +β2 +0N3LO/LO (· · ·), and β3 +0N4LO/LO (- · -). All curves are for the case of LB = 0, +αs(µ2 = 1 GeV2) ≈ 0.49. +Fig. 1 shows several lowest-order contributions to meson DAs obtained +from Eqs. (4)–(6) with the help of (3) and the Borel transform +ˆB [f(t)] (µ) = lim +t=nµ +n→∞ +(−t)n +Γ(n) +dn +dtnf(t), +ˆBeAL = − AeALB +Γ(1 − A). +(12) +These curves exhibit different behavior for the intermediate values of the +Bjorken variable x, where they decrease sequentially from LO to N4LO, +and at the endpoints, where their ratios become singular. The vicinity of +endpoints is quantitatively important for DAs of pseudoscalar and longitu- +dinally polarized vector mesons. Therefore, it makes sense to look at two +integral characteristics of the correlators—their zeroth and inverse moments, +Πn(0, 0; L) and Πn(−1, 0; L). They are formed mostly by the intermediate +and near-endpoint values of the x-dependent correlator, respectively. +1.1. The zeroth moment Πn(0, 0; L). The derivative of the zeroth mo- +ment with respect to L is proportional to the Adler function of QCD. The +corresponding exponential generating function reads +� +n⩾0 +An +n! +d +dLΠn(0, 0; L) = +eA(L−5/3) +6(1 + A)(2 + A) +� +−ψ1 +�4 + A +2 +� ++ ψ1 +�3 + A +2 +� ++ ψ1 +�2 − A +2 +� +− ψ1 +�1 − A +2 +�� +, +(13) +where ψ1 is the trigamma function. The expression (13) agrees with the cal- +culation [3] of the correlator and its anomalous dimension for n = 0, 1, 2, 3. +Also, it coincides with the Adler function D(as, L) from [4] for n = 2, 3 and +all-order D(as, L) from [5,6]. +The behavior of the Borel transform of Πn(0, 0; L) is depicted in Fig. 2. +This asymptotic series should be truncated at n = 3 where it becomes diver- +gent and bursts into factorial growth at n > 10. + +5 +0 +5 +10 +15 +20 +25 +30 +-12 +-10 +-8 +-6 +-4 +-2 +0 +2 +0 +5 +10 +15 +20 +25 +30 +0 +2 +4 +6 +8 +10 +12 +14 +Fig. 2. The ratio Rn(N) = −asβ0ˆBΠn(N, 0; L)/ˆBΠn−1(N, 0; L) for N = 0 (left +panel) and N = −1 (right panel); R0 is defined as the ratio of 2-loop and 1-loop +correlators, R0(0) = 3asCF and R0(−1) = 5asCF [7, 8]. +Blue squares are for +Rn ⩽ 1. All free parameters are the same as in Fig. 1 +1.2. The inverse moment Πn(−1, 0; L). The two generating functions for +the inverse moment can be written as +Πn(−1, 0; L) = Π′ +n(−1, 0; L) + Π′′ +n(−1, 0; L), +� +n⩾0 +An +n! +d +dLΠ′ +n(−1, 0; L) = +eA(L−5/3) +2(1 + A)(2 + A) +� +ψ1 +�2 − A +2 +� +− ψ1 +�1 − A +2 +�� +, +� +n⩾0 +An d +dLΠ′′ +n(−1, 0; L) = − 1 +A +� A +0 +daF(−1, a), +(14) +where +F(−1, a) = +Γ(4 − 2a) +6Γ(2 − a)2Γ(3 + a) +�5 + 6a − 5a2 +Γ(3 − a) ++ (1 + 2a)[γE + ψ(1 − a)] +aΓ(1 − a) +� +. +Fig. 2 illustrates the behavior of the sequence of borelized Πn(−1, 0; L) that +can be obtained with the help of (12). The series becomes factorially diver- +gent at n = 4. +2. Conclusion +We have evaluated the correlator of vector nonlocal quark currents of +order an+1 +s +βn +0 in QCD, n ⩾ 0. The lower Mellin moments of the correlator +have been calculated. The zeroth moment as well as some other fixed-order +special cases agree with previous calculations in the literature. Generating +functions for the correlator and its moments have been constructed. The +correlator at any fixed order an+1 +s +βn +0 can be expressed in terms of harmonic +polylogarithms of weight not higher than n + 2. We briefly discussed how +the higher order radiative corrections affect DA behavior. +Acknowledgements NV was supported by the Russian Science Foun- +dation grant No-18-12-00213-P. + +6 +REFERENCES +1. Mikhailov S.V., Volchanskiy N. Two-loop kite master integral for a cor- +relator of two composite vertices // J. High Energ. Phys. — 2019. — Jan. — +V. 01, no. 01. — P. 202. — arXiv:1812.02164. +2. Remiddi E., Vermaseren J.A.M. Harmonic polylogarithms // Int. J. Mod. +Phys. 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Phys. — 2021. — V. +2021. — P. 197. — arXiv:2010.03557. + diff --git a/XtAzT4oBgHgl3EQf1_4E/content/tmp_files/load_file.txt b/XtAzT4oBgHgl3EQf1_4E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ccc5cba94810d6a819bdc05a65f31f1faf4a82e8 --- /dev/null +++ b/XtAzT4oBgHgl3EQf1_4E/content/tmp_files/load_file.txt @@ -0,0 +1,307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf,len=306 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='01806v1 [hep-ph] 4 Jan 2023 Renormalon-chain contributions to two-point correlators of nonlocal quark currents S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Mikhailova,1, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Volchanskiya,b,2 a Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia b Research Institute of Physics, Southern Federal University, Prospekt Stachki 194, 344090, Rostov-na-Donu, Russia We calculate, within massless QCD, a two-point correlator of nonlocal (compos- ite) vector quark currents with arbitrary-length chains of the simplest fermion loops being inserted into gluon lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Within the large nf (or large β0) approximation, the correlator defines a perturbative contribution to the leading-twist distribution amplitudes for light mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Our results are consistent with a number of spe- cial cases in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' We consider functionals of the correlator, which are important for the phenomenology, and their properties as function series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' PACS: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='Pg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='Db;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='-t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='Bx Introduction We consider two-point correlators Πn(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) of nonlocal vector quark currents within large-β0 approximation to massless perturbative QCD1 in MS scheme, − ias π2NcCFAnΠn(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = � dDη eipη⟨0|ˆT � J†(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' x)J(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' y) � |0⟩ = n + n + n + n + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' (1) Here, L = ln(−p2/µ2) with p being an external momentum and µ being the renormalization scale and the constant A = 4 3asTFnf can be replaced by −asβ0 as prescribed by the naive nonabelization trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' (1), the nonlocal vector quark current J(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' x) is defined as the inverse Mellin transform ˆM−1 of a quark bilinear involving the Nth derivative of a quark field operator2: J(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' x) = ˆM−1J(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' N), J(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' N) = ¯d(η)˜n/ (i˜n∇)N u(η), (2) 1E-mail: mikhs@theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='jinr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='ru 2E-mail: nikolay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='volchanskiy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='com 1We work in QCD with nf = 3 massless quark flavors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Nc = 3 is the number of colors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' the Casimir invariants are CA = 3 and CF = 4/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' β0 = 11 3 CA − 4 3TF nf = 9 is the one-loop β function coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' TF = 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' as = αs/(4π) is the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 2Note that, in this paper, arguments of the Mellin transform are underlined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' f(a) = ˆMf(x) = � 1 0 dx f(x)xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 2 where x is a Bjorken fraction, η is a space-time point, ∇µ = ∂µ − igtaAa µ is the QCD covariant derivative, ˜nµ is a light-like vector, ˜n2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' In QCD, the nonlocal current (2) emerges naturally in the description of hard ex- clusive processes—its projection on a helicity-zero meson state gives twist-2 distribution amplitude (DA) of a meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' DA accumulates information about long-distance dynamics of partons constituting the meson and carrying a fraction xp of the meson momentum p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Within the approach of QCD sum rules (SR), the Borel transform ˆB of the correlator (1) determines the perturbative contributions into meson DA, DA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' LB) = ˆBas π2NcCF � n⩾0 AnΠn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L), Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = � 1 0 Πn(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) dy, (3) where LB = ln(M2 B/µ2) is the logarithm of the Borel parameter MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' In the approximation of large β0 (or nf), the pQCD part of SR is completely determined by diagrams (1) of two-loop topology with gluon lines dressed by one-loop fermion insertions—renormalon chains n = � �� � n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The generating function for the correlator Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) Let us now discuss the properties of Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L), which is the two-point correlator of one nonlocal and one local quark current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The general expression for the corresponding diagram of two-loop topology (1) with nonlocal vertices and arbitrary exponent of internal line propagator was derived in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' This “kite” diagram can be represented in terms of the hypergeometric functions 3F2(x) and 3F2(¯x), ¯x = 1−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Due to this, the sequence of Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) can be condensed as two generating functions, an exponential Π′ n and an ordinary Π′′ n: Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = Π′ n(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) + Π′′ n(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L), (4) � n⩾0 An n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' d dLΠ′ n(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = ˆS � eA(L−5/3)xA A2(1 + A)(2 + A) � −¯x(A + 4x) + 2x¯x(πA)2 cot(πA) xA sin(πA) + Ax(2¯x + A)B¯x(A, 1 − A) + 2x2¯xA2 (1 + A)2 3F2 � 1, 1, 1 + A 2 + A, 2 + A ���� x ��� , (5) 3 � n⩾0 An d dLΠ′′ n(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = − 1 A � A 0 da F(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' a), (6) where F(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' a) = 1 2a � 1 0 dy y¯y �V (x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' a) h1(a) − V (x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 0) � +(x) , (7) h1(a) = (1 − a)Γ(1 + a)Γ3(1 − a) (1 − 2a/3)(1 − 2a)Γ(1 − 2a), (8) V (x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' a) = 2 ˆS � Θ(y > x) �x y �1−a � 1 − a + 1 y − x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' (9) Here, the function h1(ε) comes from ε-dependence of the simplest quark loop in the gluon propagator (D = 4 − 2ε is the space-time dimension), V (x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' a) is a generalization of one-loop ERBL evolution-equation kernels, f(x, y)+(x) = f(x, y)−δ(x−y)f(0, y) is the plus distribution, and ˆS [f(x, y)] = f(x, y)+f(¯x, ¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Note that in the scope of this paper we are not interested in the nonlogarithmic term of the correlator, Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L = 0), since ˆB (const) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The part of the correlator that is represented as the ordinary generating function (6) is related to the counterterms in the nonlocal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' From (4)–(6), we can derive explicit coefficients of the L-expansion of the correlator Πn(x, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = (−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' n+1 � k=0 (−L)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Πk n(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' (10) The highest degree term Πn+2 n (x, 0) is equal to 0 because of the gauge sym- metry and current conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The first nonvanishing coefficient reads Πn+1 n (x, 0) = 1 2 ˆS � x ln x + δ0,n � −x ln x + 1 2x¯x �π2 3 − 5 − ln2 x ¯x ��� , (11) which is in agreement with the previous calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The following terms grow increasingly lengthy to be written out in proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Nevertheless, what (highest transcendency) types of functions they are expressed in terms of can still be specified: Πn n>0(x, 0) ∼ ˆS Li3 x + simpler polylogarithms, Πn−1 n>1(x, 0) ∼ ˆS Li4 x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' , Πk>0 n (x, 0) ∼ ˆS Hµ(x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' , µ = µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' µr : µi > 0, � µi = n − k + 3, where Hµ(x) are harmonic polylogarithms [2], Hµ(z) = � σ zm1 r� i=1 1 mµi i , |z| < 1, µ = µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' µr, σ = {∀mi ∈ N, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' r : m1 > m2 > · · · > mr > 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Left panel: LO (−), NLO (−), β0N2LO (- -), β2 0N3LO (· · ·), and β3 0N4LO (- · -) contributions to DAs for pseudoscalar or longitudinally polarized vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Right panel: the ratios NLO/LO (−), β0N2LO/LO (- -), and β2 0N3LO/LO (· · ·), and β3 0N4LO/LO (- · -).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' All curves are for the case of LB = 0, αs(µ2 = 1 GeV2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 1 shows several lowest-order contributions to meson DAs obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' (4)–(6) with the help of (3) and the Borel transform ˆB [f(t)] (µ) = lim t=nµ n→∞ (−t)n Γ(n) dn dtnf(t), ˆBeAL = − AeALB Γ(1 − A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' (12) These curves exhibit different behavior for the intermediate values of the Bjorken variable x, where they decrease sequentially from LO to N4LO, and at the endpoints, where their ratios become singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The vicinity of endpoints is quantitatively important for DAs of pseudoscalar and longitu- dinally polarized vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Therefore, it makes sense to look at two integral characteristics of the correlators—their zeroth and inverse moments, Πn(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) and Πn(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' They are formed mostly by the intermediate and near-endpoint values of the x-dependent correlator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The zeroth moment Πn(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The derivative of the zeroth mo- ment with respect to L is proportional to the Adler function of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The corresponding exponential generating function reads � n⩾0 An n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' d dLΠn(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = eA(L−5/3) 6(1 + A)(2 + A) � −ψ1 �4 + A 2 � + ψ1 �3 + A 2 � + ψ1 �2 − A 2 � − ψ1 �1 − A 2 �� , (13) where ψ1 is the trigamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The expression (13) agrees with the cal- culation [3] of the correlator and its anomalous dimension for n = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Also, it coincides with the Adler function D(as, L) from [4] for n = 2, 3 and all-order D(as, L) from [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The behavior of the Borel transform of Πn(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' This asymptotic series should be truncated at n = 3 where it becomes diver- gent and bursts into factorial growth at n > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 5 0 5 10 15 20 25 30 12 10 8 6 4 2 0 2 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The ratio Rn(N) = −asβ0ˆBΠn(N, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L)/ˆBΠn−1(N, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) for N = 0 (left panel) and N = −1 (right panel);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' R0 is defined as the ratio of 2-loop and 1-loop correlators, R0(0) = 3asCF and R0(−1) = 5asCF [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Blue squares are for Rn ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' All free parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The inverse moment Πn(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The two generating functions for the inverse moment can be written as Πn(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = Π′ n(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) + Π′′ n(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L), � n⩾0 An n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' d dLΠ′ n(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = eA(L−5/3) 2(1 + A)(2 + A) � ψ1 �2 − A 2 � − ψ1 �1 − A 2 �� , � n⩾0 An d dLΠ′′ n(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) = − 1 A � A 0 daF(−1, a), (14) where F(−1, a) = Γ(4 − 2a) 6Γ(2 − a)2Γ(3 + a) �5 + 6a − 5a2 Γ(3 − a) + (1 + 2a)[γE + ψ(1 − a)] aΓ(1 − a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 2 illustrates the behavior of the sequence of borelized Πn(−1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' L) that can be obtained with the help of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The series becomes factorially diver- gent at n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Conclusion We have evaluated the correlator of vector nonlocal quark currents of order an+1 s βn 0 in QCD, n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The lower Mellin moments of the correlator have been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The zeroth moment as well as some other fixed-order special cases agree with previous calculations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Generating functions for the correlator and its moments have been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' The correlator at any fixed order an+1 s βn 0 can be expressed in terms of harmonic polylogarithms of weight not higher than n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' We briefly discussed how the higher order radiative corrections affect DA behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Acknowledgements NV was supported by the Russian Science Foun- dation grant No-18-12-00213-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' 6 REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Mikhailov S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=', Volchanskiy N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' Two-loop kite master integral for a cor- relator of two composite vertices // J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAzT4oBgHgl3EQf1_4E/content/2301.01806v1.pdf'} +page_content=' High Energ.' metadata={'source': 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0000000000000000000000000000000000000000..43e3d83e09183706fa7e2f2bd292384e617d8f29 --- /dev/null +++ b/YNFRT4oBgHgl3EQfOTea/content/tmp_files/2301.13513v1.pdf.txt @@ -0,0 +1,1592 @@ +Highlights +Privacy Preserving Ultra-Short-term Wind Power Prediction Based on Secure Multi Party +Computation +Hang Fan,Xiaoyu Fan,Tianyi Hao,Wei Wei,Kun Chen,Guosai Wang,Xiaofeng Jia,Yidong Li,Wei Xu +• We develop a vertical privacy preserving XGBoost prediction algorithm based on the secret sharing protocol in +the pwXGBoost model. +• We design a criterion to select the suitable participant wind farm in the pwXGBoost model. +• We test the wind farms in the field data from the wind farm cluster in the Inner Mongolian. +arXiv:2301.13513v1 [cs.CR] 31 Jan 2023 + +Privacy Preserving Ultra-Short-term Wind Power Prediction Based +on Secure Multi Party Computation +Hang Fana, Xiaoyu Fanb, Tianyi Haoa, Wei Weic, Kun Chend, Guosai Wangd, Xiaofeng Jiae, +Yidong Lif and Wei Xub,∗ +aorganization=PBC School of Finance, addressline=Tsinghua University, city=Beijing, postcode=100084, country=China +borganization=Institute for Interdisciplinary Information Sciences, addressline=Tsinghua University, city=Beijing, postcode=100084, +country=China +corganization=Department of Electrical Engineering, addressline=Tsinghua University, city=Beijing, postcode=100084, country=Beijing +dorganization=Tsingjiao Information Technology Co. Ltd., addressline=Tsinghua Science and Technology Park, city=Beijing, postcode=100084, +country=Beijing +eorganization=Data Management Department, addressline=Beijing Big Data Center, city=Beijing, postcode=100044, country=Beijing +forganization=School of Computer and Information Technology, addressline=Beijing Jiaotong University, city=Beijing, postcode=100025, +country=Beijing +A R T I C L E I N F O +Keywords: +wind power prediction +privacy preserving machine learning +pwXGBoost model +secure multi party computation +A B S T R A C T +Mining the spatial and temporal correlation of wind farm output data is beneficial for enhancing +the precision of ultra-short-term wind power prediction. However, if the wind farms are owned +by separate entities, they may be reluctant to share their data directly due to privacy concerns +as well as business management regulation policies. Although cryptographic approaches have +been designed to protect privacy in the process of data sharing, it is still a challenging problem to +encrypt the original data while extracting the nonlinear relationship among multiple wind farms +in the machine learning process. This paper presents pwXGBoost, a technique based on the +machine learning tree model and secure multi-party computation (SMPC) that can successfully +extract complicated relationships while preserving data privacy. A maximum mean discrepancy +(MMD) based scheme is proposed to effectively choose adjacent candidate wind farms to par- +ticipate in the collaborative model training, therefore improving the accuracy and reducing the +burden of data acquisition. The proposed method was evaluated on real world data collected from +a cluster of wind farms in Inner Mongolia, China, demonstrating that it is capable of achieving +considerable efficiency and performance improvements while preserving privacy. +1. Introduction +1.1. Background and Motivation +The large-scale exploitation of renewable energy sources, such as wind power, has brought a large amount of +clean energy and reduced the CO2 emissions. However, the uncertainty and randomness of wind power pose serious +challenges to the operation of the power system [1]. On the electricity spot trading market, the bidding strategies of +wind farms heavily depend on the wind power predictions and severe deviations in bids are penalized. The economic +losses caused by inaccurate wind power forecasts can reach 10% of the wind farms’ electricity sales [2]. +Due to the desire for high-quality wind power prediction, there have been tremendous studies in related fields. +At present, the core concept of ultra-short-term power prediction is to mine a variety of data such as historical power +generation data and Numerical Weather Prediction (NWP) data of local wind sites through artificial intelligence and +statistical learning methods, so as to develop high-precision prediction models. At the same time, wind farms are +usually located in close proximity to each other, there is a strong spatial and temporal correlation pattern among the +sites. Utilizing this correlation pattern can considerably enhance the power prediction accuracy of wind farms. As a +result, more studies have been conducted in recent years by assuming that the data of all wind farms in the cluster can be +obtained. Through combining graph machine learning and other nonlinear methods to extract the spatial and temporal +correlation among neighboring wind farms, the prediction accuracy can be effectively improved [3, 4]. However, +historical wind power data is often owned by different companies. The long-time historical wind power can reflect the +∗Corresponding author +fanhang123456@163.com (H. Fan); weixu@mail.tsinghua.edu.cn (W. Xu) +ORCID(s): +fanhang et al.: Preprint submitted to Elsevier +Page 1 of 21 + +aShort Title of the Article +production and operation status of the wind farms in the electricity market and is confidential to each other. The NWP +data is purchased by wind farms at high cost, thus they are hesitant to share with others. Therefore, the direct sharing +of wind power data may be restricted by data management policies due to the privacy and security of the data. How +to use data from neighboring wind farms without compromising privacy is considered as the last mile and the most +difficult part of the application of spatial and temporal correlation methods in practice. +1.2. Previous Study and Literature Review +Some research is brought out to predict the wind power while preserving the original data. For this problem, +article [5] classify the solutions into three classes, namely data transformation method, decomposition method and +secure multi-party computation method. According to the definition, the data transformation method normally refer +to adding some random noise to the original data before the fitting process to protect the privacy which is called +differential privacy [6]. Although the differential privacy is successful to protect the privacy in the picture recognition +area, it is not suitable for the wind power prediction [7]. The picture recognition is a classification problem while the +wind power prediction is a regression problem which is more sensitive to the input data. Any disturb to the wind power +data can lead to a decrease of accuracy which is unacceptable for the power market and the wind farm owner. +Decomposition method regards the prediction problem as an optimization problem and decomposes it into several +sub-problems and allows each data provider to solve it separately. Carla [8] emphasizes the forecast skill improvement +due to the spatial and temporal dependencies in the time series and the business competition among wind farms. +Therefore, [8] formulates a framework which combines the data transformation methods and the alternating direction +method multipliers (ADMM). In [9], a data market even been designed to encourage the wind farms to share their data +to improve the prediction accuracy. Han [10] designed an regression market for wind power forecasting and use the +LASSO regulation as the reference for the data pricing. However, wind farm power is highly nonlinear, and the lasso +method, as a linear prediction method, is inherently difficult to capture the spatial and temporal correlation among wind +farms, so the prediction accuracy in practice is not always satisfactory. In practical wind farm power prediction tasks, +multiple nonlinear prediction methods such as machine learning, XGBoost or even neural networks and their combined +derivative models are more often used. And as stated in the article [8], privacy methods using ADMM methods for +solving cannot be directly extended to nonlinear prediction scenarios. Therefore, there is an urgent demand to explore +how to consider data privacy in nonlinear prediction models. +Secure multi-party computation in article [5] is a generalized privacy preserving computation framework [5]. This +topic is an active research field in computer science and data mining because it is compatible with non-linear operation +[11]. It calls for the fusion of classical secure multi-party computation [12], federated learning [13] and other classical +cryptography theory such as homomorphic encryption [14]. +Classical secure multi-party computing techniques include secret sharing, oblivious transmission, and garbled +circuit, which are mainly derived from the "millionaire’s problem" in 1982 [12]. In 1986, Yao proposed the theory of +the garbled circuit, which became the first general multi-party secure computing scheme [15]. After several years +of development, the classical secure multi-party computation consists of multiple cryptography protocols such as +garbled circuit [16], oblivious transmission [17] and secret sharing [18]. Garbled circuit is performed by constructing +a circuit and obfuscating the signals on the circuit, while secret sharing is performed by splitting the secret data into +multiple slices and performing computation on the slices. Because secret sharing protocol is more friendly for the +computation, most advanced privacy preserving computation platform adopt this protocol [19, 20], and it quickly +becomes a popular method in recent studies. For example, article [21] uses the secret share to realize the fully privacy +preserving distributed optimization of power system. +Federated learning is a distributed machine learning method proposed by Google in 2016 [22] that enables mul- +tiple mutually untrusted training data providers to collaboratively train machine learning models by exchanging inter- +mediate computational results such as gradients or parameters without exchanging raw data. According to the different +data distribution among participants, federation learning is generally classified into three types: horizontal federated +learning [11], vertical federated learning [11] and federated transfer learning [23]. Horizontal federation is mainly +used for sample federation between two parties with the same or similar business model, and there is a lot of feature +overlap in the data of each party, but less overlap in the number of users. Longitudinal federation is mainly used for +feature federation between two parties with different business modes but the same or similar users, with less feature +overlap but more user overlap. Federated migration is mainly used for forward learning between two parties with less +intersection of industry and users, and there is less overlap of features and users in the data of all parties. There are +many scenarios that the federated learning is used to protect the privacy. [24] used the federated learning for the volt- +fanhang et al.: Preprint submitted to Elsevier +Page 2 of 21 + +Short Title of the Article +age prediction in the local energy community. In [25], the federated fuzzy k-means is used to analyze the smart grid +meter data. Federated learning can also be used with the reinforcement learning. In [26], a federated reinforcement +learning method is designed for the peer-to-peer energy trading and the carbon allowance trading. Article [27] uses the +horizontal federated reinforcement learning to predict the wind power which can leverage the wind farms in a cluster. +However, it can not extract the wind farm spatialtemporal relationship which is implied by the wind power data at the +same time. In 2019, it was demonstrated that the gradients or parameters exchanged during federation learning can +be used to infer or even recover the original data information [28], currently, the exchange process usually requires +cryptography-based techniques (e.g., MPC) or homomorphic cryptography to avoid these risks [29]. On the other +hand, the performance of the federated learning will decrease if the data distribution of the participants are non-iid +such as the feature distribution skew, label distribution skew and quantity skew [30]. Take the wind power prediction +case for example, if the wind data of the participants do not follow similar distributions or appear to be non-iid, it is +more difficult to identify the spatial temporal correlation patterns. +Homomorphic encryption is a classical encryption method to protect data privacy by directly encrypting the plain- +text, performing various operations under the ciphertext, and finally obtaining the resulting ciphertext. Homomorphic +encryption can be classified into Fully Homomorphic, Somewhat Homomorphic, and Partially Homomorphic [31, 29] +depending on the degree of support for an unlimited number of arbitrary homomorphic operations. Homomorphic +encryption allows arbitrary computation of the ciphertext without decryption, but its performance is too slow to be- +come practical. According to the latest Fully Homomorphic computation benchmark [32] in 2022, the homomorphic +computation is orders of magnitude slower than plaintext. The limited computational speed constrains its practical +application [33]. +1.3. Contribution and Paper Organization +Although there are some works aim to preserve the privacy in the prediction process, there are still two main +problems. The first problem is the current privacy preserving method can not fully utilize the spatial and temporal +correlation while safely preserving the data privacy. Although some researchers use the federated learning method +such as the FedAVG to fusion the gradient of the neural network, it follows the horizontal data fusion method. It is more +similar to the transfer learning which is suitable when the wind power data is sufficient and horizontally partitioned +among wind farms rather than the case that the wind farm can boost his own prediction accuracy by utilizing the +spatial and temporal relationship with others. Moreover, the classic federated learning method is not safe enough for +the collaborative modeling and prediction [28]. Current privacy preserving prediction method which can extract the +spatial and temporal relationship is based on Lasso-var and it is a linear method [8]. But the spatial and temporal +relationship is highly nonlinear, and the feature extraction ability of linear method is limited. Using homomorphic +encryption and other full ciphertext computing methods can solve the nonlinear problem in the extraction of spatial +and temporal correlation, but the expensive computation cost makes it impractical [29]. The second problem is the +participant selection. If the data of each wind farm exhibits significant difference in their distributions, the non-iid +feature will effect the performance of prediction. Besides, if the number of wind farms in the collaborated power +prediction is extremely great, the communication cost will compromise the timeliness of the ultra-short-term prediction +model. If insufficient wind farms participate in the collaborative model training, spatial and temporal correlation will +not be utilized to its full potential. Therefore, we designed a method named pwXGBoost based on vertical data fusion +strategy and the secret sharing protocol which is scalable to extract the nonlinear spatial and temporal correlation pattern +of several wind farms. In the pwXGBoost model, we also borrowed ideas from personalized federation learning [34] +to screen participants for the collaborative modeling task. +The contribution of this paper is three-fold: +(1) We develop a vertical privacy preserving XGBoost prediction algorithm based on the secret sharing protocol +in the pwXGBoost model. It has the following advantages. First, it is scalable to the nonlinear data and complex +modeling of the spatial and temporal correlation compared to the renowned Lasso-var method. Second, it can realize a +lossless and secure computation of XGBoost. It is more precise than the data transformation methods and more secure +than the conventional federated learning. +(2) We design a criterion to select the suitable participant wind farm in the pwXGBoost model. In the criterion, +during the collaborative training process the maximum mean discrepancy index is adopted to assess the similarity +of the wind farm data distribution and it can select the wind farm which is most useful for the spatial and temporal +correlation extraction. +(3) We test the wind farms in the field data from the wind farm cluster in the Inner Mongolian. The data of some +fanhang et al.: Preprint submitted to Elsevier +Page 3 of 21 + +Short Title of the Article +nearby wind farms in the cluster are combined to predict the wind power of the target wind farm. The experiment +results show that the proposed pwXGBoost method is superior than all the baseline methods which only uses local +data or a linear model. The prediction time is also acceptable for the practical application. +The rest of this paper is organized as follows. Section 3 will formulate the mathematical model of the privacy +preserving ultra-short-term wind power prediction. Section 4 develops the privacy preserving XGBoost model based +on secret sharing protocol. Section 5 describes the implementation process of the privacy preserving ultra-short-term +wind power prediction. Section 6 analyzes the security of the proposed approach. The privacy preserving prediction +algorithm is tested on the wind farm cluster from Inner Mongolian and the effectiveness is validated in Section 7 +Finally. Conclusions are drawn in Section 8. +2. Preliminary +2.1. Traditional Wind Farm Power Prediction +Wind power forecast is a classical time series prediction problem which have been extensively studied. For ultra- +short-term wind power prediction, traditionally only the data of the local wind farm is used for the modeling. +푃푡+퐻 = 푓(푃푡−푀+1∶푡, 푉푡+1∶푡+푁) +(1) +Where 푃푡−푀+1∶푡 is the local historical wind power of the wind farm and 푉푡+1∶푡+푁 is the local NWP of the wind farm. +Recently, it has been recognized that exploiting the spatial and temporal correlation can improve forecast accuracy [3]. +Therefore, the wind power prediction can be modeled as follows: +푃 푖 +푡+퐻 = 푓(푃 1 +푡−푀+1∶푡, ..., 푃 푖 +푡−푀+1∶푡, ..., 푃 푛 +푡−푀+1∶푡, ..., 푉 1 +푡+1∶푡+푁, ...푉 푖 +푡+1∶푡+푁, ...푉 푛 +푡+1∶푡+푁) +(2) +Where 푃 푖 +푡−푀+1∶푡 ∈ 푅푀×1 is the historical wind power of wind farm 푖 and the step length for the prediction is 푀. +푉 푖 +푡+1∶푡+푁 ∈ 푅푁×푘 is the matrix of NWP data for wind farm 푖 and the step length for the NWP data is 푁. The variable +number of NWP data is 푘. Normally, the next 4 hour wind power are to be predicted and the time interval is 15min, +so 퐻 = 16. Function 푓 is the prediction model which can be linear model such as Lasso, neural network or XGBoost +model. In the prediction model, although only the wind power of wind farm 푖 needed to be predicted, the historical +wind power and NWP data of other nearby wind farms are used. For a wind farm cluster, if the wind farms in this +cluster are belong to the same owner, this kind of centralized prediction model is acceptable. However, when the +wind farms in the cluster are the assets of different stakeholders, direct data sharing is not appropriate. Therefore, it is +necessary to develop the privacy preserving prediction model. +2.2. A Brief Review of XGBoost +XGBoost is an ensemble of tree models to boost the performance of a single tree which is very popular in wind +power prediction[35]. For a dataset X ∈ 푅푀×푁 with 푀 samples and 푁 features. XGBoost can predict the 푖-th sample +푥푖 ∈ 푅1×푁 by using 푇 regression function as follows: +̂푦푖 = +푇∑ +푡=1 +푓푡(푥푖) +(3) +XGBoost is sequentially trained by calculating ̂푦(푡) +푖 += ̂푦(푡−1) +푖 ++ 푓푡(푥푖), where a new tree 푓푡(푥푖) is used to train the +residual of the target and the prediction in the previous iteration. For the given loss function 푙, a second-order Tylor +expansion is used to approximate it in 푡-th iteration as follows: + ≈ +푁 +∑ +푖=1 +[푙(푦푖, ̂푦(푡−1) +푖 +) + 푔푖푓푡(푥푖) + 1 +2ℎ푖푓 2 +푡 (푥푖)] + Ω(푓푡) +(4) +fanhang et al.: Preprint submitted to Elsevier +Page 4 of 21 + +Short Title of the Article +Ω(푓푡) = 훾푈 + 1 +2휆||휔||2 +(5) +Where ̂푦(푡−1) +푖 +is the current prediction results, Ω is the regulation term, 푈 is the number of leaves in the tree, 훾 and +휆 are the hyper-parameters to restrict the tree number and weights respectively. 푔푖 = 휕 ̂푦(푡−1) +푖 +푙(푦푖, ̂푦(푡−1) +푖 +) and ℎ푖 = +휕2 +̂푦(푡−1) +푖 +푙(푦푖, ̂푦(푡−1) +푖 +) are the first and second order derivative statistics of loss function. The tree model starts from 푠 single +leaf node which includes all samples. Then the node recursively splits the current samples into left and right subsets +denoted by 퐼퐿 and 퐼푅. The loss function after the split is +split ≈ 1 +2[ +(∑ +푖∈퐼퐿 푔푖)2 +∑ +푖∈퐼퐿 ℎ푖 + 휆 + +(∑ +푖∈퐼푅 푔푖)2 +∑ +푖∈퐼푅 ℎ푖 + 휆 − +(∑ +푖∈퐼퐼 푔푖)2 +∑ +푖∈퐼퐼 ℎ푖 + 휆] − 훾 +(6) +Where the best split is the one with the highest 푠푝푙푖푡. The weight 푤 of each leaf is calculated in equation (10) +푤 = − +∑ +푖∈퐼푢 푔푖 +∑ +푖∈퐼푢 ℎ푖 + 휆 +(7) +When the depth of the tree reach the highest, the training of XGBoost terminates [35]. +3. Problem Definition +Due to the privacy concern, wind farms can hardly share their data directly without any constrains. Because +once the wind power data of the wind farms are copied to other places, the risk of data abuse seems inevitable. The +privacy-preserving wind farm power prediction allows wind farms to access the data of other adjacent wind farms to +train the prediction model jointly without knowing the exact values of those data. In this section, the ultrashort-term +wind power forecast problem is formulated as follows: +푃 푖 +푡+퐻 = 푓([푃 1 +푡−푀+1∶푡], ..., 푃 푖 +푡−푀+1∶푡, ..., [푃 푛 +푡−푀+1∶푡], ..., [푉 1 +푡+1∶푡+푁], ...푉 푖 +푡+1∶푡+푁, ...[푉 푛 +푡+1∶푡+푁]) +(8) +Where [] is the secret share encryption of the variable. Through the secret share encryption, the data and computation +is only known by the owner of the data [36]. [푃 푛 +푡−푀+1] is the vector of the wind power in the secret share cipher +text of 푃 푛 +푡−푀+1 ∈ 푅푀×1 fow wind farm 푛. [푉 푛 +푡+1∶푡+푁] is the matrix in the secret share cipher text of NWP matrix +푉 푛 +푡+1∶푡+푁 ∈ 푅푁×푘 for wind farm 푛. 푀 is the step length of historical wind power used in the prediction. 푁 is the +step length of NWP data used in the prediction. Prediction function 푓 uses the cipher text of the nearby wind farms to +predict their own wind power in the next few hours. +As we know, the prediction model is constructed by the operation such as addition, multiplication, division, +compare and so on. Those operation can be also implemented in the secure share protocol to protect the privacy [19]. +The example of addition and multiplication is shown in Figure 1. +For the secure addition, two part holds number 푎 and 푏 separately. According to secret share protocol, number 푎 +is divided into two random number 푎1 and 푎2. Number 푏 is divided into two random number 푏1 and 푏2. Then 푎1 and +푏1 are sent to one computing server to get 푎1 + 푏1. In the meanwhile, 푎2 and 푏2 are sent to another computing server +to get 푎2 + 푏2. By add the number of 푎1 + 푏1 and 푎2 + 푏2, the results of 푎 + 푏 is worked out. If the two computing +server will not collude, the privacy of 푎 and 푏 is also guaranteed. The computing process is similar for the secure +multiplication. 푎1 × 푏1, 푎2 × 푏2, 푎1 × 푏2 and 푎2 × 푏1 are calculated separately on the computing server. Then the results +of 푎 × 푏 can be worked out by adding those four components. If those four computing server not collude, the privacy +can also be guaranteed. It The basic computation operation such as add, multiply and compare based on secret share +is described in the Appendix A. By utilizing the basic operation, we can construct the derivative operation such as +division, activation function and sort. By leveraging the basic operation and derivative operation, we can build the +complex machine learning method to approximate the wind power prediction function shown in equation (2). The +process is shown in Figure 2. +fanhang et al.: Preprint submitted to Elsevier +Page 5 of 21 + +Short Title of the Article +Figure 1: The Illustration of Secure Addition and Secure Multiplication +Figure 2: The Construction of Complex Machine Learning Model +4. Privacy Preserving Ultra-short-term Wind Power Prediction +4.1. The Overview of Privacy Preserving Ultra-short-term Wind Power Prediction +The basic idea of privacy preserving ultra-short-term wind power prediction is using the data of other wind farms +to enhance the accuracy of prediction model. Indeed, not only the historical wind power but also the NWP data can +be utilized in the privacy preserving training process. There are two ways of utilizing and fusing the data as shown in +Figure 3. The first one is using the historical wind power, NWP and the labels of all the wind farms to train a model +which is similar to the transfer learning [37]. It is a horizontal data fusion method. The problem for this data fusion +method is that the spatial and temporal relationship is not well considered. The second one incorporates the historical +wind power and NWP of other wind farms at the same time to predict the label which resembles a centralized prediction +method. It is a vertical data fusion method. In the wind power prediction tasks, the spatial and temporal correlation +is included in the wind power and NWP at the same time, so the vertical data fusion method is more suitable for this +case. +The participants are divided into active parties and passive parties. In wind power prediction, the active party is +the wind farm who would like to predict the wind power using the data from other wind farms. They have both feature +data and label data. The passive party is the wind farm who only has the feature data. It lends data to the active party. +However, when the active party initiated a request for a prediction, the active party need to select the appropriate wind +farms to act as the passive party. If too many wind farms take part in the training process, the massive communication +will decrease the training efficiency, and the discrepancy of the sample distribution will also effect the prediction +accuracy. If there is not enough wind farms in the training process, the spatial and temporal correlation pattern cannot +fanhang et al.: Preprint submitted to Elsevier +Page 6 of 21 + +0 +. +0 +0 +. +.. +0 +... +0 +. +0 +.. +[a] = (a1,a2) +[a] = (ai,a2) +0 +.... +0 +.. +0 +.. +0 +.. +0 +.. +0 +. +[bl = (b1,b2) +[b] = (b1, b2) +0 +.. +0 +0 +. +.... +0 +. +0 +.. +0 +... +a1 +a2 +a1 +a2 +a += +a1 ++ +a2 +b +b1 +b2 +b1 +b2 += ++ +b2 +b1 +(ai + bi) + +(a2 + b2) +a*b +aib1 ++ +a2 b2 ++ +aib2 ++ +azb1 +b += +=Basic Operation +Derivative Operation +Complex Machine +Add +Division +Learning Model +Activation +Function +Statistic +Multiply +Secret Share +Sort +Regression +Compare +ClassificationShort Title of the Article +Figure 3: Horizontal and Vertical Data Fusion Method +be fully and accurately explored. +Therefore, in our pwXGBoost model, we divide the privacy preserving ultra-short-term wind power prediction +process into two parts as shown in Figure 4. The first part is the selection of the participant wind farms (Section 4.2) and +the second part is the privacy preserving XGBoost algorithm for ultra-short-term wind power prediction (Section 4.3). +Server 1 +Server 2 +Server 3 +Server 4 +MPC +Protocol +1) The Selection of Participant +Wind Farms +… +2) Privacy Preserving XGBoost +Algorithm +𝑥! ", 𝑥# " … , 𝑥$ " +𝑥! %, 𝑥# % … , 𝑥$ % +𝑥! &, 𝑥# & … , 𝑥$ & +𝑥! ', 𝑥# ' … , 𝑥$ ' +Plaintext computation +Ciphertext computation +Data Wind farm 2 +Data Wind farm 1 +Data Wind farm 3 +Data +Target +wind farm +Data Wind farm +N-3 +Data Wind farm +N-2 +Data Wind farm +N-1 +Data Wind farm +N +Data Wind farm A +Data Wind farm B +Data +Target +wind farm +Data Wind farm C +Data Wind farm D +Figure 4: The Privacy Preserving Ultra-short-term Wind Power Prediction Process +4.2. The Selection of Participant Wind Farms +In the privacy preserving ultra-short-term wind power prediction, it is necessary to select the wind farms which +would like to engage in the data sharing. The reasonable selection of participant can increase the prediction accuracy. +Since the nearby wind farms have greater influence on the wind farm in the active party, we use the Gaussian +distance function and adjacent matrix to determine which wind farm can be used in the prediction process. To make +fanhang et al.: Preprint submitted to Elsevier +Page 7 of 21 + +feature +windpower +feature +windpower +ebel +label +P1 +NWP1 +M+1:t ++1:t+N +P1- +M+1:t +Windfarm 1 +Windfarm 1 +P1- +NWP1 +:t- +t:t+M-1 +P1 +NWP +P1 +NWP +Pt- +-1:t+N-2 +NWP1 +feature +feature +windpower +label +NWP +Windfarm 2 +P2- +t+1:t+N +M+1:t +Windfarm 2 +P2 +NWP +t:t+M-1 +:t-1 +t:t+M-1 +P2- +NWP? +P2 +1:t±2 +NWP2 +t-1:t+N-2 +feature +feature +windpower +label +Pr- +Pn +M+1:t +M+1:t +Windfarm n +Windfarm n +Pn +:t-1 +Pn +NWpn +1-1:t+2 +NWPn +Vertical Data Fusion Method +Horizontal Data Fusion MethooShort Title of the Article +full use of the spatial and temporal correlation, the wind farms with similar distribution can be selected into a group +and we use the Maximum Mean Discrepancy (MMD) to calculate the distribution distance. MMD is widely used in +the transfer learning which can measure the distance of two distribution. The multi kernel variant of MMD is used +here which maps the data distributions into a Reproducing Kernel Hibert Space (RKHS). For two distribution 푑1 and +푑2, the square of MMD between the two distributions can be calculated as +MMD2(푑1, 푑2) = ‖‖‖퐸 [휙(푑1)] − 퐸 [휙(푑2)]‖‖‖ +(9) +Where 휙(⋅) is the mapping to RKHS. In practice, the mapping is unknown and we can use the kernel trick to +replace the inner product. The result is +MMD2(푑1, 푑2) = 퐸 [퐾(푑1, 푑1)] + 퐸 [퐾(푑2, 푑2)] − 2퐸 [퐾(푑1, 푑2)] +(10) +Where 퐾(푑1, 푑2) = [휙(푥), 휙(푦)] is the desired kernel function. Because the wind farms are located in a region +and the spatial and temporal correlation is closely related to the distribution distance, the adjacent matrix is defined as +follows: +퐴푖,푗 = +{ +exp +( +− MMD2(푖,푗) +휎2 +) +, +if RMMD2(푖, 푗) ≤ 훽 ∗ mean +0, +otherwise +(11) +Where MMD2(푖, 푗) is the maximum mean discrepancy distance between wind farm 푖 and wind farm 푗, 휎 is the +standard deviation of the distance between 푛 wind farms and 훽 is the threshold. In our case, we set the half of the mean +distance as the threshold. For wind farm 푖, only the wind farms whose 퐴푖,푗 ≠ 0 are chosen as the participants. +4.3. The Privacy Preserving XGBoost Algorithm for Ultra-short-term Wind Power Prediction +XGBoost is a method which is widely used in the wind power prediction[38, 39]. There have been a number of +widely adopted XGBoost programming packages, such as XGBoost [35], LightGBM [40] and CatBoost [41]. However, +they are restricted to the centralized setting and are not applicable when data privacy is considered. Although there +are some open-source privacy-preserving machine learning framework such as Pysytft [42] and FedML [43], they do +not support XGBoost. Even though the FATE platform [44] can provide the XGBoost function, long encryption keys +are required to ensure security. However, long keys can significantly slow down the computation, and the potential of +privacy invading remains a problem. Therefore, it is necessary to build the XGBoost model based on secure multi-party +computation which can fully preserve data privacy and is also effective in practice. +For the privacy preserving XGBoost algorithm for ultra-short-term wind power prediction, it can be divided into +a training stage and a prediction stage. In the training stage, the historical wind power and NWP data of passive +parties and the active party at time 푡 is fused in a vertical way as the feature, and the wind power to be predicted is the +label. The privacy computation method is used because it can fuse the data from different sources without knowing +their values. The prediction algorithm can be trained without compromising privacy. For the prediction stage, the +encrypted historical wind power and NWP data are transmitted to the trained model and output the predicted wind +power. This part is also shown in Figure 4. +According to the secure multi party computation theory, some wind farms are selected out as the computing server +node randomly. The secret share of the data of each wind farm rather than the original data are sent to the computing +server node. In this way, if the computing server node not collude, the security is guaranteed. We will discuss the +details of the privacy preserving XGBost model based on secret sharing in Section 5. +5. Privacy Preserving XGBoost Algorithm Based on Secret Sharing +In this section, we present the design and details of privacy preserving XGBoost Algorithm based on 2-out-of-4 +secret sharing protocol. +fanhang et al.: Preprint submitted to Elsevier +Page 8 of 21 + +Short Title of the Article +Figure 5: The Training Process of Privacy Preserving XGBoost in Secret Sharing +5.1. Overall Description of Privacy Preserving XGBoost Model +The whole training process of the secure XGBoost model is shown in Figure 5. For the vertical privacy preserving +XGBoost model, there are 푚 participants take part in the computation. Here we note the active party to be party 1, and +the passive parties to be party 2, ⋯ , 푚. +5.1.1. Secure Model Training Process +The algorithm of the secure XGBoost model is shown in Algorithm 1. The model is trained among the active +party, passive party and the ciphertext backend. The input datasets of the algorithm including the feature set {퐗푘}푀×푁푘 +(1 ≤ 푘 ≤ 푚) provided by all the 푚 parties and the label set {푦}푀 owned by the active party. During the training process, +we train the 푇 decision trees sequentially. +For the 푡-th tree, at the beginning, we compute the prediction result ̂푦푖 (1 ≤ 푖 ≤ 푀) of the first 푡 − 1 trees. Then +we compute the first and second order gradients, which are 푔푖 and ℎ푖 (1 ≤ 푖 ≤ 푀) respectively. We encrypt them and +send [푔푖] and [ℎ푖] (1 ≤ 푖 ≤ 푀) to the cipher end. For each tree, we are specified the value 퐷 which is the max depth +of tree. We build the root node 푢0 at first, and set the sample space 퐼푢0 to be the set of all samples. Then we split from +the root node depth by depth. For each node at a specified depth, the active party send its sample space 퐼 to all passive +parties. Then for each party 푘, do binning on the feature dataset 퐗푘 and sample space 퐼. Encrypt the binning result +as [퐗bin +푘 +] and send it to the cipher and, and save the binning boundaries to local storage. On the cipher end, for each +party, aggregate the encrypted gradients based on the encrypted binning results, and send the encrypted [퐆푘] and [퐇푘] +to the active party. On the active party, decrypt the aggregated gradients and compute the information gain for each +party and each feature, and then decide the optimal split (푘∗, 푗∗, 푠∗). Send the optimal split to the corresponding party, +to compute the sample space of the child nodes. Repeat this process until the maximum depth is reached, and then +fanhang et al.: Preprint submitted to Elsevier +Page 9 of 21 + +Cipher +Active Party +Passive Party +Backend +Compute +gradients +Sample space +Feature +Feature +binning +Gradients +binning +(gi),(hs) +Aggregate +Aggregate +gradients +gradients +Binning result + Aggregated gradients +Node +(G)(Hn) +splitting +Splitting feature i* +and position s +Sample +Sample +splitting +splitting +Sample splitting +result I . +Leaf weight +computing +Cipher data +Finish +Plain dataShort Title of the Article +Algorithm 1 The main process of training a federated XGBoost model +Input: On party 푘 (1 ≤ 푘 ≤ 푚): {퐗푘}푀×푁푘, the feature dataset of the 푘-th party +Input: On active party: {푦}푀, the truth label dataset +Input: 푇 , number of trees; 퐷, maximum depth of tree splitting; 퐵, number of bins; 훾, minimum loss reduction for a +split; 휆, L2 regularization term +Output: The 푡-th decision tree for 1 ≤ 푡 ≤ 푇 . The main tree structure is stored on the active party, while the binning +boundaries are stored distributed across all parties. +1: for 1 ≤ 푡 ≤ 푇 do +2: +̂푦푖 ← ∑푡−1 +푖=1 푓푖(푥푖) +3: +푔푖 ← −푦푖 + +1 +1+푒− ̂푦푖 , ℎ푖 ← +푒− ̂푦푖 +(1+푒− ̂푦푖)2 +4: +Active party: Encrypt all gradients 푔푖 and ℎ푖. Send all [푔푖] and [ℎ푖] to the ciphertext side. +5: +Initialize a new tree, add the root node 푢0 to it +6: +For root node 푢0, set the sample space 퐼푢0 ← {1, 2, ⋯ , 푀} +7: +for 1 ≤ 푑 ≤ 퐷 do +8: +for each tree node on depth 푑 − 1 (parallelizable on all nodes) do +9: +Active party: For this tree node, send its sample space 퐼 to all passive parties +10: +for 1 ≤ 푘 ≤ 푀 (parallelizable on all 푘) do +11: +Party 푘: Do binning on 퐼 and 퐗푘, getting the binning result 퐗bin +푘 +and binning boundaries 푏푘,푗 for 1 ≤ +푗 ≤ 푁푘 (Algorithm 2). Encrypt 퐗bin +푘 +and send [퐗bin +푘 ] to the ciphertext side. Save the binning boundaries +푏푘,푗 to local storage for 1 ≤ 푗 ≤ 푁푘. +12: +Ciphertext Side: According to [푔푖], [ℎ푖] and [퐗bin +푘 ], compute the aggregated gradients [퐆푘] and [퐇푘] +(Algorithm 3). Send the aggregated gradients [퐆푘] and [퐇푘] to the active party. For the active party +(푘 = 1), this process can be computed in pure plain text on the active party itself instead. +13: +end for +14: +Active party: Decrypt the aggregated gradients [퐆푘] and [퐇푘] for all 푘, get 퐆푘 and 퐇푘. Compute the best +split (푘∗, 푗∗, 푠∗) of this tree node (Algorithm 4), where 푘∗ is the split party, 푗∗ is the split feature ID on +party 푘∗, and 푠∗ is the position of split point on feature 푗∗. Send the tuple (푗∗, 푠∗ to party 푘∗. (It is possible +that the split gain is not greater than 0. In that case, continue to the next node without splitting the tree.) +15: +Party 푘∗: According to (푗∗ and 푠∗), for sample vector 퐼, determine the left sample space 퐼퐿 (Algorithm +5). Send 퐼퐿 to the active party. +16: +Active Party: Compute 퐼푅 ← 퐼 − 퐼퐿. Split the current tree node into two child nodes to join the node +queue, assign 퐼퐿 and 퐼푅 to them respectively. +17: +end for +18: +end for +19: +for each leaf node 푢 in the tree do +20: +Compute the weight 푤푢 ← − +∑ +푖∈퐼푢 푔푖 +∑ +푖∈퐼푢 ℎ푖+휆. +21: +end for +22: +Add the new generated decision tree to the model. +23: end for +24: return the generated model with all trees +compute the weight of each leaf node. After that, add the new generated tree to the model. +5.1.2. Secure Gradient Computation +The local binning algorithm is shown in Algorithm 2. On each active or passive party, after getting the feature +dataset {퐗푘}푀×푁푘, process a quantile binning algorithm on each column based on the sample space 퐼. The binning +result 퐗bin +푘 +is encrypted and send to the cipher end, while the binning boundaries are saved to local storage. +The secure gradient aggregation algorithm is shown in Algorithm 3. For each feature 푗 and each bin 푏, find out +the set of samples which appear in that bin. To preserve security, the process of dividing samples into bins is still in +an encrypted computation, and the result is an encrypted 0-1 vector [flag]. After that, aggregate the gradients with the +fanhang et al.: Preprint submitted to Elsevier +Page 10 of 21 + +Short Title of the Article +Algorithm 2 Local binning algorithm on Party 푘 +Input: 퐼, sample space of the current tree node (a list composed of sample indexes) +Input: {퐗푘}푀×푁푘, feature dataset of party 푘. +Input: 퐵, number of bins. +Output: 퐗bin +푘 , binned result of party 푘; 푏푘,푗 (1 ≤ 푗 ≤ 푁푘), binning boundary vectors +1: 퐗bin +푘 +← {−1}푀×푁푘 +2: for 1 ≤ 푗 ≤ 푁푘 do +3: +Process a quantile binning on the 푗-th column of 퐗푘 based on the sample space 퐼, where the number of bins is +퐵. Assume the vector {푥bin +푗 }푀 to be the binning result, and 푏푘,푗 is the binning boundary vector. 푥bin +푗 +is a vector +of length 푚, where the values are between 0 and 푀 − 1. 푏푘,푗 is a vector of length 퐵 − 1. +4: +for 1 ≤ 푖 ≤ 푀 do +5: +if 푖 ∈ 퐼 then +6: +{퐗bin +푘 }푖,푗 ← {푥bin +푗 }푖 +7: +end if +8: +end for +9: end for +10: return 퐗bin +푘 , 푏푘,푗 (1 ≤ 푗 ≤ 푁푘) +Algorithm 3 Cipher gradient aggregation of party 푘 on the ciphertext side (for the active party, this algorithm can be +run in pure plain text on the active party) +Input: [푔푖 +] and [ℎ푖 +] (1 ≤ 푖 ≤ 푀), encrypted gradient vectors +Input: [퐗bin +푘 ]푀×푁푘, encrypted binned result of party 푘 +Output: [퐆푘 +] , [퐇푘 +], encrypted aggregated gradients for party 푘 +1: [퐆푘 +] ← [0]푁푘×퐵, [퐇푘 +] ← [0]푁푘×퐵 +2: for 1 ≤ 푗 ≤ 푁푘 do +3: +for 1 ≤ 푏 ≤ 퐵 do +4: +for 1 ≤ 푖 ≤ 푀 do +5: +[flag] ← [퐗bin +푘 ]푖,푗 = 푏 +6: +[퐆푘]푗,푏 ← [퐆푘]푗,푏 + [flag] ∗ [푔푖 +] +7: +[퐇푘]푗,푏 ← [퐇푘]푗,푏 + [flag] ∗ [ℎ푖 +] +8: +end for +9: +end for +10: end for +11: return [퐆푘 +] , [퐇푘 +] +weight [flag]. Send the aggregated gradients [퐆푘] and [퐇푘] to the active party. +5.1.3. Secure Split +The tree node splitting algorithm is shown in Algorithm 4. On the active party, after collecting all encrypted +aggregated gradients from the cipher end, decrypt them to get 퐆푘 and 퐇푘 (1 ≤ 푘 ≤ 푚). Then for each party, each +feature and each split position, compute the information gain based on aggregated gradients. Unless the information +gains are non-positive for all splits, find the optimal split (푘∗, 푗∗, 푠∗) with the best information gain 푣∗, and split the +tree node to two child nodes. +The sample splitting algorithm is shown in Algorithm 5. After party 푘∗ receives the optimal split (푗∗, 푠∗), it +compares the corresponding feature values in the sample space with the binning boundary of the optimal split, and +decide whether each sample should be allocated to the left or right child node. After getting the result, return the left +sample space 퐼퐿 to the active party. +fanhang et al.: Preprint submitted to Elsevier +Page 11 of 21 + +Short Title of the Article +Algorithm 4 Compute the optimal split point on the active party +Input: 퐼, sample space of the current tree node (a list composed of sample indexes) +Input: {퐆푘}푁푘×퐵, {퐇푘}푁푘×퐵, aggregated gradients from all parties 1 ≤ 푘 ≤ 푚 +Input: 훾, minimum loss reduction for a split; 휆, L2 regularization term +Output: 푘∗, party ID of the optimal split; 푗∗, feature ID of the optimal split on party 푘∗; 푠∗, position of the optimal +split on feature 푗∗ of party 푘∗ +1: 푘∗ ← null, 푗∗ ← null, 푠∗ ← null, 푣∗ ← 0 +2: for 1 ≤ 푘 ≤ 푚 do +3: +for 1 ≤ 푗 ≤ 푁푘 do +4: +퐺 ← ∑퐵 +푏=1{퐆푘}푗,푏, 퐻 ← ∑퐵 +푏=1{퐇푘}푗,푏 +5: +퐺퐿 ← 0, 퐺푅 ← 퐺, 퐻퐿 ← 0, 퐻푅 ← 퐻 +6: +for 1 ≤ 푠 ≤ 퐵 − 1 do +7: +퐺퐿 ← 퐺퐿 + {퐆푘}푗,푠, 퐻퐿 ← 퐻퐿 + {퐇푘}푗,푠 +8: +퐺푅 ← 퐺 − 퐺퐿, 퐻푅 ← 퐻 − 퐻퐿 +9: +푣 ← 1 +2 +( +퐺2 +퐿 +퐻퐿+휆 + +퐺2 +푅 +퐻푅+휆 − +퐺2 +퐻+휆 +) +− 훾 +10: +if 푣 > 푣∗ then +11: +푘∗ ← 푘, 푗∗ ← 푗, 푠∗ ← 푠, 푣∗ ← 푣 +12: +end if +13: +end for +14: +end for +15: end for +16: return 푘∗, 푗∗, 푠∗ +Algorithm 5 Sample splitting on party 푘opt +Input: 퐼, sample space of the current tree node (a list composed of sample indexes) +Input: 푗∗, feature ID of the optimal split on party 푘∗; 푠∗, position of the optimal split on feature 푗∗ of party 푘∗ +Input: {퐗푘∗}푀×푁푘∗ , feature dataset of party 푘∗. +Input: {푏푘∗,푗∗}퐵−1, binning boundaries of the 푗∗-th feature. +Output: 퐼퐿, sample space of the left child node +1: 퐼퐿 ← {} +2: for 푖 ∈ 퐼 do +3: +if {퐗푘∗}푖,푗∗ ≤ {푏푘∗,푗∗}푠∗ then +4: +퐼퐿 ← 퐼퐿 ∪ {푖} +5: +end if +6: end for +7: return 퐼퐿 +6. Security Analysis +Security assumptions. +For scalability and generality, we model all the participant 푁 wind farms as data providers +who agree to contribute their data for power prediction in a privacy-preserving way, and for any 푡-out-of-푛 secret +sharing protocols, we further select 푛 wind farms as the computation servers to carry the ciphertext computation, +where 푛 ≤ 푁 and 푡 < 푛. The 푡-out-of-푛 secret sharing protocol guarantees that any set of 푡 computation servers +together can reconstruct the raw data while any set less then 푡 servers learns nothing. The computation servers are +connected through secure channels, hold the secret shares of all the data providers data and execute the agreed secure +protocols. Through this model, our method can be extended to any number wind farms with willingness to share their +data and suitable for any 푡-out-of-푛 secret sharing protocols. +Similiar to other main-stream privacy-preserving applications [45, 46, 47], we define our security model as honest- +majority and semi-honest model [48] for practical performance. The above model assumes that during the ciphertext +computation, no more than a half computation servers are corrupted together (푡 < 푛 +2) and the corrupted servers will +fanhang et al.: Preprint submitted to Elsevier +Page 12 of 21 + +Short Title of the Article +follow the agreed protocol while try to learn as much information as possible about the others. Informally, a protocol +is secure in the above model if the information that the corrupted servers gained is not distinguishable as there exists +an ideal trusted third party. +Security analysis. +The computation of our method can be separated into two parts, plaintext computation and cipher- +text computation. The security of ciphertext computation is guaranteed through the modular composition theorem [49], +which offers a general way for designing complex high-level secure protocols. Firstly, we design the high-level pro- +tocol by assuming that a series of simple sub-protocols exist. Then we design each sub-protocol meeting the security +guarantee and then plug them as sub-routines in the high-level protocol. The modular composition theorem formally +stated that, if the high-level protocol can be securely evaluated its function with ideal protocols, then the security and +functionality maintained by replacing all the ideal sub-protocols with sub-routines [49]. +For all the participant wind farms, the data providers only do plaintext computation of their own data, thus intro- +ducing no privacy risk. The computation servers only see the secret shares of the raw data and carry all the ciphertext +computation. In our method, all the cross party computation are designed to be evaluated in the ciphertext, and all +the ciphertext computation logic is modular compositions of secure addition, subtraction, multiplication, comparison, +reciprocal as well as exponential, which are well-studied and commonly provided by most semi-honest MPC platforms +like [19, 20]. Thus, the security of each wind farm’s data is preserved. +Note that he participants selection stage only utilizes 2 week history data of each wind farm to select the most +similar participants in plaintext as the correlation among wind farms in a specific season is relatively stable and the +MMD calculation is computationally intensive. In practice, this method performs effectively. +7. Case Studies +The proposed privacy-preserving prediction method is tested on the field measurement data of a wind farm cluster +in Inner Mongolia, China. The privacy preserving machine learning algorithms are implemented on the PrivPy, a +general-purpose MPC platform [19] which offers a series of secure operations based on 2-out-of-4 secret sharing +protocol. All the evaluations are performed on a Kubernetes (k8s) cluster [50] that is deployed on two 64-core AMD +EPYC CPUs with 256GB RAM. Each wind farm and computing server is deployed as a separate k8s container, and +the round-trip time between each pair is approximately 0.1ms. +7.1. Data Set and Test Description +The wind power and NWP data from Jan. 1st 2021 to July 23th 2021 are recorded with a time step of 15 minutes. +The locations of 27 wind farms are shown in Figure 6. We will use a wind farm located in the center of the chosen +cluster as the target wind farm for example to illustrate the effectiveness of privacy preserving collaborative prediction +model, pwXGBoost. +To illustrate the effectiveness of the pwXGBoost, the root mean square error (RMSE) +RMSE = +√ +√ +√ +√1 +푘 +푘 +∑ +푖=1 +(푥푡푖 − ̂푥푡푖)2 +(12) +and the mean absolute error (MAE) +MAE = 1 +푘 +푘 +∑ +푖=1 +||푥푡푖 − ̂푥푡푖|| +(13) +are used as the indexes to assess the prediction accuracy on the dataset, where 푘 is the number of the samples in +the dataset. 푥푡푖 is the 푖 th wind power sample at time 푡 and ̂푥푡푖 is the predicted 푖 th wind power sample at time 푡. +7.2. The Prediction Results of the Privacy Preserving Collaborative Prediction Model +The privacy preserving XGBoost model is the method proposed in this paper for the ultra-short-term wind power +prediction. The key advantage of this method is that it can not only utilize the historical wind power and NWP data from +fanhang et al.: Preprint submitted to Elsevier +Page 13 of 21 + +Short Title of the Article +Figure 6: The Location of Wind Farms +other wind farms without breaching the privacy, but also can extract the nonlinear spatial and temporal relationship +compared to the linear method before. To illustrate the effectiveness of the method, we compare it with the methods +which only utilize the local data and those use the linear model to extract the spatial and temporal correlation. +Local_XGBoost_wo_nwp This model only uses the local historical wind power data to train the XGBoost model. +The max depth of the tree in the model is 3, learning rate is 0.3 and the tree number which is the estimator of the +XGBoost is 80. +Local_XGBoost This model only uses the local historical wind power data and NWP data to train the XGBoost +model. The max depth of the tree in the model is 3, learning rate is 0.3 and the tree number which is the estimator of +the XGBoost is 80. +Lasso_wo_nwp [8] This model can use the wind power from nearby wind farms but is the linear method. The alpha +parameter is 0.00005. +Lasso [8] This model can use the wind power and NWP data from nearby wind farms but is the linear method. The +alpha parameter is 0.00005. +pwXGBoost_wo_nwp_mmd This model can use the historical wind power from nearby wind farms. The nonlinear +spatial and temporal relationship can be extracted. The max depth of the tree in the model is 3, learning rate is 0.3 and +the tree number which is the estimator of the XGBoost is 80. +pwXGBoost_wo_mmd This model can use the historical wind power and NWP data from nearby wind farms. The +nonlinear spatial and temporal relationship can be extracted. The max depth of the tree in the model is 3, learning rate +is 0.3 and the tree number which is the estimator of the XGBoost is 80. The correlated wind farms are selected based +on the distance. +pwXGBoost This model can use the historical wind power and NWP data from nearby wind farms. The nonlinear +spatial and temporal relationship can be extracted. The max depth of the tree in the model is 3, learning rate is 0.3 +and the tree number which is the estimator of the XGBoost is 80. The correlated wind farms are selected based on the +fanhang et al.: Preprint submitted to Elsevier +Page 14 of 21 + +41.9 +41.8 +41.7 +Target Wind Farm +41.6 +latitude +41.5 +41.4 +41.3 +41.2 +109.6 +109.8 +110.0 +110.2 +110.4 +longtitudeShort Title of the Article +Table 1 +The Prediction Results of Different Models (%) +Method +1h +2h +3h +4h +RMSE +MAE +RMSE +MAE +RMSE +MAE +RMSE +MAE +Local_XGBoost_wo_nwp +4.035 +2.678 +4.876 +3.403 +5.426 +3.938 +5.554 +3.920 +Local_XGBoost +4.033 +2.673 +4.843 +3.424 +5.301 +3.758 +5.872 +4.354 +Lasso_wo_nwp +3.823 +2.462 +4.629 +3.289 +5.293 +3.729 +5.552 +4.224 +Lasso +3.817 +2.401 +4.583 +3.181 +5.243 +3.881 +5.790 +4.269 +pwXGBoost_wo_nwp_mmd +3.840 +2.480 +4.649 +3.225 +5.183 +3.662 +5.609 +4.100 +pwXGBoost_wo_mmd +3.812 +2.443 +4.536 +3.167 +5.232 +3.723 +5.553 +3.920 +pwXGBoost +3.781 +2.346 +4.409 +2.962 +4.923 +3.453 +5.325 +3.761 +MMD. The Prediction results of different models are as in Table 1. +According to the results in Table 1, the proposed pwXGBoost is better than the linear method especially when +the prediction time is longer. Because when the prediction horizontal increase, the nonlinear of the wind power also +increase and the superiority of pwXGBoost method is demonstrated. Besides, methods that take advantage of spatial +and temporal correlation patterns always performs better than those which do not exploit the correlation. It is found +that MMD is more effective when selecting the correlated wind farms than using the wind farm distance. We also +visualize the error density of the prediction model for the 4th hour in Figure 7. +Figure 7: Probability distribution of forecasting errors +fanhang et al.: Preprint submitted to Elsevier +Page 15 of 21 + +pwXGBoost +pwXGBoost_wo_mmd +20 +20 - +15 +15 +10 +10- +5 +5 +0 +0 +-0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +0.2 +0.0 +0.2 +0.4 +Lasso +Local_XGBoost +17.5 +12 +15.0 +10 +12.5 +10.0 +Isu +e +6 +7.5 +D +5.0 +4 +2.5 +2 +0.0 +0 +-0.4 +-0.2 +0.0 +0.2 +0.4 +-0.2 +0.0 +0.2 Short Title of the Article +In Figure 7, the prediction error is plotted in hist gram and fitted by kernel density estimation method. It is proved +that the prediction error of the pwXGBoost is more centralized around zero which means it has better prediction +performance. +7.3. The Analysis of the Privacy Preserving Prediction Results +The prediction results of pwXGBoost, Lasso and Local_XGBoost are compared in Figure 8. In this test, it is +obvious that the wind power prediction method which uses the information from nearby wind farms performs better +than the method only uses the local data. +Figure 8: The Comparison of Different Prediction Results +According to the prediction results in Figure 8, we can also see that the pwXGBoost model is better than the +linear method especially in scenarios with rapid changes in wind power which is crucial for the safe operation of +power system. For the linear Lasso method, although it is easy to be implemented, its ability to capture the trend and +extreme values is inferior to the nonlinear pwXGBoost. +7.4. The Correlation Analysis of the Wind Farms +In this section, we demonstrate the effectiveness of the participant selection stage. The MMD distances between +each pair of wind farms are shown in Figure 9. +We adjusted the value of the 훽 (the threshold of Equation 11), the wind farms can be divided into different groups +and different number wind farms will be selected as the participant wind farms. Larger 훽 means more wind farms will +be selected as the participant wind farms. The RMSE under different 훽 is shown in Figure 10. +The results show that when the threshold is in the range of 0.7 and 1, the number of selected wind farms are +the same and the RMSE is the lowest. When the threshold is set too low (< 0.7), fewer wind farms will participate, +and spatial and temporal correlation information is also constrained, resulting in an increase in the RMSE. On the +other hand, if the threshold is set too high (> 1.0), the various participating wind farms may introduce too diverse +distributions, also increasing the RMSE. +fanhang et al.: Preprint submitted to Elsevier +Page 16 of 21 + +0.25 +Local XGBoost +Lasso +pwXGboost +true value +0.20 +0.15 +wind power +0.10 +0.05 +0.00 +0 +25 +50 +75 +100 +125 +150 +175 +200 +TimeShort Title of the Article +Figure 9: The Correlation of Wind Farms Based on MMD +Figure 10: The RMSE Under Different MMD Threshold +fanhang et al.: Preprint submitted to Elsevier +Page 17 of 21 + +1.0 +42 +86 +- 0.8 +26 24 22 20 18 16 14 12 10 + 0.6 +0.4 + 0.2 +0.0 +81012141618 +202224260.0394- +1h +0.04600 + 2h +0.0392 +0.04575 +0.0390 +0.04550 +0.0388 +0.04525 +MSE +0.0386 +0.04500 +0.0384 - +0.04475 +0.0382 +0.04450 +0.0380 +0.04425 +0.0378 +0.04400 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +beta +beta +0.0506 +— 3h +— 4h +9 +0.0546 +0.0504 +0.0544 +0.0502 +0.0542 +0.0540 +RM +0.0498 +0.0538 +0.0496 +0.0536 +0.0494 - +0.0534 +0.0492 +0.0532 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +beta +betaShort Title of the Article +Table 2 +Time Test of Training and Prediction Process +Parties +Training Time (s) +Inference time (s) +Linear Reg +XGB +Linear Reg +XGB +5 +108.1 +1421.0 +11.2 +155.3 +10 +111.3 +2276.4 +11.3 +284.9 +15 +131.4 +3181.0 +13.2 +402.8 +20 +140.6 +3958.7 +14.2 +566.5 +25 +161.0 +4736.8 +14.2 +659.9 +27 +169.1 +5214.9 +15.2 +714.3 +7.5. Scalability and Computation Cost +The training time and prediction time of the privacy preserving ultra-short-term wind power prediction model +is also important for the practical appplication. If it consumes a lot of time when training and prediction, it is not +applicable and acceptable. Therefore, we test the consuming time of the privacy preserving XGBoost. +From Table 2, it can be seen that even when all the 27 wind farms are included in the training and prediction +process, the training time is less than 1.5h and the prediction time is less than 12min. In our prediction process, the +12 wind farms are selected based on the MMD method, the training time are reduced to less than 53 minutes and the +prediction time are less than 400s, which is acceptable in practice. +8. Conclusion +Exploiting spatial and temporal correlation is useful to improve the accuracy of the ultra-short-term wind power. +Taking data security and regulation policy issues into account, this paper proposes a privacy-preserving ultra-short- +term wind power prediction method pwXGBoost based on secret sharing protocol and carefully chosen collaborative +neighboring wind farms. The test results on the Inner Mongolian data set proved the ability of nonlinear feature +extraction compared to the linear model and the data security is guaranteed mathematically. +According to the case study results, the RMSE of the method which uses the selected wind farms to train the +prediction model is 0.4% lower than that of the method which only uses the local wind farm data in the 4 − 푡ℎ hour. In +the meanwhile, the prediction time of privacy preserving method is less than 5min. Therefore, this method is acceptable +for the application. +Besides, the secure multi party computation theory is scalable to the other nonlinear operation. Since neural +network is another widely used method in wind farm power prediction, we can also build neural network based on +secure multi party computation theory and secret share protocol. In the future, we will adopted the secret sharing +protocol to the neural network and designed the privacy preserving wind power prediction method based on neural +network to test its performance. +A. Secure Multi Party Computation +Secure Multi Party Computation (MPC) has a long history in the cryptography community, it enables a group +of data providers to jointly compute an agreed function without disclosing their data. One of the most fundamental +building blocks of MPC is secret sharing, which is the basis of most current MPC platforms [20, 19]. 푡-out-of-푛 +secret sharing splits a secret input 푥 into 푛 shares, satisfying that any 푡 shares can completely reconstruct 푥 while any +shares less than 푡 reveal nothing. For example, a commonly-used 2-out-of-3 secret sharing protocol splits 푥 into [푥] = +{[푥]1, [푥]2, [푥]3} = {(푥1, 푥2), (푥2, 푥3), (푥3, 푥1)} and let three computation servers hold the shares, respectively [20, 51]. +[푥] satisfy that 푥 ≡ ∑푗=3 +푗=1 푥푗( mod 푀) where 푀 is a large integer, usually 2푘, making 푥푗, 푖 ∈ 1, 2, 3 uniformly +distributed in a ring of ℤ2푘. It is obvious that any two servers (or more) can together construct 푥 while each single +server learns nothing. +To make MPC general (i.e., to support arbitrary functions), researchers and engineers have proposed so-called +general-purpose MPC platforms, which offers a series of basic secure operations like secure addition, subtraction, +multiplication, comparison which can be composed together to support complex functions like square-root and division +and more advanced functions like machine-learning functions (e.g., secure principal component analysis [47]). All the +fanhang et al.: Preprint submitted to Elsevier +Page 18 of 21 + +Short Title of the Article +secure basic operations are cryptographic protocols among the computation servers and preserve the privacy of the +secret input 푥 during the computation. The computation process of these secret-sharing based MPC platforms has +three stages:initialization stage, computation stage and reveal stage. +Secret share initialization stage. +For 푁 data providers, they encode the original data 푥(푖), 푖 ∈ 1, ..., 푁 into the +secret share [푥(푖)] according to the specific 푡-out-of-푛 secret sharing protocol and pass the each share to the target +computation server. In the end of this stage, each computation server 푠푗, 푗 ∈ 1, ..., 푛 will hold 푁 secret shares of +the all the data providers, {[푥(푖)]푗}푖=푁 +푖=1 . In the above 2-out-of-3 secret sharing protocol, the three servers will hold +{(푥(푖) +1 , 푥(푖) +2 )}푖=푁 +푖=1 , {(푥(푖) +2 , 푥(푖) +3 )}푖=푁 +푖=1 , {(푥(푖) +1 , 푥(푖) +3 )}푖=푁 +푖=1 , respectively. +Computation stage. +For any valid function 푓(푥푖, 푖 ∈ 1, ..., 푛) that all the data providers agreed, it will be constructed +as the modular composition of the basic secure operations, which means that in the end of each secure operation, the +computation servers will hold the secret shares of the corresponding result. For example, after the evaluation of secure +addition between secret shares [푥] and [푦], the computation servers should hold the secret share [푧] where 푧 = 푥+푦. In +this stage, all the 푛 computation servers will evaluate the composed secure operations sequentially for the final result. +Reveal stage. +After the computation stage of 푓, the computation servers will hold the secret shares of the final result +(i.e., [푓(푥푖, 푖 ∈ 1, ..., 푛)]). In this final stage, all the computation servers will pass its share to one party (predefined +in the beginning) to reconstruct 푓(푥푖, 푖 ∈ 1, ..., 푛). In fact, the reveal stage can based on the blockchain technology to +regulate the information exchange and its profit allocation [52]. +B. Secure Operations for Secret Sharing Based MPC Platforms. +In this section, we first introduce some basic secure operations briefly, showing how the secret sharing compu- +tation works. In the end, we introduce secure division operation to demonstrate how to compose basic operations for +complex functions. +Secure addition. +For two data 푥 and 푦, their secret shares are [푥] and [푦]. It is self-evidently additive homomorphic +because [푥] + [푦] = [푥 + 푦]. Each computation server can locally compute the share of the sum in the computation +process. +Secure subtraction. +In the secret sharing computation, addition and subtraction are equivalent, as 푥 minus 푦 is +equivalent to 푥 plus the opposite of 푦. Servers can firstly construct the share of [−푦] locally by computing the opposite +of its own shares as −푦 ≡ ∑푖=푛 +푖=1 −푦푖( mod 푀). Then compute [푥 − 푦] through [푥] + [−푦]. +Secure multiplication. +Computing multiplication usually requires communications among the computation servers. +Take the above 2-out-of-3 secret sharing and ABY3 [51] multiplication protocol for example, defining 푧 = 푥푦. As +푥 = (푥1 + 푥2 + 푥3)(푦1 + 푦2 + 푦3), we can define [푧] as 푧1 = 푥1푦1 + 푥1푦2 + 푥2푦1 + 훼1, 푧2 = 푥2푦2 + 푥2푦3 + 푥3푦2 + 훼2 +and 푧3 = 푥3푦3 + 푥3푦1 + 푥1푦3 + 훼3 where ∑푗=3 +푗=1 훼푗 = 0. Each computation server 푗 firstly locally compute its share +푧푗, then communicate its share to the previous server for valid secret shares (as we require each server hold two shares +from {푧1, 푧2, 푧3}). +Secure comparison. +When two numbers 푥 and 푦 are compared, 푥 < 푦 equals a Boolean indicating whether (푥 − 푦) +is negative. So the computation servers can firstly compute [푥 − 푦] based on the secure subtraction protocol and then +use the bit extraction protocol [36] to securely extract the sign bit of [푥 − 푦]. If the highest bit is 1, it means that 푥 − 푦 +is negative and therefore 푥 < 푦. +Secure division. +Comparing with other basic operations, computing non-linear functions like secure division is +more involved. As [푥] divides [푦] is equivalent to [푥] multiply [ 1 +푦], how to compute the reciprocal of [푦] (i.e., [ 1 +푦]) is +important. +As 1 +푦 is the solution of Equation 14, it is obvious that the solution 푥 = 1 +푦. +푓(푥) = 1 +푥 − 푦 = 0 +(14) +fanhang et al.: Preprint submitted to Elsevier +Page 19 of 21 + +Short Title of the Article +As we know, the Taylor expansion of 푓(푥) is as follows: +푓(푥) = 푓(푥0) + 푓 +′(푥0) +1! +(푥 − 푥0) + 푓 +′′(푥0) +2! +(푥 − 푥0)2 + ... + 푓 푛(푥0) +푛! +(푥 − 푥0)푛 +(15) +The first order approximation of 푓(푥) is used to get the solution which is the Newton-Raphson method. 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IEEE Transactions on Sustainable +Energy, 13(2):1254–1266, 2021. +fanhang et al.: Preprint submitted to Elsevier +Page 21 of 21 + diff --git a/YNFRT4oBgHgl3EQfOTea/content/tmp_files/load_file.txt b/YNFRT4oBgHgl3EQfOTea/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..714079cd256a603c8b2176e36b1bf14fd18fc563 --- /dev/null +++ b/YNFRT4oBgHgl3EQfOTea/content/tmp_files/load_file.txt @@ -0,0 +1,1012 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf,len=1011 +page_content='Highlights Privacy Preserving Ultra-Short-term Wind Power Prediction Based on Secure Multi Party Computation Hang Fan,Xiaoyu Fan,Tianyi Hao,Wei Wei,Kun Chen,Guosai Wang,Xiaofeng Jia,Yidong Li,Wei Xu We develop a vertical privacy preserving XGBoost prediction algorithm based on the secret sharing protocol in the pwXGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We design a criterion to select the suitable participant wind farm in the pwXGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We test the wind farms in the field data from the wind farm cluster in the Inner Mongolian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='13513v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='CR] 31 Jan 2023 Privacy Preserving Ultra-Short-term Wind Power Prediction Based on Secure Multi Party Computation Hang Fana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Xiaoyu Fanb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Tianyi Haoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Wei Weic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Kun Chend,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Guosai Wangd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Xiaofeng Jiae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Yidong Lif and Wei Xub,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='∗ aorganization=PBC School of Finance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' addressline=Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' city=Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' postcode=100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' country=China borganization=Institute for Interdisciplinary Information Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' addressline=Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' city=Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' postcode=100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' country=China corganization=Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' addressline=Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' city=Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' postcode=100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' country=Beijing dorganization=Tsingjiao Information Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' addressline=Tsinghua Science and Technology Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' city=Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' postcode=100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' country=Beijing eorganization=Data Management Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' addressline=Beijing Big Data Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' city=Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' postcode=100044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' country=Beijing forganization=School of Computer and Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' addressline=Beijing Jiaotong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' city=Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' postcode=100025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' country=Beijing A R T I C L E I N F O Keywords: wind power prediction privacy preserving machine learning pwXGBoost model secure multi party computation A B S T R A C T Mining the spatial and temporal correlation of wind farm output data is beneficial for enhancing the precision of ultra-short-term wind power prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, if the wind farms are owned by separate entities, they may be reluctant to share their data directly due to privacy concerns as well as business management regulation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Although cryptographic approaches have been designed to protect privacy in the process of data sharing, it is still a challenging problem to encrypt the original data while extracting the nonlinear relationship among multiple wind farms in the machine learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' This paper presents pwXGBoost, a technique based on the machine learning tree model and secure multi-party computation (SMPC) that can successfully extract complicated relationships while preserving data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' A maximum mean discrepancy (MMD) based scheme is proposed to effectively choose adjacent candidate wind farms to par- ticipate in the collaborative model training, therefore improving the accuracy and reducing the burden of data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The proposed method was evaluated on real world data collected from a cluster of wind farms in Inner Mongolia, China, demonstrating that it is capable of achieving considerable efficiency and performance improvements while preserving privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Background and Motivation The large-scale exploitation of renewable energy sources, such as wind power, has brought a large amount of clean energy and reduced the CO2 emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, the uncertainty and randomness of wind power pose serious challenges to the operation of the power system [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On the electricity spot trading market, the bidding strategies of wind farms heavily depend on the wind power predictions and severe deviations in bids are penalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The economic losses caused by inaccurate wind power forecasts can reach 10% of the wind farms’ electricity sales [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Due to the desire for high-quality wind power prediction, there have been tremendous studies in related fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' At present, the core concept of ultra-short-term power prediction is to mine a variety of data such as historical power generation data and Numerical Weather Prediction (NWP) data of local wind sites through artificial intelligence and statistical learning methods, so as to develop high-precision prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' At the same time, wind farms are usually located in close proximity to each other, there is a strong spatial and temporal correlation pattern among the sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Utilizing this correlation pattern can considerably enhance the power prediction accuracy of wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' As a result, more studies have been conducted in recent years by assuming that the data of all wind farms in the cluster can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Through combining graph machine learning and other nonlinear methods to extract the spatial and temporal correlation among neighboring wind farms, the prediction accuracy can be effectively improved [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, historical wind power data is often owned by different companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The long-time historical wind power can reflect the ∗Corresponding author fanhang123456@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='com (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Fan);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' weixu@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='cn (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Xu) ORCID(s): fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 1 of 21 aShort Title of the Article production and operation status of the wind farms in the electricity market and is confidential to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The NWP data is purchased by wind farms at high cost, thus they are hesitant to share with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, the direct sharing of wind power data may be restricted by data management policies due to the privacy and security of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' How to use data from neighboring wind farms without compromising privacy is considered as the last mile and the most difficult part of the application of spatial and temporal correlation methods in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Previous Study and Literature Review Some research is brought out to predict the wind power while preserving the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For this problem, article [5] classify the solutions into three classes, namely data transformation method, decomposition method and secure multi-party computation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to the definition, the data transformation method normally refer to adding some random noise to the original data before the fitting process to protect the privacy which is called differential privacy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Although the differential privacy is successful to protect the privacy in the picture recognition area, it is not suitable for the wind power prediction [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The picture recognition is a classification problem while the wind power prediction is a regression problem which is more sensitive to the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Any disturb to the wind power data can lead to a decrease of accuracy which is unacceptable for the power market and the wind farm owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Decomposition method regards the prediction problem as an optimization problem and decomposes it into several sub-problems and allows each data provider to solve it separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Carla [8] emphasizes the forecast skill improvement due to the spatial and temporal dependencies in the time series and the business competition among wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, [8] formulates a framework which combines the data transformation methods and the alternating direction method multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In [9], a data market even been designed to encourage the wind farms to share their data to improve the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Han [10] designed an regression market for wind power forecasting and use the LASSO regulation as the reference for the data pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, wind farm power is highly nonlinear, and the lasso method, as a linear prediction method, is inherently difficult to capture the spatial and temporal correlation among wind farms, so the prediction accuracy in practice is not always satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In practical wind farm power prediction tasks, multiple nonlinear prediction methods such as machine learning, XGBoost or even neural networks and their combined derivative models are more often used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' And as stated in the article [8], privacy methods using ADMM methods for solving cannot be directly extended to nonlinear prediction scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, there is an urgent demand to explore how to consider data privacy in nonlinear prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure multi-party computation in article [5] is a generalized privacy preserving computation framework [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' This topic is an active research field in computer science and data mining because it is compatible with non-linear operation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It calls for the fusion of classical secure multi-party computation [12], federated learning [13] and other classical cryptography theory such as homomorphic encryption [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Classical secure multi-party computing techniques include secret sharing, oblivious transmission, and garbled circuit, which are mainly derived from the "millionaire’s problem" in 1982 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In 1986, Yao proposed the theory of the garbled circuit, which became the first general multi-party secure computing scheme [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' After several years of development, the classical secure multi-party computation consists of multiple cryptography protocols such as garbled circuit [16], oblivious transmission [17] and secret sharing [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Garbled circuit is performed by constructing a circuit and obfuscating the signals on the circuit, while secret sharing is performed by splitting the secret data into multiple slices and performing computation on the slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Because secret sharing protocol is more friendly for the computation, most advanced privacy preserving computation platform adopt this protocol [19, 20], and it quickly becomes a popular method in recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For example, article [21] uses the secret share to realize the fully privacy preserving distributed optimization of power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Federated learning is a distributed machine learning method proposed by Google in 2016 [22] that enables mul- tiple mutually untrusted training data providers to collaboratively train machine learning models by exchanging inter- mediate computational results such as gradients or parameters without exchanging raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to the different data distribution among participants, federation learning is generally classified into three types: horizontal federated learning [11], vertical federated learning [11] and federated transfer learning [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Horizontal federation is mainly used for sample federation between two parties with the same or similar business model, and there is a lot of feature overlap in the data of each party, but less overlap in the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Longitudinal federation is mainly used for feature federation between two parties with different business modes but the same or similar users, with less feature overlap but more user overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Federated migration is mainly used for forward learning between two parties with less intersection of industry and users, and there is less overlap of features and users in the data of all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' There are many scenarios that the federated learning is used to protect the privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [24] used the federated learning for the volt- fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 2 of 21 Short Title of the Article age prediction in the local energy community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In [25], the federated fuzzy k-means is used to analyze the smart grid meter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Federated learning can also be used with the reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In [26], a federated reinforcement learning method is designed for the peer-to-peer energy trading and the carbon allowance trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Article [27] uses the horizontal federated reinforcement learning to predict the wind power which can leverage the wind farms in a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, it can not extract the wind farm spatialtemporal relationship which is implied by the wind power data at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In 2019, it was demonstrated that the gradients or parameters exchanged during federation learning can be used to infer or even recover the original data information [28], currently, the exchange process usually requires cryptography-based techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', MPC) or homomorphic cryptography to avoid these risks [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On the other hand, the performance of the federated learning will decrease if the data distribution of the participants are non-iid such as the feature distribution skew, label distribution skew and quantity skew [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Take the wind power prediction case for example, if the wind data of the participants do not follow similar distributions or appear to be non-iid, it is more difficult to identify the spatial temporal correlation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Homomorphic encryption is a classical encryption method to protect data privacy by directly encrypting the plain- text, performing various operations under the ciphertext, and finally obtaining the resulting ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Homomorphic encryption can be classified into Fully Homomorphic, Somewhat Homomorphic, and Partially Homomorphic [31, 29] depending on the degree of support for an unlimited number of arbitrary homomorphic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Homomorphic encryption allows arbitrary computation of the ciphertext without decryption, but its performance is too slow to be- come practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to the latest Fully Homomorphic computation benchmark [32] in 2022, the homomorphic computation is orders of magnitude slower than plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The limited computational speed constrains its practical application [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Contribution and Paper Organization Although there are some works aim to preserve the privacy in the prediction process, there are still two main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The first problem is the current privacy preserving method can not fully utilize the spatial and temporal correlation while safely preserving the data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Although some researchers use the federated learning method such as the FedAVG to fusion the gradient of the neural network, it follows the horizontal data fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is more similar to the transfer learning which is suitable when the wind power data is sufficient and horizontally partitioned among wind farms rather than the case that the wind farm can boost his own prediction accuracy by utilizing the spatial and temporal relationship with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Moreover, the classic federated learning method is not safe enough for the collaborative modeling and prediction [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Current privacy preserving prediction method which can extract the spatial and temporal relationship is based on Lasso-var and it is a linear method [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' But the spatial and temporal relationship is highly nonlinear, and the feature extraction ability of linear method is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Using homomorphic encryption and other full ciphertext computing methods can solve the nonlinear problem in the extraction of spatial and temporal correlation, but the expensive computation cost makes it impractical [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The second problem is the participant selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If the data of each wind farm exhibits significant difference in their distributions, the non-iid feature will effect the performance of prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Besides, if the number of wind farms in the collaborated power prediction is extremely great, the communication cost will compromise the timeliness of the ultra-short-term prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If insufficient wind farms participate in the collaborative model training, spatial and temporal correlation will not be utilized to its full potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, we designed a method named pwXGBoost based on vertical data fusion strategy and the secret sharing protocol which is scalable to extract the nonlinear spatial and temporal correlation pattern of several wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the pwXGBoost model, we also borrowed ideas from personalized federation learning [34] to screen participants for the collaborative modeling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The contribution of this paper is three-fold: (1) We develop a vertical privacy preserving XGBoost prediction algorithm based on the secret sharing protocol in the pwXGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It has the following advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' First, it is scalable to the nonlinear data and complex modeling of the spatial and temporal correlation compared to the renowned Lasso-var method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Second, it can realize a lossless and secure computation of XGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is more precise than the data transformation methods and more secure than the conventional federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' (2) We design a criterion to select the suitable participant wind farm in the pwXGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the criterion, during the collaborative training process the maximum mean discrepancy index is adopted to assess the similarity of the wind farm data distribution and it can select the wind farm which is most useful for the spatial and temporal correlation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' (3) We test the wind farms in the field data from the wind farm cluster in the Inner Mongolian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The data of some fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 3 of 21 Short Title of the Article nearby wind farms in the cluster are combined to predict the wind power of the target wind farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The experiment results show that the proposed pwXGBoost method is superior than all the baseline methods which only uses local data or a linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The prediction time is also acceptable for the practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Section 3 will formulate the mathematical model of the privacy preserving ultra-short-term wind power prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Section 4 develops the privacy preserving XGBoost model based on secret sharing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Section 5 describes the implementation process of the privacy preserving ultra-short-term wind power prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Section 6 analyzes the security of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The privacy preserving prediction algorithm is tested on the wind farm cluster from Inner Mongolian and the effectiveness is validated in Section 7 Finally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Conclusions are drawn in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Preliminary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Traditional Wind Farm Power Prediction Wind power forecast is a classical time series prediction problem which have been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For ultra- short-term wind power prediction, traditionally only the data of the local wind farm is used for the modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푃푡+퐻 = 푓(푃푡−푀+1∶푡, 푉푡+1∶푡+푁) (1) Where 푃푡−푀+1∶푡 is the local historical wind power of the wind farm and 푉푡+1∶푡+푁 is the local NWP of the wind farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Recently, it has been recognized that exploiting the spatial and temporal correlation can improve forecast accuracy [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, the wind power prediction can be modeled as follows: 푃 푖 푡+퐻 = 푓(푃 1 푡−푀+1∶푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푃 푖 푡−푀+1∶푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푃 푛 푡−푀+1∶푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푉 1 푡+1∶푡+푁, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푉 푖 푡+1∶푡+푁, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푉 푛 푡+1∶푡+푁) (2) Where 푃 푖 푡−푀+1∶푡 ∈ 푅푀×1 is the historical wind power of wind farm 푖 and the step length for the prediction is 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푉 푖 푡+1∶푡+푁 ∈ 푅푁×푘 is the matrix of NWP data for wind farm 푖 and the step length for the NWP data is 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The variable number of NWP data is 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Normally, the next 4 hour wind power are to be predicted and the time interval is 15min, so 퐻 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Function 푓 is the prediction model which can be linear model such as Lasso, neural network or XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the prediction model, although only the wind power of wind farm 푖 needed to be predicted, the historical wind power and NWP data of other nearby wind farms are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For a wind farm cluster, if the wind farms in this cluster are belong to the same owner, this kind of centralized prediction model is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, when the wind farms in the cluster are the assets of different stakeholders, direct data sharing is not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, it is necessary to develop the privacy preserving prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' A Brief Review of XGBoost XGBoost is an ensemble of tree models to boost the performance of a single tree which is very popular in wind power prediction[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For a dataset X ∈ 푅푀×푁 with 푀 samples and 푁 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' XGBoost can predict the 푖-th sample 푥푖 ∈ 푅1×푁 by using 푇 regression function as follows: ̂푦푖 = 푇∑ 푡=1 푓푡(푥푖) (3) XGBoost is sequentially trained by calculating ̂푦(푡) 푖 = ̂푦(푡−1) 푖 + 푓푡(푥푖), where a new tree 푓푡(푥푖) is used to train the residual of the target and the prediction in the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the given loss function 푙, a second-order Tylor expansion is used to approximate it in 푡-th iteration as follows: \ue238 ≈ 푁 ∑ 푖=1 [푙(푦푖, ̂푦(푡−1) 푖 ) + 푔푖푓푡(푥푖) + 1 2ℎ푖푓 2 푡 (푥푖)] + Ω(푓푡) (4) fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 4 of 21 Short Title of the Article Ω(푓푡) = 훾푈 + 1 2휆||휔||2 (5) Where ̂푦(푡−1) 푖 is the current prediction results, Ω is the regulation term, 푈 is the number of leaves in the tree, 훾 and 휆 are the hyper-parameters to restrict the tree number and weights respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푔푖 = 휕 ̂푦(푡−1) 푖 푙(푦푖, ̂푦(푡−1) 푖 ) and ℎ푖 = 휕2 ̂푦(푡−1) 푖 푙(푦푖, ̂푦(푡−1) 푖 ) are the first and second order derivative statistics of loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The tree model starts from 푠 single leaf node which includes all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then the node recursively splits the current samples into left and right subsets denoted by 퐼퐿 and 퐼푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The loss function after the split is \ue238split ≈ 1 2[ (∑ 푖∈퐼퐿 푔푖)2 ∑ 푖∈퐼퐿 ℎ푖 + 휆 + (∑ 푖∈퐼푅 푔푖)2 ∑ 푖∈퐼푅 ℎ푖 + 휆 − (∑ 푖∈퐼퐼 푔푖)2 ∑ 푖∈퐼퐼 ℎ푖 + 휆] − 훾 (6) Where the best split is the one with the highest \ue238푠푝푙푖푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The weight 푤 of each leaf is calculated in equation (10) 푤 = − ∑ 푖∈퐼푢 푔푖 ∑ 푖∈퐼푢 ℎ푖 + 휆 (7) When the depth of the tree reach the highest, the training of XGBoost terminates [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Problem Definition Due to the privacy concern, wind farms can hardly share their data directly without any constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Because once the wind power data of the wind farms are copied to other places, the risk of data abuse seems inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The privacy-preserving wind farm power prediction allows wind farms to access the data of other adjacent wind farms to train the prediction model jointly without knowing the exact values of those data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In this section, the ultrashort-term wind power forecast problem is formulated as follows: 푃 푖 푡+퐻 = 푓([푃 1 푡−푀+1∶푡], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푃 푖 푡−푀+1∶푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', [푃 푛 푡−푀+1∶푡], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', [푉 1 푡+1∶푡+푁], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푉 푖 푡+1∶푡+푁, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='[푉 푛 푡+1∶푡+푁]) (8) Where [] is the secret share encryption of the variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Through the secret share encryption, the data and computation is only known by the owner of the data [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [푃 푛 푡−푀+1] is the vector of the wind power in the secret share cipher text of 푃 푛 푡−푀+1 ∈ 푅푀×1 fow wind farm 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [푉 푛 푡+1∶푡+푁] is the matrix in the secret share cipher text of NWP matrix 푉 푛 푡+1∶푡+푁 ∈ 푅푁×푘 for wind farm 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푀 is the step length of historical wind power used in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푁 is the step length of NWP data used in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Prediction function 푓 uses the cipher text of the nearby wind farms to predict their own wind power in the next few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' As we know, the prediction model is constructed by the operation such as addition, multiplication, division, compare and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Those operation can be also implemented in the secure share protocol to protect the privacy [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The example of addition and multiplication is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the secure addition, two part holds number 푎 and 푏 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to secret share protocol, number 푎 is divided into two random number 푎1 and 푎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Number 푏 is divided into two random number 푏1 and 푏2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then 푎1 and 푏1 are sent to one computing server to get 푎1 + 푏1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the meanwhile, 푎2 and 푏2 are sent to another computing server to get 푎2 + 푏2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' By add the number of 푎1 + 푏1 and 푎2 + 푏2, the results of 푎 + 푏 is worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If the two computing server will not collude, the privacy of 푎 and 푏 is also guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The computing process is similar for the secure multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푎1 × 푏1, 푎2 × 푏2, 푎1 × 푏2 and 푎2 × 푏1 are calculated separately on the computing server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then the results of 푎 × 푏 can be worked out by adding those four components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If those four computing server not collude, the privacy can also be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It The basic computation operation such as add, multiply and compare based on secret share is described in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' By utilizing the basic operation, we can construct the derivative operation such as division, activation function and sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' By leveraging the basic operation and derivative operation, we can build the complex machine learning method to approximate the wind power prediction function shown in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The process is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 21 Short Title of the Article Figure 1: The Illustration of Secure Addition and Secure Multiplication Figure 2: The Construction of Complex Machine Learning Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Privacy Preserving Ultra-short-term Wind Power Prediction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Overview of Privacy Preserving Ultra-short-term Wind Power Prediction The basic idea of privacy preserving ultra-short-term wind power prediction is using the data of other wind farms to enhance the accuracy of prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Indeed, not only the historical wind power but also the NWP data can be utilized in the privacy preserving training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' There are two ways of utilizing and fusing the data as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The first one is using the historical wind power, NWP and the labels of all the wind farms to train a model which is similar to the transfer learning [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is a horizontal data fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The problem for this data fusion method is that the spatial and temporal relationship is not well considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The second one incorporates the historical wind power and NWP of other wind farms at the same time to predict the label which resembles a centralized prediction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is a vertical data fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the wind power prediction tasks, the spatial and temporal correlation is included in the wind power and NWP at the same time, so the vertical data fusion method is more suitable for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The participants are divided into active parties and passive parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In wind power prediction, the active party is the wind farm who would like to predict the wind power using the data from other wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' They have both feature data and label data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The passive party is the wind farm who only has the feature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It lends data to the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, when the active party initiated a request for a prediction, the active party need to select the appropriate wind farms to act as the passive party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If too many wind farms take part in the training process, the massive communication will decrease the training efficiency, and the discrepancy of the sample distribution will also effect the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If there is not enough wind farms in the training process, the spatial and temporal correlation pattern cannot fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 6 of 21 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. [a] = (a1,a2) [a] = (ai,a2) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [bl = (b1,b2) [b] = (b1, b2) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='. 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' a1 a2 a1 a2 a = a1 + a2 b b1 b2 b1 b2 = + b2 b1 (ai + bi) + (a2 + b2) a*b aib1 + a2 b2 + aib2 + azb1 b = =Basic Operation Derivative Operation Complex Machine Add Division Learning Model Activation Function Statistic Multiply Secret Share Sort Regression Compare ClassificationShort Title of the Article Figure 3: Horizontal and Vertical Data Fusion Method be fully and accurately explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, in our pwXGBoost model, we divide the privacy preserving ultra-short-term wind power prediction process into two parts as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The first part is the selection of the participant wind farms (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2) and the second part is the privacy preserving XGBoost algorithm for ultra-short-term wind power prediction (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Server 1 Server 2 Server 3 Server 4 MPC Protocol 1) The Selection of Participant Wind Farms … 2) Privacy Preserving XGBoost Algorithm 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' ", 𝑥# " … , 𝑥$ " 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' %, 𝑥# % … , 𝑥$ % 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' &, 𝑥# & … , 𝑥$ & 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=" ', 𝑥# ' … , 𝑥$ ' Plaintext computation Ciphertext computation Data Wind farm 2 Data Wind farm 1 Data Wind farm 3 Data Target wind farm Data Wind farm N-3 Data Wind farm N-2 Data Wind farm N-1 Data Wind farm N Data Wind farm A Data Wind farm B Data Target wind farm Data Wind farm C Data Wind farm D Figure 4: The Privacy Preserving Ultra-short-term Wind Power Prediction Process 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Selection of Participant Wind Farms In the privacy preserving ultra-short-term wind power prediction, it is necessary to select the wind farms which would like to engage in the data sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The reasonable selection of participant can increase the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Since the nearby wind farms have greater influence on the wind farm in the active party, we use the Gaussian distance function and adjacent matrix to determine which wind farm can be used in the prediction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' To make fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 7 of 21 feature windpower feature windpower ebel label P1 NWP1 M+1:t +1:t+N P1- M+1:t Windfarm 1 Windfarm 1 P1- NWP1 :t- t:t+M-1 P1 NWP P1 NWP Pt- 1:t+N-2 NWP1 feature feature windpower label NWP Windfarm 2 P2- t+1:t+N M+1:t Windfarm 2 P2 NWP t:t+M-1 :t-1 t:t+M-1 P2- NWP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' P2 1:t±2 NWP2 t-1:t+N-2 feature feature windpower label Pr- Pn M+1:t M+1:t Windfarm n Windfarm n Pn :t-1 Pn NWpn 1-1:t+2 NWPn Vertical Data Fusion Method Horizontal Data Fusion MethooShort Title of the Article full use of the spatial and temporal correlation, the wind farms with similar distribution can be selected into a group and we use the Maximum Mean Discrepancy (MMD) to calculate the distribution distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' MMD is widely used in the transfer learning which can measure the distance of two distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The multi kernel variant of MMD is used here which maps the data distributions into a Reproducing Kernel Hibert Space (RKHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For two distribution 푑1 and 푑2, the square of MMD between the two distributions can be calculated as MMD2(푑1, 푑2) = ‖‖‖퐸 [휙(푑1)] − 퐸 [휙(푑2)]‖‖‖ (9) Where 휙(⋅) is the mapping to RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In practice, the mapping is unknown and we can use the kernel trick to replace the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The result is MMD2(푑1, 푑2) = 퐸 [퐾(푑1, 푑1)] + 퐸 [퐾(푑2, 푑2)] − 2퐸 [퐾(푑1, 푑2)] (10) Where 퐾(푑1, 푑2) = [휙(푥), 휙(푦)] is the desired kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Because the wind farms are located in a region and the spatial and temporal correlation is closely related to the distribution distance, the adjacent matrix is defined as follows: 퐴푖,푗 = { exp ( − MMD2(푖,푗) 휎2 ) , if RMMD2(푖, 푗) ≤ 훽 ∗ mean 0, otherwise (11) Where MMD2(푖, 푗) is the maximum mean discrepancy distance between wind farm 푖 and wind farm 푗, 휎 is the standard deviation of the distance between 푛 wind farms and 훽 is the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In our case, we set the half of the mean distance as the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For wind farm 푖, only the wind farms whose 퐴푖,푗 ≠ 0 are chosen as the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Privacy Preserving XGBoost Algorithm for Ultra-short-term Wind Power Prediction XGBoost is a method which is widely used in the wind power prediction[38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' There have been a number of widely adopted XGBoost programming packages, such as XGBoost [35], LightGBM [40] and CatBoost [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, they are restricted to the centralized setting and are not applicable when data privacy is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Although there are some open-source privacy-preserving machine learning framework such as Pysytft [42] and FedML [43], they do not support XGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Even though the FATE platform [44] can provide the XGBoost function, long encryption keys are required to ensure security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' However, long keys can significantly slow down the computation, and the potential of privacy invading remains a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, it is necessary to build the XGBoost model based on secure multi-party computation which can fully preserve data privacy and is also effective in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the privacy preserving XGBoost algorithm for ultra-short-term wind power prediction, it can be divided into a training stage and a prediction stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the training stage, the historical wind power and NWP data of passive parties and the active party at time 푡 is fused in a vertical way as the feature, and the wind power to be predicted is the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The privacy computation method is used because it can fuse the data from different sources without knowing their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The prediction algorithm can be trained without compromising privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the prediction stage, the encrypted historical wind power and NWP data are transmitted to the trained model and output the predicted wind power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' This part is also shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to the secure multi party computation theory, some wind farms are selected out as the computing server node randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The secret share of the data of each wind farm rather than the original data are sent to the computing server node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In this way, if the computing server node not collude, the security is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We will discuss the details of the privacy preserving XGBost model based on secret sharing in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Privacy Preserving XGBoost Algorithm Based on Secret Sharing In this section, we present the design and details of privacy preserving XGBoost Algorithm based on 2-out-of-4 secret sharing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 8 of 21 Short Title of the Article Figure 5: The Training Process of Privacy Preserving XGBoost in Secret Sharing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Overall Description of Privacy Preserving XGBoost Model The whole training process of the secure XGBoost model is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the vertical privacy preserving XGBoost model, there are 푚 participants take part in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Here we note the active party to be party 1, and the passive parties to be party 2, ⋯ , 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure Model Training Process The algorithm of the secure XGBoost model is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The model is trained among the active party, passive party and the ciphertext backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The input datasets of the algorithm including the feature set {퐗푘}푀×푁푘 (1 ≤ 푘 ≤ 푚) provided by all the 푚 parties and the label set {푦}푀 owned by the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' During the training process, we train the 푇 decision trees sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the 푡-th tree, at the beginning, we compute the prediction result ̂푦푖 (1 ≤ 푖 ≤ 푀) of the first 푡 − 1 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then we compute the first and second order gradients, which are 푔푖 and ℎ푖 (1 ≤ 푖 ≤ 푀) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We encrypt them and send [푔푖] and [ℎ푖] (1 ≤ 푖 ≤ 푀) to the cipher end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For each tree, we are specified the value 퐷 which is the max depth of tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We build the root node 푢0 at first, and set the sample space 퐼푢0 to be the set of all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then we split from the root node depth by depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For each node at a specified depth, the active party send its sample space 퐼 to all passive parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then for each party 푘, do binning on the feature dataset 퐗푘 and sample space 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Encrypt the binning result as [퐗bin 푘 ] and send it to the cipher and, and save the binning boundaries to local storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On the cipher end, for each party, aggregate the encrypted gradients based on the encrypted binning results, and send the encrypted [퐆푘] and [퐇푘] to the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On the active party, decrypt the aggregated gradients and compute the information gain for each party and each feature, and then decide the optimal split (푘∗, 푗∗, 푠∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Send the optimal split to the corresponding party, to compute the sample space of the child nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Repeat this process until the maximum depth is reached, and then fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 9 of 21 Cipher Active Party Passive Party Backend Compute gradients Sample space Feature Feature binning Gradients binning (gi),(hs) Aggregate Aggregate gradients gradients Binning result Aggregated gradients Node (G)(Hn) splitting Splitting feature i* and position s Sample Sample splitting splitting Sample splitting result I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Leaf weight computing Cipher data Finish Plain dataShort Title of the Article Algorithm 1 The main process of training a federated XGBoost model Input: On party 푘 (1 ≤ 푘 ≤ 푚): {퐗푘}푀×푁푘, the feature dataset of the 푘-th party Input: On active party: {푦}푀, the truth label dataset Input: 푇 , number of trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐷, maximum depth of tree splitting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐵, number of bins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 훾, minimum loss reduction for a split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 휆, L2 regularization term Output: The 푡-th decision tree for 1 ≤ 푡 ≤ 푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The main tree structure is stored on the active party, while the binning boundaries are stored distributed across all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 1: for 1 ≤ 푡 ≤ 푇 do 2: ̂푦푖 ← ∑푡−1 푖=1 푓푖(푥푖) 3: 푔푖 ← −푦푖 + 1 1+푒− ̂푦푖 , ℎ푖 ← 푒− ̂푦푖 (1+푒− ̂푦푖)2 4: Active party: Encrypt all gradients 푔푖 and ℎ푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Send all [푔푖] and [ℎ푖] to the ciphertext side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 5: Initialize a new tree, add the root node 푢0 to it 6: For root node 푢0, set the sample space 퐼푢0 ← {1, 2, ⋯ , 푀} 7: for 1 ≤ 푑 ≤ 퐷 do 8: for each tree node on depth 푑 − 1 (parallelizable on all nodes) do 9: Active party: For this tree node, send its sample space 퐼 to all passive parties 10: for 1 ≤ 푘 ≤ 푀 (parallelizable on all 푘) do 11: Party 푘: Do binning on 퐼 and 퐗푘, getting the binning result 퐗bin 푘 and binning boundaries 푏푘,푗 for 1 ≤ 푗 ≤ 푁푘 (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Encrypt 퐗bin 푘 and send [퐗bin 푘 ] to the ciphertext side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Save the binning boundaries 푏푘,푗 to local storage for 1 ≤ 푗 ≤ 푁푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 12: Ciphertext Side: According to [푔푖], [ℎ푖] and [퐗bin 푘 ], compute the aggregated gradients [퐆푘] and [퐇푘] (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Send the aggregated gradients [퐆푘] and [퐇푘] to the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the active party (푘 = 1), this process can be computed in pure plain text on the active party itself instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 13: end for 14: Active party: Decrypt the aggregated gradients [퐆푘] and [퐇푘] for all 푘, get 퐆푘 and 퐇푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Compute the best split (푘∗, 푗∗, 푠∗) of this tree node (Algorithm 4), where 푘∗ is the split party, 푗∗ is the split feature ID on party 푘∗, and 푠∗ is the position of split point on feature 푗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Send the tuple (푗∗, 푠∗ to party 푘∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' (It is possible that the split gain is not greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In that case, continue to the next node without splitting the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=') 15: Party 푘∗: According to (푗∗ and 푠∗), for sample vector 퐼, determine the left sample space 퐼퐿 (Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Send 퐼퐿 to the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 16: Active Party: Compute 퐼푅 ← 퐼 − 퐼퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Split the current tree node into two child nodes to join the node queue, assign 퐼퐿 and 퐼푅 to them respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 17: end for 18: end for 19: for each leaf node 푢 in the tree do 20: Compute the weight 푤푢 ← − ∑ 푖∈퐼푢 푔푖 ∑ 푖∈퐼푢 ℎ푖+휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 21: end for 22: Add the new generated decision tree to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 23: end for 24: return the generated model with all trees compute the weight of each leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' After that, add the new generated tree to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure Gradient Computation The local binning algorithm is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On each active or passive party, after getting the feature dataset {퐗푘}푀×푁푘, process a quantile binning algorithm on each column based on the sample space 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The binning result 퐗bin 푘 is encrypted and send to the cipher end, while the binning boundaries are saved to local storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The secure gradient aggregation algorithm is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For each feature 푗 and each bin 푏, find out the set of samples which appear in that bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' To preserve security, the process of dividing samples into bins is still in an encrypted computation, and the result is an encrypted 0-1 vector [flag].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' After that, aggregate the gradients with the fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 10 of 21 Short Title of the Article Algorithm 2 Local binning algorithm on Party 푘 Input: 퐼, sample space of the current tree node (a list composed of sample indexes) Input: {퐗푘}푀×푁푘, feature dataset of party 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Input: 퐵, number of bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Output: 퐗bin 푘 , binned result of party 푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푏푘,푗 (1 ≤ 푗 ≤ 푁푘), binning boundary vectors 1: 퐗bin 푘 ← {−1}푀×푁푘 2: for 1 ≤ 푗 ≤ 푁푘 do 3: Process a quantile binning on the 푗-th column of 퐗푘 based on the sample space 퐼, where the number of bins is 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Assume the vector {푥bin 푗 }푀 to be the binning result, and 푏푘,푗 is the binning boundary vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푥bin 푗 is a vector of length 푚, where the values are between 0 and 푀 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푏푘,푗 is a vector of length 퐵 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 4: for 1 ≤ 푖 ≤ 푀 do 5: if 푖 ∈ 퐼 then 6: {퐗bin 푘 }푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푗 ← {푥bin 푗 }푖 7: end if 8: end for 9: end for 10: return 퐗bin 푘 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푏푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푗 (1 ≤ 푗 ≤ 푁푘) Algorithm 3 Cipher gradient aggregation of party 푘 on the ciphertext side (for the active party,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' this algorithm can be run in pure plain text on the active party) Input: [푔푖 ] and [ℎ푖 ] (1 ≤ 푖 ≤ 푀),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' encrypted gradient vectors Input: [퐗bin 푘 ]푀×푁푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' encrypted binned result of party 푘 Output: [퐆푘 ] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [퐇푘 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' encrypted aggregated gradients for party 푘 1: [퐆푘 ] ← [0]푁푘×퐵,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [퐇푘 ] ← [0]푁푘×퐵 2: for 1 ≤ 푗 ≤ 푁푘 do 3: for 1 ≤ 푏 ≤ 퐵 do 4: for 1 ≤ 푖 ≤ 푀 do 5: [flag] ← [퐗bin 푘 ]푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푗 = 푏 6: [퐆푘]푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푏 ← [퐆푘]푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푏 + [flag] ∗ [푔푖 ] 7: [퐇푘]푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푏 ← [퐇푘]푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푏 + [flag] ∗ [ℎ푖 ] 8: end for 9: end for 10: end for 11: return [퐆푘 ] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [퐇푘 ] weight [flag].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Send the aggregated gradients [퐆푘] and [퐇푘] to the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure Split The tree node splitting algorithm is shown in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On the active party, after collecting all encrypted aggregated gradients from the cipher end, decrypt them to get 퐆푘 and 퐇푘 (1 ≤ 푘 ≤ 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then for each party, each feature and each split position, compute the information gain based on aggregated gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Unless the information gains are non-positive for all splits, find the optimal split (푘∗, 푗∗, 푠∗) with the best information gain 푣∗, and split the tree node to two child nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The sample splitting algorithm is shown in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' After party 푘∗ receives the optimal split (푗∗, 푠∗), it compares the corresponding feature values in the sample space with the binning boundary of the optimal split, and decide whether each sample should be allocated to the left or right child node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' After getting the result, return the left sample space 퐼퐿 to the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 11 of 21 Short Title of the Article Algorithm 4 Compute the optimal split point on the active party Input: 퐼, sample space of the current tree node (a list composed of sample indexes) Input: {퐆푘}푁푘×퐵, {퐇푘}푁푘×퐵, aggregated gradients from all parties 1 ≤ 푘 ≤ 푚 Input: 훾, minimum loss reduction for a split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 휆, L2 regularization term Output: 푘∗, party ID of the optimal split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푗∗, feature ID of the optimal split on party 푘∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푠∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' position of the optimal split on feature 푗∗ of party 푘∗ 1: 푘∗ ← null,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푗∗ ← null,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푠∗ ← null,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푣∗ ← 0 2: for 1 ≤ 푘 ≤ 푚 do 3: for 1 ≤ 푗 ≤ 푁푘 do 4: 퐺 ← ∑퐵 푏=1{퐆푘}푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐻 ← ∑퐵 푏=1{퐇푘}푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푏 5: 퐺퐿 ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐺푅 ← 퐺,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐻퐿 ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐻푅 ← 퐻 6: for 1 ≤ 푠 ≤ 퐵 − 1 do 7: 퐺퐿 ← 퐺퐿 + {퐆푘}푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐻퐿 ← 퐻퐿 + {퐇푘}푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='푠 8: 퐺푅 ← 퐺 − 퐺퐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 퐻푅 ← 퐻 − 퐻퐿 9: 푣 ← 1 2 ( 퐺2 퐿 퐻퐿+휆 + 퐺2 푅 퐻푅+휆 − 퐺2 퐻+휆 ) − 훾 10: if 푣 > 푣∗ then 11: 푘∗ ← 푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푗∗ ← 푗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푠∗ ← 푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푣∗ ← 푣 12: end if 13: end for 14: end for 15: end for 16: return 푘∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푗∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푠∗ Algorithm 5 Sample splitting on party 푘opt Input: 퐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' sample space of the current tree node (a list composed of sample indexes) Input: 푗∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' feature ID of the optimal split on party 푘∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푠∗, position of the optimal split on feature 푗∗ of party 푘∗ Input: {퐗푘∗}푀×푁푘∗ , feature dataset of party 푘∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Input: {푏푘∗,푗∗}퐵−1, binning boundaries of the 푗∗-th feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Output: 퐼퐿, sample space of the left child node 1: 퐼퐿 ← {} 2: for 푖 ∈ 퐼 do 3: if {퐗푘∗}푖,푗∗ ≤ {푏푘∗,푗∗}푠∗ then 4: 퐼퐿 ← 퐼퐿 ∪ {푖} 5: end if 6: end for 7: return 퐼퐿 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Security Analysis Security assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For scalability and generality, we model all the participant 푁 wind farms as data providers who agree to contribute their data for power prediction in a privacy-preserving way, and for any 푡-out-of-푛 secret sharing protocols, we further select 푛 wind farms as the computation servers to carry the ciphertext computation, where 푛 ≤ 푁 and 푡 < 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The 푡-out-of-푛 secret sharing protocol guarantees that any set of 푡 computation servers together can reconstruct the raw data while any set less then 푡 servers learns nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The computation servers are connected through secure channels, hold the secret shares of all the data providers data and execute the agreed secure protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Through this model, our method can be extended to any number wind farms with willingness to share their data and suitable for any 푡-out-of-푛 secret sharing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Similiar to other main-stream privacy-preserving applications [45, 46, 47], we define our security model as honest- majority and semi-honest model [48] for practical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The above model assumes that during the ciphertext computation, no more than a half computation servers are corrupted together (푡 < 푛 2) and the corrupted servers will fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 12 of 21 Short Title of the Article follow the agreed protocol while try to learn as much information as possible about the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Informally, a protocol is secure in the above model if the information that the corrupted servers gained is not distinguishable as there exists an ideal trusted third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Security analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The computation of our method can be separated into two parts, plaintext computation and cipher- text computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The security of ciphertext computation is guaranteed through the modular composition theorem [49], which offers a general way for designing complex high-level secure protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Firstly, we design the high-level pro- tocol by assuming that a series of simple sub-protocols exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then we design each sub-protocol meeting the security guarantee and then plug them as sub-routines in the high-level protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The modular composition theorem formally stated that, if the high-level protocol can be securely evaluated its function with ideal protocols, then the security and functionality maintained by replacing all the ideal sub-protocols with sub-routines [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For all the participant wind farms, the data providers only do plaintext computation of their own data, thus intro- ducing no privacy risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The computation servers only see the secret shares of the raw data and carry all the ciphertext computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In our method, all the cross party computation are designed to be evaluated in the ciphertext, and all the ciphertext computation logic is modular compositions of secure addition, subtraction, multiplication, comparison, reciprocal as well as exponential, which are well-studied and commonly provided by most semi-honest MPC platforms like [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Thus, the security of each wind farm’s data is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Note that he participants selection stage only utilizes 2 week history data of each wind farm to select the most similar participants in plaintext as the correlation among wind farms in a specific season is relatively stable and the MMD calculation is computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In practice, this method performs effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Case Studies The proposed privacy-preserving prediction method is tested on the field measurement data of a wind farm cluster in Inner Mongolia, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The privacy preserving machine learning algorithms are implemented on the PrivPy, a general-purpose MPC platform [19] which offers a series of secure operations based on 2-out-of-4 secret sharing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' All the evaluations are performed on a Kubernetes (k8s) cluster [50] that is deployed on two 64-core AMD EPYC CPUs with 256GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Each wind farm and computing server is deployed as a separate k8s container, and the round-trip time between each pair is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Data Set and Test Description The wind power and NWP data from Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 1st 2021 to July 23th 2021 are recorded with a time step of 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The locations of 27 wind farms are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We will use a wind farm located in the center of the chosen cluster as the target wind farm for example to illustrate the effectiveness of privacy preserving collaborative prediction model, pwXGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' To illustrate the effectiveness of the pwXGBoost, the root mean square error (RMSE) RMSE = √ √ √ √1 푘 푘 ∑ 푖=1 (푥푡푖 − ̂푥푡푖)2 (12) and the mean absolute error (MAE) MAE = 1 푘 푘 ∑ 푖=1 ||푥푡푖 − ̂푥푡푖|| (13) are used as the indexes to assess the prediction accuracy on the dataset, where 푘 is the number of the samples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푥푡푖 is the 푖 th wind power sample at time 푡 and ̂푥푡푖 is the predicted 푖 th wind power sample at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Prediction Results of the Privacy Preserving Collaborative Prediction Model The privacy preserving XGBoost model is the method proposed in this paper for the ultra-short-term wind power prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The key advantage of this method is that it can not only utilize the historical wind power and NWP data from fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 13 of 21 Short Title of the Article Figure 6: The Location of Wind Farms other wind farms without breaching the privacy, but also can extract the nonlinear spatial and temporal relationship compared to the linear method before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' To illustrate the effectiveness of the method, we compare it with the methods which only utilize the local data and those use the linear model to extract the spatial and temporal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Local_XGBoost_wo_nwp This model only uses the local historical wind power data to train the XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The max depth of the tree in the model is 3, learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 and the tree number which is the estimator of the XGBoost is 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Local_XGBoost This model only uses the local historical wind power data and NWP data to train the XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The max depth of the tree in the model is 3, learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 and the tree number which is the estimator of the XGBoost is 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Lasso_wo_nwp [8] This model can use the wind power from nearby wind farms but is the linear method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The alpha parameter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='00005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Lasso [8] This model can use the wind power and NWP data from nearby wind farms but is the linear method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The alpha parameter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='00005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' pwXGBoost_wo_nwp_mmd This model can use the historical wind power from nearby wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The nonlinear spatial and temporal relationship can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The max depth of the tree in the model is 3, learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 and the tree number which is the estimator of the XGBoost is 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' pwXGBoost_wo_mmd This model can use the historical wind power and NWP data from nearby wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The nonlinear spatial and temporal relationship can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The max depth of the tree in the model is 3, learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 and the tree number which is the estimator of the XGBoost is 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The correlated wind farms are selected based on the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' pwXGBoost This model can use the historical wind power and NWP data from nearby wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The nonlinear spatial and temporal relationship can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The max depth of the tree in the model is 3, learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 and the tree number which is the estimator of the XGBoost is 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The correlated wind farms are selected based on the fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 14 of 21 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='7 Target Wind Farm 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='6 latitude 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='6 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='8 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 longtitudeShort Title of the Article Table 1 The Prediction Results of Different Models (%) Method 1h 2h 3h 4h RMSE MAE RMSE MAE RMSE MAE RMSE MAE Local_XGBoost_wo_nwp 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='035 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='678 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='876 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='403 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='426 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='938 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='554 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='920 Local_XGBoost 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='033 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='673 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='843 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='424 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='301 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='758 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='872 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='354 Lasso_wo_nwp 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='823 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='462 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='629 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='289 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='293 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='729 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='552 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='224 Lasso 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='817 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='401 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='583 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='181 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='243 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='881 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='790 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='269 pwXGBoost_wo_nwp_mmd 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='840 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='480 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='649 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='225 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='183 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='662 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='609 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='100 pwXGBoost_wo_mmd 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='812 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='443 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='536 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='167 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='232 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='723 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='553 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='920 pwXGBoost 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='781 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='346 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='409 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='962 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='923 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='453 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='325 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='761 MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Prediction results of different models are as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to the results in Table 1, the proposed pwXGBoost is better than the linear method especially when the prediction time is longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Because when the prediction horizontal increase, the nonlinear of the wind power also increase and the superiority of pwXGBoost method is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Besides, methods that take advantage of spatial and temporal correlation patterns always performs better than those which do not exploit the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is found that MMD is more effective when selecting the correlated wind farms than using the wind farm distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We also visualize the error density of the prediction model for the 4th hour in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Figure 7: Probability distribution of forecasting errors fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 15 of 21 pwXGBoost pwXGBoost_wo_mmd 20 20 - 15 15 10 10- 5 5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 Lasso Local_XGBoost 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5 12 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 Isu e 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5 D 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 Short Title of the Article In Figure 7, the prediction error is plotted in hist gram and fitted by kernel density estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is proved that the prediction error of the pwXGBoost is more centralized around zero which means it has better prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Analysis of the Privacy Preserving Prediction Results The prediction results of pwXGBoost, Lasso and Local_XGBoost are compared in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In this test, it is obvious that the wind power prediction method which uses the information from nearby wind farms performs better than the method only uses the local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Figure 8: The Comparison of Different Prediction Results According to the prediction results in Figure 8, we can also see that the pwXGBoost model is better than the linear method especially in scenarios with rapid changes in wind power which is crucial for the safe operation of power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For the linear Lasso method, although it is easy to be implemented, its ability to capture the trend and extreme values is inferior to the nonlinear pwXGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The Correlation Analysis of the Wind Farms In this section, we demonstrate the effectiveness of the participant selection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The MMD distances between each pair of wind farms are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' We adjusted the value of the 훽 (the threshold of Equation 11), the wind farms can be divided into different groups and different number wind farms will be selected as the participant wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Larger 훽 means more wind farms will be selected as the participant wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The RMSE under different 훽 is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The results show that when the threshold is in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='7 and 1, the number of selected wind farms are the same and the RMSE is the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' When the threshold is set too low (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='7), fewer wind farms will participate, and spatial and temporal correlation information is also constrained, resulting in an increase in the RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' On the other hand, if the threshold is set too high (> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='0), the various participating wind farms may introduce too diverse distributions, also increasing the RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 16 of 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='25 Local XGBoost Lasso pwXGboost true value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='15 wind power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='05 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='2 714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Scalability and Computation Cost The training time and prediction time of the privacy preserving ultra-short-term wind power prediction model is also important for the practical appplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If it consumes a lot of time when training and prediction, it is not applicable and acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, we test the consuming time of the privacy preserving XGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' From Table 2, it can be seen that even when all the 27 wind farms are included in the training and prediction process, the training time is less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='5h and the prediction time is less than 12min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In our prediction process, the 12 wind farms are selected based on the MMD method, the training time are reduced to less than 53 minutes and the prediction time are less than 400s, which is acceptable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Conclusion Exploiting spatial and temporal correlation is useful to improve the accuracy of the ultra-short-term wind power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Taking data security and regulation policy issues into account, this paper proposes a privacy-preserving ultra-short- term wind power prediction method pwXGBoost based on secret sharing protocol and carefully chosen collaborative neighboring wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The test results on the Inner Mongolian data set proved the ability of nonlinear feature extraction compared to the linear model and the data security is guaranteed mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' According to the case study results, the RMSE of the method which uses the selected wind farms to train the prediction model is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='4% lower than that of the method which only uses the local wind farm data in the 4 − 푡ℎ hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the meanwhile, the prediction time of privacy preserving method is less than 5min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Therefore, this method is acceptable for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Besides, the secure multi party computation theory is scalable to the other nonlinear operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Since neural network is another widely used method in wind farm power prediction, we can also build neural network based on secure multi party computation theory and secret share protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the future, we will adopted the secret sharing protocol to the neural network and designed the privacy preserving wind power prediction method based on neural network to test its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure Multi Party Computation Secure Multi Party Computation (MPC) has a long history in the cryptography community, it enables a group of data providers to jointly compute an agreed function without disclosing their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' One of the most fundamental building blocks of MPC is secret sharing, which is the basis of most current MPC platforms [20, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푡-out-of-푛 secret sharing splits a secret input 푥 into 푛 shares, satisfying that any 푡 shares can completely reconstruct 푥 while any shares less than 푡 reveal nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For example, a commonly-used 2-out-of-3 secret sharing protocol splits 푥 into [푥] = {[푥]1, [푥]2, [푥]3} = {(푥1, 푥2), (푥2, 푥3), (푥3, 푥1)} and let three computation servers hold the shares, respectively [20, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' [푥] satisfy that 푥 ≡ ∑푗=3 푗=1 푥푗( mod 푀) where 푀 is a large integer, usually 2푘, making 푥푗, 푖 ∈ 1, 2, 3 uniformly distributed in a ring of ℤ2푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is obvious that any two servers (or more) can together construct 푥 while each single server learns nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' To make MPC general (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', to support arbitrary functions), researchers and engineers have proposed so-called general-purpose MPC platforms, which offers a series of basic secure operations like secure addition, subtraction, multiplication, comparison which can be composed together to support complex functions like square-root and division and more advanced functions like machine-learning functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', secure principal component analysis [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' All the fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 18 of 21 Short Title of the Article secure basic operations are cryptographic protocols among the computation servers and preserve the privacy of the secret input 푥 during the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The computation process of these secret-sharing based MPC platforms has three stages:initialization stage, computation stage and reveal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secret share initialization stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For 푁 data providers, they encode the original data 푥(푖), 푖 ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푁 into the secret share [푥(푖)] according to the specific 푡-out-of-푛 secret sharing protocol and pass the each share to the target computation server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the end of this stage, each computation server 푠푗, 푗 ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푛 will hold 푁 secret shares of the all the data providers, {[푥(푖)]푗}푖=푁 푖=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the above 2-out-of-3 secret sharing protocol, the three servers will hold {(푥(푖) 1 , 푥(푖) 2 )}푖=푁 푖=1 , {(푥(푖) 2 , 푥(푖) 3 )}푖=푁 푖=1 , {(푥(푖) 1 , 푥(푖) 3 )}푖=푁 푖=1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Computation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For any valid function 푓(푥푖, 푖 ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푛) that all the data providers agreed, it will be constructed as the modular composition of the basic secure operations, which means that in the end of each secure operation, the computation servers will hold the secret shares of the corresponding result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For example, after the evaluation of secure addition between secret shares [푥] and [푦], the computation servers should hold the secret share [푧] where 푧 = 푥+푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In this stage, all the 푛 computation servers will evaluate the composed secure operations sequentially for the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Reveal stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' After the computation stage of 푓, the computation servers will hold the secret shares of the final result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', [푓(푥푖, 푖 ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푛)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In this final stage, all the computation servers will pass its share to one party (predefined in the beginning) to reconstruct 푓(푥푖, 푖 ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In fact, the reveal stage can based on the blockchain technology to regulate the information exchange and its profit allocation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure Operations for Secret Sharing Based MPC Platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In this section, we first introduce some basic secure operations briefly, showing how the secret sharing compu- tation works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the end, we introduce secure division operation to demonstrate how to compose basic operations for complex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' For two data 푥 and 푦, their secret shares are [푥] and [푦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' It is self-evidently additive homomorphic because [푥] + [푦] = [푥 + 푦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Each computation server can locally compute the share of the sum in the computation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' In the secret sharing computation, addition and subtraction are equivalent, as 푥 minus 푦 is equivalent to 푥 plus the opposite of 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Servers can firstly construct the share of [−푦] locally by computing the opposite of its own shares as −푦 ≡ ∑푖=푛 푖=1 −푦푖( mod 푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Then compute [푥 − 푦] through [푥] + [−푦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Computing multiplication usually requires communications among the computation servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Take the above 2-out-of-3 secret sharing and ABY3 [51] multiplication protocol for example, defining 푧 = 푥푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' As 푥 = (푥1 + 푥2 + 푥3)(푦1 + 푦2 + 푦3), we can define [푧] as 푧1 = 푥1푦1 + 푥1푦2 + 푥2푦1 + 훼1, 푧2 = 푥2푦2 + 푥2푦3 + 푥3푦2 + 훼2 and 푧3 = 푥3푦3 + 푥3푦1 + 푥1푦3 + 훼3 where ∑푗=3 푗=1 훼푗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Each computation server 푗 firstly locally compute its share 푧푗, then communicate its share to the previous server for valid secret shares (as we require each server hold two shares from {푧1, 푧2, 푧3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' When two numbers 푥 and 푦 are compared, 푥 < 푦 equals a Boolean indicating whether (푥 − 푦) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' So the computation servers can firstly compute [푥 − 푦] based on the secure subtraction protocol and then use the bit extraction protocol [36] to securely extract the sign bit of [푥 − 푦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' If the highest bit is 1, it means that 푥 − 푦 is negative and therefore 푥 < 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Secure division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Comparing with other basic operations, computing non-linear functions like secure division is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' As [푥] divides [푦] is equivalent to [푥] multiply [ 1 푦], how to compute the reciprocal of [푦] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=', [ 1 푦]) is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' As 1 푦 is the solution of Equation 14, it is obvious that the solution 푥 = 1 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' 푓(푥) = 1 푥 − 푦 = 0 (14) fanhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 19 of 21 Short Title of the Article As we know, the Taylor expansion of 푓(푥) is as follows: 푓(푥) = 푓(푥0) + 푓 ′(푥0) 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' (푥 − 푥0) + 푓 ′′(푥0) 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' (푥 − 푥0)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' + 푓 푛(푥0) 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' (푥 − 푥0)푛 (15) The first order approximation of 푓(푥) is used to get the solution which is the Newton-Raphson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' The approximate solution can be calculated as follows: 푥푛+1 = 푥푛 − 푓(푥푛) 푓 ′(푥푛) = 2푥푛 − 푎푥2 푛 (16) Thus, we can compose a series of secure subtraction and multiplication to update 푥푛+1 till it converges to the expected reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' References [1] Guannan He, Qixin Chen, Chongqing Kang, Qing Xia, and Kameshwar Poolla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFRT4oBgHgl3EQfOTea/content/2301.13513v1.pdf'} +page_content=' Cooperation of wind power and battery storage to provide frequency regulation in power markets.' metadata={'source': 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+{daniel007, haiz0001}@e.ntu.edu.sg, jf.carmona658@uniandes.edu.co, mano@comp.nus.edu.sg, gaocong@ntu.edu.sg +January 31, 2023 +Abstract +In text mining, topic models are a type of probabilistic gen- +erative models for inferring latent semantic topics from text +corpus. One of the most popular inference approaches to +topic models is perhaps collapsed Gibbs sampling (CGS), +which typically samples one single topic label for each ob- +served document-word pair. In this paper, we aim at im- +proving the inference of CGS for topic models. We pro- +pose to leverage state augmentation technique by maximiz- +ing the number of topic samples to infinity, and then develop +a new inference approach, called infinite latent state replica- +tion (ILR), to generate robust soft topic assignment for each +given document-word pair. Experimental results on the pub- +licly available datasets show that ILR outperforms CGS for +inference of existing established topic models. +1 +Introduction +In text mining, probabilistic topic models, such as latent +Dirichlet allocation (LDA) [1], refer to generative statisti- +cal algorithms for mining latent semantic structure of a set +of text documents. Aside from their increasing popularity +and ubiquitous adoption in text data processing, they have +been also applied to the fields such as computer vision [2] +and geographical modeling [3]. Recently, widespread appli- +cability has spurred research on developing more accurate or +efficient inference approaches to topic modeling. +Existing inference methods can be roughly grouped into +optimization and sampling categories. One of the represen- +tative optimization based approaches is the mean field vari- +ational inference (VI) [4]. VI frames the inference process +as an optimization problem, where the posterior is approxi- +mated by fitting a selected family of distributions. Unfortu- +nately, the predictive ability of VI is somewhat sacrificed for +computational efficiency [5]. On the other hand, one of the +most popular sampling based inference approaches is per- +haps the collapsed Gibbs sampling (CGS). CGS has been +shown effective and widely used to infer latent topic models +[6]. In this work, our focus is on improving the generaliza- +tion performance of CGS for inference of topic models. +Particularly, in the inference of CGS for LDA, for +each observed document-word pair, it estimates a posterior +distribution conditioned on the topic assignments of all the +other pairs, and relies on the distribution to sample one single +topic for the given pair. The model’s parameters can be then +estimated based on the sample data. Intuitively, instead of +sampling one exact topic assignment, we may yield a flexible +and more robust learning approach by sampling multiple +topics and then combining the topic samples to derive soft +topic assignment for each document-word pair. +Considering a basic situation where two samples are +drawn for the given pair, the two samples correspond to two +topic assignments, and combining the two samples would +lead to flexible search for optimal assignment by using the +two branches simultaneously. Then, as the number of sam- +ples increases, the parameter estimation would be done by +relying on more and more evidence from all of the samples. +This process is known as state augmentation for marginal +estimation. It was previously studied for improving max- +imum a posteriori estimation using Markov Chain Monte +Carlo method (MCMC) [7]. +We extend the idea, and propose to leverage the benefit +of maximizing the state augmentation by raising the number +of samples to infinity. +Then, we develop a new method, +called infinite latent state replication (ILR), to improve the +inference of topic models within Gibbs sampling framework. +As a matter of fact, the computational complexity of the +inference model often increases linearly with the number of +samples. To deal with this issue, we employ the strong law of +large numbers, and present a tractable optimization solution +that achieves the same computational complexity of CGS. +In contrast to collapsed Gibbs sampling, ILR leverages +the aggregation of whole population to estimate the density +of the posterior distribution. Thus, sampling is not required +in ILR, and the resulting inference algorithm is deterministic. +arXiv:2301.12974v1 [cs.CL] 25 Jan 2023 + +Thanks to the determinism of ILR, its convergence can be +assessed directly by checking when the parameters of the +posterior distribution stop changing. In contrast, the exact +iteration at which CGS converges is unknown, and additional +mechanisms may be required to assess its convergence. +Next, we apply ILR to inference of LDA, and observe a +significant improvement over CGS in predictive perplexity. +In addition, ILR can be also applied to inference of other +topic models that characterize the dependency between two +latent random variables, e.g., dual topic model (DTM) [8]. +This work has made the following main contributions: +• Based on Gibbs sampling framework, we develop ILR +to estimate the posterior for inference of topic models. +ILR replaces the exact assignments of latent topics by +a density given by drawing multiple topic assignments. +It is thus a less constrained learning method compared +to CGS, and may have flexibility and better chance to +improve the generalization performance for inference. +• The replication of topic assignments often results in lin- +ear increase in computational complexity. To tackle this +issue, we maximize the number of replications to infin- +ity, and transform the Gibbs sampling algorithm into a +deterministic method, whose computational complexity +is equivalent to standard Gibbs sampling algorithm. +• We conduct extensive experiments on the publicly +available benchmark datasets, and experimental results +validate the improved effectiveness of ILR over CGS +for inference of existing well-established topic models. +2 +Related Work +Along the rich history of Bayesian models, several inference +algorithms have been proposed, such as variational Bayesian +inference [1], expectation propagation [9], collapsed Gibbs +sampling [6], and belief propagation [10]. +Among these +algorithms, collapsed Gibbs sampling and variational infer- +ence are perhaps the most widely used methods due to their +effectiveness. The former, which employs a sampling strat- +egy, is known for guaranteeing the convergence to true target +posterior, while the latter, which is a theoretically backed op- +timization method, approximates the true posterior. +As far as we know, existing extensions on CGS are pri- +marily concerned with improving runtime. This is natural +given that CGS tends to result in better predictive perfor- +mance than other inference methods, but it is computation- +ally expensive. FastLDA was shown to be eight times faster +than CGS while maintaining the comparable performance +[11]. The high computational cost of CGS stems from the +O(K) time complexity incurred at every sampling step (K: +topic number). It is observed that, even given a good many +topics, words or especially infrequent words are often as- +signed only to a few topics. Then, by keeping track of these +words, FastLDA requires significantly less than K opera- +tions per sample on average. SparseLDA factorizes the pos- +terior equation into a sum of three factors. The sampling +scheme is then replaced by a uniform sampling strategy, +where the probability mass falls in one of the buckets in 90% +of the time [12]. By using an appropriate data structure, the +computation for the mass can be optimized. As a result, the +model attains an order of magnitude improvement in com- +putational complexity without affecting its predictive power +compared to CGS. Recently, AliasLDA [13] and LightLDA +[14] are able to reduce the O(K) complexity of sampling +by an efficient sampling method from discrete random vari- +ables. In the latter case, a Metropolis-Hasting step allows +sampling a topic assignment for a document-word pair in +O(1) via Walker’s alias method [15]. Then, WarpLDA im- +proved LightLDA by exploiting the memory access behavior +of other algorithms for LDA [16]. +In contrast to sampling-based inference methods, exist- +ing work on variational inference approaches mainly aims to +improve the predictive performance. One reason which may +illustrate the weakness of these methods perhaps lies in the +approximation to intractable integrals. Collapsed variational +Bayesian schemes [17] use a second-order Taylor expansion +to approximate the integrals. In addition, the zero-order in- +formation can be also used in the variational methods, and +may result in improved inference [18]. +In this work, rather than targeting the efficiency of CGS, +we aim to improve the predictive ability of CGS for inference +of latent topic models. To achieve that, we propose to lever- +age a state augmentation technique, which normally aug- +ments the probabilistic model by replicating multiple times +the latent state random variables. The technique has been +previously shown effective for improving the marginal max- +imum a posteriori estimation via the Markov chain Monte +Carlo [7]. Moreover, we show that it is possible to maxi- +mize the benefit of the state augmentation technique by rais- +ing the number of replications up to infinity. In this way, we +introduce a new deterministic algorithm to tackle the infer- +ence of topic models. Our method can lead to improved pre- +dictive performance, while it maintains equivalent computa- +tional complexity compared to the standard collapsed Gibbs +sampling algorithm. +3 +Preliminaries +Latent Dirichlet allocation (LDA), one of the most common +topic models, is able to discover K topics that pervade an un- +structured collection of documents. In LDA, we assume that +each document has a multinomial distribution over K topics +parameterized by θ. We use θd as the set of parameters for +representing the probability distribution over the set of topics +given a document d. Each topic is defined as a multinomial +distribution over the vocabulary of words parameterized by +φ. We use φk as the set of parameters of the probability over +the set of words given a topic k. Per document topic distribu- + +tion θd is generated by sampling from a Dirichlet distribution +with hyper-parameter α. Similarly, φk is generated by sam- +pling from a Dirichlet distribution with hyper-parameter β. +In latent Dirichlet allocation via collapsed Gibbs sam- +pling (LDA-CGS), to find latent topics, we use a dataset +X that consists of M observed document-word pairs xj = +(dj, wj), ∀j ∈ 1 · · · M. where |D|, and |W| are the number +of documents and the vocabulary size. Each pair is assigned +exactly to one of the topics as established by the assumption +of the model. For each jth pair, the topic assignment is gen- +erated by sampling the conditional posterior distribution of +Zj given the topic assignments of all other pairs Z−j = z−j. +Formally, we represent the conditional posterior distribution +as p(Zj|Z−j = z−j, X). The probability density of the pos- +terior distribution is given in Equation 3.1. +(3.1) +f(Zj|Z−j = z−j) ∝ (Ddj + α)(Wwj + β) +(Nj + V β) +We use the subscript Zj|Z−j to denote the density of the +posterior distribution p(Zj|Z−j, X). The Ddj, Wwj, and Nj +are count vectors computed using the indicator function 1(): +Ddj = +M +� +m=1 +m̸=j +1(dj = dm) ∗ ¯ +zm ; Wwj = +M +� +m=1 +m̸=j +1(wj = wm) ∗ ¯ +zm +Nj = +M +� +m=1 +m̸=j +¯ +zm +(3.2) +where ¯ +zm refers to the realization of the random variable +Zm. We use the bar notation to represent a one-hot encoding +representation. +Hence, Ddj denotes the vector of topic +assignments for document dj, Wwj denotes the vector of +topic assignments for word token wj, and Nj represents the +vector of topic assignments over the entire set of document- +word pairs. This representation facilitates the comparison of +Equation 3.1 with the posterior density proposed in Section +4. +The collapsed Gibbs sampling approach to LDA (LDA- +CGS) is summarized in Algorithm 1. Note that, at iteration +i, the computation for the posterior density in Line 5 not +only depends on the topic assignments of all the pairs whose +index values are smaller than j, as denoted by zi +j . +Thus, we denote the density at the ith iteration for the jth +record to be f(Zi +j|Zi +j = zi−1 +>j ). +The convergence of LDA-CGS can be obtained when +the topic assignments zj for each pair are drawn from the +marginal distributions p(Zj). +Unfortunately, since we do +not have an analytical form for p(Zj), we cannot measure +exactly at which iteration the samples are drawn from p(Zj). +Algorithm 1 Collapsed Gibbs Sampling for LDA +1: Input: Hyper-parameters K, α, β, dataset X +2: Initialize z0 +j using a uniform distribution, and set initial +iteration i = 1 +3: repeat +4: +for j = 1 · · · M do +5: +Compute f(Zi +j|Zi +j += zi−1 +>j ) from +Equation3.1 +6: +Draw zi +j from f(Zi +j|Zi +j = zi−1 +>j ) +7: +i = i + 1 +8: until Convergence +9: Collect samples and estimate ˆθ and ˆφ as shown in [6] +10: return ˆθ, ˆφ +4 +Infinite Latent State Replication Inference +When applying collapsed Gibbs sampling to LDA, for +the jth document-word pair, we compute the posterior +p(Zj|Z−j = z−j) by using the samples z−j obtained for +all other pairs. Then, we draw ONE single sample zj from +this distribution, and continue to compute the posterior for +the next document-word pair p(Zj+1|Z−(j+1) = z−(j+1)). +In collapsed Gibbs sampling, the posterior distributions are +updated until the convergence is reached. Table 1 shows a +toy example of applying collapsed Gibbs sampling to infer- +ence of LDA, where the number of topics is K = 3. For each +of M observed document-word pairs (dj, wj), a topic label +(1, 2, or 3) is assigned by relying on the posterior probability +distribution (last column). +Table 1: A toy example of applying collapsed Gibbs sampling to +inference of LDA (K = 3) +j +(dj, wj) +zj +Posterior prob. +1 +(d1, w1) +2 +[ 0.2 0.3 0.5 ] +... +... +... +... +M +(dM, wM) +2 +[ 0.2 0.4 0.4] +In contrast, the replication of latent states augments the +collapsed Gibbs sampling algorithm by repeatedly drawing +R samples from each posterior distribution to compute the +newly updated posterior distributions. +The replication of +latent space has been previously studied to improve the +parameter estimate in maximum a posteriori estimation by +using Markov Chain Monte Carlo estimations (MCMC). In +MCMC, the replications of the latent space actually provide +more evidence about the search path, which can lead to a +more robust exploration of the parameter space [7]. +To our knowledge, the replication of latent states +does not affect the global optima of the original model. +The augmented model with R replications described by +p′(Θ, Z(1), ..Z(R), X) is the Rth power of the model that + +does not use replications [19], as shown below. +(4.3) +p′(Θ, Z(1), ..Z(R), X) ∝ +R +� +r=1 +p(Θ, Z(r), X) +In latent replication inference, the replications are de- +fined as multiple (R) copies of the latent variables: +Z(r) +j += 1 +RZj, ∀j ∈ 1 · · · M, ∀r ∈ 1 · · · R +(4.4) +A linear transformation can be adopted to maintain the +expected value of the random variables Zj, ∀j ∈ 1 · · · M, +without affecting the optima of the model. However, this +transformation is limited to models, where the latent random +variable follows a multinomial distribution. +Table 2: +A toy example of applying R += 4 replications in +collapsed Gibbs sampling to inference of LDA (K = 3). The topic +assignment is performed repeatedly for R times for each observed +document-word pair (separated by dotted lines). +j +(dj, wj) +z(r) +j +Posterior prob. (Equation 4.5) +1 +(d1, w1) +2 +[ 0.22 0.28 0.5 ] +1 +(d1, w1) +3 +[ 0.22 0.28 0.5 ] +1 +(d1, w1) +1 +[ 0.22 0.28 0.5 ] +1 +(d1, w1) +3 +[ 0.22 0.28 0.5 ] +... +... +... +... +M +(dM, wM) +2 +[ 0.2 0.39 0.41] +M +(dM, wM) +3 +[ 0.2 0.39 0.41] +M +(dM, wM) +2 +[ 0.2 0.39 0.41] +M +(dM, wM) +3 +[ 0.2 0.39 0.41] +We then employ multiple replications of latent states +in collapsed Gibbs sampling for inference of LDA. For- +mally, in CGS, we draw one single sample from each +posterior, and then the computation of the jth posterior, +p(Zj|Z−j=z−j), ∀j ∈ 1..M, relies on ONE point estimate +from each of the other posteriors, i.e., zi, ∀i ∈ 1..M, i ̸= j. +In contrast, in latent state replication inference, we draw +multiple (R) samples from each posterior, and then the com- +putation of the jth posterior would leverage MULTIPLE point +estimates from other posteriors, i.e., z(r) +i +, ∀i ∈ 1..M, ∀r ∈ +R, i ̸= j. Therefore, the combination of multiple point esti- +mates for a posterior serves to approximate the density of the +posterior. Then, taking into account the replicas, the jth pos- +terior corresponds to: P(Zj|Z(1) +−j =z(1) +−j , · · · , Z(R) +−j =z(R) +−j ), +where R is the number of replications for estimates. +Table 2 shows a toy example of applying R = 4 replica- +tions of latent states in collapsed Gibbs sampling to inference +of LDA (K = 3). Conventionally, for a given document- +word pair, e.g., (d1, w1), CGS simply uses one sample, e.g., +z1 +1 = 2, as the estimate of the categorical posterior distri- +bution [0.220.280.5], which estimates the posterior using a +histogram with proportions given by [0.001.000.00]. In con- +trast, the state replication inference uses R = 4 point esti- +mates which can be aggregated in a histogram with propor- +tions given by [0.250.250.5] (See the first group in Table 2). +Clearly, the estimate based on the four replications approx- +imates the posterior [0.220.280.5] with higher fidelity. This +may suggest that the replication of latent states empowers the +inference process with a more informative path, which can +be exploited by Gibbs chain to converge to a better optima. +Intuitively, while CGS uses a hard topic assignment given by +the point estimate, the state replication inference leverages a +soft topic assignment, i.e., splitting the assignment along the +K coordinates of the latent space. +p′(Zj|Z(1) +−j ..Z(R) +−j , X) ∝ +� � +p′(θ, φ, Z(r) +−j Z(R), X) dθ dφ += +|D| +� +m +� +Γ(� +k αk) +�K +k Γ(αk) +K +� +k +θ +αk−1+ 1 +R +�R +r ck(r),m,∗ +m,k +dθm× +K +� +k +� +Γ(� +j βj) +�K +k Γ(βk) +|W | +� +j +φ +βk−1+ 1 +R +�R +r ck(r),m,∗ +k,j +dφk += +|D| +� +m +�K +k Γ( 1 +R +� +r ck(r),m,∗ + αk) +Γ(� +k +1 +R +� +r ck(r),m,∗ + αk) × +K +� +k +�|W | +j +Γ( 1 +R +� +r ck(r),∗,j + βj) +Γ(� +j +1 +R +� +r cj(r),∗,j + βj) +∝ +�� +r c−(a,b) +z(r) +a,b,a,∗ + αz(r) +a,b +� +× +�� +r c−(a,b) +z(r) +a,b,∗,ya,b + βya,b +� +� +r c−(a,b) +z(r) +a,b,∗,∗ + � +j βj +(4.5) +It is worth noting that, when applying latent state repli- +cations within CGS framework, the posterior probability is +the same for each replica of a given document-word pair. If +we employ R latent replications for inference, the posterior +probability can be computed by following the steps shown in +Equation 4.5, some relevant counts are defined as follows. +ck′,d′,w′ = +M +� +m=1 +1(zm = k′ ∧ dm = d′ ∧ wm = w′) +c−j +k′,d′,w′ = +M +� +m=1,m̸=j +1(zm = k′ ∧ dm = d′ ∧ wm = w′) +To facilitate the comparison between standard model +and state replication model, Equation 4.5 may be rewritten +in the same form as Equation 3.2: +Ddj = +R +� +r +M +� +m=1 +m̸=j +1(dj = x(d) +m ) ∗ z(r) +m +R +(4.6) +Wwj = +R +� +r +M +� +m=1 +m̸=j +1(wj = x(w) +m ) ∗ z(r) +m +R +; Nj = +R +� +r +M +� +m=1 +m̸=j +z(r) +m +R + +Previous studies have shown that the predictive power +of applying latent state replication to parameter estimation +can be improved with increasing the number of replications +[19], [7]. Thus, we propose to maximize the number of latent +state replications to improve the inference of LDA. As the +number of replications tends toward infinity, in terms of the +law of large numbers, we can infer that the proportion of +the replications obtained by using a categorical distribution +will converge to the probability mass of the categorical +distribution. Mathematically, for all k ∈ 1..K: +(4.7) +lim +R→∞ +1 +R +R +� +r=1 +1(zr=k) = p′(Zj = k|Z(1) +−j ..Z(R) +−j , X) = κk +j , +where κk +j is one of the K parameters of the posterior distri- +bution. We define κj as the vector that holds the parameters +of the posterior κj = [κ1 +j, ..κK +j ]. Since the posterior is a +categorical distribution, this vector corresponds to the prob- +ability mass function of the posterior distribution. +The update of the Gibbs sampler for infinite latent +replications can be then obtained by raising R to infinity in +Equation 4, and the result is shown in Equation 4.8. +Ddj= +M +� +m=1 +m̸=j +1(dj = x(d) +m ) ∗ κk +m ; Wwj= +M +� +m=1 +m̸=j +1(wj = x(w) +m ) ∗ κk +m +Nj = +M +� +m=1 +m̸=j +κk +m +(4.8) +Generally, from a generative perspective, it is computa- +tionally unfeasible to sample an infinite number of times. As +a matter of fact, by applying the law of large numbers, we +can obtain updates that use vector additions without the need +for sampling. A comparison between Table 1 and 2 shows +that the topic assignment in state replication inference does +not correspond to a single label, but instead, corresponds to +a mixture of labels. +As the number of assigned labels (replications) in- +creases towards infinity, the presented updates drive the +Gibbs chain to convergence by using the whole probability +mass from all of the M posterior distributions. As a con- +sequence, the parameters of the inference algorithm are not +estimated from samples, but instead are obtained directly +by using Equation 4. Different from sampling algorithms, +Equation 4 will yield the same result on every iteration after +convergence. +The deterministic inference driven by the soft topic as- +signment mechanism allows for better exploration at the op- +timal region than the estimations based on random samples. +As a result, ILR achieves significant improvements in gener- +alization performance in terms of predictive perplexity (See +Section 6). +As for computational cost, the parameter updates for +each of the document-word pairs has a time complexity of +O(K), which achieves the same cost as collapsed Gibbs +sampler. This is actually not a big issue, as we focus on the +improvement of predictive power of CGS under a theoretical +framework. +Moreover, we study the applicability of the proposed in- +finite latent replication inference to topic models that char- +acterize dependency between latent random variables. +5 +Applicability to Inference of Dual Topic Model +The dual topic model (DTM) takes as input a co-occurrence +data matrix that underlies the inter-relationship between row +and column variables, e.g., user-location matrix where each +entry means the number of times a user (row) visits a +location (column) [8]. Then, it identifies the row topics Z +or distribution over the rows of the co-occurrence matrix, +the column topics Y or distribution over the columns of the +matrix, and the joint distribution of both column and row +topics. +The collapsed Gibbs sampling inference for the jth row- +column pair is given by Equation 5.9: +f(Zj, Yj|Z−j=z−j, Y−j=Y−j) ∝ +(Ddj + βr)⊺(Wwj + βc) +(N r +j + Rβr)⊺(N c +j + Cβc) ⊙ (Pj + α), +(5.9) +where R and C are the numbers of rows and columns in the +input matrix respectively. βr and βc are hyper-parameters +used to generate Z and Y respectively, and the following are +count vectors computed using the indicator function 1(): +Ddj = +M +� +m=1 +m̸=j +1(dj = x(d) +m ) ∗ ¯ +zm ; Wwj = +M +� +m=1 +m̸=j +1(wj = x(w) +m ) ∗ ¯ +ym +Pj= +M +� +m=1 +m̸=j +¯ +zm +⊺. ¯ +ym ; N r +j = +M +� +m=1 +m̸=j +¯ +zm ; N c +j = +M +� +m=1 +m̸=j +¯ +ym +(5.10) +Note that +¯ +zmT . ¯ +ym is a convenient representation for +encoding one unit to a matrix, where rows have the same +dimension as ¯ +zm, and the columns have the same dimension +as ¯ +ym. The ⊙ refers to element-wise multiplication. +Table 3 shows a toy example of applying collapsed +Gibbs sampling with latent state replications to inference of +DTM, where the numbers of latent row and column topics +are Kr = 2 and Kc = 2, respectively. +According to the law of large numbers, the proportion of +the joint pairs obtained by the replicas z(r) +j +and z(r) +j +would +converge to the value of the parameters of joint posterior +probability. For all kr ∈ 1..Kr, ∀kc ∈ 1..Kc: +lim +R→∞ +1 +R +R +� +r=1 +1(zr=kr, yr=kc) = +(5.11) +p(Zj=kr, Yj=kc|Z(1) +−j , Y (1) +−j ..Z(R) +−j , Y (R) +−j )=κkr,kc +j + +Table 3: A toy example of applying collapsed Gibbs sampling with +latent state replications to inference of DTM (Kr = 2, Kc = 2). +j +(dj, wj) +z(r) +j +z(r) +j +Posterior prob. +1 +(d1, w1) +1 +1 +� 0.5 +0 +0 +0.5 +� +1 +(d1, w1) +2 +2 +� 0.5 +0 +0 +0.5 +� +1 +(d1, w1) +2 +2 +� 0.5 +0 +0 +0.5 +� +1 +(d1, w1) +1 +1 +� 0.5 +0 +0 +0.5 +� +... +... +... +... +M +(dM, wM) +1 +2 +� 0.2 +0.3 +0.4 +0.1 +� +M +(dM, wM) +2 +1 +� 0.2 +0.3 +0.4 +0.1 +� +M +(dM, wM) +1 +2 +� 0.2 +0.3 +0.4 +0.1 +� +M +(dM, wM) +1 +1 +� 0.2 +0.3 +0.4 +0.1 +� +The κ corresponds to a matrix that holds the parame- +ters of the bivariate categorical distribution. Following the +same procedure as in LDA, we first derive the case for R +replications, and then drive R up to infinity to derive the ILR +inference method applied to DTM. The resulting factors are +shown as follows. +Ddj = +M +� +m=1 +m̸=j +1(dj = x(d) +m ) ∗ κm ; Wwj = +M +� +m=1 +m̸=j +1(wj = x(w) +m ) ∗ κm +Pj= +M +� +m=1 +m̸=j +κm ; N r +j = +M +� +m=1 +m̸=j +κm ; N c +j = +M +� +m=1 +m̸=j +κm +(5.12) +Note that for DTM, the factors are computed from +matrix additions in contrast to the vector additions found for +LDA. +6 +Experiments +6.1 +Datasets We used three publicly available datasets to +evaluate the proposed method ILR for inference of topic +models, i.e., NIPS:1[20], Lastfm 2, and Movielens 3 datasets. +The NIPS dataset comes from the Neural Information Pro- +cessing Systems proceedings from 1987 to 2015. The dataset +consists of 11,040,357 records with 11,463 words, which are +generated by 5,811 authors. The Lastfm dataset contains +1https://archive.ics.uci.edu/ml/datasets/NIPS+ +Conference+Papers+1987-2015 +2http://www.dtic.upf.edu/˜ocelma/ +MusicRecommendationDataset/lastfm-360K.html +3https://grouplens.org/datasets/movielens/ +145,534,518 records, which are generated based on the tu- +ples (user, artist, plays) of 360,000 users. Movielens is a +popular benchmark dataset for movie rating prediction and +recommendation. We used the Movielens 1M dataset that +consists of 445,094 ratings generated by the most popular +1,223 users on 1,214 movies. +We conducted three types of experiments on the +datasets, i.e., predictive perplexity, coherence of latent topic +detection, and sensitivity of inference to hyper-parameter +setting. +6.2 +Predictive Perplexity Results Predictive perplexity is +widely used to evaluate the generalization ability of a learn- +ing algorithm. In this section, we use the perplexity metric to +evaluate the proposed method ILR for inference of two well- +established topic models LDA and DTM, and compare it +with the most commonly used inference algorithm collapsed +Gibbs sampling (CGS). +6.2.1 +Predictive Perplexity of LDA We first evaluate the +performance of inference of LDA by using ILR against +CGS. In this experiment, the number of topics K was tuned +amongst {5, 10, 20, 50}. The hyper-parameter α was set as +0.5, while β was empirically determined to be 0.9, 0.8, and +0.5 on NIPS, Lastfm, and Movielens, respectively. We held +out 40% of each dataset for testing, and run the two inference +methods for 100 iterations. We report the average perplexity +results on five trials under the setting. +Figure 1 shows the predictive perplexity results of LDA +inferred by ILR and CGS on the Lastfm, NIPS, and Movie- +lens datasets, respectively (The lower, the better). Overall, +the proposed method ILR (solid line) clearly outperforms +CGS (dashed line) for inference of LDA across all the values +of K for each of the three datasets. Note that both inference +methods use almost same time for running. +In addition, there is an exception for ILR given K = +50 on the NIPS dataset. +As the iterations progress, the +perplexity of LDA via ILR drops sharply, and then increases +gradually. This is perhaps due to the choice of K being too +large for ILR to learn on this dataset. This case has been also +found by Blei et al. [21], when they evaluated the predictive +power of topic models versus the parameter K. The finding +may suggest that appropriate choice of hyper-parameters is +advised. In practice, we can employ cross-validation on a +small development dataset to find the suitable values of the +hyper-parameters. +We conclude that, given appropriate choice of hyper- +parameters, ILR consistently outperforms CGS for inference +of LDA in terms of predictive capability. In addition, the +speed of convergence of both algorithms are comparable, +however, ILR typically tends to lower bound the perplexity +curves of CGS in most cases. +We also observe that our +method shows a monotonic decrease in the perplexity, and + +Figure 1: Predictive perplexity of LDA inferred by ILR and CGS for various values of K on the three datasets. +only when the curve breaks this condition, ILR will typically +complete its training. +This evidence may be potentially +exploited as a source for hyper-parameter tuning. In CGS, +this is not always the case, as the effect of random sampling +may not satisfy the monotonic decrease of the predictive +perplexity. +6.2.2 +Predictive Perplexity of DTM Next, we further +evaluate ILR against CGS for the inference of the dual topic +models (DTM) using the predictive perplexity on the three +benchmark datasets. In DTM, to estimate the probability +of a document-word pair p(w, D), we use the estimated +parameters θ, φr, φc as follows: +p(w, D) = +Kr +� +kr=1 +Kc +� +kc=1 +p(w|φkr)p(D|φkc)p(kr, kc|θ) +In this experiment, we run a basic grid search ranging +over the Kr and Kc values of 10 and 15. +We set the +hyper-parameters α, βr, and βc as 0.5. Figure 2 present the +predictive perplexity of DTM as a function of the number +of iterations for various values of Kr and Kc. +Overall, +ILR again outperforms CGS for inference of the topic model +DTM for various combinations of the Kr and Kc values on +the three datasets. +It is worth noting that both inference methods ILR and +CGS take a few iterations before they begin to learn. This +is because DTM has been trying to simultaneously assign +the values of two latent random variables. We also observed +that the amount of iterations needed for the methods to +start learning depends on the initialization. +A uniform +initialization typically translates into more iterations taken +by the methods to start learning. A good initialization may +be to randomly recommend some parameters for θ, which +can be then used to generate initial topic or distribution +assignments of the latent variables. +6.3 +Topic Coherence We use topic coherence to evaluate +the proposed method ILR for inference of LDA in latent +topic detection. The automatic topic coherence has been well +studied, and the normalized point-wise mutual information +(NPMI), point-wise mutual information (PMI), and pairwise +log-conditional probability (LCP) are three common metrics +which have been shown to correlate positively with human +judgment for topic coherence evaluation [22]. To compute +the coherence of each of detected topics, we used the co- +occurrence statistics of the top N most likely words of the +topic in the corpus. In the experiment, we fixed the hyper- +parameters α and β to 0.5, and the number of topics K to +25. We varied the values of N (10, 20, 50) to study the effect +of selecting different sets of the most likely words on the +coherence of detected latent topics. +Table 4 shows the topic coherence results of LDA in- +ferred by ILR and CGS in terms of LCP, PMI, and NPMI +(The higher, the better). Overall, ILR improves CGS for in- +ferring latent topics of LDA according to the mean coher- +ence scores. All the metrics show that ILR results in better +performance with the exception of PMI on the Movielens +dataset. But this is not a big issue, as PMI tends to assign +high weights to infrequent words in a corpus, and is known +to be not as reliable as other metrics. +The improvement of ILR over CGS remains as N +increases. The topic coherence scores of both ILR and CGS +drop a little with growing the value of N, but this actually +agrees well with expectation. We also observe that the mean + +Lastfm +3000 +Perplexity +2800 +ILR K:5 +2600 +CGS K:5 +2400 +ILR K:10 +2200 +redictive +CGS K:10 +2000 +1800 +1600 +1400 +1200 +0 +20 +40 +60 +80 +100NIPS +1620 +Predictive Perplexity +1600 +ILR K:5 +1580 +CGS K:5 +1560 +ILR K:10 +1540 +CGS K:10 +1520 +1500 +1480 +1460 +0 +20 +40 +60 +80 +100Movielens +2000 +ILR K:5 +1800 +CGS K:5 +1600 +ILR K:10 +Predictive F +CGS K:10 +1400 +1200 +1000 +0 +20 +40 +60 +80 +1003000 +Perplexity +ILR K:20 +2500 +CGS K:20 +ILR K:50 +Predictive +2000 +CGS K:50 +1500 +1000 +0 +20 +40 +60 +80 +100 +Iterations1640 +Perplexity +1620 +ILR K:20 +1600 +CGS K:20 +1580 +ILR K:50 +Predictive +1560 +CGS K:50 +1540 +1520 +1500 +1480 +0 +20 +40 +60 +80 +100 +Iterations2000 +Perplexity +1800 +ILR K:20 +CGS K:20 +1600 +ILR K:50 +Predictive +1400 +CGS K:50 +1200 +1000 +.................................................... +800 +0 +20 +40 +60 +80 +100 +IterationsFigure 2: Predictive perplexity of DTM inferred by ILR and CGS for various values of K on the three datasets. +Table 4: Automatic topic coherence results of LDA inferred by ILR +and CGS, where the ILR results are highlighted in grey. +N +LCP +PMI +NPMI +mean +max , min +mean +max , min +mean +max , min +NIPS +10 +CGS -6.662 -6.08 , -6.96 +0.841 1.20 , 0.55 +0.057 0.09 , 0.04 +ILR +-6.588 -6.17 , -6.93 0.942 1.28 , 0.59 +0.063 0.08 , 0.04 +20 +CGS -6.682 -6.14 , -6.92 +0.806 1.03 , 0.55 +0.054 0.07 , 0.04 +ILR +-6.619 -6.14 , -6.98 0.879 1.15 , 0.58 +0.059 0.07 , 0.04 +50 +CGS -6.687 -6.17 , -6.86 +0.759 0.91 , 0.56 +0.051 0.06 , 0.04 +ILR +-6.641 -6.16 , -6.93 0.801 0.98 , 0.58 +0.054 0.06 , 0.04 +Lastfm +10 +CGS -5.031 -4.56 , -5.79 +1.618 2.54 , 0.94 +0.128 0.19 , 0.08 +ILR +-4.993 -4.52 , -5.70 1.692 2.38 , 1.15 +0.133 0.18 , 0.09 +20 +CGS -5.205 -4.79 , -6.02 +1.631 2.49 , 1.02 +0.125 0.18 , 0.08 +ILR +-5.236 -4.78 , -6.04 1.689 2.33 , 1.13 +0.129 0.18 , 0.09 +50 +CGS -5.572 -5.17 , -6.32 +1.644 2.36 , 1.02 +0.119 0.17 , 0.07 +ILR +-5.601 -5.17 , -6.56 1.692 2.30 , 1.06 +0.122 0.17 , 0.08 +Movielens +10 +CGS -6.182 -5.65 , -6.77 +0.732 1.16 , 0.59 +0.053 0.09 , 0.04 +ILR +-5.669 -5.56 , -6.02 0.689 1.09 , 0.55 +0.054 0.08 , 0.04 +20 +CGS -6.198 -5.74 , -6.66 +0.724 1.08 , 0.63 +0.053 0.08 , 0.04 +ILR +-5.699 -5.57 , -6.04 0.688 1.09 , 0.55 +0.053 0.08 , 0.04 +50 +CGS -6.231 -5.78 , -6.57 +0.723 1.01 , 0.63 +0.052 0.07 , 0.04 +ILR +-5.774 -5.63 , -6.21 0.682 1.02 , 0.56 +0.052 0.07 , 0.04 +NPMI scores are similar, for example, on the Movielens +dataset, even given different values of N, this is perhaps +due to the fact that the NPMI metric involves an additional +normalization step. +6.4 +Hyper-parameter Sensitivity In this section, +we +evaluate ILR against CGS according to the sensitivity of in- +ference of topic models, and We compute the predictive per- +plexity as a function of various hyper-parameter settings. In +this experiment, we applied CGS and ILR to the inference +of LDA. We varied the values of α and β from 0.01 to 0.5, +given fixed number of latent topics K = 25. In addition, we +let the methods run for 500 iterations to guarantee a fair as- +sessment. Table 5 presents the predictive perplexity results +of LDA inferred by ILR and CGS on the Movielens dataset +(The lower, the better). +Table 5: Sensitivity of inference of LDA to various values of hyper- +parameters α and β on the Movielens dataset. +α \ β +Method +0.01 +0.05 +0.10 +0.25 +0.50 +0.01 +CGS +981.7 +958.4 +944.6 +940.2 +934.6 +ILR +910.7 +906.8 +904.7 +901.8 +899.7 +0.05 +CGS +957.9 +932.0 +922.6 +921.1 +920.8 +ILR +906.8 +904.2 +902.6 +900.3 +898.3 +0.10 +CGS +945.0 +924.8 +920.6 +911.2 +910.4 +ILR +904.3 +902.1 +901.0 +899.3 +897.4 +0.25 +CGS +937.1 +918.2 +911.2 +908.1 +906.3 +ILR +900.6 +899.0 +898.0 +897.3 +895.7 +0.50 +CGS +925.1 +914.0 +909.3 +904.7 +905.2 +ILR +898.2 +896.8 +895.9 +895.3 +896.3 +Given the same combination of the values of α and +β, ILR significantly improves CGS for inference of LDA. +Surprisingly, we observe that, across all the combinations of +the hyper-parameter values, the worst inference perplexity + +3000 +Perplexity +CGSKr:10Kc:15 +2500 +ILRKr:10Kc:15 +2000 +CGSKr:15Kc:15 +Predictive +1500 +ILRKr:15Kc:15 +1000 +500 +0 +50 +100 +150 +200 +250 +300 +350 +400 +Iterations1600 +Perplexity +1550 +CGSKr:10Kc:15 +1500 +ILRKr:10Kc:15 +1450 +CGSKr:15Kc:15 +redictive +1400 +ILRKr:15Kc:15 +1350 +1300 +1250 +1200 +0 +100 +200 +300 +400 +500 +600 +Iterations1350 +lexity +1300 +CGS Kr:10 Kc:15 +1250 +ILRKr:10Kc:15 +Perpl +1200 +1150 +CGSKr:15Kc:15 +1100 +ILRKr:15Kc:15 +1050 +1000 +950 +900 +0 +100 +200 +300 +400 +500 +600 +700 +IterationsLastfm +3000 +CGS Kr:10 Kc:10 +2500 +LRKr:10Kc:10 +2000 +CGSKr:15Kc:10 +Predictive F +ILRKr:15Kc:10 +1500 +1000 +500 +0 +50 +100 +150 +200 +250 +300 +350 +400NIPS +1600 +Perplexity +1550 +CGSKr:10Kc:10 +1500 +LRKr:10Kc:10 +1450 +CGSKr:15Kc:10 +Predictive F +1400 +ILRKr:15Kc:10 +1350 +1300 +1250 +..................... +1200 +0 +100 +200 +300 +400 +500 +600Movielens +1350 +1300 +lexity +CGS Kr:10 Kc:10 +1250 +LRKr:10Kc:10 +Perpl +1200 +CGS Kr:15 Kc:10 +1150 +Predictive +1100 +ILRKr:15Kc:10 +1050 +1000 +950 +.................................... +900 +0 +100 +200 +300 +400 +500 +600 +700of LDA via ILR (910.7 when α = β = 0.01) is comparable +with the best perplexity of LDA via CGS (904.7 when α = +0.5, β = 0.25). Interestingly, either ILR or CGS improves +the inference, as the value of the parameter α or β increases. +Moreover, we observe that the inference of LDA via +CGS is more sensitive to the choice of values of hyper- +parameters. The gap between the maximum and minimum +perplexity scores is 77.0 for CGS, while the gap is only +15.4 for ILR, as shown in Table 5. +This agrees with +expectation, as CGS is known to be sensitive to hyper- +parameter settings. In contrast, the experimental results do +not show that the selection of hyper-parameters leads to +the significant difference on the inference of LDA by the +proposed method ILR. +7 +Conclusion +In this paper, we have presented an infinite latent state repli- +cation (ILR) algorithm, which leads to a deterministic ap- +proach to inference of topic models within Gibbs sampling +framework. ILR benefits from the state augmentation for +marginal estimation, and casts a given topic model to a +tractable model with soft topic assignments. The flexibility +in soft assignments results in improved generalization per- +formance for inferring topics. We applied ILR to inference +of two well-established topic models LDA and DTM. Ex- +perimental results on real-world datasets validate that ILR +outperforms CGS for the inference in terms of topic coher- +ence and predictive perplexity, and the results hold despite +various settings of hyper-parameters. +References +[1] D. M. Blei, A. Y. Ng, and M. I. Jordan, “Latent dirichlet +allocation,” Journal of Machine Learning Research, vol. 3, +pp. 993–1022, Mar. 2003. +[2] J. Sivic, B. C. Russell, A. A. Efros, A. Zisserman, and +W. T. Freeman, “Discovering object categories in image +collections,” in Proc. of the Tenth International Conference +on Computer Vision, 2005. +[3] L. Hong, A. Ahmed, S. Gurumurthy, A. J. Smola, and +K. 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Baldwin, “Machine reading +tea leaves: Automatically evaluating topic coherence and +topic model quality,” in Proc. of the 14th Conference of +the European Chapter of the Association for Computational +Linguistics, pp. 530–539, 2014. + diff --git a/ZNFOT4oBgHgl3EQf-jSE/content/tmp_files/load_file.txt b/ZNFOT4oBgHgl3EQf-jSE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b2d68fffc3896fb9c307cafa08ca036e7e39cca --- /dev/null +++ b/ZNFOT4oBgHgl3EQf-jSE/content/tmp_files/load_file.txt @@ -0,0 +1,1116 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf,len=1115 +page_content='Improving the Inference of Topic Models via Infinite Latent State Replications Daniel Rugeles1, Zhen Hai1, Juan Felipe Carmona 2 Manoranjan Dash 3, Gao Cong1, 1 School of Computer Science and Engineering, Nanyang Technological University 2 Departamento de Matem´aticas, Universidad de los Andes, Colombia 3 School of Computing, National University of Singapore {daniel007, haiz0001}@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='sg, jf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='carmona658@uniandes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='co, mano@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='sg, gaocong@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='sg January 31, 2023 Abstract In text mining, topic models are a type of probabilistic gen- erative models for inferring latent semantic topics from text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' One of the most popular inference approaches to topic models is perhaps collapsed Gibbs sampling (CGS), which typically samples one single topic label for each ob- served document-word pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this paper, we aim at im- proving the inference of CGS for topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We pro- pose to leverage state augmentation technique by maximiz- ing the number of topic samples to infinity, and then develop a new inference approach, called infinite latent state replica- tion (ILR), to generate robust soft topic assignment for each given document-word pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Experimental results on the pub- licly available datasets show that ILR outperforms CGS for inference of existing established topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 1 Introduction In text mining, probabilistic topic models, such as latent Dirichlet allocation (LDA) [1], refer to generative statisti- cal algorithms for mining latent semantic structure of a set of text documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Aside from their increasing popularity and ubiquitous adoption in text data processing, they have been also applied to the fields such as computer vision [2] and geographical modeling [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Recently, widespread appli- cability has spurred research on developing more accurate or efficient inference approaches to topic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Existing inference methods can be roughly grouped into optimization and sampling categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' One of the represen- tative optimization based approaches is the mean field vari- ational inference (VI) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' VI frames the inference process as an optimization problem, where the posterior is approxi- mated by fitting a selected family of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Unfortu- nately, the predictive ability of VI is somewhat sacrificed for computational efficiency [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' On the other hand, one of the most popular sampling based inference approaches is per- haps the collapsed Gibbs sampling (CGS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' CGS has been shown effective and widely used to infer latent topic models [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this work, our focus is on improving the generaliza- tion performance of CGS for inference of topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Particularly, in the inference of CGS for LDA, for each observed document-word pair, it estimates a posterior distribution conditioned on the topic assignments of all the other pairs, and relies on the distribution to sample one single topic for the given pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The model’s parameters can be then estimated based on the sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Intuitively, instead of sampling one exact topic assignment, we may yield a flexible and more robust learning approach by sampling multiple topics and then combining the topic samples to derive soft topic assignment for each document-word pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Considering a basic situation where two samples are drawn for the given pair, the two samples correspond to two topic assignments, and combining the two samples would lead to flexible search for optimal assignment by using the two branches simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, as the number of sam- ples increases, the parameter estimation would be done by relying on more and more evidence from all of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This process is known as state augmentation for marginal estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' It was previously studied for improving max- imum a posteriori estimation using Markov Chain Monte Carlo method (MCMC) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We extend the idea, and propose to leverage the benefit of maximizing the state augmentation by raising the number of samples to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, we develop a new method, called infinite latent state replication (ILR), to improve the inference of topic models within Gibbs sampling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As a matter of fact, the computational complexity of the inference model often increases linearly with the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' To deal with this issue, we employ the strong law of large numbers, and present a tractable optimization solution that achieves the same computational complexity of CGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In contrast to collapsed Gibbs sampling, ILR leverages the aggregation of whole population to estimate the density of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Thus, sampling is not required in ILR, and the resulting inference algorithm is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='12974v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='CL] 25 Jan 2023 Thanks to the determinism of ILR, its convergence can be assessed directly by checking when the parameters of the posterior distribution stop changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In contrast, the exact iteration at which CGS converges is unknown, and additional mechanisms may be required to assess its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Next, we apply ILR to inference of LDA, and observe a significant improvement over CGS in predictive perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In addition, ILR can be also applied to inference of other topic models that characterize the dependency between two latent random variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', dual topic model (DTM) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This work has made the following main contributions: Based on Gibbs sampling framework, we develop ILR to estimate the posterior for inference of topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' ILR replaces the exact assignments of latent topics by a density given by drawing multiple topic assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' It is thus a less constrained learning method compared to CGS, and may have flexibility and better chance to improve the generalization performance for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The replication of topic assignments often results in lin- ear increase in computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' To tackle this issue, we maximize the number of replications to infin- ity, and transform the Gibbs sampling algorithm into a deterministic method, whose computational complexity is equivalent to standard Gibbs sampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We conduct extensive experiments on the publicly available benchmark datasets, and experimental results validate the improved effectiveness of ILR over CGS for inference of existing well-established topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 2 Related Work Along the rich history of Bayesian models, several inference algorithms have been proposed, such as variational Bayesian inference [1], expectation propagation [9], collapsed Gibbs sampling [6], and belief propagation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Among these algorithms, collapsed Gibbs sampling and variational infer- ence are perhaps the most widely used methods due to their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The former, which employs a sampling strat- egy, is known for guaranteeing the convergence to true target posterior, while the latter, which is a theoretically backed op- timization method, approximates the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As far as we know, existing extensions on CGS are pri- marily concerned with improving runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This is natural given that CGS tends to result in better predictive perfor- mance than other inference methods, but it is computation- ally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' FastLDA was shown to be eight times faster than CGS while maintaining the comparable performance [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The high computational cost of CGS stems from the O(K) time complexity incurred at every sampling step (K: topic number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' It is observed that, even given a good many topics, words or especially infrequent words are often as- signed only to a few topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, by keeping track of these words, FastLDA requires significantly less than K opera- tions per sample on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' SparseLDA factorizes the pos- terior equation into a sum of three factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The sampling scheme is then replaced by a uniform sampling strategy, where the probability mass falls in one of the buckets in 90% of the time [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' By using an appropriate data structure, the computation for the mass can be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As a result, the model attains an order of magnitude improvement in com- putational complexity without affecting its predictive power compared to CGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Recently, AliasLDA [13] and LightLDA [14] are able to reduce the O(K) complexity of sampling by an efficient sampling method from discrete random vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In the latter case, a Metropolis-Hasting step allows sampling a topic assignment for a document-word pair in O(1) via Walker’s alias method [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, WarpLDA im- proved LightLDA by exploiting the memory access behavior of other algorithms for LDA [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In contrast to sampling-based inference methods, exist- ing work on variational inference approaches mainly aims to improve the predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' One reason which may illustrate the weakness of these methods perhaps lies in the approximation to intractable integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Collapsed variational Bayesian schemes [17] use a second-order Taylor expansion to approximate the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In addition, the zero-order in- formation can be also used in the variational methods, and may result in improved inference [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this work, rather than targeting the efficiency of CGS, we aim to improve the predictive ability of CGS for inference of latent topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' To achieve that, we propose to lever- age a state augmentation technique, which normally aug- ments the probabilistic model by replicating multiple times the latent state random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The technique has been previously shown effective for improving the marginal max- imum a posteriori estimation via the Markov chain Monte Carlo [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Moreover, we show that it is possible to maxi- mize the benefit of the state augmentation technique by rais- ing the number of replications up to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this way, we introduce a new deterministic algorithm to tackle the infer- ence of topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Our method can lead to improved pre- dictive performance, while it maintains equivalent computa- tional complexity compared to the standard collapsed Gibbs sampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 3 Preliminaries Latent Dirichlet allocation (LDA), one of the most common topic models, is able to discover K topics that pervade an un- structured collection of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In LDA, we assume that each document has a multinomial distribution over K topics parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We use θd as the set of parameters for representing the probability distribution over the set of topics given a document d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Each topic is defined as a multinomial distribution over the vocabulary of words parameterized by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We use φk as the set of parameters of the probability over the set of words given a topic k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Per document topic distribu- tion θd is generated by sampling from a Dirichlet distribution with hyper-parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Similarly, φk is generated by sam- pling from a Dirichlet distribution with hyper-parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In latent Dirichlet allocation via collapsed Gibbs sam- pling (LDA-CGS), to find latent topics, we use a dataset X that consists of M observed document-word pairs xj = (dj, wj), ∀j ∈ 1 · · · M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' where |D|, and |W| are the number of documents and the vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Each pair is assigned exactly to one of the topics as established by the assumption of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' For each jth pair, the topic assignment is gen- erated by sampling the conditional posterior distribution of Zj given the topic assignments of all other pairs Z−j = z−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Formally, we represent the conditional posterior distribution as p(Zj|Z−j = z−j, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The probability density of the pos- terior distribution is given in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1) f(Zj|Z−j = z−j) ∝ (Ddj + α)(Wwj + β) (Nj + V β) We use the subscript Zj|Z−j to denote the density of the posterior distribution p(Zj|Z−j, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The Ddj, Wwj, and Nj are count vectors computed using the indicator function 1(): Ddj = M � m=1 m̸=j 1(dj = dm) ∗ ¯ zm ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Wwj = M � m=1 m̸=j 1(wj = wm) ∗ ¯ zm Nj = M � m=1 m̸=j ¯ zm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2) where ¯ zm refers to the realization of the random variable Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We use the bar notation to represent a one-hot encoding representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Hence, Ddj denotes the vector of topic assignments for document dj, Wwj denotes the vector of topic assignments for word token wj, and Nj represents the vector of topic assignments over the entire set of document- word pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This representation facilitates the comparison of Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 with the posterior density proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The collapsed Gibbs sampling approach to LDA (LDA- CGS) is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Note that, at iteration i, the computation for the posterior density in Line 5 not only depends on the topic assignments of all the pairs whose index values are smaller than j, as denoted by zi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Thus, we denote the density at the ith iteration for the jth record to be f(Zi j|Zi j = zi−1 >j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The convergence of LDA-CGS can be obtained when the topic assignments zj for each pair are drawn from the marginal distributions p(Zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Unfortunately, since we do not have an analytical form for p(Zj), we cannot measure exactly at which iteration the samples are drawn from p(Zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Algorithm 1 Collapsed Gibbs Sampling for LDA 1: Input: Hyper-parameters K, α, β, dataset X 2: Initialize z0 j using a uniform distribution, and set initial iteration i = 1 3: repeat 4: for j = 1 · · · M do 5: Compute f(Zi j|Zi j = zi−1 >j ) from Equation3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 6: Draw zi j from f(Zi j|Zi j = zi−1 >j ) 7: i = i + 1 8: until Convergence 9: Collect samples and estimate ˆθ and ˆφ as shown in [6] 10: return ˆθ, ˆφ 4 Infinite Latent State Replication Inference When applying collapsed Gibbs sampling to LDA, for the jth document-word pair, we compute the posterior p(Zj|Z−j = z−j) by using the samples z−j obtained for all other pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, we draw ONE single sample zj from this distribution, and continue to compute the posterior for the next document-word pair p(Zj+1|Z−(j+1) = z−(j+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In collapsed Gibbs sampling, the posterior distributions are updated until the convergence is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 1 shows a toy example of applying collapsed Gibbs sampling to infer- ence of LDA, where the number of topics is K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' For each of M observed document-word pairs (dj, wj), a topic label (1, 2, or 3) is assigned by relying on the posterior probability distribution (last column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 1: A toy example of applying collapsed Gibbs sampling to inference of LDA (K = 3) j (dj, wj) zj Posterior prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 1 (d1, w1) 2 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' M (dM, wM) 2 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4] In contrast, the replication of latent states augments the collapsed Gibbs sampling algorithm by repeatedly drawing R samples from each posterior distribution to compute the newly updated posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The replication of latent space has been previously studied to improve the parameter estimate in maximum a posteriori estimation by using Markov Chain Monte Carlo estimations (MCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In MCMC, the replications of the latent space actually provide more evidence about the search path, which can lead to a more robust exploration of the parameter space [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' To our knowledge, the replication of latent states does not affect the global optima of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The augmented model with R replications described by p′(Θ, Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.Z(R), X) is the Rth power of the model that does not use replications [19], as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3) p′(Θ, Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.Z(R), X) ∝ R � r=1 p(Θ, Z(r), X) In latent replication inference, the replications are de- fined as multiple (R) copies of the latent variables: Z(r) j = 1 RZj, ∀j ∈ 1 · · · M, ∀r ∈ 1 · · · R (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4) A linear transformation can be adopted to maintain the expected value of the random variables Zj, ∀j ∈ 1 · · · M, without affecting the optima of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' However, this transformation is limited to models, where the latent random variable follows a multinomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 2: A toy example of applying R = 4 replications in collapsed Gibbs sampling to inference of LDA (K = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The topic assignment is performed repeatedly for R times for each observed document-word pair (separated by dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' j (dj, wj) z(r) j Posterior prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' (Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5) 1 (d1, w1) 2 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 ] 1 (d1, w1) 3 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 ] 1 (d1, w1) 1 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 ] 1 (d1, w1) 3 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' M (dM, wM) 2 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='41] M (dM, wM) 3 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='41] M (dM, wM) 2 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='41] M (dM, wM) 3 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='41] We then employ multiple replications of latent states in collapsed Gibbs sampling for inference of LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' For- mally, in CGS, we draw one single sample from each posterior, and then the computation of the jth posterior, p(Zj|Z−j=z−j), ∀j ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.M, relies on ONE point estimate from each of the other posteriors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', zi, ∀i ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.M, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In contrast, in latent state replication inference, we draw multiple (R) samples from each posterior, and then the com- putation of the jth posterior would leverage MULTIPLE point estimates from other posteriors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', z(r) i , ∀i ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.M, ∀r ∈ R, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Therefore, the combination of multiple point esti- mates for a posterior serves to approximate the density of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, taking into account the replicas, the jth pos- terior corresponds to: P(Zj|Z(1) −j =z(1) −j , · · · , Z(R) −j =z(R) −j ), where R is the number of replications for estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 2 shows a toy example of applying R = 4 replica- tions of latent states in collapsed Gibbs sampling to inference of LDA (K = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Conventionally, for a given document- word pair, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', (d1, w1), CGS simply uses one sample, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', z1 1 = 2, as the estimate of the categorical posterior distri- bution [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5], which estimates the posterior using a histogram with proportions given by [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In con- trast, the state replication inference uses R = 4 point esti- mates which can be aggregated in a histogram with propor- tions given by [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5] (See the first group in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Clearly, the estimate based on the four replications approx- imates the posterior [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5] with higher fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This may suggest that the replication of latent states empowers the inference process with a more informative path, which can be exploited by Gibbs chain to converge to a better optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Intuitively, while CGS uses a hard topic assignment given by the point estimate, the state replication inference leverages a soft topic assignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', splitting the assignment along the K coordinates of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' p′(Zj|Z(1) −j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Z(R) −j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' X) ∝ � � p′(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Z(r) −j Z(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' X) dθ dφ = |D| � m � Γ(� k αk) �K k Γ(αk) K � k θ αk−1+ 1 R �R r ck(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='k dθm× K � k � Γ(� j βj) �K k Γ(βk) |W | � j φ βk−1+ 1 R �R r ck(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='j dφk = |D| � m �K k Γ( 1 R � r ck(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗ + αk) Γ(� k 1 R � r ck(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗ + αk) × K � k �|W | j Γ( 1 R � r ck(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='j + βj) Γ(� j 1 R � r cj(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='j + βj) ∝ �� r c−(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b) z(r) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗ + αz(r) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b � × �� r c−(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b) z(r) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b + βya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b � � r c−(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b) z(r) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='∗ + � j βj (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5) It is worth noting that, when applying latent state repli- cations within CGS framework, the posterior probability is the same for each replica of a given document-word pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' If we employ R latent replications for inference, the posterior probability can be computed by following the steps shown in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5, some relevant counts are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' ck′,d′,w′ = M � m=1 1(zm = k′ ∧ dm = d′ ∧ wm = w′) c−j k′,d′,w′ = M � m=1,m̸=j 1(zm = k′ ∧ dm = d′ ∧ wm = w′) To facilitate the comparison between standard model and state replication model, Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 may be rewritten in the same form as Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2: Ddj = R � r M � m=1 m̸=j 1(dj = x(d) m ) ∗ z(r) m R (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6) Wwj = R � r M � m=1 m̸=j 1(wj = x(w) m ) ∗ z(r) m R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Nj = R � r M � m=1 m̸=j z(r) m R Previous studies have shown that the predictive power of applying latent state replication to parameter estimation can be improved with increasing the number of replications [19], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Thus, we propose to maximize the number of latent state replications to improve the inference of LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As the number of replications tends toward infinity, in terms of the law of large numbers, we can infer that the proportion of the replications obtained by using a categorical distribution will converge to the probability mass of the categorical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Mathematically, for all k ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.K: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7) lim R→∞ 1 R R � r=1 1(zr=k) = p′(Zj = k|Z(1) −j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.Z(R) −j , X) = κk j , where κk j is one of the K parameters of the posterior distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We define κj as the vector that holds the parameters of the posterior κj = [κ1 j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.κK j ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Since the posterior is a categorical distribution, this vector corresponds to the prob- ability mass function of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The update of the Gibbs sampler for infinite latent replications can be then obtained by raising R to infinity in Equation 4, and the result is shown in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Ddj= M � m=1 m̸=j 1(dj = x(d) m ) ∗ κk m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Wwj= M � m=1 m̸=j 1(wj = x(w) m ) ∗ κk m Nj = M � m=1 m̸=j κk m (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8) Generally, from a generative perspective, it is computa- tionally unfeasible to sample an infinite number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As a matter of fact, by applying the law of large numbers, we can obtain updates that use vector additions without the need for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' A comparison between Table 1 and 2 shows that the topic assignment in state replication inference does not correspond to a single label, but instead, corresponds to a mixture of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As the number of assigned labels (replications) in- creases towards infinity, the presented updates drive the Gibbs chain to convergence by using the whole probability mass from all of the M posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As a con- sequence, the parameters of the inference algorithm are not estimated from samples, but instead are obtained directly by using Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Different from sampling algorithms, Equation 4 will yield the same result on every iteration after convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The deterministic inference driven by the soft topic as- signment mechanism allows for better exploration at the op- timal region than the estimations based on random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As a result, ILR achieves significant improvements in gener- alization performance in terms of predictive perplexity (See Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As for computational cost, the parameter updates for each of the document-word pairs has a time complexity of O(K), which achieves the same cost as collapsed Gibbs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This is actually not a big issue, as we focus on the improvement of predictive power of CGS under a theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Moreover, we study the applicability of the proposed in- finite latent replication inference to topic models that char- acterize dependency between latent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 5 Applicability to Inference of Dual Topic Model The dual topic model (DTM) takes as input a co-occurrence data matrix that underlies the inter-relationship between row and column variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', user-location matrix where each entry means the number of times a user (row) visits a location (column) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Then, it identifies the row topics Z or distribution over the rows of the co-occurrence matrix, the column topics Y or distribution over the columns of the matrix, and the joint distribution of both column and row topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The collapsed Gibbs sampling inference for the jth row- column pair is given by Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='9: f(Zj, Yj|Z−j=z−j, Y−j=Y−j) ∝ (Ddj + βr)⊺(Wwj + βc) (N r j + Rβr)⊺(N c j + Cβc) ⊙ (Pj + α), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='9) where R and C are the numbers of rows and columns in the input matrix respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' βr and βc are hyper-parameters used to generate Z and Y respectively, and the following are count vectors computed using the indicator function 1(): Ddj = M � m=1 m̸=j 1(dj = x(d) m ) ∗ ¯ zm ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Wwj = M � m=1 m̸=j 1(wj = x(w) m ) ∗ ¯ ym Pj= M � m=1 m̸=j ¯ zm ⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' ¯ ym ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' N r j = M � m=1 m̸=j ¯ zm ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' N c j = M � m=1 m̸=j ¯ ym (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='10) Note that ¯ zmT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' ¯ ym is a convenient representation for encoding one unit to a matrix, where rows have the same dimension as ¯ zm, and the columns have the same dimension as ¯ ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The ⊙ refers to element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 3 shows a toy example of applying collapsed Gibbs sampling with latent state replications to inference of DTM, where the numbers of latent row and column topics are Kr = 2 and Kc = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' According to the law of large numbers, the proportion of the joint pairs obtained by the replicas z(r) j and z(r) j would converge to the value of the parameters of joint posterior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' For all kr ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.Kr, ∀kc ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.Kc: lim R→∞ 1 R R � r=1 1(zr=kr, yr=kc) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='11) p(Zj=kr, Yj=kc|Z(1) −j , Y (1) −j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.Z(R) −j , Y (R) −j )=κkr,kc j Table 3: A toy example of applying collapsed Gibbs sampling with latent state replications to inference of DTM (Kr = 2, Kc = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' j (dj, wj) z(r) j z(r) j Posterior prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 1 (d1, w1) 1 1 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 � 1 (d1, w1) 2 2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 � 1 (d1, w1) 2 2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 � 1 (d1, w1) 1 1 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' M (dM, wM) 1 2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 � M (dM, wM) 2 1 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 � M (dM, wM) 1 2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 � M (dM, wM) 1 1 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 � The κ corresponds to a matrix that holds the parame- ters of the bivariate categorical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Following the same procedure as in LDA, we first derive the case for R replications, and then drive R up to infinity to derive the ILR inference method applied to DTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The resulting factors are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Ddj = M � m=1 m̸=j 1(dj = x(d) m ) ∗ κm ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Wwj = M � m=1 m̸=j 1(wj = x(w) m ) ∗ κm Pj= M � m=1 m̸=j κm ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' N r j = M � m=1 m̸=j κm ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' N c j = M � m=1 m̸=j κm (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='12) Note that for DTM, the factors are computed from matrix additions in contrast to the vector additions found for LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 6 Experiments 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 Datasets We used three publicly available datasets to evaluate the proposed method ILR for inference of topic models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', NIPS:1[20], Lastfm 2, and Movielens 3 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The NIPS dataset comes from the Neural Information Pro- cessing Systems proceedings from 1987 to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The dataset consists of 11,040,357 records with 11,463 words, which are generated by 5,811 authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The Lastfm dataset contains 1https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='edu/ml/datasets/NIPS+ Conference+Papers+1987-2015 2http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='dtic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='upf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='edu/˜ocelma/ MusicRecommendationDataset/lastfm-360K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='html 3https://grouplens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='org/datasets/movielens/ 145,534,518 records, which are generated based on the tu- ples (user, artist, plays) of 360,000 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Movielens is a popular benchmark dataset for movie rating prediction and recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We used the Movielens 1M dataset that consists of 445,094 ratings generated by the most popular 1,223 users on 1,214 movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We conducted three types of experiments on the datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=', predictive perplexity, coherence of latent topic detection, and sensitivity of inference to hyper-parameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 Predictive Perplexity Results Predictive perplexity is widely used to evaluate the generalization ability of a learn- ing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this section, we use the perplexity metric to evaluate the proposed method ILR for inference of two well- established topic models LDA and DTM, and compare it with the most commonly used inference algorithm collapsed Gibbs sampling (CGS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 Predictive Perplexity of LDA We first evaluate the performance of inference of LDA by using ILR against CGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this experiment, the number of topics K was tuned amongst {5, 10, 20, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The hyper-parameter α was set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5, while β was empirically determined to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5 on NIPS, Lastfm, and Movielens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We held out 40% of each dataset for testing, and run the two inference methods for 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We report the average perplexity results on five trials under the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Figure 1 shows the predictive perplexity results of LDA inferred by ILR and CGS on the Lastfm, NIPS, and Movie- lens datasets, respectively (The lower, the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Overall, the proposed method ILR (solid line) clearly outperforms CGS (dashed line) for inference of LDA across all the values of K for each of the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Note that both inference methods use almost same time for running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In addition, there is an exception for ILR given K = 50 on the NIPS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' As the iterations progress, the perplexity of LDA via ILR drops sharply, and then increases gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This is perhaps due to the choice of K being too large for ILR to learn on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This case has been also found by Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' [21], when they evaluated the predictive power of topic models versus the parameter K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The finding may suggest that appropriate choice of hyper-parameters is advised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In practice, we can employ cross-validation on a small development dataset to find the suitable values of the hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We conclude that, given appropriate choice of hyper- parameters, ILR consistently outperforms CGS for inference of LDA in terms of predictive capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In addition, the speed of convergence of both algorithms are comparable, however, ILR typically tends to lower bound the perplexity curves of CGS in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We also observe that our method shows a monotonic decrease in the perplexity, and Figure 1: Predictive perplexity of LDA inferred by ILR and CGS for various values of K on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' only when the curve breaks this condition, ILR will typically complete its training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This evidence may be potentially exploited as a source for hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In CGS, this is not always the case, as the effect of random sampling may not satisfy the monotonic decrease of the predictive perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 Predictive Perplexity of DTM Next, we further evaluate ILR against CGS for the inference of the dual topic models (DTM) using the predictive perplexity on the three benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In DTM, to estimate the probability of a document-word pair p(w, D), we use the estimated parameters θ, φr, φc as follows: p(w, D) = Kr � kr=1 Kc � kc=1 p(w|φkr)p(D|φkc)p(kr, kc|θ) In this experiment, we run a basic grid search ranging over the Kr and Kc values of 10 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We set the hyper-parameters α, βr, and βc as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Figure 2 present the predictive perplexity of DTM as a function of the number of iterations for various values of Kr and Kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Overall, ILR again outperforms CGS for inference of the topic model DTM for various combinations of the Kr and Kc values on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' It is worth noting that both inference methods ILR and CGS take a few iterations before they begin to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This is because DTM has been trying to simultaneously assign the values of two latent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We also observed that the amount of iterations needed for the methods to start learning depends on the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' A uniform initialization typically translates into more iterations taken by the methods to start learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' A good initialization may be to randomly recommend some parameters for θ, which can be then used to generate initial topic or distribution assignments of the latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 Topic Coherence We use topic coherence to evaluate the proposed method ILR for inference of LDA in latent topic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The automatic topic coherence has been well studied, and the normalized point-wise mutual information (NPMI), point-wise mutual information (PMI), and pairwise log-conditional probability (LCP) are three common metrics which have been shown to correlate positively with human judgment for topic coherence evaluation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' To compute the coherence of each of detected topics, we used the co- occurrence statistics of the top N most likely words of the topic in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In the experiment, we fixed the hyper- parameters α and β to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5, and the number of topics K to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We varied the values of N (10, 20, 50) to study the effect of selecting different sets of the most likely words on the coherence of detected latent topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 4 shows the topic coherence results of LDA in- ferred by ILR and CGS in terms of LCP, PMI, and NPMI (The higher, the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Overall, ILR improves CGS for in- ferring latent topics of LDA according to the mean coher- ence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' All the metrics show that ILR results in better performance with the exception of PMI on the Movielens dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' But this is not a big issue, as PMI tends to assign high weights to infrequent words in a corpus, and is known to be not as reliable as other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The improvement of ILR over CGS remains as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The topic coherence scores of both ILR and CGS drop a little with growing the value of N, but this actually agrees well with expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We also observe that the mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Lastfm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='. 800 0 20 40 60 80 100 IterationsFigure 2: Predictive perplexity of DTM inferred by ILR and CGS for various values of K on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 4: Automatic topic coherence results of LDA inferred by ILR and CGS, where the ILR results are highlighted in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' N LCP PMI NPMI mean max , min mean max , min mean max , min NIPS 10 CGS -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='662 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='841 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='20 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='09 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 ILR 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='588 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='17 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='942 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='28 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 20 CGS -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='682 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='14 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='806 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='03 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='07 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 ILR 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='619 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='14 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='879 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='15 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='07 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 50 CGS -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='687 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='17 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='91 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='06 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 ILR 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='641 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='16 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='98 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='06 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 Lastfm 10 CGS -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='031 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='56 , -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='618 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='54 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='19 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 ILR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='993 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='52 , -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='692 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='38 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='18 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='09 20 CGS -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='205 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='79 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='631 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='49 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='18 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 ILR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='236 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='78 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='689 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='33 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='18 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='09 50 CGS -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='572 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='17 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='644 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='36 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='17 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='07 ILR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='601 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='17 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='692 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='30 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='17 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 Movielens 10 CGS -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='182 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='65 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='732 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='16 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='09 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 ILR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='669 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='56 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='689 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='09 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 20 CGS -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='198 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='74 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='724 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 ILR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='699 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='57 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='688 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='09 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='08 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 50 CGS -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='231 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='78 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='723 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='01 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='07 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 ILR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='774 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='63 , -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='682 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='02 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='07 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='04 NPMI scores are similar, for example, on the Movielens dataset, even given different values of N, this is perhaps due to the fact that the NPMI metric involves an additional normalization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 Hyper-parameter Sensitivity In this section, we evaluate ILR against CGS according to the sensitivity of in- ference of topic models, and We compute the predictive per- plexity as a function of various hyper-parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In this experiment, we applied CGS and ILR to the inference of LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We varied the values of α and β from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5, given fixed number of latent topics K = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In addition, we let the methods run for 500 iterations to guarantee a fair as- sessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 5 presents the predictive perplexity results of LDA inferred by ILR and CGS on the Movielens dataset (The lower, the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Table 5: Sensitivity of inference of LDA to various values of hyper- parameters α and β on the Movielens dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' α \\ β Method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='01 CGS 981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6 940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6 ILR 910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8 899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='05 CGS 957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='9 932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6 921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8 ILR 906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='10 CGS 945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8 920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6 911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 ILR 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='25 CGS 937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 ILR 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='6 899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='50 CGS 925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1 914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 ILR 898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2 896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='8 895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='9 895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3 Given the same combination of the values of α and β, ILR significantly improves CGS for inference of LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Surprisingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' we observe that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' across all the combinations of the hyper-parameter values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' the worst inference perplexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Perplexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='CGSKr:10Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ILRKr:10Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='CGSKr:15Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Predictive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ILRKr:15Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Iterations1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Perplexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1550 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='CGSKr:10Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ILRKr:10Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='CGSKr:15Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='redictive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='ILRKr:15Kc:15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Iterations1350 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='400NIPS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='Perplexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1550 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='CGSKr:10Kc:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1500 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='1250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 1200 0 100 200 300 400 500 600Movielens 1350 1300 lexity CGS Kr:10 Kc:10 1250 LRKr:10Kc:10 Perpl 1200 CGS Kr:15 Kc:10 1150 Predictive 1100 ILRKr:15Kc:10 1050 1000 950 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='. 900 0 100 200 300 400 500 600 700of LDA via ILR (910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 when α = β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='01) is comparable with the best perplexity of LDA via CGS (904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='7 when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='5, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Interestingly, either ILR or CGS improves the inference, as the value of the parameter α or β increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Moreover, we observe that the inference of LDA via CGS is more sensitive to the choice of values of hyper- parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The gap between the maximum and minimum perplexity scores is 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='0 for CGS, while the gap is only 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content='4 for ILR, as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' This agrees with expectation, as CGS is known to be sensitive to hyper- parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' In contrast, the experimental results do not show that the selection of hyper-parameters leads to the significant difference on the inference of LDA by the proposed method ILR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 7 Conclusion In this paper, we have presented an infinite latent state repli- cation (ILR) algorithm, which leads to a deterministic ap- proach to inference of topic models within Gibbs sampling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' ILR benefits from the state augmentation for marginal estimation, and casts a given topic model to a tractable model with soft topic assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' The flexibility in soft assignments results in improved generalization per- formance for inferring topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' We applied ILR to inference of two well-established topic models LDA and DTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Ex- perimental results on real-world datasets validate that ILR outperforms CGS for the inference in terms of topic coher- ence and predictive perplexity, and the results hold despite various settings of hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Blei, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Newman, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' Baldwin, “Machine reading tea leaves: Automatically evaluating topic coherence and topic model quality,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' of the 14th Conference of the European Chapter of the Association for Computational Linguistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} +page_content=' 530–539, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFOT4oBgHgl3EQf-jSE/content/2301.12974v1.pdf'} diff --git a/ZdAzT4oBgHgl3EQfK_sm/content/tmp_files/2301.01105v1.pdf.txt b/ZdAzT4oBgHgl3EQfK_sm/content/tmp_files/2301.01105v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9c8ab8711498fe04df7377c32ce285d6fa2dc5d --- /dev/null +++ b/ZdAzT4oBgHgl3EQfK_sm/content/tmp_files/2301.01105v1.pdf.txt @@ -0,0 +1,770 @@ +arXiv:2301.01105v1 [math.AC] 3 Jan 2023 +MODULES OF FINITE INJECTIVE DIMENSION +MOHSEN ASGHARZADEH +Abstract. We study some aspects of distinguished finitely generated modules of finite injec- +tive dimension. Following Bass’ conjecture, this implies that the ring is Cohen-Macaulay. We +derive some properties of rings from them. Our list of properties contains the reduced, domain, +normality, and regularity (complete-intersection, Gorensteiness, etcetera) of rings. +Contents +1. +Introduction +1 +2. +A family of ideals of small height +3 +3. +Modules with few generators +6 +4. +Auslander’s zero-divisor conjecture +9 +5. +Almost complete-intersection +11 +6. +Matlis and canonical module +13 +References +15 +1. Introduction +Bass [9] conjectured that the existence of a finitely generated module of finite injective dimension +imposes some restrictions on the ring, namely the Cohen-Macaulay property. Peskine-Szpiro [23] +proved Bass’ conjecture in many cases, and the conjecture has been settled by Roberts [24] in the +full setting. Despite this, it is referred to as Bass’ conjecture. The work of Peskine-Szpiro contains +a lot of related beautiful results. In particular, they searched for more restrictions on the ring if +one focuses on very special modules of finite injective dimension. We also note that Matlis was +interested in the ideals of injective dimension one. These are our motivations to do this note. +Let p be in Spec(R), and suppose pdR(p) < ∞, equivalently pdR(R/p) < ∞. Peskine-Szpiro +[23, Corollary II.3.3] proved that R is an integral domain. It is easy to see the ring is domain if +idR(R/p) < ∞. A natural question arises: +Question 1.1. What can say about R if idR(p) < ∞? +Some of the results from Section 2 are summarized in the following item: +2020 Mathematics Subject Classification. 13C14; 13H10; 13D22. +Key words and phrases. Cohen-Macualay rings; injective dimension; integral domains; regular (Gorenstein) rings; +Auslander’s zero-divisor; regular sequences. +1 + +2 +M. ASGHARZADEH +1) Suppose that idR(p) < ∞ for all p ∈ Assh(R). Then R is reduced. +2) Suppose min(R) is singleton. If idR(p) < ∞ for some p ∈ Spec(R), then R is domain. +3) Suppose that idR(p) < ∞ for all p ∈ Spec(R) of height at most one. Then R is normal. +4) Let R be excellent UFD containing k and of dimension > 3. Suppose that idR(p) < ∞ for +all p ∈ Spec(R) of height two. Then R is regular. +We also present a quasi-version of the reduced and normal items. For example, we show: +Corollary 1.2. Suppose that idR(p) < ∞ for some p ∈ Spec(R) of positive height. Then R is +quasi-normal. +Suppose we allow more ideals are of finite injective dimension. +For instance, suppose that +idR(a) < ∞ for all ideals of big-height at most two. Then, we apply our displaced results plus a +famous result of Auslander [6] to show R is regular. +In Section 3 we focus on modules of finite injective dimensions with few generators. In fact, +Vaconcelos asked in [27, Page 51] “how do Gorenstein rings arise?”, and he studied the question +by looking at certain modules with few number of generators, see [27, 2.40]. Our first result in §3 +is as follows: +Proposition 1.3. Let (R, m) be quasi-normal, and let M be such that +i) idR(M) < ∞, +ii) M is reflexive, +iii) µ(M) ≤ 2. +Then R is Gorenstein. +As an application, we extend Proposition 1.3 to the case of four generated modules over rings +of type at most two. Also, we present some few comments to §2. Then, we slightly extend [27, +2.40] to more general setting. +Suppose M is a nonzero tor-rigid module, Auslander [7, 4.3] proved each M-regular sequence is +an R-regular sequence. As modules of finite projective dimension are not in general tor-rigid, he +conjectured (proved by Roberts [25, 6.2.3]) that the above property is true for modules of finite +projective dimension. This is referred to as Auslander’s zero-divisor conjecture. In Section 4 we +present the injective analogue of Auslander’s zero-divisor conjecture. We do this via two different +methods. We use these to reconstruct Auslander’s zero-divisor conjecture over Cohen-Macaulay +rings. Also, this enables us to reprove some things from §2. +In Section 5 we deal with +Question 1.4. (See [15]) Suppose idR(I/I2) is finite. Is I generated by a regular sequence? +We present two cases that the answer is positive. Namely, in the almost complete-intersection +case. This immediately recovers the main result of [17] by Kunz. +Recall that [27, 2.38] suggests canonical module is an important source to produce, in a construc- +tive way, modules of finite injective dimension. In fact, Matlis applied this to study the following +over 1-dimensional integral domains: + +MODULES OF FINITE INJECTIVE DIMENSION +3 +i) When is Q/R ։ ER(k)? +ii) When is Q/I = ER(k)? +In order to see these are related to modules of finite injective dimension, recall that ii) is equivalent +to ideals of finite injective dimension. In Section 6 we extend them to the higher dimensional +situation. It may be nice to recall that R is called quasi-Gorenstein, if Hdim(R) +m +(R) ∼= ER(k), +so Matlis was interested on this and its semi-form as well. Finally, we extend an observation of +Auslander [5] about cohomology of Q/R and present its converse part, see Observation 6.6. +For all unexplained notation and definitions see the book [10]. +2. A family of ideals of small height +In this note (R, m, k) is a commutative noetherian local ring, and modules are finitely gener- +ated, otherwise specialized. The notation pdR(−) (resp. idR(−)) stands for the projective (resp. +injective) dimension of (−). +Notation 2.1. We denote the set of all associated prime ideals by AssR(−). +Proposition 2.2. Suppose min(R) is singleton. If for some p ∈ Spec(R) one has idR(p) < ∞, +then R is an integral domain. +Proof. Following Bass’ conjecture (see [10, Section 9.6]) R is Cohen-Macaulay. +In particular, +Ass(R) = min(R). Suppose min(R) = {P}. By definition, there is an x ∈ R such that (0 :R x) = P. +Suppose x ∈ m. Also, recall that +x ∈ Zd(R) = +� +q∈Ass(R) +q = +� +q∈min(R) +q = P = (0 : x). +In other words, x2 = 0. Since min(R) is singleton, P ⊆ p. Since idR(p) < ∞, and due to [10, +Proposition 3.1.9], we know that idRp(pRp) < ∞. According to Auslander-Buchsbaum-Serre, Rp +is regular (see [10, Theorem 2.2.7]). Since Rp is regular its localizations are as well. In particular, +RP = (Rp)P Rp is regular. Recall that x ∈ P. Since x/1 ∈ PRP , it should be zero. There is an +y ∈ R \ P such that xy = 0. Since xy = 0 we have y ∈ (0 : x) = P. This contradiction says that +x /∈ m. Since the ring is local, x is unit. From this P = (0 : x) = 0. Consequently, R is an integral +domain. +□ +We denote the set of all associated prime ideal p with the property that dim R/p = dim R, by +AsshR(−). +Proposition 2.3. Suppose that idR(p) < ∞ for all p ∈ Assh(R). Then R is reduced. +Proof. Recall that R is Cohen-Macaulay. In particular, R is equidimensional. Let +Assh(R) = Ass(R) = min(R) = {p1, . . . , pt}. +In view of Proposition 2.2 we may assume t > 1. Let P ∈ min(�t +i=1 pi). There is an i such that +pi ⊆ P. It follows that P is minimal over pi. Thus, P = pi. Since idR(pi) < ∞, idRpi(piRpi) < ∞. + +4 +M. ASGHARZADEH +According to Auslander-Buchsbaum-Serre, Rpi is regular. Then +( +t� +i=1 +pi)RP ⊆ PRP = 0. +Since min(−) is the minimal part of Supp(−), we proved that +Supp( +t� +i=1 +pi) = ∅, +and so �t +i=1 pi = 0. Therefore, R is reduced. +□ +Notation 2.4. By (Gi) (resp. (Ri)) we mean Rp is Gorenstein (resp. regular) for all p ∈ Spec(R) +of height at most i. +Recall that a module M satisfies (Si) if depth(Mp) ≥ min{i, dim(Mp)} for all p ∈ Spec(R). +Proposition 2.5. Suppose that idR(p) < ∞ for some p ∈ Assh(R). Then R is quasi-reduced. +Proof. Following Bass’ conjecture, R is Cohen-Macaulay. In particular, R satisfies Serre’s condition +(S1). Let q ∈ Ass(R) = min(R). We have two possibilities i) q = p or ii) q ̸= p. +i) In the first case, Rq is a field. To see this, recall that idRq(qRq) = idRp(pRp) is finite, +because idR(p) < ∞. This in turn implies that Rq is zero-dimensional local and regular. +From this, Rq is a field. +ii) In the second case, Rq is Gorenstein. Indeed, we have Rq = pRq. Consequently, idRq(Rq) = +idRq(pRq), and recall that the later is finite, because p is of finite injective dimension. By +this, Rq is Gorenstein. +In both cases, Rq is Gorenstein. It remains to note that (S1) and (G0) imply the quasi-reduced +condition. +□ +Proposition 2.6. Suppose that idR(p) < ∞ for all p ∈ Spec(R) of height at most one. Then R is +normal. +Proof. Recall from the assumptions that R satisfies Serre’s condition (S2). Let p ∈ Spec(R) be +of height at most one. Since idR(p) < ∞, idRp(pRp) < ∞. According to Auslander-Buchsbaum- +Serre, Rp is regular. So, R satisfies (R1). It remains to note that if a ring satisfies Serre’s condition +(S2) and (R1) is normal. +□ +We cite [26] for basics of quasi-normal rings. +Proposition 2.7. Suppose that idR(p) < ∞ for some p ∈ Spec(R) of positive height. Then R is +quasi-normal. +Proof. As Proposition 2.6, the ring R satisfies (S2). Let q ∈ Spec(R) be of height at most one. We +may assume that p in not maximal, otherwise the ring becomes regular. We have two possibilities +i) q ⊆ p or ii) q ⊈ p. + +MODULES OF FINITE INJECTIVE DIMENSION +5 +i) We localize the ring at p and by using the assumption idR(p) < ∞, we know that Rp is +regular. Since q ⊆ p, we have Rq ∼= (Rp)qRp. Since localization of a regular ring is regular +we deduce that Rq is regular. So, R satisfies (R1). +ii) Suppose q ⊈ p. We claim that p ⊈ q. Indeed, suppose not. Then p ⊆ q. Due to the +assumption ht(p) ≥ 1, and the fact that ht(q) ≤ 1, we deduce that p = q. This contradicts +the assumption from ii). So, p ⊈ q. This gives us the natural isomorphism pRq ∼= Rq. +Since idR(p) < ∞, its localizations are as well. In particular, idRq(Rq) = idRq(pRq) < ∞. +By definition Rq is Gorenstein. So, R satisfies (G1). +In both cases R is (G1). +It remains to note that if a ring satisfies (S2) and (G1), then it is +quasi-normal. +□ +Corollary 2.8. Suppose that idR(p) < ∞ for some p ∈ Spec(R). Then R is quasi-reduced. +Proof. If p is in Assh(R), then the desired claim is subject of Proposition 2.5. So, without loss of +generality, we may assume that dim(R) > 0 and that p ∈ Spec(R) \ Assh(R). By Bass’ conjecture, +R is Cohen-Macaulay. Since the ring is equi-dimensional we have ht(p) > 0. It remains to apply +Proposition 2.7. +□ +Proposition 2.9. Let (R, m, k) be excellent UFD containing k and of dimension > 3. Suppose +that idR(p) < ∞ for all p ∈ Spec(R) of height two. Then R is regular. +Proof. This is straightforward to see R satisfies Serre’s condition (S3) and (R2). Let M := Syz2(k). +It is reflexive and locally free over the punctured spectrum. According to the Miller’s paper [21], +there is a free module F and a prime ideal p of height at most two that fits in the following +Bourbaki sequence +0 −→ F −→ M −→ p −→ 0 +(∗) +Recall that R is Cohen-Macaulay. Thanks to a result of Murthy [10, 3.3.19] R is Gorenstein. This +shows that idR(F) < ∞. We apply this in (∗) to see idR(M) is finite. Thus, idR(k) < ∞. Thanks +to Auslander-Buchsbaum-Serre, R is regular. +□ +What about rings of mixed characteristic? More formally, we ask: +Problem 2.10. Find a mixed characteristic analogue of Bourbaki sequence and prime ideal?* +If we allow more ideals are of finite injective dimension, then we present the following 3 positive +answers: +Proposition 2.11. Let (R, m) be local. Suppose that idR(a) < ∞ for all ideals of big-height at +most two. Then R is regular. +Proof. By Proposition 2.6 R is normal. We assume that dim R > 0. Thanks to Bass, R is Cohen- +Macaulay. For any nonzero x ∈ m, (xR) is height-unmixed and of height one. By assumption +*We are interested on it, even if the ring is regular. + +6 +M. ASGHARZADEH +idR(xR) is finite. +Since R is domain, xR ∼= R. +Thus, idR(R) < ∞. +In other words, R is +Gorenstein. This allows us to assume that pdR(a) < ∞ for all ideal a of big-height at most two. +This plus the normality and Cohen-Macaulay enable us to apply a result of Auslander [6, Corollary +5], and deduce that R is regular. +□ +Proposition 2.12. Let (R, m, k) be UFD. Suppose that idR(a) < ∞ for all unmixed ideal of height +two. Then R is regular. +Proof. We may assume that dim(R) > 1. Let x, y be a parameter sequence. It is height unmixed. +By assumption, idR(x, y) < ∞. Again, Bass’ conjecture says that the ring is Cohen-Macaulay. By +the mentioned result of Murthy, R is Gorenstein. So, pdR(a) < ∞ for all unmixed ideal of height +two. It remains to apply the discussion below [6, Corollary 5]. +□ +Here, is another variation of Proposition 2.11: +Observation 2.13. Suppose any two generated ideal of R is of finite injective dimension. Then +R is a Gorenstein UFD. +Proof. The ring R := (1) is of finite injective dimension. So, R is Gorenstein. * Over Gorenstein +rings, finite injective dimension implies finite projective dimension (see [10, 3.1.25]). Now, result +is clear by [11, 5.3]. +□ +3. Modules with few generators +We start by the following easy exercise: +Fact 3.1. Let R be a local domain. Suppose there is a cyclically presented module M of finite +injective dimension. Then R is Gorenstein. +Proof. Let M = R/xR for some x. Set i := idR(M) < ∞ and look at 0 → R +x +−→ R → M → 0. +This gives Exti+1 +R (k, R) +x +−→ Exti+1 +R (k, R) → Exti+1 +R (k, M) = 0. By Nakayama, Exti+1 +R (k, R) = 0, +and so idR(R) < ∞. By definition, R is Gorenstein. +□ +The following is nontrivial: +Fact 3.2. (Peskine-Szpiro) Suppose there is a cyclic module M of finite injective dimension. Then +R is Gorenstein. +Corollary 3.3. Suppose R is Gorenstein. If for some p ∈ Spec(R) one has idR(p) < ∞, then R +is an integral domain. +Proof. We have pdR(p) < ∞. Then R is an integral domain. +□ +Corollary 3.4. Suppose for some principal ideal p ∈ Spec(R) one has idR(p) < ∞. Then R is an +integral domain. +*Alternatively, use [23, II.5.5]. + +MODULES OF FINITE INJECTIVE DIMENSION +7 +Proof. Thanks to Fact 3.2, R is Gorenstein. By Corollary 3.3, R is an integral domain. +□ +Here, we present a new proof of [27, Theorem 5.2] and [22, Theorem 2.11(iv)]. +Corollary 3.5. Let I ⊳ R be nonzero. If idR(R/I) < ∞ then I has an R-regular element. +Proof. Thanks to Fact 3.2, R is Gorenstein. By [10, 3.1.25], pdR(R/I) < ∞. Equivalently, I +has a finite free resolution. By a famous result of Burch [10, Corollary 1.4.7] I contains a regular +element. +□ +Notation 3.6. By µ(−) we mean the minimal number of elements that needs to generates a finitely +generated module (−). +Proposition 3.7. Let (R, m) be quasi-normal, and let M be such that +i) idR(M) < ∞, +ii) M is reflexive, +iii) µ(M) ≤ 2. +Then R is Gorenstein. +Proof. By Fact 3.2 we may assume that M is not cyclic. In the light of Bass’ conjecture, R is +Cohen-Macaulay. Let d := dim(R), and x := x1, . . . , xd be a system of parameter. In the light of +[23, II.5.4] we have +Extd +R(HomR(k, R/xR), M) ∼= k ⊗R +M +xM +∼= M +mM . +The notation r(R) stands for type of R. We deduce by this that +⊕r(R) Extd +R(k, M) ∼= k ⊕ k. +Let us denote µi(M) for the ith Bass number of M, i.e., the number of copies of ER(k) in the +minimal injective resolution of M. Following this terminology, we deduce µd(M)r(R) = 2. In the +case r(R) = 1, there is nothing to proof, as the rings of type one are Gorenstein. So, without loss +of generality, r(R) = 2. This implies that µd(M) = 1. Since M is torsion-free, it is of maximal +dimension. Then, in view of [13, 3.4.c] we observe that M is maximal Cohen-Macaulay. Without +loss of generality we may and do assume that the ring is complete. Over Cohen-Macaulay and +complete ring the canonical module exists, as it is unique, we denoted it by ωR. +Since M is +maximal Cohen-Macaulay and of finite injective dimension, we use a result of Sharp [22] to present +the identification M = ⊕ωR (also, see [10, 3.3.28]). Since µ(M) = 2 and µ(ωR) = r(R) = 2, we +deduce that M = ωR. By (−)∗ we mean HomR(−, R). Thanks to [16, Appendix], there is an exact +sequence +0 −→ ω∗ −→ R2 −→ ωR −→ 0 +(∗) +Recall that Ext1 +R(R2, R) = 0, and apply HomR(−, R) to the displayed short exact sequence (∗), +gives us the following: + +8 +M. ASGHARZADEH +0 −−−−→ +ω∗ +−−−−→ R2 −−−−→ ω∗∗ −−−−→ Ext1 +R(ωR, R) −−−−→ 0 += +� += +� += +� +0 −−−−→ +ω∗ +−−−−→ R2 −−−−→ +ω +−−−−→ +0 +From this we conclude that Ext1 +R(ωR, R) = 0. Now, we are going to apply [2, 5.3] and to deduce +that R is Gorenstein. +□ +Proposition 3.8. Let (R, m) be quasi-normal of type at most two, and let M be an indecomposable +module such that +i) idR(M) < ∞, +ii) M is reflexive, +iii) µ(M) ≤ 4. +Then R is Gorenstein. +Proof. We may assume that µ(M) > 3. +Also, we may assume that r(R) = 2. +By the above +argument, 2µd(M) = µd(M)r(R) = µ(M). From this, µ(M) is even. In fact µ(M) = 4 and so +µd(M) = 2. Recall that M is (S2). According to [1, Theorem 3] we know M is Cohen-Macaulay, +and so maximal Cohen-Macaulay. Since it is of finite injective dimension, M = ⊕ωR. Since M is +indecomposable, M = ωR. But, +4 = µ(M) = µ(ωR) = r(R) = 2. +This contradiction completes the proof. +□ +Corollary 3.9. Let (R, m) be quasi-normal of type two, and let M be a module such that idR(M) < +∞. Then µ(M) is even. +The following is dual to [25, 6.2.4]: +Observation 3.10. Suppose there is a finite length module A of finite injective dimension. Then +R is Cohen-Macaulay. +Proof. We may assume the ring is complete. By Matlis, M := HomR(A, ER(k)) is of finite length +and of finite flat dimension. By [14], pdR(M) < ∞. Since ℓ(M ⊗ R) < ∞, and by intersection +theorem (see for example [10, Theorem 9.4.5]), dim R ≤ pdR(M) = depth(R) − depth(M) = +depth(R), as claimed. +□ +The following slightly extends [25, 6.2.4]: +Observation 3.11. Suppose there is an artinian module A of finite projective dimension. Then +R is Cohen-Macaulay. +Proof. This is similar to the previous item. +□ +The following drops the Cohen-Macaulay assumption from [27, Proposition 2.40]: + +MODULES OF FINITE INJECTIVE DIMENSION +9 +Corollary 3.12. Let (R, m) be a 2-dimensional ring, I an ideal of grade two and finite projective +dimension. Then r(R/I) = r(R)(µ(I) − 1). +Proof. Recall from 2 = dim R ≥ ht(I) ≥ grade(I, R) = 2 that I is m-primary. From 0 → I → R → +R/I → 0 we deduce that the artinian module R/I is of finite projective dimension. By Observation +3.11, R is Cohen-Macaulay. +The exact sequence . . . → Hi +m(I) → Hi +m(R) → Hi +m(R/I) → . . . +gives us depth(I) = 1. +By the Cohen-Macaulay property and Auslander-Buchsbaum formula, +pdR(I) = depth(R) − depth(I) = 1. These allow us to apply [27, Proposition 2.40] and conclude +r(R/I) = r(R)(µ(I) − 1). +□ +Peskine-Szpiro [23, Corollary II.5.7] study the case µ(I) = 2. Here, is the case µ(I) = 3: +Corollary 3.13. Let (R, m) be a 2-dimensional ring, I an ideal of grade two and finite injective +dimension. If µ(I) = 3, then: +i) R is Gorenstein, +ii) R/I is Cohen-Macaulay and of type two. +Proof. i) Recall that R is Cohen-Macaulay. +Let x = x1, x2 be a parameter sequence. +Then +Ext2 +R(HomR(k, R/xR), I) ∼= +I +mI . This gives r(R)µ2(I) = 3. Recall that depth(I) = 1 < 2 = dim(I). +Thanks to [13, Proposition 3.8] we know that µ2(I) ≥ 2. Plugging this in the previous equality +yields that r(R) = 1, i.e., R is Gorenstein. +ii) Since R/I is zero-dimensional, it is Cohen-Macaulay. Due to the first item we know R is +Gorenstein. This yields that pdR(I) < ∞. By Corollary 3.12 R/I is of type two. +□ +4. Auslander’s zero-divisor conjecture +As a basic tool, we prepare a version of Auslander’s zero-divisor conjecture for modules of finite +injective dimension. This has its own importance. +Proposition 4.1. Let M be finitely generated, and of finite injective dimension. Then any M- +regular element is R-regular. +Proof. By [12, Proposition 3.6] it is enough to show that +grade(p, M) ≤ grade(p, R) +∀p ∈ Spec(R). +Following Bass’ conjecture, R is Cohen-Macaulay. Then +grade(p, M) +(1) +≤ depth(Mp) +(2) +≤ dim(Mp) +(3) +≤ idRp(Mp) +(4) += depth(Rp) +(5) += grade(p, R), +where + +10 +M. ASGHARZADEH +1): see [18, Exercise 16.5]; +2): see [10, Proposition 1.2.12]; +3), 4): these are in [10, Theorem 3.1.17]; +5): see [10, Theorem 2.1.3(b)] and recall that R is Cohen-Macaulay. +The proof is now complete. +□ +Let us extend Proposition 4.1 by a new argument: +Proposition 4.2. Let M be finitely generated, and of finite injective dimension. Then any M- +regular element is R-regular. +Proof. Let x ∈ m be M-regular. Then x is regular over M ⊗R �R, as the completion is flat. It easy +to see id � +R(M ⊗R �R) < ∞. In sum, �R is Cohen-Macaulay and accepts the canonical module. Recall +that +Ass � +R +� +Hom � +R(ω � +R, M ⊗R �R) +� +(1) += Supp(ω � +R) ∩ Ass � +R(M ⊗R �R) +(2) += Spec( �R) ∩ Ass � +R(M ⊗R �R) +(3) += Ass � +R(M ⊗R �R), +where +1): see [10, 1.2.27]; +2): see [10, Theorem 3.3.5(b)]; +3): trivial. +Since +Zd(Hom � +R(ω � +R, M ⊗R �R)) += ∪q∈Ass(Hom � +R(ω � +R,M⊗R � +R))q += ∪q∈Ass(M⊗R � +R)q += Zd(M ⊗R �R), +and x /∈ Zd(M ⊗R �R), we deduce that x is regular over Hom � +R(ω � +R, M ⊗R �R). It is easy to see +from [22] that pd � +R +� +Hom � +R(ω � +R, M ⊗R �R) +� +is finite. Also, see [10, 9.6.5(b)]. Following Auslander’s +zero-divisor, x is �R-regular, see [10, Theorem 9.4.7]. Since R ⊆ �R, x is R-regular. This is what we +want to prove. +□ +Let us present a modern proof of Proposition 2.2: +Corollary 4.3. Suppose min(R) is singleton. If for some p ∈ Spec(R) one has idR(p) < ∞, then +R is an integral domain. +Proof. We know R is Cohen-Macaulay, and in particular, min(R) = Ass(R). Let min(R) = {P}. +Let π : R → Rp be the natural localization map, sending r to r/1. Let r ∈ ker(π). By definition, +there is x ∈ R \ p such that rx = 0. We claim that x is p-regular. Indeed, if not, then +x ∈ Zd(p) = ∪q∈Ass(p)q ⊆ ∪q∈Ass(R)q = P +Since min(R) = {P}, we see P ⊆ p. So, x ∈ p. This contradiction shows that x is p-sequence. +Now, we apply the assumption along with Proposition 4.2 to observe that x is R-sequence. From + +MODULES OF FINITE INJECTIVE DIMENSION +11 +this r = 0, i.e., π is injective. Now recall from idR(p) < ∞ that Rp is regular and local. Regular +local rings are domain (see [10, Proposition 2.2.3]). From this, Rp is an integral domain. Since +R ⊆ Rp we get to the desired claim. +□ +Let us reprove Auslander’s zero-divisor conjecture, over Cohen-Macaulay rings: +Observation 4.4. Let (R, m) be Cohen-Macaulay, M be finitely generated with pdR(M) < ∞. If +x is M-regular, then x is R-regular. +Proof. We may and do assume that R is complete. In particular, ωR exists. The assumption +pdR(M) < ∞ implies that idR(M ⊗R ωR) < ∞. We proved in the previous proposition that +Ass(M ⊗R ωR) = Ass(HomR(ωR, M ⊗R ωR)). We are going to use [22, Theorem 2.9] to deduce +that HomR(ωR, M ⊗R ωR) ∼= M. Combining this along with the previous observation, we know +that Ass(M ⊗R ωR) = Ass(M). So, x is (M ⊗R ωR)-regular. Thanks to Proposition 4.1, x is +R-regular, as claimed. +□ +Remark 4.5. i) Recall that [4] asks the computation of associated prime ideals of tensor product, +as a sample, we proved in the setting of Observation 4.4 that +Ass(M ⊗R ωR) = Ass(M). +In particular, we reproved [4, Corollary 4.6] by a different argument. +ii) By symmetry, and in the proof of Observation 4.4, one may use [4, Corollary 4.6] instead of +[22, Theorem 2.9]. +Recall from [7] module T is said to be tor-rigid if there is a non-negative integer n such that for +every finitely generated R-module M, vanishing of TorR +n (T, M) implies TorR +n+i(T, M) vanishes for +all i ≥ 0. By Auslander [7, 4.3], this implies the zero-divisor property. +Remark 4.6. Despite to above useful duality between modules of finite projective dimension and +modules of finite injective dimension, let us present a different situation. It is easy to see modules of +projective dimension one are tor-rigid. But modules of injective dimension one are not necessarily +tor-rigid. This may appear even over 1-dimensional integral domains of type two. For instance, +let R := k[[x3, x4, x5]]. Then (x3, x4) is of injective dimension one, but not tor-rigid. +For more connections to these topics, we cite [4]. +5. Almost complete-intersection +In this section (S, n) is regular, I ⊳ S is an ideal, and R := S/I. We study the following: +Question 5.1. (See [15]) Suppose idR(I/I2) is finite. Is I generated by a regular sequence? +Here, we support it by the following two observations: +Observation 5.2. If R is zero-dimensional, then I is generated by a regular sequence. + +12 +M. ASGHARZADEH +Proof. The R-module I/I2 is injective, as injective dimension is bounded by depth. By Matlis +decomposition, I/I2 = ⊕ωR. Now, let n ≥ 0 be such that mn ̸= 0 but mn+1 = 0. Then +mn(I + I2) = (mnI + I2) ⊆ mn+1 + I2 = 0. +From this, mnωR = 0. As ωR is faithful, we conclude that n = 0. In other words, R is field. So, I +is generated by a regular sequence. +□ +Corollary 5.3. Question 5.1 reduces to locally complete-intersection ideals over the punctured +spectrum. +Proof. We proceed by induction on d := dim R. The case d = 0 is subject of Observation 5.2. +So, we may assume d > 0 and suppose the desired claim holds for rings of dimension < d. The +assumptions of Question 5.1 behave well with respect to localization for all q ∈ Var(I) \ {n}, and +note that dim(Rq) < d. These allow us to apply the inductive step and assume in addition that I +is locally complete-intersection over the punctured spectrum. This completes the proof. +□ +Observation 5.4. Suppose I is prime, µ(I) ≤ ht(I) + 1 and idR(I/I2) < ∞. Then I is generated +by a regular sequence. +Proof. By Bass’ conjecture, R is Cohen-Macaulay. Then KR = ωR is of finite injective dimension. +Suppose on the way of contradiction that µ(I) = ht(I)+ 1. In the light of [20, Proposition 1] there +is an exact sequence +0 −→ KR −→ Rn −→ I/I2 −→ 0, +which show that idR(R) < ∞. Over Gorenstein rings, finite injective dimension implies finite +projective dimension. So, pdR(I/I2) is finite. Follows from [28] that I is generated by a regular +sequence, as claimed. +□ +Let us present a new proof of [17]: +Theorem 5.5. (Kunz) Almost complete intersection domains are not Gorenstein rings. +Proof. We adopt the previous notation. Suppose on the way of contradiction that R is Gorenstein. +Then from 0 → KR → Rn → I/I2 → 0, and finiteness of idR(R), we conclude that idR(I/I2) < ∞. +By the previous result, ht(I) = µ(I), a contradiction. +□ +Corollary 5.6. Suppose I is prime and generated by 3 elements. If idR(I/I2) < ∞, then I is +generated by a regular sequence. +Proof. We have ht(I) ≤ 3. In the case ht(I) = 1 there is nothing to prove, because it becomes +principal as S is UFD. In the case ht(I) = 3 the claim is in [10, 1.2.21]. Then we may assume that +ht(I) = 2. The desired claim follows by Observation 5.4. +□ +Observation 5.7. If R is Gorenstein and idR(I/I2) < ∞, then I is generated by a regular +sequence. + +MODULES OF FINITE INJECTIVE DIMENSION +13 +Proof. Since R is Gorenstein, pdR(I/I2) < ∞. In view of [8, main result] I generated by a regular +sequence. +□ +For more connections to [15], we cite [3]. +6. Matlis and canonical module +A ring R of dimension d := dim(R) is called quasi-Gorenstein, if Hd +m(R) ∼= ER(k). When is +Hd +m(R) ։ ER(k) surjective? The following may regard as a higher version of [19, 15.17]: +Proposition 6.1. Suppose R is complete, generically Gorenstein and Cohen-Macaulay. +Then +Hdim R +m +(R) ։ ER(k) is surjective. +Proof. Let d := dim(R). Since the ring is complete and Cohen-Macaulay, we know ωR exists. Since +R is generically Gorenstein, ωR ⊳ R. Suppose first that d := dim R = 0. Then R is Gorenstein, +and so ωR = ER(k) = R. To see Hd +m(R) ։ ER(k), it remains to note that Hd +m(R) = R. So, +we may assume that d > 0. +Either ωR = R, or ωR ⊳ R is of height one. +Without loss of +the generality we may assume that ωR ⊳ R is of height one. The Cohen-Macaulay assumption +says 1 = ht(ωR) = grade(ωR, R). Let x ∈ ωR be a regular element. This gives an embedding +R ֒→ Rx ⊆ ωR. Apply HomR(−, ER(k)) to it, and use local duality, we observe that +ω∨ +−−−−→ +R∨ +−−−−→ 0 +� +∼= +� +Hd +m(R) −−−−→ ER(k). +In other words, the induced map Hd +m(R) ։ ER(k) is surjective, as claimed. +□ +Proposition 6.2. d := dim(R). Suppose Hd +m(R) ։ ER(k). If d = 0, then R is Gorenstein. +Proof. Recall that +Hd +m(R) −−−−→ ER(k) −−−−→ 0 += +� += +� +R +f +−−−−→ ER(k). +Let I := ker(f). So, R/I = ER(k). Recall that +R +⊆ +−→ E(R) = ⊕ER(k) = ⊕R/I. +From this, I = 0. Thus, R = ER(k). Consequently, R is Gorenstein. +□ +Let R be a 1-dimensional integral domain and let Q be the fraction field. Matlis proved that +Q/ωR = ER(k). Here, is the higher version (recall that Q/ωR = H1 +m(ωR)): +Proposition 6.3. Suppose R is complete, integral domain and Cohen-Macaulay. +Let I be a +nonzero Cohen-Macaulay ideal. Let d := dim(R). The following are equivalent: +i) idR(I) < ∞, +ii) I = ωR, + +14 +M. ASGHARZADEH +iii) Hd +m(I) = ER(k). +Proof. i) ⇒ ii): Since dim I = d, the ideal I becomes maximal Cohen-Macaulay. The desired +claim is in [10, 3.3.28]. +ii) ⇒ iii): Due to the right exactness, Hd +m(ωR) = Hd +m(R)⊗RωR. Recall that Hd +m(ωR) is artinian, +and apply Matlis duality to it, +Hd +m(ωR) += Hd +m(ωR)∨∨ += HomR +� +Hd +m(R) ⊗R ωR, ER(k) +�∨ += HomR +� +ωR, HomR(Hd +m(R), ER(k) +�∨ += HomR(ωR, ωR)∨ += R∨ += ER(k). +iii) ⇒ i): Let D(−) := HomR(−, ωR). We apply local duality along with the assumption to deduce +that: +D(I) = Hd +m(I)∨ = ER(k)∨ = �R = R. +Over maximal Cohen-Macaulay, D2(−) = Id(−). Then I ∼= D2(I) ∼= D(R) ∼= ωR. Since ωR is of +finite injective dimension, I is as well. +□ +Corollary 6.4. Let R be a 1-dimensional integral domain and let Q be the fraction field. Let I ⊳R +be nonzero. Then Q/I = ER(k) iff I = ωR. +The following was proved by Matlis in 1-dimensional complete Gorenstein domain case, and +by Auslander [5] in the m-adic complete case. Since being complete in m-adic topology implies +completeness in R-topology, so it extends both: +Observation 6.5. Let R be a quasi-local domain complete in R-topology. Then Ext1 +R(Q/R, R) ∼= +R. +Proof. Completeness in R-topology implies Ext1 +R(Q, R) = 0. Also, HomR(Q, R) = 0. The sequence +0 → R → Q → Q/R → 0 implies +0 = HomR(Q, R) → HomR(R, R) → Ext1 +R(Q/R, R) → Ext1 +R(Q, R) = 0, +i.e., the claim follows. +□ +Noetherian local rings are reduced* by Krull’s intersection theorem. So, the following is converse +to Auslander [5]: +Observation 6.6. Let R be a quasi-local reduced domain such that Ext1 +R(Q/R, R) ∼= R. Then R +is complete in R-topology. +*Sorry, we used “reduced” for different concepts. + +MODULES OF FINITE INJECTIVE DIMENSION +15 +Proof. The sequence 0 → R → Q → Q/R → 0 implies +0 = HomR(Q, R) → HomR(R, R) → Ext1 +R(Q/R, R) → Ext1 +R(Q, R) → Ext1 +R(R, R) = 0. +Recall that Ext1 +R(Q, R) = ⊕IQ is a Q-vector space. We localize the above sequence and use the +assumption Ext1 +R(Q/R, R) ∼= R to deduce that Ext1 +R(Q, R) = ⊕IQ = 0. The reduced assumption +implies that 0 → R → lim +←−r∈R R/rR → Ext1 +R(Q, R) → 0 is exact. Since Ext1 +R(Q, R) = 0, we have +R = lim +←−r∈R R/rR, as claimed. +□ +References +[1] Y. Aoyama, Complete local (Sn−1) rings of type n ≥ 3 are Cohen-Macaulay, Proc. Japan Acad. Ser. A 70 +(1994), no. 3, 80–83. +[2] M. Asgharzadeh, Finite support of tensor products, arXiv:1902.10509 [math.AC]. +[3] M. Asgharzadeh, Reflexivity revisited, arXiv:1812.00830 [math.AC]. +[4] M. Asgharzadeh, A note on Cohen-Macaulay descent, arXiv:2011.04525 [math.AC]. +[5] M. Auslander, Comments on the functor Ext, Topology 8 (1969), 151–166. +[6] M. Auslander, Remarks on a theorem of Bourbaki, Nagoya Math. J. 27 (1966), 361–369. +[7] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961) 631-647. +[8] Benjamin Briggs, Vasconcelos’ conjecture on the conormal module, Invent. Math. 227 (2022), no. 1, 415–428. +[9] H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. +[10] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge +University Press, Cambridge, 1993. +[11] David A. Buchsbaum, David Eisenbud, Some structure theorems for finite free resolutions, Advances in Math. +12 (1974), 84–139. +[12] Olgur Celikbas, Uyen Le, Hiroki Matsui, On the depth and reflexivity of tensor products, J. Algebra 606 (2022), +916–932. +[13] H.-B. Foxby, On the µi in a minimal injective resolution II, Math. Scand. 41 (1977), 19-44. +[14] M. Raynaud and L. Gruson, Crit`eres de platitude et de projectivit´e. Techniques de ”platification” d’un module, +Invent. Math. 13 (1971), 1–89. +[15] Rafael Holanda, Cleto B. Miranda-Neto, Vanishing of (co)homology, freeness criteria, and the Auslander- +Reiten conjecture for Cohen-Macaulay Burch rings, arXiv:2212.05521. +[16] D. Hanes and C. Huneke, Some criteria for the Gorenstein property, J. Pure Appl. Algebra 201 (2005), no. +1-3, 4–16. +[17] E. Kunz, Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111–115. +[18] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Math, 8, (1986). +[19] E. Matlis, 1-dimensional Cohen-Macaulay Rings. Lecture Notes in Mathematics, Vol. 327. Springer-Verlag, +Berlin-New York, 1973. +[20] T. Matsuoka, On almost complete intersections, Manuscripta Math. 21 (1977), 329-340. +[21] Matthew Miller, Bourbaki’s theorem and prime ideals, J. Algebra 64 (1980), no. 1, 29–36. +[22] Rodney Y.Sharp, Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings, +Proc. London Math. Soc. (3) 25 (1972), 303–328. +[23] Christian Peskine and Lucien Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. IHES. 42 +(1973), 47–119. +[24] P. Roberts, Le th´eor`eme d’ intersection, C. R. Acad. Sci. Paris Ser. I Math., 304 (1987), 177-180. +[25] P. Roberts, +Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics, vol. 133, +Cambridge University Press, Cambridge (1998). +[26] Wolmer V. Vasconcelos, Quasi-normal rings, Illinois J. Math. 14 (1970), 268–273. + +16 +M. ASGHARZADEH +[27] Wolmer V. Vasconcelos, Divisor theory in module categories, North-Holland Mathematics Studies 14 (North- +Holland Publishing Co., Amsterdam, 1974). +[28] Wolmer V. Vasconcelos, On the homology of I/I2, Comm. Algebra 6 (1978), no. 17, 1801-1809. +M. Asgharzadeh +Email address: mohsenasgharzadeh@gmail.com + diff --git a/ZdAzT4oBgHgl3EQfK_sm/content/tmp_files/load_file.txt b/ZdAzT4oBgHgl3EQfK_sm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de2b024b59721010e0164c0f78d8a20a3f018631 --- /dev/null +++ b/ZdAzT4oBgHgl3EQfK_sm/content/tmp_files/load_file.txt @@ -0,0 +1,869 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf,len=868 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='01105v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='AC] 3 Jan 2023 MODULES OF FINITE INJECTIVE DIMENSION MOHSEN ASGHARZADEH Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We study some aspects of distinguished finitely generated modules of finite injec- tive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Following Bass’ conjecture, this implies that the ring is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We derive some properties of rings from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Our list of properties contains the reduced, domain, normality, and regularity (complete-intersection, Gorensteiness, etcetera) of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' A family of ideals of small height 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Modules with few generators 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Auslander’s zero-divisor conjecture 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Almost complete-intersection 11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Matlis and canonical module 13 References 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Introduction Bass [9] conjectured that the existence of a finitely generated module of finite injective dimension imposes some restrictions on the ring, namely the Cohen-Macaulay property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Peskine-Szpiro [23] proved Bass’ conjecture in many cases, and the conjecture has been settled by Roberts [24] in the full setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Despite this, it is referred to as Bass’ conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The work of Peskine-Szpiro contains a lot of related beautiful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, they searched for more restrictions on the ring if one focuses on very special modules of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We also note that Matlis was interested in the ideals of injective dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' These are our motivations to do this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let p be in Spec(R), and suppose pdR(p) < ∞, equivalently pdR(R/p) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Peskine-Szpiro [23, Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3] proved that R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It is easy to see the ring is domain if idR(R/p) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' A natural question arises: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' What can say about R if idR(p) < ∞?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Some of the results from Section 2 are summarized in the following item: 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 13C14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 13H10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 13D22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Cohen-Macualay rings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' injective dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' integral domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' regular (Gorenstein) rings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Auslander’s zero-divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' regular sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH 1) Suppose that idR(p) < ∞ for all p ∈ Assh(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 2) Suppose min(R) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If idR(p) < ∞ for some p ∈ Spec(R), then R is domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 3) Suppose that idR(p) < ∞ for all p ∈ Spec(R) of height at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 4) Let R be excellent UFD containing k and of dimension > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for all p ∈ Spec(R) of height two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We also present a quasi-version of the reduced and normal items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' For example, we show: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for some p ∈ Spec(R) of positive height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is quasi-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose we allow more ideals are of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' For instance, suppose that idR(a) < ∞ for all ideals of big-height at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then, we apply our displaced results plus a famous result of Auslander [6] to show R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In Section 3 we focus on modules of finite injective dimensions with few generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In fact, Vaconcelos asked in [27, Page 51] “how do Gorenstein rings arise?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=', and he studied the question by looking at certain modules with few number of generators, see [27, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Our first result in §3 is as follows: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be quasi-normal, and let M be such that i) idR(M) < ∞, ii) M is reflexive, iii) µ(M) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' As an application, we extend Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3 to the case of four generated modules over rings of type at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Also, we present some few comments to §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then, we slightly extend [27, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='40] to more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose M is a nonzero tor-rigid module, Auslander [7, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3] proved each M-regular sequence is an R-regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' As modules of finite projective dimension are not in general tor-rigid, he conjectured (proved by Roberts [25, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3]) that the above property is true for modules of finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This is referred to as Auslander’s zero-divisor conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In Section 4 we present the injective analogue of Auslander’s zero-divisor conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We do this via two different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We use these to reconstruct Auslander’s zero-divisor conjecture over Cohen-Macaulay rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Also, this enables us to reprove some things from §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In Section 5 we deal with Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' (See [15]) Suppose idR(I/I2) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Is I generated by a regular sequence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We present two cases that the answer is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Namely, in the almost complete-intersection case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This immediately recovers the main result of [17] by Kunz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that [27, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='38] suggests canonical module is an important source to produce, in a construc- tive way, modules of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In fact, Matlis applied this to study the following over 1-dimensional integral domains: MODULES OF FINITE INJECTIVE DIMENSION 3 i) When is Q/R ։ ER(k)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ii) When is Q/I = ER(k)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In order to see these are related to modules of finite injective dimension, recall that ii) is equivalent to ideals of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In Section 6 we extend them to the higher dimensional situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It may be nice to recall that R is called quasi-Gorenstein, if Hdim(R) m (R) ∼= ER(k), so Matlis was interested on this and its semi-form as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Finally, we extend an observation of Auslander [5] about cohomology of Q/R and present its converse part, see Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' For all unexplained notation and definitions see the book [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' A family of ideals of small height In this note (R, m, k) is a commutative noetherian local ring, and modules are finitely gener- ated, otherwise specialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The notation pdR(−) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' idR(−)) stands for the projective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' injective) dimension of (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We denote the set of all associated prime ideals by AssR(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose min(R) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If for some p ∈ Spec(R) one has idR(p) < ∞, then R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Following Bass’ conjecture (see [10, Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6]) R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, Ass(R) = min(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose min(R) = {P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By definition, there is an x ∈ R such that (0 :R x) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Also, recall that x ∈ Zd(R) = � q∈Ass(R) q = � q∈min(R) q = P = (0 : x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In other words, x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since min(R) is singleton, P ⊆ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since idR(p) < ∞, and due to [10, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='9], we know that idRp(pRp) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' According to Auslander-Buchsbaum-Serre, Rp is regular (see [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since Rp is regular its localizations are as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, RP = (Rp)P Rp is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that x ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since x/1 ∈ PRP , it should be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' There is an y ∈ R \\ P such that xy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since xy = 0 we have y ∈ (0 : x) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This contradiction says that x /∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since the ring is local, x is unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From this P = (0 : x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Consequently, R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ We denote the set of all associated prime ideal p with the property that dim R/p = dim R, by AsshR(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for all p ∈ Assh(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, R is equidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let Assh(R) = Ass(R) = min(R) = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' , pt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In view of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2 we may assume t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let P ∈ min(�t i=1 pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' There is an i such that pi ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It follows that P is minimal over pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thus, P = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since idR(pi) < ∞, idRpi(piRpi) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH According to Auslander-Buchsbaum-Serre, Rpi is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then ( t� i=1 pi)RP ⊆ PRP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since min(−) is the minimal part of Supp(−), we proved that Supp( t� i=1 pi) = ∅, and so �t i=1 pi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Therefore, R is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By (Gi) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' (Ri)) we mean Rp is Gorenstein (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' regular) for all p ∈ Spec(R) of height at most i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that a module M satisfies (Si) if depth(Mp) ≥ min{i, dim(Mp)} for all p ∈ Spec(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for some p ∈ Assh(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is quasi-reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Following Bass’ conjecture, R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, R satisfies Serre’s condition (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let q ∈ Ass(R) = min(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We have two possibilities i) q = p or ii) q ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' i) In the first case, Rq is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' To see this, recall that idRq(qRq) = idRp(pRp) is finite, because idR(p) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This in turn implies that Rq is zero-dimensional local and regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From this, Rq is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ii) In the second case, Rq is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Indeed, we have Rq = pRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Consequently, idRq(Rq) = idRq(pRq), and recall that the later is finite, because p is of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By this, Rq is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In both cases, Rq is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It remains to note that (S1) and (G0) imply the quasi-reduced condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for all p ∈ Spec(R) of height at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall from the assumptions that R satisfies Serre’s condition (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let p ∈ Spec(R) be of height at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since idR(p) < ∞, idRp(pRp) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' According to Auslander-Buchsbaum- Serre, Rp is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, R satisfies (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It remains to note that if a ring satisfies Serre’s condition (S2) and (R1) is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ We cite [26] for basics of quasi-normal rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for some p ∈ Spec(R) of positive height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is quasi-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' As Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6, the ring R satisfies (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let q ∈ Spec(R) be of height at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We may assume that p in not maximal, otherwise the ring becomes regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We have two possibilities i) q ⊆ p or ii) q ⊈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' MODULES OF FINITE INJECTIVE DIMENSION 5 i) We localize the ring at p and by using the assumption idR(p) < ∞, we know that Rp is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since q ⊆ p, we have Rq ∼= (Rp)qRp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since localization of a regular ring is regular we deduce that Rq is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, R satisfies (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ii) Suppose q ⊈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We claim that p ⊈ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Indeed, suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then p ⊆ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Due to the assumption ht(p) ≥ 1, and the fact that ht(q) ≤ 1, we deduce that p = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This contradicts the assumption from ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, p ⊈ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This gives us the natural isomorphism pRq ∼= Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since idR(p) < ∞, its localizations are as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, idRq(Rq) = idRq(pRq) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By definition Rq is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, R satisfies (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In both cases R is (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It remains to note that if a ring satisfies (S2) and (G1), then it is quasi-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for some p ∈ Spec(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is quasi-reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If p is in Assh(R), then the desired claim is subject of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, without loss of generality, we may assume that dim(R) > 0 and that p ∈ Spec(R) \\ Assh(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Bass’ conjecture, R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since the ring is equi-dimensional we have ht(p) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It remains to apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m, k) be excellent UFD containing k and of dimension > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(p) < ∞ for all p ∈ Spec(R) of height two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This is straightforward to see R satisfies Serre’s condition (S3) and (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let M := Syz2(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It is reflexive and locally free over the punctured spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' According to the Miller’s paper [21], there is a free module F and a prime ideal p of height at most two that fits in the following Bourbaki sequence 0 −→ F −→ M −→ p −→ 0 (∗) Recall that R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to a result of Murthy [10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='19] R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This shows that idR(F) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We apply this in (∗) to see idR(M) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thus, idR(k) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to Auslander-Buchsbaum-Serre, R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ What about rings of mixed characteristic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' More formally, we ask: Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Find a mixed characteristic analogue of Bourbaki sequence and prime ideal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' * If we allow more ideals are of finite injective dimension, then we present the following 3 positive answers: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(a) < ∞ for all ideals of big-height at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6 R is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We assume that dim R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to Bass, R is Cohen- Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' For any nonzero x ∈ m, (xR) is height-unmixed and of height one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By assumption We are interested on it, even if the ring is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH idR(xR) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since R is domain, xR ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thus, idR(R) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In other words, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This allows us to assume that pdR(a) < ∞ for all ideal a of big-height at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This plus the normality and Cohen-Macaulay enable us to apply a result of Auslander [6, Corollary 5], and deduce that R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m, k) be UFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose that idR(a) < ∞ for all unmixed ideal of height two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We may assume that dim(R) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let x, y be a parameter sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It is height unmixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By assumption, idR(x, y) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Again, Bass’ conjecture says that the ring is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By the mentioned result of Murthy, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, pdR(a) < ∞ for all unmixed ideal of height two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It remains to apply the discussion below [6, Corollary 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Here, is another variation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='11: Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose any two generated ideal of R is of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is a Gorenstein UFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The ring R := (1) is of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' * Over Gorenstein rings, finite injective dimension implies finite projective dimension (see [10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Now, result is clear by [11, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Modules with few generators We start by the following easy exercise: Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let R be a local domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose there is a cyclically presented module M of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let M = R/xR for some x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Set i := idR(M) < ∞ and look at 0 → R x −→ R → M → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This gives Exti+1 R (k, R) x −→ Exti+1 R (k, R) → Exti+1 R (k, M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Nakayama, Exti+1 R (k, R) = 0, and so idR(R) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By definition, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ The following is nontrivial: Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' (Peskine-Szpiro) Suppose there is a cyclic module M of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If for some p ∈ Spec(R) one has idR(p) < ∞, then R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We have pdR(p) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose for some principal ideal p ∈ Spec(R) one has idR(p) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Alternatively, use [23, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' MODULES OF FINITE INJECTIVE DIMENSION 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3, R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Here, we present a new proof of [27, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2] and [22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='11(iv)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let I ⊳ R be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If idR(R/I) < ∞ then I has an R-regular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By [10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='25], pdR(R/I) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Equivalently, I has a finite free resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By a famous result of Burch [10, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7] I contains a regular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By µ(−) we mean the minimal number of elements that needs to generates a finitely generated module (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be quasi-normal, and let M be such that i) idR(M) < ∞, ii) M is reflexive, iii) µ(M) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2 we may assume that M is not cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In the light of Bass’ conjecture, R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let d := dim(R), and x := x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' , xd be a system of parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In the light of [23, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4] we have Extd R(HomR(k, R/xR), M) ∼= k ⊗R M xM ∼= M mM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The notation r(R) stands for type of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We deduce by this that ⊕r(R) Extd R(k, M) ∼= k ⊕ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let us denote µi(M) for the ith Bass number of M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=', the number of copies of ER(k) in the minimal injective resolution of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Following this terminology, we deduce µd(M)r(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In the case r(R) = 1, there is nothing to proof, as the rings of type one are Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, without loss of generality, r(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This implies that µd(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since M is torsion-free, it is of maximal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then, in view of [13, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='c] we observe that M is maximal Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Without loss of generality we may and do assume that the ring is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Over Cohen-Macaulay and complete ring the canonical module exists, as it is unique, we denoted it by ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since M is maximal Cohen-Macaulay and of finite injective dimension, we use a result of Sharp [22] to present the identification M = ⊕ωR (also, see [10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since µ(M) = 2 and µ(ωR) = r(R) = 2, we deduce that M = ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By (−)∗ we mean HomR(−, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to [16, Appendix], there is an exact sequence 0 −→ ω∗ −→ R2 −→ ωR −→ 0 (∗) Recall that Ext1 R(R2, R) = 0, and apply HomR(−, R) to the displayed short exact sequence (∗), gives us the following: 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH 0 −−−−→ ω∗ −−−−→ R2 −−−−→ ω∗∗ −−−−→ Ext1 R(ωR, R) −−−−→ 0 = �\uf8e6\uf8e6 = �\uf8e6\uf8e6 = �\uf8e6\uf8e6 0 −−−−→ ω∗ −−−−→ R2 −−−−→ ω −−−−→ 0 From this we conclude that Ext1 R(ωR, R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Now, we are going to apply [2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3] and to deduce that R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be quasi-normal of type at most two, and let M be an indecomposable module such that i) idR(M) < ∞, ii) M is reflexive, iii) µ(M) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We may assume that µ(M) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Also, we may assume that r(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By the above argument, 2µd(M) = µd(M)r(R) = µ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From this, µ(M) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In fact µ(M) = 4 and so µd(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that M is (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' According to [1, Theorem 3] we know M is Cohen-Macaulay, and so maximal Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since it is of finite injective dimension, M = ⊕ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since M is indecomposable, M = ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' But, 4 = µ(M) = µ(ωR) = r(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be quasi-normal of type two, and let M be a module such that idR(M) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then µ(M) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The following is dual to [25, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4]: Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose there is a finite length module A of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We may assume the ring is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Matlis, M := HomR(A, ER(k)) is of finite length and of finite flat dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By [14], pdR(M) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since ℓ(M ⊗ R) < ∞, and by intersection theorem (see for example [10, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5]), dim R ≤ pdR(M) = depth(R) − depth(M) = depth(R), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ The following slightly extends [25, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4]: Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose there is an artinian module A of finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This is similar to the previous item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ The following drops the Cohen-Macaulay assumption from [27, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='40]: MODULES OF FINITE INJECTIVE DIMENSION 9 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be a 2-dimensional ring, I an ideal of grade two and finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then r(R/I) = r(R)(µ(I) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall from 2 = dim R ≥ ht(I) ≥ grade(I, R) = 2 that I is m-primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From 0 → I → R → R/I → 0 we deduce that the artinian module R/I is of finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='11, R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The exact sequence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' → Hi m(I) → Hi m(R) → Hi m(R/I) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' gives us depth(I) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By the Cohen-Macaulay property and Auslander-Buchsbaum formula, pdR(I) = depth(R) − depth(I) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' These allow us to apply [27, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='40] and conclude r(R/I) = r(R)(µ(I) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Peskine-Szpiro [23, Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7] study the case µ(I) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Here, is the case µ(I) = 3: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be a 2-dimensional ring, I an ideal of grade two and finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If µ(I) = 3, then: i) R is Gorenstein, ii) R/I is Cohen-Macaulay and of type two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' i) Recall that R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let x = x1, x2 be a parameter sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then Ext2 R(HomR(k, R/xR), I) ∼= I mI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This gives r(R)µ2(I) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that depth(I) = 1 < 2 = dim(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='8] we know that µ2(I) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Plugging this in the previous equality yields that r(R) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=', R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ii) Since R/I is zero-dimensional, it is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Due to the first item we know R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This yields that pdR(I) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='12 R/I is of type two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Auslander’s zero-divisor conjecture As a basic tool, we prepare a version of Auslander’s zero-divisor conjecture for modules of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This has its own importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let M be finitely generated, and of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then any M- regular element is R-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By [12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6] it is enough to show that grade(p, M) ≤ grade(p, R) ∀p ∈ Spec(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Following Bass’ conjecture, R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then grade(p, M) (1) ≤ depth(Mp) (2) ≤ dim(Mp) (3) ≤ idRp(Mp) (4) = depth(Rp) (5) = grade(p, R), where 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH 1): see [18, Exercise 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 2): see [10, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 3), 4): these are in [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 5): see [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3(b)] and recall that R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Let us extend Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1 by a new argument: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let M be finitely generated, and of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then any M- regular element is R-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let x ∈ m be M-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then x is regular over M ⊗R �R, as the completion is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It easy to see id � R(M ⊗R �R) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In sum, �R is Cohen-Macaulay and accepts the canonical module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that Ass � R � Hom � R(ω � R, M ⊗R �R) � (1) = Supp(ω � R) ∩ Ass � R(M ⊗R �R) (2) = Spec( �R) ∩ Ass � R(M ⊗R �R) (3) = Ass � R(M ⊗R �R), where 1): see [10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 2): see [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5(b)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 3): trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since Zd(Hom � R(ω � R, M ⊗R �R)) = ∪q∈Ass(Hom � R(ω � R,M⊗R � R))q = ∪q∈Ass(M⊗R � R)q = Zd(M ⊗R �R), and x /∈ Zd(M ⊗R �R), we deduce that x is regular over Hom � R(ω � R, M ⊗R �R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It is easy to see from [22] that pd � R � Hom � R(ω � R, M ⊗R �R) � is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Also, see [10, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Following Auslander’s zero-divisor, x is �R-regular, see [10, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since R ⊆ �R, x is R-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This is what we want to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Let us present a modern proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose min(R) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If for some p ∈ Spec(R) one has idR(p) < ∞, then R is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We know R is Cohen-Macaulay, and in particular, min(R) = Ass(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let min(R) = {P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let π : R → Rp be the natural localization map, sending r to r/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let r ∈ ker(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By definition, there is x ∈ R \\ p such that rx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We claim that x is p-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Indeed, if not, then x ∈ Zd(p) = ∪q∈Ass(p)q ⊆ ∪q∈Ass(R)q = P Since min(R) = {P}, we see P ⊆ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, x ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This contradiction shows that x is p-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Now, we apply the assumption along with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2 to observe that x is R-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From MODULES OF FINITE INJECTIVE DIMENSION 11 this r = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=', π is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Now recall from idR(p) < ∞ that Rp is regular and local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Regular local rings are domain (see [10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From this, Rp is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since R ⊆ Rp we get to the desired claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Let us reprove Auslander’s zero-divisor conjecture, over Cohen-Macaulay rings: Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let (R, m) be Cohen-Macaulay, M be finitely generated with pdR(M) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If x is M-regular, then x is R-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We may and do assume that R is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, ωR exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The assumption pdR(M) < ∞ implies that idR(M ⊗R ωR) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We proved in the previous proposition that Ass(M ⊗R ωR) = Ass(HomR(ωR, M ⊗R ωR)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We are going to use [22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='9] to deduce that HomR(ωR, M ⊗R ωR) ∼= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Combining this along with the previous observation, we know that Ass(M ⊗R ωR) = Ass(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, x is (M ⊗R ωR)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thanks to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1, x is R-regular, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' i) Recall that [4] asks the computation of associated prime ideals of tensor product, as a sample, we proved in the setting of Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4 that Ass(M ⊗R ωR) = Ass(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In particular, we reproved [4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6] by a different argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ii) By symmetry, and in the proof of Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4, one may use [4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6] instead of [22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall from [7] module T is said to be tor-rigid if there is a non-negative integer n such that for every finitely generated R-module M, vanishing of TorR n (T, M) implies TorR n+i(T, M) vanishes for all i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Auslander [7, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3], this implies the zero-divisor property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Despite to above useful duality between modules of finite projective dimension and modules of finite injective dimension, let us present a different situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' It is easy to see modules of projective dimension one are tor-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' But modules of injective dimension one are not necessarily tor-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This may appear even over 1-dimensional integral domains of type two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' For instance, let R := k[[x3, x4, x5]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then (x3, x4) is of injective dimension one, but not tor-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' For more connections to these topics, we cite [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Almost complete-intersection In this section (S, n) is regular, I ⊳ S is an ideal, and R := S/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We study the following: Question 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' (See [15]) Suppose idR(I/I2) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Is I generated by a regular sequence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Here, we support it by the following two observations: Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If R is zero-dimensional, then I is generated by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The R-module I/I2 is injective, as injective dimension is bounded by depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Matlis decomposition, I/I2 = ⊕ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Now, let n ≥ 0 be such that mn ̸= 0 but mn+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then mn(I + I2) = (mnI + I2) ⊆ mn+1 + I2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From this, mnωR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' As ωR is faithful, we conclude that n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In other words, R is field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, I is generated by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Question 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1 reduces to locally complete-intersection ideals over the punctured spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We proceed by induction on d := dim R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The case d = 0 is subject of Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, we may assume d > 0 and suppose the desired claim holds for rings of dimension < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The assumptions of Question 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1 behave well with respect to localization for all q ∈ Var(I) \\ {n}, and note that dim(Rq) < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' These allow us to apply the inductive step and assume in addition that I is locally complete-intersection over the punctured spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose I is prime, µ(I) ≤ ht(I) + 1 and idR(I/I2) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then I is generated by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By Bass’ conjecture, R is Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then KR = ωR is of finite injective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose on the way of contradiction that µ(I) = ht(I)+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In the light of [20, Proposition 1] there is an exact sequence 0 −→ KR −→ Rn −→ I/I2 −→ 0, which show that idR(R) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Over Gorenstein rings, finite injective dimension implies finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, pdR(I/I2) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Follows from [28] that I is generated by a regular sequence, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Let us present a new proof of [17]: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' (Kunz) Almost complete intersection domains are not Gorenstein rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We adopt the previous notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose on the way of contradiction that R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then from 0 → KR → Rn → I/I2 → 0, and finiteness of idR(R), we conclude that idR(I/I2) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' By the previous result, ht(I) = µ(I), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose I is prime and generated by 3 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If idR(I/I2) < ∞, then I is generated by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We have ht(I) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In the case ht(I) = 1 there is nothing to prove, because it becomes principal as S is UFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In the case ht(I) = 3 the claim is in [10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then we may assume that ht(I) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The desired claim follows by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If R is Gorenstein and idR(I/I2) < ∞, then I is generated by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' MODULES OF FINITE INJECTIVE DIMENSION 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since R is Gorenstein, pdR(I/I2) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In view of [8, main result] I generated by a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ For more connections to [15], we cite [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Matlis and canonical module A ring R of dimension d := dim(R) is called quasi-Gorenstein, if Hd m(R) ∼= ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' When is Hd m(R) ։ ER(k) surjective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The following may regard as a higher version of [19, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='17]: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose R is complete, generically Gorenstein and Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then Hdim R m (R) ։ ER(k) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let d := dim(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since the ring is complete and Cohen-Macaulay, we know ωR exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since R is generically Gorenstein, ωR ⊳ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose first that d := dim R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is Gorenstein, and so ωR = ER(k) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' To see Hd m(R) ։ ER(k), it remains to note that Hd m(R) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, we may assume that d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Either ωR = R, or ωR ⊳ R is of height one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Without loss of the generality we may assume that ωR ⊳ R is of height one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The Cohen-Macaulay assumption says 1 = ht(ωR) = grade(ωR, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let x ∈ ωR be a regular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' This gives an embedding R ֒→ Rx ⊆ ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Apply HomR(−, ER(k)) to it, and use local duality, we observe that ω∨ −−−−→ R∨ −−−−→ 0 �\uf8e6\uf8e6 ∼= �\uf8e6\uf8e6 Hd m(R) −−−−→ ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' In other words, the induced map Hd m(R) ։ ER(k) is surjective, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' d := dim(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose Hd m(R) ։ ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' If d = 0, then R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that Hd m(R) −−−−→ ER(k) −−−−→ 0 = �\uf8e6\uf8e6 = �\uf8e6\uf8e6 R f −−−−→ ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let I := ker(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, R/I = ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that R ⊆ −→ E(R) = ⊕ER(k) = ⊕R/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' From this, I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Thus, R = ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Consequently, R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Let R be a 1-dimensional integral domain and let Q be the fraction field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Matlis proved that Q/ωR = ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Here, is the higher version (recall that Q/ωR = H1 m(ωR)): Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Suppose R is complete, integral domain and Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let I be a nonzero Cohen-Macaulay ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let d := dim(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The following are equivalent: i) idR(I) < ∞, ii) I = ωR, 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ASGHARZADEH iii) Hd m(I) = ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' i) ⇒ ii): Since dim I = d, the ideal I becomes maximal Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The desired claim is in [10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' ii) ⇒ iii): Due to the right exactness, Hd m(ωR) = Hd m(R)⊗RωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that Hd m(ωR) is artinian, and apply Matlis duality to it, Hd m(ωR) = Hd m(ωR)∨∨ = HomR � Hd m(R) ⊗R ωR, ER(k) �∨ = HomR � ωR, HomR(Hd m(R), ER(k) �∨ = HomR(ωR, ωR)∨ = R∨ = ER(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' iii) ⇒ i): Let D(−) := HomR(−, ωR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We apply local duality along with the assumption to deduce that: D(I) = Hd m(I)∨ = ER(k)∨ = �R = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Over maximal Cohen-Macaulay, D2(−) = Id(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then I ∼= D2(I) ∼= D(R) ∼= ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since ωR is of finite injective dimension, I is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let R be a 1-dimensional integral domain and let Q be the fraction field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let I ⊳R be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then Q/I = ER(k) iff I = ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The following was proved by Matlis in 1-dimensional complete Gorenstein domain case, and by Auslander [5] in the m-adic complete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since being complete in m-adic topology implies completeness in R-topology, so it extends both: Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let R be a quasi-local domain complete in R-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then Ext1 R(Q/R, R) ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Completeness in R-topology implies Ext1 R(Q, R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Also, HomR(Q, R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The sequence 0 → R → Q → Q/R → 0 implies 0 = HomR(Q, R) → HomR(R, R) → Ext1 R(Q/R, R) → Ext1 R(Q, R) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=', the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ Noetherian local rings are reduced* by Krull’s intersection theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' So, the following is converse to Auslander [5]: Observation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Let R be a quasi-local reduced domain such that Ext1 R(Q/R, R) ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Then R is complete in R-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Sorry, we used “reduced” for different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' MODULES OF FINITE INJECTIVE DIMENSION 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The sequence 0 → R → Q → Q/R → 0 implies 0 = HomR(Q, R) → HomR(R, R) → Ext1 R(Q/R, R) → Ext1 R(Q, R) → Ext1 R(R, R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Recall that Ext1 R(Q, R) = ⊕IQ is a Q-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' We localize the above sequence and use the assumption Ext1 R(Q/R, R) ∼= R to deduce that Ext1 R(Q, R) = ⊕IQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' The reduced assumption implies that 0 → R → lim ←−r∈R R/rR → Ext1 R(Q, R) → 0 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Since Ext1 R(Q, R) = 0, we have R = lim ←−r∈R R/rR, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' □ References [1] Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Vasconcelos, On the homology of I/I2, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Algebra 6 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' 17, 1801-1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content=' Asgharzadeh Email address: mohsenasgharzadeh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAzT4oBgHgl3EQfK_sm/content/2301.01105v1.pdf'} diff --git a/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf b/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e6ffa2b8b2d3950e55d5433027c180cdf5d82615 --- /dev/null +++ b/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8b97ae00eb24717c27fa4cf2461aac451f2b125a983ab302ca7b05a4b9563e26 +size 1186848 diff --git a/_dE1T4oBgHgl3EQfDAIA/vector_store/index.faiss b/_dE1T4oBgHgl3EQfDAIA/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1b34e9345ad59cfb62cfb224c4c22b9bc52430eb --- /dev/null +++ b/_dE1T4oBgHgl3EQfDAIA/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:15718d61c2c006542b28a42cd07f304b6f6efb7513b688f57c6f64ee809fbc76 +size 8650797 diff --git 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We determine a set of necessary conditions on a partition-indexed family of com- +plex numbers to be the “highest coefficients” of a positive and symmetric multi-faced universal +product; i.e. the product associated with a multi-faced version noncommutative stochastic inde- +pendence, such as bi-freeness. The highest coefficients of a universal product are the weights of +the moment-cumulant relation for its associated independence. We show that these conditions +are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate +a (automatically unique) symmetric universal product with the prescribed highest coefficients. +Furthermore, we give a quite explicit description of such families of coefficients, thereby produc- +ing a list of candidates that must contain all positive symmetric universal products. We discover +in this way four (three up to trivial face-swapping) previously unknown moment-cumulant re- +lations that give rise to symmetric universal products; to decide whether they are positive, and +thus give rise to independences which can be used in an operator algebraic framework, remains +an open problem. +1. Introduction +At the latest with Voiculescu’s invention of freeness [Voi85], it became aparent that the “obvious” +extension of classical stochastic independence, tensor independence, is not the only and not always +the most suitable concept in inherently noncommutative situations. In fact, Boolean independence +(not yet under this name) has already featured much earlier in the work of von Waldenfels [vW73, +vW75]. Those “noncommutative independences” share many properties with classical stochastic +independence and tensor independence. +In particular, under the assumption of independence, +mixed moments are uniquely determined and can be calculated from marginal moments (also +giving rise to an associated convolution product for probability measures on the real line). Another +interesting independence is monotone independence, which was discovored by Muraki [Mur01]; this +is a non-symmetric independence relation. +An extremely useful tool when dealing with random variables which have all moments are the +corresponding cumulants. The theory of free cumulants, linearizing free additive convolution, was +initiated by Speicher [Spe94], see also the book by Nica and Speicher [NS06]. Boolean cumulants +were formalized by Speicher and Woroudi [SW97]. Understanding the monotone cumulants took +a bit longer, many questions were answered by Hasebe and Saigo [HS11]. The problem in the +monotone case is that independence is not in general characterized by vanishing of mixed cumu- +lants. This is directly related to the non-symmetric nature, as becomes apparent when interpreting +moment-cumulant relations via exponential and logarithm maps, as is done in a related but different +setting by Manzel and Schürmann [MS17] (Hopf algebraic) or Ebrahimi-Fard and Patras [EFP15] +(shuffle-algebraic); non-zero mixed cumulants can appear in the Campbell-Baker-Hausdorff series. +Since the work of Speicher [Spe97], Ben Ghorbal and Schürmann [BGS02], and Muraki [Mur02, +Mur03], we know that the five independence relations for noncommutative random variables, ten- +sor, free, Boolean, monotone and antimonotone independence, are indeed very special. For these +The work of both authors was supported by German Research Foundation (DFG) grant no. 397960675. The +work of MG was carried out during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme at NTNU +Trondheim, as a guest researcher at Saarland University in the scope of the SFB-TRR 195, and as a postdoctoral +scientific employee at University of Greifswald. The work of PV was partially carried out as a PhD student and +scientific employee at University of Greifswald. +1 +arXiv:2301.01816v1 [math.FA] 4 Jan 2023 + +2 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +independences, the joint distribution of independent random variables is obtained from the mar- +ginal distributions by means of a “universal product”, i.e. a product operation which fulfills a +number of natural conditions, including associativity and universality (i.e. in a specific sense not +dependent on the concrete realization of the noncommutative random variables) and a “factor- +ization for length 2”-condition; and they are the only ones with this property.1 Replacing that +“factorization for length 2”-condition by a positivity condition, a decade later, Muraki [Mur13] +proved a similar result with a much simpler proof, while at the same time using a much bet- +ter motivated assumption, namely that the product operation restricts to a product operation +for states on augmented ∗-algebras.2 This kind of positivity is also the right condition to study +quantum Lévy processes on dual groups in the sense of Ben Ghorbal and Schürmann [BGS05], +see also [SV14], where Schoenberg correspondence between convolution semigroups of states and +conditionally positive generators is proved in this context. In 2014, Voiculescu [Voi14] introduced +a new nontrivial extension of free independence, bifreeness, for sequences of pairs of random vari- +ables, or pairs of faces as Voiculescu called the general underlying framework. Taking up on this +idea, more examples of 2-faced or, more generally, multi-faced independences have been discovered +[Liu19, Liu18, GS19, GHS20, Ger17]. The general theory of multi-faced universal products from +which those independences can be obtained was established by Manzel and Schürmann [MS17]. It +turned out that not all of the examples fulfill the natural positivity condition. Positivity is still +enough to assure Schoenberg correspondence in this generalized setting, see [Ger21]. In an effort to +classify positive multi-faced universal products, two routes have been taken. In [GHU21], Gerhold, +Hasebe, Ulrich completely classified 2-faced universal products which have a natural representa- +tion on the tensor product or the free product Hilbert space of the GNS spaces of the factors. +In Varšo’s PhD thesis [Var21], he proved that there are at most 12 two-faced universal products +which fulfill additional assumptions of symmetry and a “combinatorial” moment cumulant relation +(i.e. determined by a subset of all two-faced partitions, where more generally weights on two-faced +partitions can appear).3 In this article we present, simplify, and extend those results of [Var21]. +A single-faced independence can trivially be regarded a two-faced independence, and every +two-faced independence is a certain kind of mixture of two-single-faced independences. However, +neither do those two single-faced independences determine the two-faced independence, nor is +it obvious that any combination of single-faced independences can be combined in any way to +form a two-faced independence.4 The main result of this article is to present a family of two- +faced symmetric universal products such that every positive symmetric two-faced universal product +must belong to that family, we call them candidates. +This is achieved in three steps. +First, +we prove necessary conditions for a family of weights on ordered partitions to be the highest +coefficients of a positive multi-faced universal product (Theorem 6.1); second, we determine all +permutation invariant weights (= weights on non-ordered partitions) which fulfill those properties +(Corollary 7.11), we call such weights here admissable; third, we prove that admissable weights +are always the highest coefficients of a (uniquely determined) symmetric multi-faced universal +product (Theorem 8.2). The family of candidates consists of (identifying an independence with its +underlying universal product, and disregarding the difference between a 2-faced independence and +its image under swapping the faces) +• 2-faced continuous 1-parameter deformations of free, tensor and bifree independence (pos- +itivity is proved in [GHU21]), +• a tensor-free independence (positivity is not known), +• a new free-free and a new tensor-tensor independence, different from the trivial ones, +bifreeness, and their deformations (positivity is not known), +1Speicher [Spe97] proved that there are only three universal calculation rules for mixed moments in the sym- +metric case. Ben Ghorbal and Schürmann [BGS02] axiomatized independences via universal products and showed +equivalence to universal calculation rules. Muraki [Mur02, Mur03] extended the results to the non-symmetric setting. +2In the purely algebraic context, i.e. without positivity, Muraki’s classification was slightly extended by Gerhold +and Lachs in [GL15], showing that there is a non-symmetric deformation of Boolean independence. +3In [Var21], it was also noticed for the first time the possibility that the moment cumulant relation of a positive +universal product might not need to be of combinatorial form, which was indeed confirmed in [GHU21]. +4Note that the study of another kind of mixture of single-faced independences was initiated by Młotkowski +[Mło04] and received again more attention after work Speicher and Wysozcański [SW16] and Ebrahimi-Fard, Patras +and Speicher [EFPS18] on the corresponding cumulants; this approach is closely related to graph products of groups +and the corresponding universal products are not associative. + +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +3 +• tensor-boolean, free-boolean and boolean independence; positivity for those is also covered +in [GHU21], for free-boolean it was first shown by Liu [Liu19] and for boolean independence +positivity is of course well-known. +We call the independences which are not realized in [GHU21], i.e. those whose positivity is yet +unknown, exceptional. +We prove many of the preliminary results for the general symmetric multi-faced case. Theo- +rem 6.1, where we find necessary conditions on weights to arise as highest coefficients of a uni- +versal product is even formulated for not necessarily symmetric products and could be used as a +starting point for a more general classification including multi-faced universal products based on +monotone independence, such as for example bimonotone independence (of type II) as defined in +[Ger17, GHS20]. +It easily follows from the main result that there are no non-trivial positive and symmetric +trace preserving universal products (Remark 7.12) and that tensor independence and bifreeness +are the only two positive symmetric 2-faced independences which allow to define a convolution of +probability measures on R2 (Remark 7.13). +Among our additional results, we characterize when a positive symmetric multi-faced universal +product is unit preserving (Theorem 9.7), i.e. when it can be defined consistently for arbitrary +unital algebras (in the other cases, the product operation is only defined for linear functionals on +augmented algebras). This is indeed the case for the three continuous families and the four (three +up to swapping the faces) exceptional cases. Furthermore, we establish a simplified mixed moment +formula for the special combinatorial case where the highest coefficients are only 0 or 1, so that +the moment cumulant relation is simply governed by a specific set of partitions (Theorem 8.4). +The outline of the article is as follows. In Sections 2 to 5, we introduce the basic concepts, in +particular multi-faced partitions and moment cumulant relations for a family of weights. We also +extract along the way the relevant special cases of Manzel and Schürmann’s cumulant theory for +universal products. In Section 6 we prove the necessary conditions for a family of weights to be the +highest coefficients of a positive multi-faced universal product (symmetric or not). In Section 7 we +show that those necessary conditions allow us to obtain a concrete list of candidates for symmetric +and positive two-faced universal products. In Section 8 we prove that in the symmetric case the +conditions are sufficient to reconstruct a universal product in the algebraic sense (with a simplified +formula in the combinatorial case), but it remains open whether these universal products are +automatically positive. Finally, we characterize in Section 9 which universal products in our list +are unit preserving. In Section 10 we name four tasks which have to be completed in order to +achieve a complete classification of positive multi-faced universal products. +2. Preliminaries and notation +We will have to deal a lot with tuples of all kinds, so we introduce some useful notation. Let +X and Y be arbitrary sets. For any natural number n, denote by [n] the set {1, . . . , n}. For +an n-tuple t = +� +t(1), . . . , t(n) +� +∈ Xn and a subset I = {i1 < . . . < ik} ⊂ [n], we define the +restricted tuple t ↾ I := +� +t(i1), . . . , t(ik) +� +. Two tuples t ∈ Xn, s ∈ Y n of the same length may +be combined to form the tuple t × s ∈ (X × Y )n with (t × s)(i) = +� +t(i), s(i) +� +, and conversely, +every tuple in (X × Y )n is of that form. The set of n-tuples of arbitrary length n is denoted +X∗ = � +n∈N0 Xn. When a set X does not carry any multiplicative structure, we might use the +word notation, t(1) . . . t(n) := +� +t(1), . . . , t(n) +� +∈ Xn. The entries of a tuple t might be written ti +instead of t(i) from time to time; or we might use t as a shorthand for (t1, . . . , tn) without further +comment when the ti have been around before. +An algebra means an associative C algebra, not necessarily unital. The free product of algebras +A1, A2 is denoted A1 ⊔ A2, reminding of the fact that this is the coproduct in the category of alge- +bras: for arbitrary algebra homomorphisms hi : Ai → B, there is a unique algebra homomorphism +h1 ⊔ h2 : A1 ⊔ A2 → B with h1 ⊔ h2 ↾ Ai = hi. +For a vector space V , we denote by T0(V ) = � +n∈N V ⊗n the (non-unital) free algebra over V . +We will identify T0(V1 ⊕V2) = T0(V1)⊔T(V2) without further commenting. The unital free algebra +is denoted T(V ) = � +n∈N0 V ⊗n, and is this unital algebra is the unitization of T0(V ). +Let F be a fixed finite set, whose elements we call faces or colors. We could of course assume +F = [m] for m ∈ N, but since there will be a lot of integers around, we prefer to use more abstract +symbols. Typical choices for |F| = 2 are F = {L, R} (for left and right) or F = {◦, •}. We typically +use the symbol • for an arbitrary element of F. + +4 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +A multi-faced algebra is an algebra A that is freely generated by given subalgebras A•, • ∈ F +(the faces of A), i.e. the canonical algebra homomorphism � +•∈F A• → A is an isomorphism; this +is indicated by writing A = � +•∈F A•. +A multi-faced algebra homomorphism j : A → B is an +algebra homomorphism between multi-faced algebras A, B with j(Ak) ⊂ Bk. We consider the +free product of multi-faced algebras again a multi-faced algebra with faces (A ⊔ B)• := A• ⊔ B•. +Note that the free product of multi-faced algebras is the coproduct in the category AlgF of multi- +faced algebras with multi-faced algebra homomorphisms, i.e. for every pair of multi-faced algebra +homomorphisms ji : Ai → B there is a unique multi-faced algebra homomorphism j1⊔j2 : A1⊔A2 → +B restricting to ji on Ai, respectively for i = 1, 2. We use the same symbol ⊔ to denote the canonical +homomorphism j1 ⊔ j2 : A1 ⊔ A2 → B1 ⊔ B2 when ji : Ai → Bi, it should always be clear from the +context which codomain is meant. +A multi-faced ∗-algebra is a multi-faced algebra with an involution such that each face is a +∗-subalgebra. Of course, the free product of multi-faced ∗-algebras is again a multi-faced ∗-algebra +in the obvious way and the free product of multi-faced ∗-homomorphisms is a ∗-homomorphism. +We say that a linear functional ϕ: A → C defined on a multi-faced ∗-algebra is a restricted state +if its unital extension to the unitization of A is a state (or, equivalently, positive). +3. Universal products +Definition 3.1 (Cf. [Ger21, Rem. 3.4]). A multi-faced universal product is a binary product +operation for linear functionals on multi-faced algebras which associates with functionals ϕ1, ϕ2 on +multi-faced algebras A1, A2, respectively, a functional ϕ1 ⊙ ϕ2 on A1 ⊔ A2 such that +• (ϕ1 ◦ j1) ⊙ (ϕ2 ◦ j2) = (ϕ1 ⊙ ϕ2) ◦ (j1 ⊔ j2) for all multi-faced algebra homomorphisms +ji : Bi → Ai (universality) +• (ϕ1 ⊙ ϕ2) ⊙ ϕ3 = ϕ1 ⊙ (ϕ2 ⊙ ϕ3) (associativity) +• (ϕ1 ⊙ ϕ2) ↾ Ai = ϕi (restriction property). +The product is called +• symmetric if ϕ1 ⊙ ϕ2 = ϕ2 ⊙ ϕ1, +• positive if the product of restricted states on multi-faced ∗-algebras is a restricted state on +the free product ∗-algebra. +Note that we made several implicit identifications between isomorphic free products in the last +definition. For a more detailed discussion see [Ger21]. +Let A1, . . . , Ak be multi-faced algebras and A = A1 ⊔ . . . ⊔ Ak (i.e. we identify the Ai with +subalgebras of their free product). For s = b × f ∈ ([k] × F)n, we denote +As := +� +a1 . . . an ∈ A : ai ∈ Af(i) +b(i) +� +. +Note that the As are not necessarily pairwise disjoint (consecutive repitition of the same letter in +the word s leads to a subset). Elements of [k]n are referred to as block structures and elements of +Fn are called face structures. +Given a multi-faced universal product ⊙, we define its linearized part as +ϕ1 ⊡ · · · ⊡ ϕk(a) := +∂k +∂t1 . . . ∂tk +(t1ϕ1) ⊙ . . . ⊙ (tkϕk)(a) +���� +t=0 +(that this expression is well-defined should be understood as part of the following theorem). +Theorem 3.2 (Adjusted and simplified from [MS17, Th. 4.2, Rem. 4.3, 4.4]). +Let ⊙ be a positive multi-faced universal product. +Then there are unique coefficients αs (s ∈ +([k] × F)n) such that, for all linear functionals ϕj : Aj → C on multi-faced algebras Aj (j ∈ [k]) +and all a ∈ As, +ϕ1 ⊡ · · · ⊡ ϕk(a) = αs · ϕ1 +� +� +→ +� +b(j)=1 +aj +� +� . . . ϕk +� +� +→ +� +b(j)=k +aj +� +� . +(1) +(The symbol +→ +� indicates that the product is to be taken in the same order as the factors aj appear +in a the tensor product.) The αs are called highest coefficients of ⊙. 5 +5In the complete expansion of the universal product, also lower coefficients would appear, according to terms +that are not multilinear in the ϕi. + +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +5 +Proof. First assume that s ∈ ([k] × F)n is alternating, i.e. s(i) ̸= s(i + 1) for i = 1, . . . , n − 1. By +[MS17, Rem. 4.3], the formula given in [MS17, Th. 4.2] can be applied, and due to the “linearization” +we performed only those terms survive that are linear in each ϕj. For a positive universal product, +[MS17, Rem. 4.3] implies that there is only one such term, corresponding to the “right-ordered +highest coefficient” (i.e. the aj are multiplied in the same order in which they appear as factors in +a) associated with s, denoted αs in this article. +If s is not alternating, then we define αs := α�s where �s is the alternating tuple obtained +from s merging repeating entries into one. By universality it is obvious that (1) extends to all +s ∈ ([k] × F)∗. +□ +Definition 3.3. A multi-faced restricted state ϕ: A → C is called trivially multi-faced if there are +∗-isomorphisms j•1,•2 : A•1 → A•2 with ϕ ↾ A•2 = ϕ•1 ◦ j•1,•2 for all •1, •2 ∈ F. +Lemma 3.4. Let ⊙ be a positive multi-faced universal product. Then, for every s ∈ ([k] × F)∗, +there are trivially multi-faced restricted states ϕ1, . . . , ϕk and an element a ∈ As with αs = ϕ1 ⊡ +· · · ⊡ ϕk(a). +Proof. Let s = b×f ∈ ([k]×F)n. Define A• +j := C and Aj := � +•∈F A• +j. Then ϕj = � +•∈F id: Aj → +C is a state, in particular a restricted state, and trivially multi-faced. Put a• +i := 1 for all i ∈ [k] +and all • ∈ F. Now it is easy to see that ϕi(a•1 +i . . . a•m +i +) = 1 for all m ∈ N and •ℓ ∈ F (ℓ ∈ [m]). +With a := af(1) +b(1) . . . af(n) +b(n), the claim now follows from Theorem 3.2. +□ +4. Partitions +In general, a multi-faced set is a set X together with a map f : X → F, the face structure of X. +The subsets X• := f −1({•}) are called the faces of X. A multi-faced subset of X is just a subset +of the underlying set viewed as a multi-faced set with respect to the restricted face structure. +In this article, we only deal with multi-faced sets whose underlying set X is finite and totally +ordered. Any word f = f(1) . . . f(n) ∈ F∗ defines such a finite and totally ordered multi-faced +set Xf = [n] with face structure k �→ f(k) (which we identify with the word f). +Conversely, +we associate with a finite totally ordered multi-faced set X = ({x1, . . . , xn}, f) the word |X| := +f(x1) . . . f(xn) ∈ F∗. We choose this on first sight odd notation because the word f plays the same +role as the number of elements of a set plays in the single-faced case in the moment-cumulant +formulas we are aiming at. +Let X be a multi-faced set and ∼ an equivalence relation such that +• the equivalence classes are intervals, +• f is constant on equivalence classes. +Then we understand the quotient X/∼ as a multi-faced set with the induced total order and face +map. +Example 4.1. We briefly discuss the two situations that will appear several times in this article. +(1) Let f ∈ Fn be a face and ∼ the equivalence relation on [n] that identifies two neighboring +points k, k + 1 in the same face, i.e. f(k) = f(k + 1). In this case we write f/(k ∼ k + 1) +for the quotient Xf/ ∼ and denote its elements i instead of {i} for the trivial equivalence +classes of i ∈ [n] \ {k, k + 1} and {k, k + 1} for the two-element equivalence class of k and +k + 1. +(2) Let X be an arbitrary multi-faced set and ∼ the equivalence relation whose equivalence +classes are the maximal intervals on which f is constant. We then call the quotient Xred := +X/∼ the reduction of X. In the reduction, neighboring points will always have different +faces, so that no further quotienting is possible. +A partition of a multi-faced set X is a collection of multi-faced subsets whose underlying sets +form a set partition. The set of all partitions of a multi-faced set X is denoted P(X). An ordered +partition of X is a partition of X together with a total order between the blocks. The set of all +ordered parititions is denoted P<(X). +For a word f ∈ F∗, we put P(f) := P(Xf) and P<(f) := P<(Xf). We also denote +P := +� +f∈F∗ +P(f), +P< := +� +f∈F∗ +P(f) +Example 4.2. Let F = {◦, •} and consider f = ◦••◦• ∈ F∗. Then π = {V1, V2} with V1 = +{1, 3, 4}, V2 = {2, 5} is an element of P(f) and we have |V1| = ◦•◦, |V2| = ••. This can be nicely + +6 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +drawn as an arc diagram, π = +. In the following we will not distinguish between a partition +and its arc diagram. In this article, we only use arc-diagrams to denote partitions in P, i.e. without +a block-order; the height of the blocks is completely arbitrary. +P(f) is a partially ordered set by the order of reverse refinement. The maximum and minimum +of P(f) are denoted 1f and 0f, respectively, i.e. 1f is the one-block partition and in 0f all blocks +are singletons. +There is a canonical bijection between P(X/∼) and the set of π ∈ P(X) such that equivalent +points of X lie in the same block of π. +For a multi-faced partition π, consider the equivalence relation ∼ on the underlying multi-faced +set X whose equivalence classes are the maximal intervals of X on which f is constant and which lie +completely inside one block of π. We define the reduction of π as the induced multi-faced partition +πred on X/∼. Then πred will not have neighboring legs that are in the same face and in the same +block. +For a multi-faced set X, we define its mirror image X as the same set with the same face +structure, but order reversed. For π ∈ P(X), we put π ∈ P(X) as the same set partition as π. +Finally, we introduce a notation for uniting blocks. Let π = {B1 < . . . < Bk} ∈ P<(X) with +blocks Bi, Bi+1 that are nearest neighbors for the order on π. Then we define πBi⌣Bi+1 := {B1 < +. . . < Bi−1 < Bi ∪ Bi+1 < . . . < Bk}. Similarly, for π ∈ P(f) and arbitrary blocks B1, B2 ∈ π, +πB1⌣B2 := π \ {B1, B2} ∪ {B1 ∪ B2}. +Let fi ∈ Fmi, i ∈ [n], be face structures and f their concatenation, i.e. f(m1 + . . . + mi−1 + ℓ) = +fi(ℓ) for all i ∈ [n], ℓ ∈ [mi]. Given partitions πi ∈ P(fi), we define their concatenation as the +partition π ∈ P(f) which has for every block V ∈ πi with i ∈ [n] a block �V := {ℓ : ℓ+�i−1 +j=1 mj ∈ V }. +Roughly speaking, π restricts to πi on the legs corresponding to fi. +5. Moment-cumulant relations +Definition 5.1. A family of complex numbers α = (απ)π∈P< is called (family of) weights on +ordered partitions, a family α = (απ)π∈P is called (family of) weights on partitions. Weights on +(ordered) partitions are called monic if απ = 1 for every one-block partition. +For a family of numbers +αs : s ∈ ([k] × F)n, k, n ∈ N +(as it is for example obtained from a universal product by Theorem 3.2) and π = {B1 < . . . < Bk} ∈ +P<(f) an ordered multi-faced partition with k blocks, we define sπ ∈ ([k] × F)n via sπ(i) := (κ, •) +if i ∈ Bκ and f(i) = • and put +απ := αsπ. +In this way, we associate with each universal product a family of weights on ordered partitions, and +we say that the weights of a universal product are its highest coefficients. Note that such weights +are always monic. +We say that weights on ordered partitions α are invariant under permutation of blocks if +α{B1<...1 +απ logα(ϕ)⊗|π|(xπ). +Note that often expα and logα are interpreted as a bijections between linear functionals on T(V ) +vanishing on 1 and unital linear functionals on T(V ) by extending ψ and exp(ψ) accordingly. +We use the following conventions. +• If the weights α come from a universal product ⊙, we write exp⊙ := expα and log⊙ := logα. +• If (x• +i )i∈I• form a basis of V •, we identify T(V ) and T0(V ) with the noncommutative +(unital or non-unital) polynomial algebras C⟨x• +i : i ∈ I•, • ∈ F⟩ and C⟨x• +i : i ∈ I•, • ∈ F⟩0, +respectively. +Definition 5.7. Let A be a multi-faced algebra and ϕ: A → C a linear functional. We define +ˆA := T0 +�� +•∈F A•� +and ˆϕ := ϕ◦µ, where µ: T0 +�� +•∈F A•� +→ A is the canonical homomorphism. +Observation 5.8. Let α be monic weights. Let furthermore ϕ: A → C be a linear functional on a +multi-faced algebra A and a = (a• +i ∈ A• : i ∈ I•, • ∈ F) a family of elements. With the notations +from the previous definitions, for X = x•1 +i1 ⊗ . . . ⊗ x•n +in , it holds that +ˆϕ(a•1 +i1 ⊗ . . . ⊗ a•n +in ) = ma(x•1 +i1 ⊗ . . . ⊗ x•n +in ) +and +logα ˆϕ(a•1 +i1 ⊗ . . . ⊗ a•n +in ) = ca(x•1 +i1 ⊗ . . . ⊗ x•n +in ). + +8 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +Theorem 5.9 (Adjusted and simplified from [MS17, Th. 7.2]). +A positive and symmetric universal product is uniquely determined by its highest coefficients. More +precisely, for a = a1 . . . an with ai ∈ Af(i) +b(i) so that a1 ⊗ . . . ⊗ an ∈ T0 +�� +i∈[2],•∈F A• +i +� += ˆA1 ⊔ ˆA2, +ϕ1 ⊙ ϕ2(a1 . . . an) = exp⊙ +� +log⊙( ˆϕ1) ⊕ log⊙( ˆϕ2) +� +(a1 ⊗ . . . ⊗ an); +here we use the direct sum as a shorthand notation +ψ1 ⊕ ψ2(b1 ⊗ . . . ⊗ bk) := +� +� +� +� +� +ψ1(b1 ⊗ . . . ⊗ bk) +∀i : bi ∈ A• +1, +ψ2(b1 ⊗ . . . ⊗ bk) +∀i : bi ∈ A• +2, +0 +∃i, j : bi ∈ A• +1, bj ∈ A• +2. +Proof. We only explain why this is a special case of [MS17, Th. 7.2]. Since ⊙ is positive, their are +no wrong-ordered highest coefficients. In the symmetric case, the exponential and logarithm map +used in [MS17] coincide with the maps of Definition 5.6 and are therefore determined by the highest +coefficients. Since ⊙ is symmetric, the second ingredient which is in general needed to determine +the universal product, namely the nth order cumulant Lie algebra, is trivial for all n. +□ +Proposition 5.10. Let ⊙ be a symmetric universal product with highest coefficients α. Then +α-independence is equivalent to ⊙-independence. +Proof. In the α-cumulants are ⊙-cumulants and we know that mixed ⊙-cumulants of ⊙-independent +subalgebras vanish from [MS17, Theorem 7.1] (note that the assumption of ⊙ being symmetric is +essential here). +□ +6. Highest coefficients: necessary conditions +The question we wish to answer is the following: under which conditions on the weights α is +there a universal product ⊙ with highest coefficients α? In this section we find a set of necessary +conditions. +Theorem 6.1. Let ⊙ be a positive multi-faced universal product. Then the highest coefficients +fulfill: +(i) α1f = 1 for all f ∈ F∗. +(ii) α({{1},{2}},f) = 1 for every f ∈ F2. +(iii) απ = αred(π). +(iv) Suppose π ∈ P<(f) has blocks B1 < B2 that are nearest neighbors for the order of π +and have neighboring legs in the same face, i.e. there exist i ∈ B1, j ∈ B2, |i − j| = 1, +f(i) = f(j). Then +απ = απB1⌣B2 · α{B1 i) +every summand will contain a factor F = ψi(. . . arar+1 . . .) because the two factors ar, ar+1 are +from the same block and face and therefore have to be treated as one. Summands with more +factors containing ψi vanish in the linearization procedure. Therefore, we obtain +∂k +∂t1 . . . ∂tk +� +(t1ϕ1) ⊙ . . . ⊙ +� +(tiϕi) ⊙ (ti+1ϕi+1) +� +⊙ . . . ⊙ (tkϕk) +� +(a) +���� +t=0 += απB1⌣B2 +∂2 +∂ti∂ti+1 +� +(tiϕi) ⊙ (ti+1ϕi+1) +� +(. . . arar+1 . . .) +���� +t=0 += απB1⌣B2 · α{B1 2 blocks. Starting similar as before, we +can without loss of generality assume that B1 = {1} and f(1) = f(2). In that case, we find +that απ = απB1⌣B2 · αB1,B2 = 1. +(2) Clearly, it is enough to prove the claim for n = 2. We prove the claim by induction on the +number of blocks |π|. If |π| = 2, then |π1| = |π2| = 1 and the three partitions are interval +partitions, in particular απ = 1 = απ1απ2. If |π| > 2, then |π1| > 1 or |π2| > 1. In case +|π1| > 2, let 1 ∈ B1 ∈ π1. We can assume without loss of generality that 2 ∈ B2 belongs +to a different block B1 ̸= B2 ∈ π1 and f(1) = f(2); if those conditions are not met, it +does not change the coefficient to change the color of the first leg to match the color of the +second leg and merge them into one until we are in the described situation. Now we find +απ1 = απ1B1⌣B2 · α{B1,B2} and απ = απB1⌣B2 · α{B1,B2}. Of course, |πB1⌣B2| = |π| − 1, +so we may assume that the statement holds for πB1⌣B2 which is the concatenation of +π1B1⌣B2 and π2. Altogether, +απ = απB1⌣B2 · α{B1,B2} = απ1B1⌣B2 · απ2 · α{B1,B2} = απ1απ2. +If |π2| > 1, we argue analogously, but we have to change the color of the last leg. +(3) We have απ = ασαB1,B2, and α{B1,B2} = 1 by (1). +(4) Decompose π into a concatenation of irreducible π1, . . . , πn. For each πi, the outer legs +can be collapsed because we can change the color of the first and last leg. After collapsing +the outer legs, the obtained partitions ˜πi can be concatenated again to obtain ˜π ∈ P. +□ +It is worth noting, that in the proof of (2), we need invariance of the coefficients under changing +both extremal legs. For example the weights associated with bi-Boolean independence defined in +[GS19] do not share this property. +Lemma 7.3. Two admissable families coincide if and only if they coincide on 2-block partitions. +Proof. Assume that (απ), (βπ) are admissable families with ασ = βσ for all 2-block partitions σ. +By definition, the value on 1-block partitions is 1. Given an n-block partition π with n > 2, we +alternatingly +• change the color of the first leg to match the color of the second leg, cf. (v), +• combine the first two legs into one if they belong to the same block, cf. (iii), +to obtain a partition ˜π such that the first two legs of π have the same color but belong to different +blocks B1, B2. Then απ = α˜π, βπ = β˜π by definition of admissable weights. Using (iv), we then +have απ = απB1⌣B2 · α{B1,B2} and βπ = βπB1⌣B2 · β{B1,B2}, where πB1⌣B2 is an (n − 1)-block +partition and {B1, B2} is a 2-block partition. We can iterate the procedure until we obtain απ, βπ +as products of coefficients of the same sequence of 2-block partitions, thus proving the claim. +□ +Corollary 7.4. Two admissable families coincide if and only if they coincide on 2-block partitions +of at most four legs. +Proof. If a two-block partition has more than 4 legs, simply split the third leg to obtain its co- +efficient as a product of two coefficients of two-block partitions with a strictly smaller number of +legs. This process eventually allows to express each two-block coefficient as a product of two-block +coefficients with at most 4 legs. +□ +Definition 7.5. We introduce shorthand notations for the basic coefficients, where ◦, • ∈ F: +ν◦ := α +� +� +, +ν• := α +� +� +, +ν◦• := α +� +� +ξ◦ := α +� +� +, +ξ• := α +� +� +ξ◦• := α +� +� +Corollary 7.6. Two admissable families coincide if and only if they have the same basic coeffi- +cients. +Proof. A two-block partition with at most four legs is either an interval partition (in which case its +coefficient is 1) or it can be reduced by changing color and combining legs to one of the partitions +that define the basic coefficients. +□ +Lemma 7.7. We have the following relations between the basic coefficients for all ◦, • ∈ F. +(1) ν2 +◦ = ν◦, ξ2 +◦ = ξ◦ , i.e. ν◦, ξ◦ ∈ {0, 1}. + +12 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +(2) tν◦ = t for t ∈ {ν◦•, ξ◦, ξ◦•}, i.e. ν◦ = 0 =⇒ ν◦• = ξ◦ = ξ◦• = 0 +(3) |t|2t = t for t ∈ {ν◦•, ξ◦•}, i.e. ν◦•, ξ◦• ∈ {0} ∪ T +(4) ν◦•ξ◦ = ξ◦•ξ◦,i.e. ξ◦ = 1 =⇒ ν◦• = ξ◦• +(5) ν◦•ξ◦• = ν◦•ξ◦•ξ◦, i.e. ξ◦ = 0 =⇒ ν◦• = 0 or ξ◦• = 0 +Proof. +(1) This follows as in the single-faced case, see [Spe97]. Alternatively, this follows easily as a +special case ◦ = • from the items below. +(2) Consider +. Split the inner ◦-leg and merge it’s copy with the outer block to obtain +α +� +� += α +� +� +α +� +� +. The other cases work analogously. +(3) First note that α +� +� += α +� +� += |ν◦•|2. This leads to +|ν◦•|2ν◦• = α +� +� +ν◦• = α +� +� += ν◦•ν◦ = ν◦•. +Similarly, α +� +� += α +� +� += |ξ◦•|2 and hence +|ξ◦•|2ξ◦• = α +� +� +ξ◦• = α +� +� += ξ◦•ν◦ = ξ◦•. +(4) This follows from +ν◦•ξ◦ = α +� +� += α +� +� +ν◦◦ = α +� +� +ν◦ = ξ◦•ξ◦ν◦ = ξ◦•ξ◦ +(5) Reusing parts of the calculation above, we find +ν◦•ξ◦• = α +� +� += ν◦•α +� +� += ν◦•ξ◦•ξ◦ +□ +Corollary 7.8. Two admissable sets of partitions coincide if and only if they +have the same intersection with +� +, +, +, +, +, +: ◦, • ∈ F +� +. +Furthermore, for an admissable set Π we have the following implications: +(1) If Π contains at least one of the partitions +, +, +, then it contains +. +(2) If Π contains at least one of the partitions +, +, +, then it contains +. +(3) If Π contains two of the basic partitions +, +, +, then Π ⊃ A◦•. +(4) If Π contains two of the basic partitions +, +, +, then Π ⊃ A◦•. +From now on, we restrict to the two-faced case, F = {◦, •}. +Definition 7.9. A two-faced partition π is called +• interval partition all legs are outer or, equivalently, if all its blocks are intervals; I◦• denotes +the set of all interval partitions, +• noncrossing if for all i < j < k < ℓ and blocks B, C ∈ π, +i, k ∈ B, j, ℓ ∈ C =⇒ B = C; +NC◦• denotes the set of all noncrossing partitions, +• binoncrossing if for all i < j < k < ℓ and blocks B ̸= C ∈ π, +i, k ∈ B, j, ℓ ∈ C =⇒ f(j) ̸= f(k), +i, ℓ ∈ B, j, k ∈ C =⇒ f(j) = f(k); +• interval-noncrossing if it is noncrossing and all ◦-legs are outer I◦NC• denotes the set of +all interval-noncrossing partitions, +• noncrossing-interval if it is interval-noncrossing after swapping the colors ◦ and •; NC◦I• +denotes the set of all noncrossing-interval partitions, +• interval-arbitrary if all ◦-legs are outer +• arbitrary-interval if it is interval-arbitrary after swapping the colors ◦ and •; A◦I• denotes +the set of all arbitrary-interval partitions, +• noncrossing-arbitrary if every block B that contains an inner ◦-leg is a monochrome interval +NC◦A• denotes the set of all noncrossing-arbitrary partitions, +• arbitrary-noncrossing if it is noncrossing-arbitrary after swapping the colors ◦ and •; +A◦NC• denotes the set of all arbitrary-interval partitions, + +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +13 +A◦• += +� +, +� += +� +, +� +NC◦• += +� +� +pC◦• += +� +, +� +biNC◦• += +� +� +NC◦A• += +� +, +� +A◦NC• += +� +, +� +I◦A• += +� +� +pNC◦• += +� +, +� +A◦I• += +� +� +I◦NC• += +� +� +NC◦I• += +� +� +I◦• +=�� +Figure 1. Hasse diagram of all two-colored admissable sets of partitions. +• pure noncrossing if it is noncrossing and all inner blocks are monochrome; pNC◦• denotes +the set of all pure noncrossing partitions, +• pure crossing if connected inner legs have the same color; pC◦• denotes the set of all pure +noncrossing partitions, +• arbitrary without any conditions; the set of all bipartitions is also denoted A◦•. +Theorem 7.10. There are exactly 12 admissable sets of partitions (9 if we identify a set with the +one obtained by simply swapping the two colors), namely those given in Definition 7.9. Figure 1 +displays their respective containment by means of a Hasse diagram and gives minimal generating +sets of 2-block partitions. +Proof. We know that a set obtained from a positive symmetric two-faced universal product is +automatically admissable. Of course, swapping the two colors turns an admissable set into an +admissable set. This helps to settle admissability of a large number of sets in the diagram: +• The sets I◦,•, NC◦,•, A◦,• are the sets of interval, noncrossing, and all partitions (ignoring +the colors), and thus are known to come from the trivially two-faced Boolean, free and +tensor universal product, respectively. Swapping the colors does not change these sets of +partitions. +• The set NC◦I• is the set of noncrossing-interval partitions, which originates from free- +boolean independence [Liu19]. Swapping the colors leads to the set I◦NC•. +• The set A◦I• comes from tensor-boolean independence [GHU21]. Swapping the colors leads +to the set I◦A•. +• The set biNC is the set of binoncrossing partitions, it comes from bifree independence +[CNS15, Voi14]. Swapping the colors does not change the set. +We are left with the sets of pure crossing and pure noncrossing partitions and with the sets of +noncrossing-arbitrary and arbitrary-noncrossing partitions, where again by swapping the colors it +is enough to deal with the noncrossing-arbitrary ones. All properties are easily verfied. +The theorem now follows from the fact that each admissable set is uniquely determined by which +basic two-block partitions have nonzero coefficients, and from the implications in Corollary 7.8. +□ +Corollary 7.11. Let ⊙ be a positive symmetric 2-faced universal product. Then +• either it is one of the products obtained in [GHU21] + +14 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +• or its highest coefficients are given by the indicator function of one of the admissable sets +of partitions NC◦A•, A◦NC•, pNC◦•, pC◦•. +Proof. If ⊙ is a positive symmetric universal product, then its highest coefficients form an admiss- +able family of weights. If all the basic coefficients are 0 or 1, the family must be given by the +indicator function of one of the admissable sets of partitions and all except the mentioned for are +identified as positive products in [GHU21]. If one of the basic coefficients is not 0 or 1, there are +only three possibilities, in each of which the universal product has been found to be positive in +[GHU21]: +• ν◦• = ξ◦• = q ∈ T \ {1}, in this case all other basic coefficients are forced to be equal +to 1; by comparison of the basic coefficients, the corresponding universal product is the +deformed tensor product, +• ν◦• = q ∈ T \ {1}, ξ◦• = 0; in this case, the product must coincide with the deformed free +product, +• ν◦• = 0, ξ◦• = q ∈ T\{1}; in this case, the product must coincide with the deformed bifree +product. +□ +Remark 7.12. A remarkable property of freeness is that the free product of traces is again a +trace. We cannot expect such a behaviour for any non-trivial multi-faced independence. Indeed, +this would force the highest coefficients to be invariant under cyclic permutations, and since we +may change the color of the first leg, we could change the color of every leg without changing the +coefficient. +Remark 7.13. Bifreeness allows to define a convolution for probability measures on R2. This +comes from the fact that for bifree pairs (a◦ +1, a• +2), (b◦ +1, b• +2) one always has commutativity of a◦ +1 with +b• +2 and of a• +2 with b◦ +1. Consequently, a◦ +1 + b◦ +1 commutes with a• +2 + b• +2 whenever a◦ +1, a• +2 commute and +b◦ +1, b• +2 commute. If independent variables in different faces commute, one must have ξ◦• = 1, which +is only the case for tensor and bifree independence. +Remark 7.14. There are other interesting symmetric two-faced universal products which are not +positive, for example the bi-Boolean product. It seems very well possible to do a classification +under slightly relaxed conditions, only assuming that one is allowed to change the color of the first +leg and not assuming any mirror symmetry (recall that we used changing the color on both sides +to show that highest coefficients for all interval partitions are 1). However, it is not clear how to +motivate those properties when one does not aim for positivity. For the construction of a universal +product in the algebraic sense (see Section 8), conditions (v) and (vi) are not necessary at all. +8. Reconstruction of universal products from highest coefficients +In this section we prove that every admissable family leads to a unique universal product. In par- +ticular, we can associate universal products with the admissable sets NC◦A•, A◦NC•, pNC◦•, pC◦•. +However, it remains an open problem at the moment to decide whether or not those universal prod- +ucts are positive. +In the following proofs we use several times the relation between α-weighted logarithm and +α-cumulants as given in Observation 5.8. +Lemma 8.1. Suppose that the weights (απ)π∈P are admissable. Fix a family of elements a = (a• +i : +• ∈ F, i ∈ I•) ⊂ A in an algebraic probability space (A, Φ) such that [n] is the disjoint union of +the I•. Put f(i) := • if i ∈ I• and assume that f(k) = f(k + 1) = • ∈ F. We expand the family a +to a family ˜a by replacing I• with +˜I• := I• ∪ {˜ı}, +a• +˜ı := a• +ka• +k+1. +For X := xf(1) +1 +. . . xf(n) +n +, ˜X := xf(i) +1 +. . . xf(k−1) +k−1 +x• +˜ı xf(k+2) +k+2 +. . . xf(n) +n +the moments and cumulants ac- +cording to Definition 5.4 fulfill m˜a( ˜X) = ma(X) and +c˜a( ˜X) = ca(X) + +� +σ={A,B}∈P(f) +k∈A̸=B∋k+1 +ασca(X ↾ A)ca(X ↾ B). +Proof. The claimed equality for the moments is obvious. The claim for the cumulants is proved +by induction on n. For n = 2, i.e. X = x• +1x• +2 , we have +c˜a(x• +˜ı ) = m˜a(x• +˜ı ) = ma(x• +1x• +2) = ca(x1x2) + α +� +� +ca(x• +1)ca(x• +2). + +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +15 +For general n, we can use the moment-cumulant relations for ma(X) = m˜a( ˜X) and obtain +ma(X) = +� +π∈P(f) +απ +� +B∈π +ca(X ↾ B) += +� +π∈P(f) +k,k+1∈ ˆ +B∈π +απca(X ↾ ˆB) +� +B∈π\{ ˆ +B} +ca(X ↾ B) ++ +� +ρ∈P(f) +B1,B2∈ρ +k∈B1̸=B2∋k+1 +αρca(X ↾ B1)ca(X ↾ B2) +� +B∈ρ\{B1,B2} +ca(X ↾ B) += +� +π∈P(f) +k,k+1∈ ˆ +B∈π +απ +� +ca(X ↾ ˆB) + +� +{B1,B2}∈P( ˆ +B) +k∈B1̸=B2∋k+1 +α{B1,B2}ca(X ↾ B1)ca(X ↾ B2) +� +· +� +B∈π\{ ˆ +B} +ca(X ↾ B) += ca(X) + +� +{A,B}∈P(f) +k∈A̸=B∋k+1 +α{A,B}ca(X ↾ A)ca(X ↾ B) ++ +� +1f ̸=π∈P(f) +k,k+1∈ ˆ +B∈π +απ +� +ca(X ↾ ˆB) + +� +{B1,B2}∈P( ˆ +B) +k∈B1,k+1∈B2 +α{B1,B2}ca(X ↾ B1)ca(X ↾ B2) +� +· +� +B∈π\{ ˆ +B} +ca(X ↾ B) +where we used αρ = απα{B1,B2} for π = ρB1⌣B2. On the other hand, with ˜f ∈ F[n]/(k∼k+1), +˜f({k, k + 1}) = f(k) := f(k + 1) and ˜f(ℓ) := f(ℓ) for ℓ ̸= {k, k + 1}, +m˜a( ˜X) = +� +σ∈P(˜f) +ασ +� +B∈σ +c˜a( ˜X ↾ B) += +� +σ∈P(˜f) +{k,k+1}∈ ˜ +B +ασc˜a( ˜X ↾ ˜B) +� +B∈σ\{ ˜ +B} +ca(X ↾ B) += c˜a( ˜X) + +� +1˜f ̸=σ∈P(˜f) +{k,k+1}∈ ˜ +B +ασc˜a( ˜X ↾ ˜B) +� +B∈σ\{ ˜ +B} +ca(X ↾ B). +Recall that there is a canonical bijection between partitions σ ∈ P(˜f) and partitions π ∈ P(f) +with k, k + 1 in the same block ˆB ∈ π. Also, the highest coefficients ασ and απ agree under this +bijection by Theorem 6.1 (iii). Using the induction hypothesis on ca(X ↾ ˆB) finishes the proof. +□ +Theorem 8.2. Suppose that the weights (απ)π∈P are admissable. +Then there exists a unique +symmetric universal product with highest coefficients (απ)π∈P. +Proof. The uniqueness statement is proved in [MS17, Th. 7.2], see Theorem 5.9. +Let ϕk : Ak → C be linear functionals on 2-faced algebras (k = 1, 2). Recall Definition 5.6 +of expα and logα and Definition 5.7, which sets the notation for lifting ϕk to linear functionals +ˆϕk = ϕk ◦ µk on the tensor algebras T0(� +•∈F A• +k). We simply write exp := expα and log := logα +in the following. We define +ϕ1 �⊙ϕ2 := exp +� +log( ˆϕ1) ⊕ log( ˆϕ2) +� +∈ T0 +�� +•∈F +A• +1 ⊕ A• +2 +�′ +. +(6) +The main task is now to prove that ϕ1 �⊙ϕ2 vanishes on the ideal I := ker(µ1 ⊔ µ2) in +T0 +�� +•∈F +A• +1 ⊕ A• +2 +� += +� +•∈F +T0(A• +1) ⊔ T0(A• +2) + +16 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +(i.e. the ideal generated by the relations a ⊗ b = ab for a, b ∈ A• +j), so that ϕ1 �⊙ϕ2 descends to a +functional ϕ1 ⊙ ϕ2 on the quotient +A1 ⊔ A2 = T0 +�� +•∈F +A• +1 ⊕ A• +2 +� +/I. +Let a1 . . . an with aℓ ∈ Af(ℓ) +b(ℓ) such that ak and ak+1 lie in the same direct summand, ak, ak+1 ∈ A• +j. +Define ˜f ∈ F[n]/(k∼k+1), ˜f({k, k + 1}) = f(k) = f(k + 1) and ˜f(ℓ) = f(ℓ) for ℓ ̸= {k, k + 1}. +With a := (a1, . . . , an) and X := x1 . . . xn ∈ C⟨x1, . . . , xn⟩, ˜a := (a1, . . . , akak+1, . . . an) and +˜X := x1 . . . x{k,k+1}xn ∈ C⟨x1, . . . , x{k,k+1}, xn⟩ we have +• for B ⊂ {1, . . . , k, k + 1, . . . , n} with aℓ ∈ Ai for all ℓ ∈ B +log ˆϕi(a ↾ B) = ca(X ↾ B), +• for B ⊂ {1, . . . , {k, k + 1}, . . . , n} with aℓ ∈ Ai for all ℓ ∈ B +log ˆϕi(˜a ↾ B) = c˜a( ˜X ↾ B) +With this in hand, we calculate +exp(log ˆϕ1 ⊕ log ˆϕ2)(a1 ⊗ . . . ⊗ akak+1 ⊗ . . . ⊗ an) += +� +π∈P(˜f) +απ(log ˆϕ1 ⊕ log ˆϕ2)⊗|π|(aπ) += +� +π∈P(˜f) +{k,k+1}∈ ˜ +B∈π +απc˜a( ˜X ↾ ˜B) +� +B∈π\{ ˜ +B} +c˜a( ˜X ↾ B) +and, on the other hand, +exp(log ˆϕ1 ⊕ log ˆϕ2)(a1 ⊗ . . . ⊗ ak ⊗ ak+1 ⊗ . . . ⊗ an) += +� +π∈P(f) +k,k+1∈ ˆ +B∈π +απ log ˆϕ1(a ˆ +B) +� +B∈π\{ ˆ +B} +log ˆϕiB(aB) ++ +� +σ∈P(f) +B1,B2∈σ +k∈B1̸=B2∋k+1 +ασ log ˆϕ1(aB1) log ˆϕ1(aB2) +� +B∈σ\{B1,B2} +log ˆϕiB(aB) += +� +π∈P(f) +k,k+1∈ ˆ +B∈π +απ +� +�ca(X ↾ ˆB) + +� +B1 ˙∪B2= ˆ +B +α{B1,B2}ca(X ↾ B1)ca(X ↾ B2) +� +� +� +B∈π\{ ˆ +B} +ca(X ↾ B) +using απα{B1,B2} = ασ when σ = π \ { ˆB} ∪ {B1, B2}, i.e. π = σB1⌣B2. The two expressions agree +by Lemma 8.1 and, therefore, we have a well-defined map ϕ1 ⊙ ϕ2(a1 . . . an) = ϕ1 ⊙ ϕ2(a1 ⊗ . . . ⊗ +an + I) := ϕ1 �⊙ ϕ2(a1 ⊗ . . . ⊗ an). +The verification that ⊙ is indeed a symmetric universal product is left to the reader; note that +by well-definedness we can choose the same exp and log map when checking associativity, so that +this reduces to associativity of the direct sum. A detailed proof can also be found in [Var21] for +the case of the weights being 0 or 1. +To check that the highest coefficients of ⊙ are indeed given by α, it is enough to consider +products of two functionals ϕ1, ϕ2. For a1 . . . an ∈ A1 ⊔ A2, ai ∈ Af(i) +ε(i), Bk = {i : ε(i) = k}, we find +∂2 +∂t1∂t2 +(t1ϕ1) ⊙ (t2ϕ2)(a1 . . . an) +���� +t=0 += +� +π∈P(f) +∂2 +∂t1∂t2 +απ(log t1 ˆϕ1 ⊕ log t2 ˆϕ2)⊗|π|(aπ) +���� +t=0 += +∂2 +∂t1∂t2 +α{B1,B2} log t1 ˆϕ1(aB1) log t2 ˆϕ2(aB2) +���� +t=0 += α{B1,B2}ϕ1(aB1)ϕ2(aB2) +as needed.8 +□ +8The notation aB = +→ +� +i∈Bai, aπ = � +B∈π aB refers to a1 ⊗ . . . ⊗ an, but note that by well-definedness the +choice of decomposition of a1 . . . an ∈ A1 ⊔ A2 as a tensor in T0(� +•∈F A• +1 ⊕ A• +2) does not influence the result! + +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +17 +The formula to compute mixed moments can be considerably simplified in the special case where +the the highest coefficients are only 0 or 1. +Definition 8.3. We say that a symmetric universal product is combinatorial with partition set Π +if its highest coefficients are all either 0 or 1 and Π = {π ∈ P : απ = 1}. +Theorem 8.4. Let ⊙ be a combinatorial universal product with admissable partition set Π one of +the 12 sets of Theorem 7.10. Furthermore, let ϕℓ be a linear functional on a multi-faced algebra +Aℓ (ℓ = 1, . . . , k), and a = a1 . . . an ∈ As for s = b × f ∈ ([k] × F)n. Denote +• π ∈ P(f) the multi-faced partition with blocks Bk := {i : b(i) = k} (whenever non-empty), +• Π≤π := {σ ∈ Π : σ ≤ π} the set of refinements of π inside Π ∩ P(f), +• S the set of maximal elements of Π≤π (i.e. coarsest refinements of π inside Π), +• ∧R is the maximal common refinement of partitions in R ⊂ Π ∩ P(f), ∧∅ := 1f, +• Φ := ϕ1 ⊙ . . . ⊙ ϕk, +• ˆΦ the lift of Φ to T0 +� +� +ℓ∈[k],•∈F +A• +ℓ +� +. +Then +Φ(a) = +� +∅̸=R⊆S +(−1)#R−1 ˆΦ⊗|∧R|(a∧R). +Proof. Put Ψ := log ˆΦ = log ˆϕ1 ⊕ . . . ⊕ log ˆϕk. The key observation is that a refinement σ of a +partition ρ ∈ Π belongs to Π if and only if σ ↾ B ∈ Π for all blocks B ∈ ρ; this can be easily seen +for each of the 12 admissable sets of partitions individually. Using the moment cumulant formula +on each block of ∧R and the observation on refinements just made, we find +� +R⊆S +(−1)#R ˆΦ⊗|∧R|(a∧R) = +� +R⊆S +� +σ≤∧R +σ∈Π +(−1)#RΨ⊗|σ|(aσ) +(7) +(equality of the summands for R = ∅ will be discussed below.) Now, the same partition σ ∈ Π +can of course be a refinement of ∧R for different R ⊆ S. Denote T(σ) := {ρ ∈ S : σ ≤ ρ} and +n(σ) := #T(σ). Then σ ≤ ∧R if and only if R ⊆ T(σ), and for every k ∈ {0, 1, . . . , n(σ)} there are +�n(σ) +k +� +many such R with #R = k. If n(σ) = 0, i.e. if σ is not a refinement of π, then Ψ⊗|σ|(aσ) = 0 +because mixed cumulants vanish. This leads to +RHS of (7) = +� +σ∈Π≤π +n(σ) +� +k=0 +(−1)k +�n(σ) +k +� +� +�� +� +=0 +Ψ⊗|σ|(aσ) = 0. +Recall that we defined ∧∅ := 1f, so that +Φ(a) = ˆΦ(a1f ) = +� +σ∈P(f)∩Π +Ψ⊗|σ|(aσ) = +� +σ≤1f +σ∈Π +Ψ⊗|σ|(aσ); +this confirms that the choice is consistent with (7), and it also shows that the statement of the +theorem is equivalent to LHS of (7) = 0. +□ +Example 8.5. Let ⊙ be the universal product associated with NC◦A•. Then +has set of +coarsest refinements S = { +, +} in NC◦A• with ∧S = +, leading to +ϕ ⊙ ψ(a◦ +1b• +1a• +2a◦ +3b• +2) = ϕ(a◦ +1a• +2)ϕ(a◦ +3)ψ(b• +1b• +2) + ϕ(a◦ +1a• +2a◦ +3)ψ(b• +1)ψ(b• +2) − ϕ(a◦ +1a• +2)ϕ(a◦ +3)ψ(b• +1)ψ(b• +2) +for all ϕ ∈ A′, ψ ∈ B′, a• +i ∈ A• (i = 1, 2, 3), and b• +k ∈ B• (k = 1, 2). +9. Unit preserving universal products +In [DAGSV22], Diaz-Aguilera, Gaxiola, Santos, and Vargas characterize when the moment +cumulant relation associated with weights on partitions leads to independent constants, finding +this to be the case if and only if the weights do not change when removing or inserting a singleton +from or to the partition. Manzel and Schürmann discuss in [MS17, Rem. 3.1] the relation between +universal products in the category of multi-faced algebras and in the category of multi-faced unital +algebras and observe that while a product for the unital category always gives rise to a product +for the non-unital category, the other way round requires a condition, namely that the universal +product respects the units or is unit preserving as we prefer to write in this article. In this section +we briefly review universal products in the category of multi-faced unital algebras, define what + +18 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +exactly it means to be unit preserving, generalize the definition of singleton inductive weights to +the multi-faced setting, and finally characterize unit preserving symmetric universal product as +those whose highest coefficients are singleton inductive. +In the category of unital algebras with unital algebra homomorphisms, the coproduct is given +by the unital free product, which can be constructed from the non-unital free product as +A1 ⊔ +1 A2 := (A1 ⊔ A2)/⟨1A1 − 1A2⟩; +here ⟨·⟩ denotes the generated two-sided ideal. +Definition 9.1. +• A multi-faced unital algebra is a unital algebra A with unital subalgebras A•, • ∈ F, such +that the canonical unital algebra homomorphism � +1 •∈F A• → A is an isomorphism, in +which case we write A = � +1 •∈F A•. +• A multi-faced unital algebra homomorphism is a unital algebra homomorphism which maps +face into face. +• The unital free product of multi-faced unital algebras A1, A2 is a multi-faced unital algebra +with (A1 ⊔ +1 A2)• := A• +1 ⊔ +1 A• +2. +• A linear functional φ: A → C on a multi-faced unital algebra is unital if φ(1A) = 1. +Multi-faced unital algebras with multi-faced unital algebra homomorphisms form a category, +in which ⊔ +1 is a coproduct. One can adapt Definition 3.1 to the unital situation and obtains the +following. +Definition 9.2. A universal product in the category of multi-faced unital algebras is a binary +product operation for unital linear functionals on multi-faced unital algebras which associates with +unital functionals φ1, φ2 on multi-faced unital algebras A1, A2, respectively, a unital functional +φ1 ⊙ φ2 on A1 ⊔ +1 A2 such that +• (φ1 ◦j1)⊙(φ2 ◦j2) = (φ1 ⊙φ2)◦(j1 ⊔ +1 j2) for all multi-faced unital algebra homomorphisms +ji : Bi → Ai (universality) +• (φ1 ⊙ φ2) ⊙ φ3 = φ1 ⊙ (φ2 ⊙ φ3) (associativity) +• (φ1 ⊙ φ2) ↾ Ai = φi (restriction property). +As Manzel and Schürmann noticed in [MS17, Rem. 3.1], every universal product ˜⊙ in the +category of multi-faced unital algebras gives rise to a universal product in the sense of Definition 3.1, +simply putting +ϕ1 ⊙ ϕ2 := ˜ϕ1 ˜⊙ ˜ϕ2 ↾ A1 ⊔ A2 ⊂ ˜A1 ⊔ +1 ˜A2 +where ˜Ai denotes the unitization of a multi-faced algebra and ˜ϕi the unital extension of a linear +functional. +Conversely, if a universal product ⊙ in the non-unital case is given, one would like to define +φ1 ˜⊙ φ2(p(a)) := ϕ1 ⊙ ϕ2(a) +(8) +with the following conventions: +• Ai := � +•∈F A• +i , so that Ai ∼= Ai/IAi with IAi := ⟨1◦ − 1• : ◦, • ∈ F⟩ ⊂ A, +• pAi : Ai → Ai denotes the canonical homomorphism, +• ϕi := φi ◦ pAi : Ai → C to A, i.e. ϕi(a) := φi(a + IAi), +• p: A1 ⊔ A2 → A1 ⊔ +1 A2 denotes the canonical homomorphism. +Definition 9.3. A universal product is unit preserving (or respects units if, whenever A1, A2 are +multifaced algebras with each A• +i unital and ϕi a linear functional on Ai which vanishes on the +ideal ⟨1◦ +i −1• +i : ◦, • ∈ F⟩ ⊂ Ai and such that ϕi ↾A• +i is unital for every • ∈ F, then ϕ1 ⊙ϕ2 vanishes +on the ideal ⟨1◦ +i − 1• +j : i, j ∈ [2], ◦, • ∈ F⟩ ⊂ A1 ⊔ A2 and ϕ1 ⊙ ϕ2 ↾ A• +i is unital for every i ∈ [2], +• ∈ F. +Remark 9.4. A multi-faced universal product is unit preserving if and only if (8) is well-defined, +in which case it yields a universal product in the category of multi-faced unital algebras [MS17, +Rem. 3.1]. Since Manzel and Schürmann do not give a definition of “respecting units”, let us briefly +check that Definition 9.3 captures what they mean. +Assume that ⊙ is unit preserving. The ϕi = φi ◦ pAi in (8) are linear functionals on Ai, vanish +on ker pAi = IAi = ⟨1◦ +i − 1• +i : ◦, • ∈ F⟩ ⊂ Ai and fulfill ϕi(1• +i ) = φi(1i) = 1. Therefore, we may +conclude that ϕ1 ⊙ϕ2 vanishes on the ideal ⟨1◦ +i −1• +j : i, j ∈ [2], ◦, • ∈ F⟩ ⊂ A1 ⊔A2, which coincides +with the kernel of the canonical homomorphism p: A1 ⊔ A2 → A1 ⊔ +1 A2. This means that there is + +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +19 +a well-defined linear functional φ1 ˜⊙ φ2 with ϕ1 ⊙ ϕ2 = φ1 ˜⊙ φ2 ◦ p. This functional is also unital +because φ1 ˜⊙ φ2(1) = φ1 ˜⊙ φ2(p(1• +i )) = ϕi(1• +i ) = 1. +We leave the rest of the simple, but notationally cumbersome proof of the claim (in particular +universality and associativity of ˜⊙) to the interested reader. +In the following we will need often remove a singleton block B = {s} from a partition π ∋ B. +While consistent use of notation would dictate to write π \ {B} = π \ {{s}}, we will prefer to write +π \ {s} for better legibility. +Definition 9.5 (multi-faced version of [DAGSV22, Def. 3.2]). A family of weights (απ)π∈P is +singleton inductive if απ = απ\{s} for every singleton block {s} ∈ π. +Lemma 9.6 (multi-faced version of [DAGSV22, Th. 3.2]). Let α be monic, singleton inductive +weights. Suppose that A = � +•∈F A• is a multi-faced algebra such that each face A• is unital (with +unit 1•) and that ϕ fulfills ϕ(1•) = 1 and ϕ vanishes on the ideal ⟨1◦ − 1• : ◦, • ∈ F⟩ ⊂ A. Then +logα ˆϕ(a1 ⊗ . . . ⊗ an) = 0 whenever n > 1 and as = 1• for some s ∈ [n], • ∈ F; +here ˆϕ is the lift of ϕ to a T0 +�� +•∈F A•� +. +Proof. We prove the claim by induction. For n = 2 and arbitrary ◦, • ∈ F, +logα ˆϕ(1◦ ⊗ a•) = ϕ(1◦a•) − ϕ(1◦)ϕ(a•) = ϕ(1•a•) − ϕ(a•) = 0 +and analogously logα ˆϕ(a◦ ⊗ 1•) = 0. Now assume the statement holds for all 1 < m < n and +consider a = a1 ⊗ . . . ⊗ an with ai ∈ Af(i), as = 1f(s). Note that ϕ(a1 . . . an) = ϕ(a1 . . . ˇas . . . an) +(here ˇas means omission of the factor) and logα ˆϕ(as) = logα ˆϕ(1•) = ϕ(1•) = 1. We find +logα ˆϕ(a1 ⊗ . . . ⊗ an) = ϕ(a1 . . . an) − +� +π∈P(f)\{1f } +απ(logα ˆϕ)⊗|π|(aπ) += ϕ(a1 . . . an) − +� +{s}∈π∈P(f) +απ(logα ˆϕ)⊗|π|(aπ) − +� +{s}/∈π∈P(f)\{1f } +απ(logα ˆϕ)⊗|π|(aπ) +� +�� +� +=0 by induction hypothesis += ϕ(a1 . . . ˇas . . . an) − +� +{s}∈π∈P(f) +απ\{s}(logα ˆϕ)⊗|π|−1(aπ\{s}) logα ˆϕ(as) += ϕ(a1 . . . ˇas . . . an) − +� +σ∈P(f↾[n]\{s}) +ασ(logα ˆϕ)⊗|σ|(aσ) = 0, +where we used that the weights are singleton inductive as well as the moment cumulant relation +for a1 . . . ˇas . . . an. +□ +Theorem 9.7. For a multi-faced positive symmetric universal product ⊙, the following are equiv- +alent. +(1) ⊙ is unit preserving, +(2) ν◦ = 1 for all ◦ ∈ F, +(3) the highest coefficients of ⊙ are singleton inductive. +Proof. Let ⊙ be unit preserving. To calculate ν◦ = α +� +� +, we can assume that also the extremal +legs are ◦-legs. We can therefore ignore the faces and calculate, as in the single-faced case, ϕ1 ⊙ +ϕ2(aba′) = ν◦ϕ1(aa′)ϕ2(b) + βϕ1(a)ϕ1(a′)ϕ2(b) for all a, a′ ∈ A◦ +1, b ∈ A◦ +2, with some universal +constant β. Suppose that ϕ1, ϕ2 are as in Definition 9.3 and furthermore b = 1◦ +2, ϕ1(a) = ϕ1(a′) = +0, ϕ1(aa′) = 1, then +ϕ1 ⊙ ϕ2(a1◦ +2a′) = ϕ1 ⊙ ϕ2(a1◦ +1a′) = ϕ1 ⊙ ϕ2(aa′) = ϕ1(aa′) = 1 +because ⊙ preserves units. We also have ϕ2(b) = ϕ2(1◦ +2) = 1. Putting everything together, ν◦ = 1. +A simple induction on the number of blocks shows that απ = ν◦ · απ\{s} whenever π ∈ P has +a singleton block {s} ∈ π of color ◦. Therefore, ν◦ = 1 for all ◦ ∈ F implies that the highest +coefficients are singleton inductive. +Now assume that the highest coefficients of a positive symmetric universal product are singleton +inductive. Let A1, A2 be multi-faced algebras with unital faces, s = b × f ∈ ([2] × F)n, ai ∈ Af(i) +b(i) +for i ∈ [n], ˆa = a1 ⊗ . . . ⊗ an, a = a1 . . . an and as = 1f(s) +b(s). Then, with log := log⊙, +ϕ1 ⊙ ϕ2(a) = +� +π∈P(f) +απ(log ˆϕ1 ⊕ log ˆϕ2)⊗|π|(ˆaπ) = +� +{s}∈π∈P(f) +απ(log ˆϕ1 ⊕ log ˆϕ2)⊗|π|(ˆaπ) + +20 +CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH +by Lemma 9.6. Because α is singleton inductive, απ = απ\{s}. Also, for any π with {s} ∈ π, we +have +(log ˆϕ1 ⊕ log ˆϕ2)⊗|π|(ˆaπ) = (log ˆϕ1 ⊕ log ˆϕ2)⊗|π\{s}|(ˆaπ\{s}) log ˆϕi(as) +� +�� +� +=1 +. +Therefore, ϕ1 ⊙ ϕ2(a1 . . . an) = ϕ1 ⊙ ϕ2(a1 . . . ˇas . . . an). This calculation works for any i ∈ [2] and +• ∈ F, so the statement follows. +□ +Corollary 9.8. A two-faced positive symmetric universal product is unit preserving if and only if +its associated set of partitions contains pNC◦•. +10. Summary and outlook +We found conditions on weights that are necessarily satisfied by the highest coefficients of a +positive two-faced universal product. In the symmetric case, we showed that weights which fulfill +these conditions are always the highest coefficients of a uniquely determined universal product. +We could also determine all families of weights which fulfill these conditions, thereby providing a +list of candidates for positive symmetric universal products. +We hope that the methods developed in this work will eventually lead to a complete classification +of positive multi-faced universal products. To that end, the following problems will have to be +overcome: +• Prove or disprove positivity of the “exceptional cases” which do not admit a representation +on free or tensor product. +• Extend the classification of admissable weights to more than two faces. +• Extend the classification of admissable weights to the non-symmetric case. +• Extend the reconstruction theorem to the non-symmetric case. This might be significantly +more difficult because the cumulants have to be combined using the Campbell-Baker- +Hausdorff formula instead of just the direct sum. +Acknowledgements +We are grateful to Michael Schürmann for numerous fruitful discussions in the course of this re- +search. MG thanks Moritz Weber and Roland Speicher for stimulating discussions and atmosphere +during his stay in Saarbrücken. +References +[BGS02] +A. Ben Ghorbal and M. Schürmann. Non-commutative notions of stochastic independence. Math. Proc. +Cambridge Philos. Soc., 133(3):531–561, 2002. doi:10.1017/S0305004102006072. 1, 2 +[BGS05] +A. Ben Ghorbal and M. Schürmann. Quantum Lévy processes on dual groups. Math. Z., 251(1):147–165, +2005. doi:10.1007/s00209-005-0793-x. 2, 9 +[CNS15] +I. Charlesworth, B. Nelson, and P. Skoufranis. On two-faced families of non-commutative random +variables. Canad. J. Math., 67(6):1290–1325, 2015. doi:10.4153/CJM-2015-002-6. 13 +[DAGSV22] C. Diaz-Aguilera, T. Gaxiola, J. 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Available from http://www.numdam.org/item/ +SPS_1975__9__565_0/. 1 + diff --git a/aNAzT4oBgHgl3EQf2f7B/content/tmp_files/load_file.txt b/aNAzT4oBgHgl3EQf2f7B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d6163bd9cd5b3f1f4a345d7105c2902a37d5a40 --- /dev/null +++ b/aNAzT4oBgHgl3EQf2f7B/content/tmp_files/load_file.txt @@ -0,0 +1,1540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf,len=1539 +page_content='TOWARDS A CLASSIFICATION OF MULTI-FACED INDEPENDENCES: A COMBINATORIAL APPROACH MALTE GERHOLDa & PHILIPP VARŠOb a\ue9d9 0000-0003-4029-1108 Department of Mathematical Sciences, NTNU Trondheim b\ue9d9 0000-0001-9199-2516 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We determine a set of necessary conditions on a partition-indexed family of com- plex numbers to be the “highest coefficients” of a positive and symmetric multi-faced universal product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' the product associated with a multi-faced version noncommutative stochastic inde- pendence, such as bi-freeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The highest coefficients of a universal product are the weights of the moment-cumulant relation for its associated independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We show that these conditions are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate a (automatically unique) symmetric universal product with the prescribed highest coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Furthermore, we give a quite explicit description of such families of coefficients, thereby produc- ing a list of candidates that must contain all positive symmetric universal products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We discover in this way four (three up to trivial face-swapping) previously unknown moment-cumulant re- lations that give rise to symmetric universal products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' to decide whether they are positive, and thus give rise to independences which can be used in an operator algebraic framework, remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Introduction At the latest with Voiculescu’s invention of freeness [Voi85], it became aparent that the “obvious” extension of classical stochastic independence, tensor independence, is not the only and not always the most suitable concept in inherently noncommutative situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In fact, Boolean independence (not yet under this name) has already featured much earlier in the work of von Waldenfels [vW73, vW75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Those “noncommutative independences” share many properties with classical stochastic independence and tensor independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In particular, under the assumption of independence, mixed moments are uniquely determined and can be calculated from marginal moments (also giving rise to an associated convolution product for probability measures on the real line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Another interesting independence is monotone independence, which was discovored by Muraki [Mur01];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' this is a non-symmetric independence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' An extremely useful tool when dealing with random variables which have all moments are the corresponding cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The theory of free cumulants, linearizing free additive convolution, was initiated by Speicher [Spe94], see also the book by Nica and Speicher [NS06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Boolean cumulants were formalized by Speicher and Woroudi [SW97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Understanding the monotone cumulants took a bit longer, many questions were answered by Hasebe and Saigo [HS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The problem in the monotone case is that independence is not in general characterized by vanishing of mixed cumu- lants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' This is directly related to the non-symmetric nature, as becomes apparent when interpreting moment-cumulant relations via exponential and logarithm maps, as is done in a related but different setting by Manzel and Schürmann [MS17] (Hopf algebraic) or Ebrahimi-Fard and Patras [EFP15] (shuffle-algebraic);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' non-zero mixed cumulants can appear in the Campbell-Baker-Hausdorff series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Since the work of Speicher [Spe97], Ben Ghorbal and Schürmann [BGS02], and Muraki [Mur02, Mur03], we know that the five independence relations for noncommutative random variables, ten- sor, free, Boolean, monotone and antimonotone independence, are indeed very special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For these The work of both authors was supported by German Research Foundation (DFG) grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 397960675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The work of MG was carried out during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme at NTNU Trondheim, as a guest researcher at Saarland University in the scope of the SFB-TRR 195, and as a postdoctoral scientific employee at University of Greifswald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The work of PV was partially carried out as a PhD student and scientific employee at University of Greifswald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='01816v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='FA] 4 Jan 2023 2 CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH independences, the joint distribution of independent random variables is obtained from the mar- ginal distributions by means of a “universal product”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' a product operation which fulfills a number of natural conditions, including associativity and universality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' in a specific sense not dependent on the concrete realization of the noncommutative random variables) and a “factor- ization for length 2”-condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' and they are the only ones with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='1 Replacing that “factorization for length 2”-condition by a positivity condition, a decade later, Muraki [Mur13] proved a similar result with a much simpler proof, while at the same time using a much bet- ter motivated assumption, namely that the product operation restricts to a product operation for states on augmented ∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2 This kind of positivity is also the right condition to study quantum Lévy processes on dual groups in the sense of Ben Ghorbal and Schürmann [BGS05], see also [SV14], where Schoenberg correspondence between convolution semigroups of states and conditionally positive generators is proved in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In 2014, Voiculescu [Voi14] introduced a new nontrivial extension of free independence, bifreeness, for sequences of pairs of random vari- ables, or pairs of faces as Voiculescu called the general underlying framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Taking up on this idea, more examples of 2-faced or, more generally, multi-faced independences have been discovered [Liu19, Liu18, GS19, GHS20, Ger17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The general theory of multi-faced universal products from which those independences can be obtained was established by Manzel and Schürmann [MS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' It turned out that not all of the examples fulfill the natural positivity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Positivity is still enough to assure Schoenberg correspondence in this generalized setting, see [Ger21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In an effort to classify positive multi-faced universal products, two routes have been taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In [GHU21], Gerhold, Hasebe, Ulrich completely classified 2-faced universal products which have a natural representa- tion on the tensor product or the free product Hilbert space of the GNS spaces of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In Varšo’s PhD thesis [Var21], he proved that there are at most 12 two-faced universal products which fulfill additional assumptions of symmetry and a “combinatorial” moment cumulant relation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' determined by a subset of all two-faced partitions, where more generally weights on two-faced partitions can appear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='3 In this article we present, simplify, and extend those results of [Var21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A single-faced independence can trivially be regarded a two-faced independence, and every two-faced independence is a certain kind of mixture of two-single-faced independences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' However, neither do those two single-faced independences determine the two-faced independence, nor is it obvious that any combination of single-faced independences can be combined in any way to form a two-faced independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='4 The main result of this article is to present a family of two- faced symmetric universal products such that every positive symmetric two-faced universal product must belong to that family, we call them candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' This is achieved in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' First, we prove necessary conditions for a family of weights on ordered partitions to be the highest coefficients of a positive multi-faced universal product (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' second, we determine all permutation invariant weights (= weights on non-ordered partitions) which fulfill those properties (Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='11), we call such weights here admissable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' third, we prove that admissable weights are always the highest coefficients of a (uniquely determined) symmetric multi-faced universal product (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The family of candidates consists of (identifying an independence with its underlying universal product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' and disregarding the difference between a 2-faced independence and its image under swapping the faces) 2-faced continuous 1-parameter deformations of free,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' tensor and bifree independence (pos- itivity is proved in [GHU21]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' a tensor-free independence (positivity is not known),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' a new free-free and a new tensor-tensor independence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' different from the trivial ones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' bifreeness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' and their deformations (positivity is not known),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 1Speicher [Spe97] proved that there are only three universal calculation rules for mixed moments in the sym- metric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Ben Ghorbal and Schürmann [BGS02] axiomatized independences via universal products and showed equivalence to universal calculation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Muraki [Mur02, Mur03] extended the results to the non-symmetric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 2In the purely algebraic context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' without positivity, Muraki’s classification was slightly extended by Gerhold and Lachs in [GL15], showing that there is a non-symmetric deformation of Boolean independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 3In [Var21], it was also noticed for the first time the possibility that the moment cumulant relation of a positive universal product might not need to be of combinatorial form, which was indeed confirmed in [GHU21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4Note that the study of another kind of mixture of single-faced independences was initiated by Młotkowski [Mło04] and received again more attention after work Speicher and Wysozcański [SW16] and Ebrahimi-Fard, Patras and Speicher [EFPS18] on the corresponding cumulants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' this approach is closely related to graph products of groups and the corresponding universal products are not associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH 3 tensor-boolean, free-boolean and boolean independence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' positivity for those is also covered in [GHU21], for free-boolean it was first shown by Liu [Liu19] and for boolean independence positivity is of course well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We call the independences which are not realized in [GHU21], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' those whose positivity is yet unknown, exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We prove many of the preliminary results for the general symmetric multi-faced case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='1, where we find necessary conditions on weights to arise as highest coefficients of a uni- versal product is even formulated for not necessarily symmetric products and could be used as a starting point for a more general classification including multi-faced universal products based on monotone independence, such as for example bimonotone independence (of type II) as defined in [Ger17, GHS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' It easily follows from the main result that there are no non-trivial positive and symmetric trace preserving universal products (Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='12) and that tensor independence and bifreeness are the only two positive symmetric 2-faced independences which allow to define a convolution of probability measures on R2 (Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Among our additional results, we characterize when a positive symmetric multi-faced universal product is unit preserving (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' when it can be defined consistently for arbitrary unital algebras (in the other cases, the product operation is only defined for linear functionals on augmented algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' This is indeed the case for the three continuous families and the four (three up to swapping the faces) exceptional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Furthermore, we establish a simplified mixed moment formula for the special combinatorial case where the highest coefficients are only 0 or 1, so that the moment cumulant relation is simply governed by a specific set of partitions (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The outline of the article is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In Sections 2 to 5, we introduce the basic concepts, in particular multi-faced partitions and moment cumulant relations for a family of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We also extract along the way the relevant special cases of Manzel and Schürmann’s cumulant theory for universal products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In Section 6 we prove the necessary conditions for a family of weights to be the highest coefficients of a positive multi-faced universal product (symmetric or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In Section 7 we show that those necessary conditions allow us to obtain a concrete list of candidates for symmetric and positive two-faced universal products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In Section 8 we prove that in the symmetric case the conditions are sufficient to reconstruct a universal product in the algebraic sense (with a simplified formula in the combinatorial case), but it remains open whether these universal products are automatically positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Finally, we characterize in Section 9 which universal products in our list are unit preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In Section 10 we name four tasks which have to be completed in order to achieve a complete classification of positive multi-faced universal products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Preliminaries and notation We will have to deal a lot with tuples of all kinds, so we introduce some useful notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let X and Y be arbitrary sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For any natural number n, denote by [n] the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For an n-tuple t = � t(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , t(n) � ∈ Xn and a subset I = {i1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' < ik} ⊂ [n], we define the restricted tuple t ↾ I := � t(i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , t(ik) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Two tuples t ∈ Xn, s ∈ Y n of the same length may be combined to form the tuple t × s ∈ (X × Y )n with (t × s)(i) = � t(i), s(i) � , and conversely, every tuple in (X × Y )n is of that form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The set of n-tuples of arbitrary length n is denoted X∗ = � n∈N0 Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' When a set X does not carry any multiplicative structure, we might use the word notation, t(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' t(n) := � t(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , t(n) � ∈ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The entries of a tuple t might be written ti instead of t(i) from time to time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' or we might use t as a shorthand for (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , tn) without further comment when the ti have been around before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' An algebra means an associative C algebra, not necessarily unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The free product of algebras A1, A2 is denoted A1 ⊔ A2, reminding of the fact that this is the coproduct in the category of alge- bras: for arbitrary algebra homomorphisms hi : Ai → B, there is a unique algebra homomorphism h1 ⊔ h2 : A1 ⊔ A2 → B with h1 ⊔ h2 ↾ Ai = hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a vector space V , we denote by T0(V ) = � n∈N V ⊗n the (non-unital) free algebra over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We will identify T0(V1 ⊕V2) = T0(V1)⊔T(V2) without further commenting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The unital free algebra is denoted T(V ) = � n∈N0 V ⊗n, and is this unital algebra is the unitization of T0(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let F be a fixed finite set, whose elements we call faces or colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We could of course assume F = [m] for m ∈ N, but since there will be a lot of integers around, we prefer to use more abstract symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Typical choices for |F| = 2 are F = {L, R} (for left and right) or F = {◦, •}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We typically use the symbol • for an arbitrary element of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4 CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH A multi-faced algebra is an algebra A that is freely generated by given subalgebras A•, • ∈ F (the faces of A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' the canonical algebra homomorphism � ∈F A• → A is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' this is indicated by writing A = � ∈F A•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A multi-faced algebra homomorphism j : A → B is an algebra homomorphism between multi-faced algebras A, B with j(Ak) ⊂ Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We consider the free product of multi-faced algebras again a multi-faced algebra with faces (A ⊔ B)• := A• ⊔ B•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Note that the free product of multi-faced algebras is the coproduct in the category AlgF of multi- faced algebras with multi-faced algebra homomorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' for every pair of multi-faced algebra homomorphisms ji : Ai → B there is a unique multi-faced algebra homomorphism j1⊔j2 : A1⊔A2 → B restricting to ji on Ai, respectively for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We use the same symbol ⊔ to denote the canonical homomorphism j1 ⊔ j2 : A1 ⊔ A2 → B1 ⊔ B2 when ji : Ai → Bi, it should always be clear from the context which codomain is meant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A multi-faced ∗-algebra is a multi-faced algebra with an involution such that each face is a ∗-subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Of course, the free product of multi-faced ∗-algebras is again a multi-faced ∗-algebra in the obvious way and the free product of multi-faced ∗-homomorphisms is a ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We say that a linear functional ϕ: A → C defined on a multi-faced ∗-algebra is a restricted state if its unital extension to the unitization of A is a state (or, equivalently, positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Universal products Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='1 (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' [Ger21, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A multi-faced universal product is a binary product operation for linear functionals on multi-faced algebras which associates with functionals ϕ1, ϕ2 on multi-faced algebras A1, A2, respectively, a functional ϕ1 ⊙ ϕ2 on A1 ⊔ A2 such that (ϕ1 ◦ j1) ⊙ (ϕ2 ◦ j2) = (ϕ1 ⊙ ϕ2) ◦ (j1 ⊔ j2) for all multi-faced algebra homomorphisms ji : Bi → Ai (universality) (ϕ1 ⊙ ϕ2) ⊙ ϕ3 = ϕ1 ⊙ (ϕ2 ⊙ ϕ3) (associativity) (ϕ1 ⊙ ϕ2) ↾ Ai = ϕi (restriction property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The product is called symmetric if ϕ1 ⊙ ϕ2 = ϕ2 ⊙ ϕ1, positive if the product of restricted states on multi-faced ∗-algebras is a restricted state on the free product ∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Note that we made several implicit identifications between isomorphic free products in the last definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a more detailed discussion see [Ger21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , Ak be multi-faced algebras and A = A1 ⊔ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' ⊔ Ak (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' we identify the Ai with subalgebras of their free product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For s = b × f ∈ ([k] × F)n, we denote As := � a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' an ∈ A : ai ∈ Af(i) b(i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Note that the As are not necessarily pairwise disjoint (consecutive repitition of the same letter in the word s leads to a subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Elements of [k]n are referred to as block structures and elements of Fn are called face structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Given a multi-faced universal product ⊙, we define its linearized part as ϕ1 ⊡ · · · ⊡ ϕk(a) := ∂k ∂t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' ∂tk (t1ϕ1) ⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' ⊙ (tkϕk)(a) ���� t=0 (that this expression is well-defined should be understood as part of the following theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2 (Adjusted and simplified from [MS17, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let ⊙ be a positive multi-faced universal product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then there are unique coefficients αs (s ∈ ([k] × F)n) such that, for all linear functionals ϕj : Aj → C on multi-faced algebras Aj (j ∈ [k]) and all a ∈ As, ϕ1 ⊡ · · · ⊡ ϕk(a) = αs · ϕ1 � � → � b(j)=1 aj � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' ϕk � � → � b(j)=k aj � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' (1) (The symbol → � indicates that the product is to be taken in the same order as the factors aj appear in a the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=') The αs are called highest coefficients of ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 5 5In the complete expansion of the universal product, also lower coefficients would appear, according to terms that are not multilinear in the ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' First assume that s ∈ ([k] × F)n is alternating, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' s(i) ̸= s(i + 1) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' By [MS17, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='3], the formula given in [MS17, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2] can be applied, and due to the “linearization” we performed only those terms survive that are linear in each ϕj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a positive universal product, [MS17, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='3] implies that there is only one such term, corresponding to the “right-ordered highest coefficient” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' the aj are multiplied in the same order in which they appear as factors in a) associated with s, denoted αs in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' If s is not alternating, then we define αs := α�s where �s is the alternating tuple obtained from s merging repeating entries into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' By universality it is obvious that (1) extends to all s ∈ ([k] × F)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A multi-faced restricted state ϕ: A → C is called trivially multi-faced if there are ∗-isomorphisms j•1,•2 : A•1 → A•2 with ϕ ↾ A•2 = ϕ•1 ◦ j•1,•2 for all •1, •2 ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let ⊙ be a positive multi-faced universal product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then, for every s ∈ ([k] × F)∗, there are trivially multi-faced restricted states ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , ϕk and an element a ∈ As with αs = ϕ1 ⊡ · · ⊡ ϕk(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let s = b×f ∈ ([k]×F)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Define A• j := C and Aj := � ∈F A• j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then ϕj = � ∈F id: Aj → C is a state, in particular a restricted state, and trivially multi-faced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Put a• i := 1 for all i ∈ [k] and all • ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Now it is easy to see that ϕi(a•1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' a•m i ) = 1 for all m ∈ N and •ℓ ∈ F (ℓ ∈ [m]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' With a := af(1) b(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' af(n) b(n), the claim now follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Partitions In general, a multi-faced set is a set X together with a map f : X → F, the face structure of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The subsets X• := f −1({•}) are called the faces of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A multi-faced subset of X is just a subset of the underlying set viewed as a multi-faced set with respect to the restricted face structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In this article, we only deal with multi-faced sets whose underlying set X is finite and totally ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Any word f = f(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' f(n) ∈ F∗ defines such a finite and totally ordered multi-faced set Xf = [n] with face structure k �→ f(k) (which we identify with the word f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Conversely, we associate with a finite totally ordered multi-faced set X = ({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' , xn}, f) the word |X| := f(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' f(xn) ∈ F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We choose this on first sight odd notation because the word f plays the same role as the number of elements of a set plays in the single-faced case in the moment-cumulant formulas we are aiming at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let X be a multi-faced set and ∼ an equivalence relation such that the equivalence classes are intervals, f is constant on equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then we understand the quotient X/∼ as a multi-faced set with the induced total order and face map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We briefly discuss the two situations that will appear several times in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' (1) Let f ∈ Fn be a face and ∼ the equivalence relation on [n] that identifies two neighboring points k, k + 1 in the same face, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' f(k) = f(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In this case we write f/(k ∼ k + 1) for the quotient Xf/ ∼ and denote its elements i instead of {i} for the trivial equivalence classes of i ∈ [n] \\ {k, k + 1} and {k, k + 1} for the two-element equivalence class of k and k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' (2) Let X be an arbitrary multi-faced set and ∼ the equivalence relation whose equivalence classes are the maximal intervals on which f is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We then call the quotient Xred := X/∼ the reduction of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In the reduction, neighboring points will always have different faces, so that no further quotienting is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A partition of a multi-faced set X is a collection of multi-faced subsets whose underlying sets form a set partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The set of all partitions of a multi-faced set X is denoted P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' An ordered partition of X is a partition of X together with a total order between the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The set of all ordered parititions is denoted P<(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a word f ∈ F∗, we put P(f) := P(Xf) and P<(f) := P<(Xf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We also denote P := � f∈F∗ P(f), P< := � f∈F∗ P(f) Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let F = {◦, •} and consider f = ◦••◦• ∈ F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then π = {V1, V2} with V1 = {1, 3, 4}, V2 = {2, 5} is an element of P(f) and we have |V1| = ◦•◦, |V2| = ••.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' This can be nicely 6 CLASSIFICATION OF MULTI-FACED INDEPENDENCES: COMBINATORIAL APPROACH drawn as an arc diagram, π = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In the following we will not distinguish between a partition and its arc diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In this article, we only use arc-diagrams to denote partitions in P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' without a block-order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' the height of the blocks is completely arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' P(f) is a partially ordered set by the order of reverse refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' The maximum and minimum of P(f) are denoted 1f and 0f, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 1f is the one-block partition and in 0f all blocks are singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' There is a canonical bijection between P(X/∼) and the set of π ∈ P(X) such that equivalent points of X lie in the same block of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a multi-faced partition π, consider the equivalence relation ∼ on the underlying multi-faced set X whose equivalence classes are the maximal intervals of X on which f is constant and which lie completely inside one block of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We define the reduction of π as the induced multi-faced partition πred on X/∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then πred will not have neighboring legs that are in the same face and in the same block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a multi-faced set X, we define its mirror image X as the same set with the same face structure, but order reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For π ∈ P(X), we put π ∈ P(X) as the same set partition as π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Finally, we introduce a notation for uniting blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let π = {B1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' < Bk} ∈ P<(X) with blocks Bi, Bi+1 that are nearest neighbors for the order on π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Then we define πBi⌣Bi+1 := {B1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' < Bi−1 < Bi ∪ Bi+1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' < Bk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Similarly, for π ∈ P(f) and arbitrary blocks B1, B2 ∈ π, πB1⌣B2 := π \\ {B1, B2} ∪ {B1 ∪ B2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Let fi ∈ Fmi, i ∈ [n], be face structures and f their concatenation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' f(m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' + mi−1 + ℓ) = fi(ℓ) for all i ∈ [n], ℓ ∈ [mi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Given partitions πi ∈ P(fi), we define their concatenation as the partition π ∈ P(f) which has for every block V ∈ πi with i ∈ [n] a block �V := {ℓ : ℓ+�i−1 j=1 mj ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Roughly speaking, π restricts to πi on the legs corresponding to fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Moment-cumulant relations Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' A family of complex numbers α = (απ)π∈P< is called (family of) weights on ordered partitions, a family α = (απ)π∈P is called (family of) weights on partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Weights on (ordered) partitions are called monic if απ = 1 for every one-block partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' For a family of numbers αs : s ∈ ([k] × F)n, k, n ∈ N (as it is for example obtained from a universal product by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='2) and π = {B1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' < Bk} ∈ P<(f) an ordered multi-faced partition with k blocks, we define sπ ∈ ([k] × F)n via sπ(i) := (κ, •) if i ∈ Bκ and f(i) = • and put απ := αsπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' In this way, we associate with each universal product a family of weights on ordered partitions, and we say that the weights of a universal product are its highest coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' Note that such weights are always monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content=' We say that weights on ordered partitions α are invariant under permutation of blocks if α{B1<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf'} +page_content='