Add files using upload-large-folder tool

-NAzT4oBgHgl3EQf_P6F/content/tmp_files/2301.01945v1.pdf.txt

@@ -0,0 +1,3160 @@
+Optimizing density-functional simulations for two-dimensional metals
+Kameyab Raza Abidi and Pekka Koskinen∗
+NanoScience Center, Department of Physics, University of Jyv¨askyl¨a, 40014 Jyv¨askyl¨a, Finland
+(Dated: January 6, 2023)
+Unlike covalent two-dimensional (2D) materials like graphene, 2D metals have non-layered struc-
+tures due to their non-directional, metallic bonding. While experiments on 2D metals are still scarce
+and challenging, density-functional theory (DFT) provides an ideal approach to predict their basic
+properties and assist in their design. However, DFT methods have been rarely benchmarked against
+metallic bonding at low dimensions. Therefore, to identify optimal DFT attributes for a desired
+accuracy, we systematically benchmark exchange-correlation functionals from LDA to hybrids and
+basis sets from plane waves to local basis with different pseudopotentials. With 1D chain, 2D hon-
+eycomb, 2D square, 2D hexagonal, and 3D bulk metallic systems, we compare the DFT attributes
+using bond lengths, cohesive energies, elastic constants, densities of states, and computational costs.
+Although today most DFT studies on 2D metals use plane waves, our comparisons reveal that local
+basis with often-used PBE exchange-correlation is well sufficient for most purposes, while plane
+waves and hybrid functionals bring limited improvement compared to the greatly increased compu-
+tational cost. These results ease the demands for generating DFT data for better interaction with
+experiments and for data-driven discoveries of 2D metals incorporating machine learning algorithms.
+I.
+INTRODUCTION
+The discovery of graphene nearly two decades ago
+sparked an entire new research field of two-dimensional
+(2D) materials [1]. The 2D materials pedigree has ex-
+panded ever since, thanks to unique properties and vi-
+sions for novel applications [2–5].
+Most 2D materials
+are covalently bound and have layered structures eas-
+ily exfoliable from three-dimensional (3D) bulk matter
+[6, 7]. However, in contrast to directional covalent bond-
+ing, non-directional metallic bonding prefers large coor-
+dination numbers, which renders low-dimensional metal
+structures energetically unfavourable. Despite this pref-
+erence for large coordination, in 2014 atomically thin sta-
+ble iron patches were discovered in graphene pores [8].
+This discovery has been followed by rapid progress in re-
+search on 2D metals and alloys, making 2D metals a full
+member the 2D materials family [9–14].
+The wavering stability of 2D metals makes experi-
+ments challenging, whereby research relies heavily on
+computations. A reasonable description of metallic bond-
+ing requires electronic structure simulations, which has
+made the density-functional theory (DFT) [15, 16] the
+workhorse method for modeling 2D metals [17–31]. Most
+DFT studies have chosen plane wave (PW) basis sets [32]
+and the non-empirical Perdew-Burke-Ernzerhof (PBE)
+exchange-correlation functional [33].
+These choices for
+DFT attributes are plausible in the context of delocal-
+ized electrons in periodic systems that are still lacking
+experimental data. However, DFT attributes have not
+been systematically benchmarked for metallic bonding
+at low dimensions. It is not certain whether these stan-
+dard choices are efficient and accurate enough or they if
+simply waste computational resources.
+∗ pekka.j.koskinen@jyu.fi
+The DFT attributes consist of few central choices. The
+first choice is the flavor of exchange-correlation (xc) func-
+tional, the level of which is of central importance for con-
+sistent results. A functional performing well in some sys-
+tems may perform poorly in others. Here we make use
+of several xc-functionals to obtain a systematic picture
+of their performance in low-dimensional metallic bond-
+ing [34]. The second choice is the type of basis function.
+Plane waves are suitable for periodic systems, whose elec-
+trons fill out the entire simulation cell. Unfortunately,
+the non-periodic directions of low-dimensional systems
+require large vacuum regions that make PW simulations
+inefficient compared to modeling bulk. Thus, an addi-
+tional choice in PW simulations is an optimum size of the
+vacuum. In this respect, PW and grid-based DFT share
+the same challenges [35, 36]. Another alternative for ba-
+sis is linear combination of atomic orbitals (LCAO), and
+controlling its size provides a powerful handle to trade
+between accuracy and efficiency [37].
+The choice of basis type has implications beyond mere
+accuracy. For example, PW is not suitable for studying
+electron transport using nonequilibrium Green’s function
+method in nanoscaled devices [38]. In addition, with the
+coming of data science and machine learning in materials
+science, lots of consistent DFT data is required for ma-
+chine learning -enabled 2D metals studies [39–43]. This
+efficiency demand calls for a critical examination of the
+necessity of PW method to model metallic bonding in
+low dimensions.
+Third choice for periodic systems is the number of k-
+points along periodic directions for the desired accuracy.
+Fourth choice is the level of Fermi-broadening of elec-
+tronic states, which is partly a physical choice but mostly
+a necessity for rapid convergence of the self-consistent it-
+eration of the electron density. In practice, there are a
+plethora of other choices to make for numerical stability
+and speedup, but they are often chosen as default val-
+ues that have been previously fine-tuned for each DFT
+arXiv:2301.01945v1  [cond-mat.mtrl-sci]  5 Jan 2023
+
+2
+FIG. 1. Schematics of the systems with different dimensional-
+ities and coordination numbers C: 1D chain (C = 2), 2D hon-
+eycomb (C = 3), 2D square (C = 4), 2D hexagonal (C = 6),
+and 3D bulk (C = 12). The quadrilaterals show the simula-
+tion cells.
+code. In this article, we consider the above-mentioned
+choices of DFT attributes regarding xc-functionals, basis
+sets, vacuum, k-point sampling, and Fermi-broadening,
+and juxtapose their performance against various prop-
+erties of selected low-dimensional metal systems.
+The
+selected systems include a one-dimensional chain (coor-
+dination number C = 2), three two-dimensional lattices
+(C = 3, 4, and 6), and a 3D bulk (C = 12) (Figure 1).
+These systems enable comparative analysis of the perfor-
+mance of DFT attributes in various dimensions. Being
+low-dimensional systems, these structures are prone to
+various symmetry-breaking deformations, such as out-of-
+plane buckling in 2D or Peierls distortions in 1D [26, 44].
+However, in order to enable unambiguous comparison of
+the effect of dimensionality and coordination and avoid
+making unfounded conclusions based on incomplete set
+of deformations, we retain our focus on these ideal, non-
+deformed systems.
+We also compare the performance
+and speed of DFT to the density-functional tight-binding
+(DFTB) method, which is the next-in-line approximation
+to DFT [45]. One of our main conclusions is that, for
+general purposes, DFT-LCAO can be chosen over the de-
+fault DFT-PW without compromising accuracy, a choice
+which enables simulating transport and helps generating
+DFT data more effortlessly.
+Our treatise will advance
+DFT modeling of 2D metals and help boosting the inter-
+action with experiments.
+II.
+COMPUTATIONAL METHODS
+The basic idea DFT is to use the variational principle
+to generate exact ground state energy and density for the
+systems of interest [15]. The ground state energy E is a
+functional of the electron density (n),
+E[n] = T[n] + Eext[n] + EH[n] + Exc[n] ,
+(1)
+where T[n] is the Kohn-Sham kinetic energy for the fic-
+titious non-interacting electron system, Eext[n] is the ex-
+TABLE I. Exchange-correlation functionals used in this work.
+Functional and its family
+Refs.
+Local Density Approximation (LDA)
+[15, 55]
+Generalized Gradient Approximation (GGA)
+[56]
+RPBE
+[57]
+PW91
+[58, 59]
+PBE
+[33]
+Hybrid Functionals
+[60]
+B3LYP
+[61]
+PBE0
+[62]
+HSE03 (screening ω = 0.15 Bohr−1)
+[63]
+HSE06 (screening ω = 0.11 Bohr−1)
+[64]
+ternal potential energy, EH[n] is the Hartree energy, and
+Exc[n] is the exchange-correlation energy. The xc term
+attempts to capture the complex features of many-body
+quantum mechanics, and a variety of approximate xc
+functionals have been developed for different purposes
+[34].
+As a result, the quality of xc functional mostly
+determines the quality of the results.
+Here, using the
+QuantumATK (S-2021.06) DFT implementation [46], we
+explore the set of eight xc functionals ranging from local
+density approximation to hybrid functionals (Table I).
+We used two types of basis sets, plane waves and
+LCAOs. The wave-function energy cutoff for plane waves
+was 800 eV. Cutoff needed no separate analysis for low-
+dimensional metals, because it depends only on element
+and pseudopotential [47].
+For LCAOs, we used three
+variants: LCAO-M(edium), LCAO-H(igh), and LCAO-
+U(ltra). These variants derive from the numerical basis
+sets of the FHI-aims package [48], but are further opti-
+mized for computational speed of the LCAO calculator.
+For example, for Ag the radial functions for Medium ba-
+sis are 3s/2p/1d (14), for High 4s/3p/5d/1f (35), and
+for Ultra 4s/3p/5d/2f/1g (51), with brackets displaying
+the total number of orbitals per atom [37, 48]. Local ba-
+sis sets were used in conjunction with norm-conserving
+PseudoDojo pseudopotentials [49].
+Further, the total energy convergence criteria for self-
+consistent electron density was ≤ 10−7eV. System ge-
+ometries were optimized to forces below 1 meV�A
+−1 and
+stresses below 0.3 meV�A
+−3 using the LBFGS [50] algo-
+rithm.
+The k-points were sampled by the Monkhorst-
+Pack method [51]. All calculations were spin-polarized
+and the initial guess for lattice parameters were adopted
+from the Atlas of 2D metals [20].
+To complement the results with various DFT at-
+tributes with wider context, we analyzed the systems
+with Ag also with DFTB method at the level of self-
+consistent charge [45, 52]. The Ag parametrizations were
+taken from earlier studies [53, 54].
+
+(a) 1D Chain
+(b) 2D Honeycomb (hc)
+(c) 2D Square (sq)
+OODD
+(d) 2D Hexagonal (hex)
+(e) 3D Bulk
+X3
+III.
+RESULTS AND DISCUSSION
+A.
+Convergence Analysis
+We made various systematic convergence analyses for
+the group of coinage metals Cu, Ag, and Au [17–19].
+Computational and experimental studies have shown
+that the free-standing monolayer patches of these met-
+als are stabilized by graphene pores [13, 22, 24, 31]. The
+analyses were done using PBE xc-functional [33], projec-
+tor augmented waves (PAW) for core electrons [65], and
+plane waves for valence electrons.
+a.
+k-point convergence:
+The k-point convergence
+was studied using the 2D systems with a converged vac-
+uum of 15 �A in the non-periodic direction (as confirmed
+below).
+The total energy is practically converged at
+30 × 30 × 1 k-point sampling, and we define the energy
+tolerance using this value,
+∆E = ENk×Nk×1 − E30×30×1 .
+(2)
+Apart from rapid convergence at very few k-points, the
+convergence is exponential. Chosen relative energy tol-
+erance can therefore be approximated by
+log δ = A1 + B1L ,
+(3)
+where δ =| ∆E | /E3D is an (approximate) relative en-
+ergy tolerance, the ratio between energy tolerance to the
+3D cohesive energy E3D [66]. The length L = acNk, the
+product of simulation box length and the number of k-
+points in corresponding direction, is the maximum period
+of the Bloch wave function. Using L as the convergence
+parameter helps identifying the required k-point sam-
+pling for variable simulation cell sizes in later research.
+The k-point convergence is not monotonic; more k-
+points does not necessarily mean better accuracy (Fig-
+ure 2). However, for different system symmetries and cell
+shapes and sizes, the ansatz (3) works satisfactorily. Lin-
+ear regression analysis to the data gives the parameters
+A1 = −1.29 and B1 = −0.036 �A
+−1 (Figure 2). Inverting
+Eq. (3), we can obtain an optimal number of k-points for
+given simulation cell size ac and desired accuracy δ as
+Nk(δ) = ceil
+�L(δ)
+ac
+�
+,
+(4)
+where ceil(x) = ⌈x⌉ maps x to the least integer greater
+than or equal to x. For instance, with relative accuracy
+δ = 10−3 one obtains the Nk = ⌈47 ˚A/ac⌉, suggesting
+Γ-point calculations for 4.7-nm-sized simulation cells. In
+subsequent analyses, we use Nk = 13, suggesting ∼ δ =
+10−2.5...−3 relative tolerance.
+b.
+Vacuum convergence:
+Using plane waves requires
+periodicity in all directions, regardless of system dimen-
+sions. Low-dimensional systems need therefore a large
+vacuum region in the non-periodic direction to avoid spu-
+rious interactions with periodic images of the system.
+Larger vacuum means more volume and computational
+FIG. 2. The k-point convergence of total energy for 2D sys-
+tems made of coinage metals. δ is the relative energy toler-
+ance and L is the maximum period of the Bloch function [cf.
+Eq.(4)]. The linear fit refers to Eq. (3).
+cost, implying a need to minimize the vacuum without
+affecting the energy. For a complete picture, we inves-
+tigate vacuum convergence not only in 2D systems and
+but also in 1D chains and free atoms.
+We normalize atoms’ dimensions by their van der
+Waals radii RvdW and consider the normalized vacuum
+Lnorm = Lvac/RvdW , where Lvac is the vacuum along the
+non-periodic direction (i.e., the separation between peri-
+odic images.) The total energy is practically converged
+at 8-˚A vacuum, and we define the energy tolerance as
+∆E = E(Lvac) − E(8 �A) and relative energy tolerance
+again as δ = ∆E/E3D. The tolerance converges roughly
+exponentially, log δ = A2 + B2Lnorm (Figure 3). Conse-
+quently, the vacuum for a desired relative energy accu-
+racy for a given element can be estimated from
+Lvac(δ) = RvdW
+(log δ − A2)
+B2
+,
+(5)
+where the parameters A2 = 2.38 and B2 = −1.65 were
+obtained by linear regression. For instance, the relative
+tolerance δ = 10−3 requires Lvac = 3.3 × RvdW .
+In
+subsequent analysis, if not said otherwise, we will use
+Lvac = 10 �A, which for Ag means δ = 10−4.2, in rough
+alignment with k-point convergence.
+Still, such a single estimate is indicative at best. The
+vacuum convergence follows roughly the coordination
+number, free atom converging the slowest, hexagonal sys-
+tem the fastest (Figure 3).
+This suggests that for a
+given element the vacuum should be set by the lowest-
+coordinated atom—or by the free atom to be on the
+safe side. After all, a modest 16 % increase in vacuum
+(Lnorm = 2.5 → 3.0) may increase the relative accuracy
+by an order of magnitude. Thus, a single fit as above
+is not the best guideline and the vacuum convergence is
+best considered by case basis, especially in the presence
+
+Au
+100
+Au.
+Cu.
+Ag.
+SO
+SC
+Auhc
+10-1
+Linear fit
+10-2
+10-3
+10-5
+10-6
+10-7
+10-8
+20
+40
+60
+80
+100
+120
+0
+140
+L (A)4
+FIG. 3. Vacuum convergence of the total energy for 1D and
+2D systems made of coinage metals. δ is the relative energy
+tolerance and Lnorm is vacuum normalized in terms of van der
+Waals radii. Free atom vacuum convergences are added for
+comparison.
+of possible charge transfer.
+B.
+Effect of Fermi broadening
+In principle the Fermi-broadening is a physical param-
+eter intimately linked to the electronic temperature T;
+in practice it is frequently used as a technical parameter
+to accelerate the self-consistency convergence. The tech-
+nical attitude towards broadening is evident in available
+methods other than the Fermi-function. Computational
+literature shows a plethora of different values for Fermi-
+broadening, but its effect is rarely discussed in detail. For
+insulators and semiconductors the broadening is inconse-
+quential, but for metals it matters. In this section, we
+want to investigate its effect on the energetics systemat-
+ically, for sheer completeness and future reference.
+Ideally, broadening should be chosen to enable rapid
+convergence without conflicting too much with other con-
+vergence parameters. We investigated the effect of broad-
+ening by increasing the electronic temperature T from
+10−5 K to 1000 K and looked at the energy difference
+∆E(T) = E(T) − E(10−5 K).
+(6)
+The temperature 10−5 K was the smallest that enabled
+robust convergence for all systems. Vacuum was 15 ˚A
+for all systems. As a result, 1D systems were most sensi-
+tive to the broadening, 3D bulk systems were least sen-
+sitive (Figure 4). This result is plausible, because the
+density of states is the smallest for 1D systems. In 2D
+and 3D systems there are more k-points, density of states
+at Fermi-level is greater, and state occupations average
+over a larger set of states, consequently diminishing the
+influence of broadening. The 2D systems show energy
+FIG. 4. The effect of electronic temperature on the cohesion
+energy of coinage metals in different dimensions.
+variation around ∼ 10 meV upon increasing temperature
+to 1000 K, corresponding to 86 meV energy broaden-
+ing (Figure 4). For the remainder of the calculations in
+this article, we used the electronic temperature of 580 K
+(�=0.05 eV).
+C.
+Performance of exchange-correlation functionals
+We investigated the performance of xc functionals by
+first fixing certain attributes.
+To eliminate uncertain-
+ties from an insufficient description of valence electrons,
+we used the most complete PW basis set and the PAW
+potential to describe the core electrons.
+We used the
+converged number of k-points and size of vacuum from
+previous analysis, as well as the recently adopted 0.05 eV
+broadening. With these choices, we may concentrate on
+the performance of xc-functionals without worrying too
+much about artifacts from other sources.
+We also investigate xc functionals by using only Ag
+systems. By belonging to the same group, the coinage
+metals follow similar trends and it is reasonable to expect
+other metals to follow the trends of Ag. Still, we do not
+claim Ag displays completely universal trends, for there
+are elements that have complex many-body effects even
+beyond the capabilities of DFT.
+In the following, we compare the xc-functional perfor-
+mance against bond lengths, cohesive energies, and elas-
+tic moduli of all 1D, 2D, and 3D systems. The electronic
+structure is compared in terms of later-introduced char-
+acteristic figures related to the density of states at the
+Fermi-level.
+a.
+Cohesive Energies:
+The cohesive energy was de-
+fined as
+Ecoh = Efree − E/N ,
+(7)
+
+100
+10
+Ag1D
+Ag
+AuFree
+AuD
+Auhex
+ny
+AU
+CuiD
+CuFree
+hex
+10-4
+Linear fit
+1.5
+2.0
+2.5
+3.0
+3.5
+ norm0
+.5
+(Aa) 
+△E
+-10
+-15
+3D
+2D
+1D
+-20
+400
+0
+200
+600
+800
+1000
+Electronic temperature
+(K5
+FIG. 5. The cohesive energies of optimized 1D, 2D (hc, sq,
+and hex), and 3D systems of Ag with different xc-functionals.
+where E is the energy of the system with N atoms and
+Efree is the energy of free atom calculated by placing it
+inside a 15-�A cube.
+All functionals display similar trends, cohesive energy
+increasing monotonically from 1D to 3D bulk (Figure 5).
+Yet the quantitative differences are visible. LDA displays
+its well-known tendency to overestimate cohesive ener-
+gies. The 3D bulk cohesion shoots over the experimental
+value by 23 % [66]. GGA functionals work significantly
+better, where PW91 and PBE are now off by approx-
+imately ≈ 13 − 14 %.
+In contrast, RPBE shows con-
+siderable underbinding and even less accurate cohesion
+than LDA. Among hybrid functionals, the performance
+of screened exchange HSE03 and HSE06 is better than
+PBE0, which still suffers from the spurious Coulomb in-
+teraction. B3LYP describes cohesion poorly and is out-
+performed by practically all other functionals, and should
+be avoided while modeling 2D metals—a conclusion not
+surprising in the light of previous observations [67]. In
+addition, convergence of free atom with B3LYP was dif-
+ficult and required loosening the convergence criterion to
+≤ 10−6 eV (loosening had an insignificant effect on the
+cohesion of Figure 5). As a rule, GGA and hybrid func-
+tionals outperform LDA, but a hybrid functionals do not
+necessarily outperform GGA. PW91 and PBE appear as
+still as fair choices for robust energetics for general pur-
+poses.
+b.
+Dimensionality-dependence of energetics:
+In 2D
+metal modeling, the coordination of single metal atoms
+can range from C ∼ 1 to C ∼ 6 and occasionally beyond.
+The computational method should therefore capture cor-
+rectly the relative energetics of atoms at different coor-
+dination numbers. In other words, the cohesion should
+increase with the coordination number with an appropri-
+ate dependence.
+Our ansatz for the C-dependence for
+FIG. 6. Trends of low-dimensional energetics with different
+xc-functionals. The fitted scaling exponent γ is plotted for
+different xc-functionals; smaller γ means that energy depends
+less linearly on the coordination number [see Eq.(7)].
+the cohesion Ecoh is
+Ecoh(C) = E3D
+coh × (C/12)γ ,
+(8)
+where E3D
+coh is the 3D bulk cohesion and γ is an expo-
+nent that quantifies the coordination- or dimensionality-
+dependence of the cohesion energy. The ansatz has the
+correct asymptotic limits [Ecoh(0) = 0 and Ecoh(12) =
+E3D
+coh] and suffices for our purposes in this article. (We
+tested also more refined ansatzes, but the conclusions
+remained the same.) The exponent γ was obtained by
+fitting the Eq. (8) for energies from each functional.
+As the result, LDA and all GGA and HSE function-
+als show roughly the same γ, the same dimensionality-
+dependence in energetics (Figure 6). Especially the de-
+pendencies in different GGAs are nearly identical. Only
+the dependencies in B3LYP and PBE0 are clear out-
+liers, PBE0 showing more linear dependence on C (γ
+closer to one) and B3LYP showing more non-linear de-
+pendence on C (γ further away from one). Interestingly,
+although LDA badly overestimates the absolute cohe-
+sion energies, the dimensionality-dependence lies some-
+where in between GGAs and HSE functionals. In conclu-
+sion, GGA-PBE appears to capture the dimensionality-
+dependence of energetics comparably well and be still a
+serious competitor to the far more costly HSE function-
+als.
+c.
+Bond Lengths:
+The bond lengths were obtained
+directly from the optimized lattice constants (Figure 7).
+In accordance with overbinding, LDA functional shows
+small bond lengths. In 3D, the functionals PW91, PBE,
+PBE0, HSE03, and HSE06 are underbinding and show
+1 − 2 % too large bond lengths.
+PBE0 shows short-
+est bonds among hybrid functionals, and B3LYP shows
+longest bonds among all functionals. Nearly all function-
+als show monotonic increase of bond length with coor-
+
+4.0
+Ag3D
+Ag.
+Ag
+0
+hex
+3.5
+3.0 -
+(eV)
+2.5
+Cohesive energy (
+2.0
+1.5
+1.0-
+0.5-
+0.0
+LDA
+RPBE
+PW91
+PBE
+B3LYP
+PBEO
+HSE03
+HSE06
+Exchange-correlation functional0.46
+0.44 -
+0.42 -
+0.40 -
+0.38 -
+0.36 -
+0.34 -
+0.32 -
+0.30
+LDA
+RPBE
+PW91
+PBE
+B3LYP
+PBEO
+HSE03HSE06
+Exchange-correlation functional6
+dination number. Only LDA functional is an exception:
+it has a slightly smaller bond length for 2D hexagonal
+lattice than for 1D chain.
+d.
+Elastic constants (theory recap):
+Due to colorful
+practices in the notations of low-dimensional elasticity,
+and to avoid any confusion, we wish to define explicitly
+the elastic constants presented in this article.
+Within the linear elastic regime the stresses {σi} and
+strains {εi} (i = 1 . . . 6) satisfy the generalized Hooke’s
+law
+σi =
+6
+�
+j=1
+Cijεj ,
+(9)
+where Cij are elastic constants and expressed as a 6 × 6
+matrix and ε1 = εxx, ε2 = εyy, ε3 = εzz, ε4 = 2εyz, ε5 =
+2εxz, ε6 = 2εxy, when following the Voigt notation. We
+adapted the formalism of Refs. [68–72] to evaluate the
+elastic constants for 1D, 2D and 3D systems.
+In 3D, the strain tensor is
+ϵ3D =
+�
+�
+ε1
+ε6/2 ε5/2
+ε6/2
+ε2
+ε4/2
+ε5/2 ε4/2
+ε3
+�
+� .
+(10)
+The elastic constants are obtained by applying selected
+strains {εi} to the equilibrium simulation cell and by
+calculating the partial derivatives
+Cij = ∂2∆U
+∂εi∂εj
+.
+(11)
+Here ∆U(εi) = U(εi)−U(0) is the elastic energy density
+per unit volume, where U(εi) is the energy density at
+strain εi. For a system with cubic symmetry, the energy
+FIG. 7. Optimized bond lengths of 1D, 2D (hc, sq, and hex),
+and 3D systems of Ag with different xc-functionals
+density is
+∆U(εi) =1
+2
+�
+C11ε2
+1 + C11ε2
+2 + C11ε2
+3 + C12ε1ε2 + C12ε1ε3
++C12ε2ε1 + C12ε2ε3 + C12ε3ε1 + C12ε3ε2
++C44ε2
+4 + C44ε2
+5 + C44ε2
+6
+�
+.
+(12)
+For 2D systems, the strain tensor is
+ϵ2D =
+�
+ε1
+ε6/2
+ε6/2
+ε2
+�
+.
+(13)
+Again, the elastic constants are obtained by applying se-
+lected strains {εi} to the equilibrium simulation cell and
+by calculating the partial derivatives
+Cij = ∂2∆U
+∂εi∂εj
+(14)
+Here ∆U(εi) = U(εi) − U(0) is the energy density per
+unit area, where U(εi) is the energy density at strain εi.
+For a system with square symmetry, the energy density
+is
+∆U(εi) =1
+2(C11ε2
+1 + C22ε2
+2 + 2C12ε1ε2 + 2C16ε1ε6
++2C26ε2ε6 + C66ε2
+6)
+(15)
+and all three elastic constants C11, C12 and C66 are in-
+dependent. However, for a hexagonal system, only con-
+stants C11 and C12 are independent and C66 = (C11 −
+C12)/2.
+Finally, for 1D systems, the strain-tensor matrix is sim-
+ply ϵ1D = (ε1). Yet again, the elastic constant is obtained
+by applying the strain ε1 to the equilibrium simulation
+cell and by taking the partial derivative
+C1 = ∂2∆U
+∂2ε1
+.
+(16)
+Here ∆U(εi) = U(εi) − U(0) is the energy density per
+unit length, where U(εi) is the energy density at strain
+εi. In other words,
+∆U(ε1) = 1
+2C11ε2
+1 .
+(17)
+Table II summarizes the formulae for the elastic con-
+stants and their relations. Note that the elastic constants
+in different dimensions have also different units: they are
+GPa for 3D, GPa nm for 2D, and GPa nm2 for 1D (GPa
+nm3−D or eV/˚AD in short, where D is the dimensional-
+ity).
+e.
+Elastic
+constants
+(results):
+Functionals
+show
+similar trends for bulk moduli, but there are quantita-
+tive differences (Figure 8a).
+We remind that because
+the elastic moduli in different dimensions have different
+units, the trend with respect to the coordination num-
+ber can be compared only between different 2D lattices.
+LDA overestimates the bulk moduli systematically, for
+3D bulk by almost 40 %. Only for 1D chain the modulus
+
+Aghc
+Ag3D
+3.0 -
+bso
+2.90 -
+Bond length (A)
+2.80
+2.70
+2.60
+2.50
+LDA
+RPBE
+PW91
+PBE
+B3LYP
+HSE03
+HSE06
+PBEO
+Exchange-correlation functional7
+FIG. 8. Elastic properties of low-dimensional systems of Ag
+with different xc-functionals. Bulk moduli (a) and Young’s
+moduli (b) are shown for all systems, shear moduli (c) and
+Poisson’s ratio (d) are shown only for 3D and stable 2D sys-
+tems. Units for moduli are GPa nm3−D, where D is the sys-
+tem dimensionality.
+TABLE II. Formulae for Bulk Modulus (K), Shear-modulus
+(G), Young’s modulus (Y), and Poisson’s ratio (µ) for the
+systems in Fig. 1.
+System
+K
+G
+Y
+µ
+1D
+C11
+-
+K
+-
+2Dhex/hc
+C11+C12
+2
+C11−C12
+2
+4KG
+K+G
+K−G
+K+G
+2Dsq
+C11+C12
+2
+C66
+C2
+11−C2
+12
+C11
+C11
+C12
+3D
+C11+2C12
+3
+3C44+C11−C12
+5
+9KG
+3K+G
+3K−2G
+2(3K+G)
+is in line with HSE06.
+Among GGAs, the bulk mod-
+uli of PW91 and PBE are nearly the same. The hybrid
+functionals have fairly similar performance, with B3LYP
+again showing a striking exception, especially related to
+1D modulus. These observations in bulk moduli apply
+also to Young’s moduli (Figure 8b). Only GGAs show
+somewhat larger stiffness and the trends in 2D moduli
+for B3LYP and PBE0 are different.
+The shear modulus and Poisson’s ratio are defined only
+for 2D and 3D systems (Figures 8c and d). Moreover,
+shear modulus is not reported for the 2D square lattice
+due to instability against shear deformations. In addi-
+tion, some deformations with PBE0 and B3LYP resulted
+in consistent numerical errors, forcing us to omit shear
+and Young’s modulus as well Poisson ratio for these func-
+tionals.
+In summary, the most consistent behavior in
+elastic moduli is displayed by HSE and GGA function-
+als. LDA, B3LYP and PBE0 functionals suffer from both
+numerical challenges and deviant trends at least in some
+elastic properties.
+f.
+Electronic structure (density of states):
+To com-
+plement pure energetic and geometric properties, we now
+extend our investigations to electronic structure proper-
+ties. Electronic structure is a complex topic with many
+features. To reduce complexity and extract trends, we in-
+vestigate the electronic structure simply in terms of the
+density of states DOS(ϵ) and its projections DOSl(ϵ) to
+s (l = 0), p (l = 1), and d (l = 2) angular momen-
+tum states. In addition, we focus only on energies at the
+vicinity of the Fermi-level ϵ = ϵF .
+Consequently, we define the quantities
+Nl =
+� ∞
+−∞
+DOSl (ϵ) g (ϵ) dϵ
+(18)
+that give the number of l-type orbitals surrounding the
+Fermi-level. The DOS is also normalized by the number
+of atoms in the simulation cell. The envelope function
+g(ϵ) has a Gaussian form
+g (ϵ) = exp
+�
+−1
+2
+�ϵ − ϵf
+σ
+�2�
+(19)
+
+160 -
+I AgiD
+Agsg
+1 Ag3D
+Aghc 
+Aghex
+(a)
+140 -
+120
+80 -
+60 -
+20 -
+115
+b
+100
+80 -
+Young's modulus
+40 -
+20 -
+C)
+40
+30 -
+Shear modulus
+20 -
+-01
+d)
+os'O
+ratio
+0.45 
+s
+0.30 -
+HSE03HSE06
+Exchange-correlation functional8
+FIG. 9. Effect of xc functional on the electronic structure of
+low-dimensional metals made of Ag. Heatmap visualizes the
+number of s-type states (Ns), p-type states (Np), d-type states
+(Nd), and the total number of states (Nt) within a ∼ 1 eV
+energy window around the Fermi-level [see Eq.(18)].
+and we used σ = 1 eV energy window around ϵF .
+In general, the s-orbital contribution decreases with in-
+creasing coordination number for all xc functionals (Fig-
+ure 9).
+In 1D the main contribution comes from s-
+orbitals, followed by p- and d-orbitals for all functionals.
+In 2D this order is rearranged to p > s > d. In 3D this
+same trend is retained by all hybrid functionals.
+The
+LDA, PW91, and PBE have very similar orbital contri-
+bution ordering. For all xc functionals, the p contribu-
+tion is the largest for honeycomb, smallest for 1D, and
+smallest for hexagonal among 2D systems. The ordering
+of Np with respect to different coordination number is
+the same for GGAs, PBE0, and B3LYP. For HSE03 and
+HSE06 all Nl are very similar. The d-orbital contribu-
+tions follow trend similar to s-orbitals. The value of Nd
+is the highest for LDA and the lowest for PBE0 for all
+systems; the most visible difference is the generally low
+Nd of all hybrid functionals, especially in 1D.
+Regarding the total DOS, all GGAs produce nearly
+identical Nt, apart from 3D bulk in RPBE. The total
+DOS from hybrids differs somewhat from the LDA and
+GGA functionals. HSE functionals show similar Nt for
+C = 6 and 12 systems, but differ in other systems. Over-
+all, trends in the total densities are inconsistent for LDA
+and PBE0 functionals, but somewhat consistent among
+GGA as well as B3LYP and HSE functionals.
+g.
+Conclusions on xc functionals:
+To summarize,
+PW91 and PBE perform similarly for forces, energies,
+and densities of states, while RPBE shows underbinding,
+smaller bond lengths, and smaller elastic constants. LDA
+is inferior to GGA practically in all respects. Among hy-
+brid functionals, the performances of HSE03 and HSE06
+aligned in all respects. B3LYP failed to improve GGA in
+terms of accuracy in the lattice constants and cohesive
+energies, even if its electronic structures resembled those
+of HSE functionals. Cohesion energy displayed congruent
+dimensionality-dependencies, apart from visibly differing
+dependencies by B3LYP and PBE0 functionals.
+Before reaching ultimate conclusions, however, we have
+to consider the computational cost (Table III). As ex-
+pected by the nonlocal character of the hybrid function-
+als, already minimal-cell systems require 2 − 3 orders
+of magnitude more computational time for hybrids than
+for LDA and GGA, and for larger systems the differ-
+ence would increase even further. Considering the low
+computational cost, GGA functionals perform extremely
+well compared to hybrid functionals, compared even to
+the most robust HSE family. To conclude, unless the low-
+dimensional metals are studied for very specific purposes,
+the standard PBE indeed remains the preferred weapon
+of choice for low-dimensional metals modeling.
+TABLE III. Computational cost of different xc-functionals:
+Time in seconds to calculate the energy of minimal-cell sys-
+tems using 24 cores. The cell has one atom for all systems
+except for 2D honeycomb.
+LDA RPBE PW91 PBE B3LYP PBE0 HSE03 HSE06
+1D
+39
+39
+44
+43
+476
+1360
+491
+1897
+hc
+49
+59
+62
+58
+16786 20937
+18662
+15006
+sq
+18
+24
+23
+22
+1469
+1739
+1535
+1493
+hex
+16
+19
+20
+17
+1454
+1800
+1698
+1675
+3D
+14
+18
+19
+17
+88553 41352
+38802
+38704
+D.
+Performance of different basis sets
+In this section, we choose PBE xc functional and repeat
+the systematics of the previous section while this time
+varying the basis set. The converged plane wave basis
+gives the best results that provide the reference assessing
+the performance of the three LCAO basis sets Medium,
+High, and Ultra introduced in Section II.
+To obtain a broader context, we compared the DFT-
+LCAO with DFTB method, which uses a minimal local
+basis and contains approximations speeding up the cal-
+culations. Here we used the parameters available for Ag
+developed earlier [53, 54]. However, parametrization can
+be done in different ways, and one should not consider
+these results as unique and absolute representation of
+DFTB.
+a.
+Cohesive Energies:
+The LCAO-U and LCAO-H
+produce cohesive energies very close to those of PW (Fig-
+ure 10). LCAO-M overbinds slightly in comparison, but
+the accuracy for 2D systems is still 3 − 4 % compared to
+PW. The dependence of cohesion on coordination num-
+ber is reproduced with all basis sets, and differences are
+difficult to see on absolute scale. DFTB follows similar
+behavior, but shows significant overbinding, especially
+for 3D bulk.
+
+1.35
+Ag3D
+Aghex
+Agsq
+1.20
+Aghc
+Agid
+1.05
+Ag3D.
+Aghex -
+0.90
+Aghc -
+0.75
+Agid.
+Ag3D
+Aghex
+0.60
+Agsq
+Aghc
+0.45
+AgiD
+Ag3D
+0.30
+Aghex -
+Agsq-
+0.15
+Aghc -
+Agid:
+0.00
+LDA
+RPBE
+PW91
+PBE
+B3LYP
+PBEO
+HSE03
+HSE06
+Exchange-correlation functional9
+FIG. 10. Cohesive energies of optimized 1D, 2D (hc, sq, and
+hex), and 3D systems made of Ag with different basis sets.
+Bars on the left show DFTB results with minimal basis for
+comparison.
+b.
+Dimensionality-dependence
+of
+energetics:
+As
+with xc functionals, we investigate how basis set affects
+the dependence of energetics on coordination number.
+Again this dependence is analyzed via the scaling
+exponent γ in Eq. (8) fitted to the cohesive energies as
+a function of C.
+Compared to PW, the dependence on C becomes sys-
+tematically more linear as we move from Ultra to High
+and ultimately to Medium basis (Figure 11). However,
+still the Medium basis reproduces γ to within 5 % accu-
+racy compared to PW basis. Even DFTB compares well
+in the overall coordination-dependence, although there
+are visible problems in capturing the DFT trends for
+2D systems (the green bars for DFTB in Figure 10).
+However, to state the main point, the choice of basis in-
+fluences dimensionality-dependence of energetics far less
+than xc functional: note that Figs. 6 and 11 have the
+same scale in γ.
+c.
+Bond Lengths:
+The LCAO-U and LCAO-H bond
+lengths are very similar, accurate to within 0.77 % com-
+pared to PW (Figure 12). All LCAO variants overesti-
+mate all bonds, LCAO-M having the lowest performance
+with 1.6 % too long bonds. DFTB no longer captures the
+DFT trends in coordination-dependence. The 1D chain
+bond length is larger than honeycomb and the 2D bonds
+vary wildly, even if the C-ordering still remains correct.
+d.
+Elastic constants and moduli:
+For 1D and 2D sys-
+tems, elastic moduli have minor dependence on basis set
+(Figure 13). The largest deviation from PW occurs for
+3D bulk, for all LCAO variants.
+This deviation likely
+stems from the better space-filling character of PW ba-
+sis. Moreover, although performing well in cohesion and
+bond lengths, LCAO-M performs poorly in all elastic
+properties. LCAO-U is close to PW in all respects, and
+FIG. 11. Trends of low-dimensional energetics with different
+basis sets. The fitted scaling exponent γ is plotted for different
+basis sets; smaller γ means that energy depends less linearly
+on the coordination number [see Eq.(7)]. The vertical scale is
+the same as in Fig. 6.
+FIG. 12. Bond lengths of optimized 1D, 2D (hc, sq, and hex),
+and 3D systems made of Ag with different basis sets.
+LCAO-M captures all the same trends, even if with some
+quantitative differences. These results suggest that, ex-
+cept perhaps for LCAO-M, LCAO basis can be reliable
+for studying mechanical properties of low-dimensional
+metallic systems.
+The LCAO variant -dependency of
+elastic properties is even smaller than the changes upon
+switching from GGA to hybrid functional (compare Figs.
+8 and 13).
+In comparison, DFTB shows both trend differences
+and large absolute differences compared to DFT-LCAO
+(Figure 13). For example, the 1D elastic modulus is over-
+estimated by a factor of ∼ 5.
+Even the trend within
+2D systems was not reproduced. It appears that the Ag
+parametrization should be revised for more reliable me-
+
+Ag3D
+Ag.
+4.0
+1
+bss
+Shex
+3.5
+3.0 -
+2.5
+2.0.
+1.5
+1.0.
+0.5
+0.0
+DFTB
+LCAO-M
+LCAO-H
+LCAO-U
+PW
+Basis set0.46
+0.44 -
+0.42 -
+0.40 -
+0.38
+0.36 -
+0.34 -
+0.32
+0.30
+DFTB
+LCAO-M
+LCAO-H
+LCAO-U
+PW
+Basis setAgh
+Ag
+Shex
+S3D
+Shc
+bss
+3.0
+2.90
+Bond length (A)
+2.80
+2.70
+2.60
+2.50
+DFTB
+LCAO-M
+LCAO-H
+LCAO-U
+PW
+Basis set10
+chanical properties of low-dimensional Ag systems.
+e.
+Electronic structure (density of states):
+Also the
+electronic structure from LCAO is compared here against
+PW results,
+using the indicator numbers given by
+Eq. (18).
+For 2D structures PW gives orbital contri-
+butions in order p > s > d (Figure 14).
+For LCAO
+this trend shuffles to s > d > p, that is, the p contri-
+bution diminishes for all LCAO variants.
+For 1D sys-
+tem the orbital ordering for PW and LCAO basis re-
+mains the same. However, still all basis sets—including
+minimal-basis DFTB—show consistent C-dependence in
+orbital contributions around the Fermi-level. LCAO-H
+and LCAO-U results align better, while LCAO-M re-
+sults are different in some respects. In summary, the C-
+dependence of the total DOS in 2D metals is reproduced
+by LCAO to a fair degree, but the orbital contributions
+are different.
+f.
+Conclusions on basis sets:
+To conclude, LCAO
+basis competes extremely well with PW for studying
+energetic and geometric properties of low-dimensional
+metal systems. Even elastic moduli are reproduced rea-
+sonably well by LCAO-H and LCAO-U basis, compared
+to converged PW basis. The performance of LCAO-M
+basis was notably modest, regarding elastic properties
+and also the details of electronic structure.
+The or-
+bital breakup of the electronic structures at the vicin-
+ity of Fermi-level for PW and LCAO variants differed
+markedly.
+Regarding DFTB, the Ag parametrizations clearly re-
+quire revisiting. The cohesive energies are too large, bond
+lengths are both large and small, and elastic moduli are
+close to arbitrary. Still many of the qualitative trends
+regarding C-dependence were reproduced reliably.
+However, before again reaching ultimate conclusions,
+we have to consider the computational cost with differ-
+ent basis (Table IV). The cost was investigated by simula-
+tion cells with 32−64 atoms and a couple of dozen cores.
+The comparison is thus by no means unique or absolute,
+but it does give a rough inkling of the computational de-
+mands. As expected, DFTB outspeeds DFT by one to
+three orders of magnitude. Within DFT, switching from
+LCAO-M to LCAO-U results in cost increases from a fac-
+tor of two (1D) up to a factor of ∼ 15 (3D). Especially
+for low-dimensional systems LCAOs are faster than PW,
+nearly by two orders of magnitude.
+For 3D bulk PW
+is very competitive against LCAO due to lacking vac-
+uum region; here LCAO-U is even slower than PW. In
+conclusion, unless very high accuracy is of central impor-
+tance, LCAO has demonstrated a fair accuracy in most
+properties and should be prioritized over PW due to its
+superior efficiency. Even LCAO-M basis can be consid-
+ered for simulations where the improved speed wins over
+lost accuracy.
+FIG. 13. Elastic properties of low-dimensional systems of Ag
+with different basis sets. Bulk moduli (a) and Young’s moduli
+(b) are shown for all systems, shear moduli (c) and Poisson’s
+ratio (d) are shown only for 3D and stable 2D systems. Units
+for moduli are GPa nm3−D, where D is the system dimen-
+sionality.
+
+Agid
+Ag3D
+Aghex 
+Aghc
+Agsq
+(a)
+120 -
+100
+ Bulk modulus
+08
+60 -
+40 -
+20-
+0
+(b)
+140 -
+120 -
+80
+Young's r
+F 09
+40 -
+20
+0
+(c)
+50 -
+40 -
+modulus
+Shear r
+1
+20
+0
+(d)
+Fos'O
+0.35 -
+LCAO-M
+LCAO-H
+LCAO-U
+PW
+DFTB
+Basis set11
+FIG. 14.
+Effect of basis set on the electronic structure of
+low-dimensional metals made of Ag. Heatmap visualizes the
+number of s-type states (Ns), p-type states (Np), d-type states
+(Nd), and total number of states (Nt) within a ∼ 1 eV energy
+window around the Fermi-level [see Eq.(19)].
+TABLE IV. Computational cost of different basis sets: Time
+in seconds to calculate the energy of systems using 24 cores.
+The parenthesis contain the number of atoms in the supercell.
+Systems
+DFTB
+LCAO-M
+LCAO-H
+LCAO-U
+PW
+1D (32)
+10
+175
+265
+310
+11890
+2D hc (64)
+20
+215
+355
+610
+13120
+2D sq (64)
+18
+190
+300
+500
+12370
+2D hex(64)
+17
+130
+290
+655
+6885
+3D (64)
+19
+145
+855
+2220
+2050
+E.
+Combined scanning of xc functionals and basis
+sets
+Above we investigated xc functionals (with PW basis)
+and basis sets (with PBE functional) separately. How-
+ever, the performance of xc functionals and basis sets
+can be coupled.
+We therefore complement our analy-
+sis by combined scanning of different xc functionals with
+different basis sets. The bond lengths, cohesive energies,
+elastic constants, and orbital contributions to DOS ob-
+tained at different basis set-xc functional -combinations
+are shown in Tables V, VI, and VII in the Appendix.
+For LDA, the choice of basis set did not affect the co-
+hesion dependence on C (Table V). Changing the basis
+set from PW to LCAO increases the cohesive energy for
+C ≥ 4 and decreases it for C = 1 and 3.
+Decreasing
+the LCAO size also decreases the cohesion, as expected
+in the light of variational principle. Bond lengths with
+PW, LCAO-U and LCAO-H basis are nearly equal. With
+LCAO-M bonds are longer for all systems. The elastic
+properties are nearly basis-independent, with the notable
+exception of LCAO-M (Table VI). Most sensitive to the
+choice of basis is the electronic structure; all LCAO vari-
+ants show the same trend, which however differs signifi-
+cantly from PW (Table VII).
+For GGAs, the performance remains robust upon re-
+ducing the size of the basis set. In fact, the observations
+in Subsection III D with PBE are representative for other
+GGAs as well. Switching PW to LCAO-U or LCAO-H
+changes bond lengths and cohesive energies less than 1 %;
+less robust LCAO-M decreases cohesive energies by 4 %
+and increases bond lengths by ≈ 1.5 % (Table V). Basis
+set sensitivity is the smallest for PW91 and the largest
+for RPBE. Elastic constants follow the accuracy trends
+similar to those of energetics and geometric properties.
+PBE shows some basis set sensitivity, especially for the
+bulk moduli of 2D systems (Table VI).
+For hybrid functionals, the matters are less systematic.
+Using LCAO-M in conjunction with unscreened B3LYP
+and PBE0 functionals results in significant overbinding;
+bond lengths are underestimated by more than 10 % (Ta-
+ble V). With LCAO-H and LCAO-U basis sets the same
+xc functionals underestimate bonds only by ≈ 2 %, while
+increase cohesive energies by ≤ 24 %. B3LYP and PBE0
+are thus extremely sensitive to the quality of LCAO ba-
+sis. Moreover, B3LYP and PBE0 are unable to produce
+elastic moduli due to persistent numerical errors. In con-
+trast, the screened HSE functionals produced robust ge-
+ometries, energetics and elastic properties upon changing
+the size of the LCAO basis. The robustness was even
+better than with PW91 and PBE, although admittedly
+at a considerable computational cost. The orbital con-
+tributions to DOS with PW and LCAO basis were dif-
+ferent; the same effect was observed for PBE functional
+(Figure 14). Among different LCAO variants, LCAO-H
+and LCAO-U show similar orbital contributions for all
+systems. In addition to energetic and geometric prop-
+erties, the peculiarities of B3LYP and PBE0 functionals
+are observable also in electronic properties (Table VII).
+In general, hybrid functionals in conjunction with LCAO-
+H and LCAO-U basis requires prohibitive computational
+resources even for single atom.
+F.
+The effect of DFT implementation
+In addition to DFT attributes, it is important also
+to be able to rely on the DFT implementation itself.
+For completeness, therefore, we briefly discuss the mag-
+nitude of differences related to the numerical imple-
+mentation of DFT. We calculated the cohesive ener-
+gies, bond lengths, and elastic moduli also with the
+GPAW code, using plane wave basis with the same
+800 eV energy cutoff and default parameters [36]. The
+QuantumATK/GPAW cohesive energies were 1.1671 eV /
+1.1661 eV (1D), 1.5062 eV / 1.5054 eV (2D hc), 1.8293 eV
+/ 1.8286 eV (2D sq), 2.0583 eV / 2.0570 eV (2D hex),
+2.5326 eV / 2.5323 eV (3D), bond lenghts 2.6480 ˚A/
+2.6501 ˚A (1D), 2.6700 ˚A/ 2.6682 ˚A (2D hc), 2.6998 ˚A/
+2.700567 ˚A
+(2D
+sq),
+2.7877 ˚A/
+2.7894 ˚A
+(2D
+hex),
+
+1.80
+Ag3D
+Aghex
+1.60
+Aghc
+Agid
+1.40
+Ag3D
+Aghex
+1.20
+ABsq
+Aghc
+1.00
+Agid
+Ag3D
+Aghex
+0.80
+Agsq
+Aghc
+0.60
+Agid
+Ag3D
+0.40
+Aghex
+0.20
+Aghc
+Agid
+0.00
+LCAO-U
+PW
+DFTB
+LCAO-M
+LCAO-H
+Basis set12
+2.9301 ˚A/ 2.9305 ˚A (3D), and bulk moduli 18.32 GPa nm2
+/18.73 GPa nm2 (1D), 17.20 GPa nm / 17.21 GPa nm (2D
+hc), 31.46 GPa nm / 31.26 GPa nm (2D sq), 38.07 GPa nm
+/ 37.79 GPa nm (2D hex), 92.03 GPa / 90.37 GPa (3D).
+Thus, default parameters without tuning give code-
+related differences in cohesive energies ≲ 1.3 meV, in
+bond lengths ≲ 0.002 ˚A, and in bulk moduli ≲ 1 % (2D
+systems) or ≃ 2% (1D and 3D systems). Although the
+comparison used the PBE functional and plane waves, it
+is reasonable to suspect the level of differences to remain
+similar also for other functionals and basis sets. Over-
+all, code-related differences remain considerably smaller
+than the differences originating from physical attributes.
+IV.
+SUMMARY AND CONCLUSION
+In summary, we investigated the performance of vari-
+ous DFT attributes in the modeling of low-dimensional
+elemental metals.
+For future reference, the number of
+k-points, the size of the vacuum region, and the magni-
+tude of Fermi-broadening were given tolerance-dependent
+rules of thumb. Such rules help choosing combinations
+of attributes that result in commensurate accuracies.
+The most robust against the choice of basis set
+was HSE06, followed by HSE03, PBE, PW91, RPBE
+and LDA. The B3LYP produced inaccurate cohesions
+and bond lengths—with the highest computational cost.
+Only the electronic structure in B3LYP was in line with
+other hybrid functionals.
+The energetics, geometries, and elastic properties with
+PW, LCAO-U, and LCAO-H basis sets were in over-
+all good agreement.
+The greatest disparities between
+PW and LCAO methods resided in the orbital contribu-
+tions to the DOS, although in the total DOS they were
+moderated.
+On a general level, LCAO-U and LCAO-
+H performed similarly at different xc functionals; there-
+fore, for general purposes, LCAO-H should be preferred
+over LCAO-U due to superior efficiency (Table IV). The
+LCAO-M basis worked varyingly well in many respects,
+except when used in conjunction with B3LYP and PBE0
+functionals.
+To conclude, in the research of metallic bonding at
+low dimensions, the best value for a given cost is proba-
+bly given by semi-local PW91 and PBE xc functionals in
+conjunction with moderately-sized LCAO-U or LCAO-
+H basis sets.
+These results are encouraging for doing
+large-scale, high-throughput DFT simulations to gener-
+ate data for machine learning algorithms. In comparison,
+DFTB is a very speedy method and is capable of simu-
+lations unaccessible by DFT [73–75], but the quality of
+parametrization needs to be ensured first. We hope that
+our results and gentle recommendations help lifting 2D
+metal research to new heights, expedite better interac-
+tion with experiments, and feed machine learning algo-
+rithms with quality data to drive further discoveries in
+low-dimensional metals.
+ACKNOWLEDGMENTS
+We acknowledge the Finnish Grid and Cloud Infras-
+tructure (FGCI) for computational resources.
+[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D.-e. Jiang,
+Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.
+Firsov, Science 306, 666 (2004).
+[2] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V.
+Khotkevich, S. V. Morozov, and A. K. Geim, Proceedings
+of the National Academy of Sciences 102, 10451 (2005).
+[3] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,
+and M. S. Strano, Nature nanotechnology 7, 699 (2012).
+[4] V. Shanmugam, R. A. Mensah, K. Babu, S. Gawusu,
+A. Chanda, Y. Tu, R. E. Neisiany, M. F¨orsth, G. Sas, and
+O. Das, Particle & Particle Systems Characterization 39,
+2200031 (2022).
+[5] X. Zhang, A. Chen, L. Chen, and Z. Zhou, Advanced
+Energy Materials 12, 2003841 (2022).
+[6] S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev,
+and A. Kis, Nature Reviews Materials 2, 1 (2017).
+[7] D. Geng and H. Y. Yang, Advanced Materials 30,
+1800865 (2018).
+[8] J. Zhao, Q. Deng, A. Bachmatiuk, G. Sandeep, A. Popov,
+J. Eckert, and M. H. R¨ummeli, Science 343, 1228 (2014).
+[9] Y. Ma, B. Li, and S. Yang, Mater. Chem. Front. 2, 456
+(2018).
+[10] Y. Chen, Z. Fan, Z. Zhang, W. Niu, C. Li, N. Yang,
+B. Chen, and H. Zhang, Chemical Reviews 118, 6409
+(2018).
+[11] T. Wang, M. Park, Q. Yu, J. Zhang, and Y. Yang, Ma-
+terials Today Advances 8, 100092 (2020).
+[12] H. Q. Ta, R. G. Mendes, Y. Liu, X. Yang, J. Luo,
+A. Bachmatiuk, T. Gemming, M. Zeng, L. Fu, L. Liu, and
+M. H. R¨ummeli, Advanced Science 8, 2100619 (2021).
+[13] G. Zagler, M. Reticcioli, C. Mangler, D. Scheinecker,
+C. Franchini, and J. Kotakoski, 2D Materials 7, 045017
+(2020).
+[14] X. Wang, C. Chen, B. Jiang, H. Duan, and K. Du, Acta
+Materialia 229, 117844 (2022).
+[15] P. Hohenberg and W. Kohn, Physical Review 136, B864
+(1964).
+[16] W. Kohn and L. J. Sham, Physical Review 140, A1133
+(1965).
+[17] L.-M. Yang, T. Frauenheim, and E. Ganz, Physical
+Chemistry Chemical Physics 17, 19695 (2015).
+[18] L.-M. Yang, M. Dornfeld, T. Frauenheim, and E. Ganz,
+Physical Chemistry Chemical Physics 17, 26036 (2015).
+[19] L.-M. Yang, T. Frauenheim, and E. Ganz, Journal of
+Nanomaterials 2016, 8429510 (2016).
+[20] J. Nevalaita and P. Koskinen, Physical Review B 97,
+035411 (2018).
+[21] J. Nevalaita and P. Koskinen, Physical Review B 98,
+115433 (2018).
+[22] J. Nevalaita and P. Koskinen, Nanoscale 11, 22019
+
+13
+(2019).
+[23] J. Nevalaita and P. Koskinen, AIP Advances 10, 065327
+(2020).
+[24] S. Ono, Physical Review B 102, 165424 (2020).
+[25] Y. Ren, L. Hu, Y. Shao, Y. Hu, L. Huang, and X. Shi,
+Journal of Materials Chemistry C 9, 4554 (2021).
+[26] S. Ono, Physical Review Materials 5, 104004 (2021).
+[27] B. Anam and N. Gaston, Journal of Physics: Condensed
+Matter 33, 125901 (2021).
+[28] P. Kapoor, M. Sharma, and P. Ahluwalia, Physica
+E: Low-dimensional Systems and Nanostructures 131,
+114745 (2021).
+[29] A. Kutana, T. Altalhi, Q. Ruan, J.-J. Zhang, E. S. Penev,
+and B. I. Yakobson, Nanoscale Adv. 4, 1408 (2022).
+[30] A. A. Sangolkar and R. Pawar, Physica status solidi (b)
+259, 2100489 (2022).
+[31] A. A. Sangolkar, S. Jha, and R. Pawar, Advanced Theory
+and Simulations 5, 2200057 (2022).
+[32] G. Kresse and J. Furthm¨uller, Physical Review B 54,
+11169 (1996).
+[33] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re-
+view Letter 77, 3865 (1996).
+[34] S. Lehtola, C. Steigemann, M. J. Oliveira, and M. A.
+Marques, SoftwareX 7, 1 (2018).
+[35] E. L. Briggs, D. J. Sullivan, and J. Bernholc, Physical
+Review B 54, 14362 (1996).
+[36] J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen,
+M. Du�lak,
+L. Ferrighi,
+J. Gavnholt,
+C. Glinsvad,
+V.
+Haikola,
+H.
+A.
+Hansen,
+H.
+H.
+Kristoffersen,
+M. Kuisma, A. H. Larsen, L. Lehtovaara, M. Ljung-
+berg,
+O. Lopez-Acevedo,
+P. G. Moses,
+J. Ojanen,
+T. Olsen, V. Petzold, N. A. Romero, J. Stausholm-
+Møller, M. Strange, G. A. Tritsaris, M. Vanin, M. Walter,
+B. Hammer, H. H¨akkinen, G. K. H. Madsen, R. M. Niem-
+inen, J. K. Nørskov, M. Puska, T. T. Rantala, J. Schiøtz,
+K. S. Thygesen, and K. W. Jacobsen, Journal of Physics:
+Condensed Matter 22, 253202 (2010).
+[37] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Jun-
+quera, P. Ordej´on, and D. S´anchez-Portal, Journal of
+Physics: Condensed Matter 14, 2745 (2002).
+[38] M. Brandbyge, J.-L. Mozos, P. Ordej´on, J. Taylor, and
+K. Stokbro, Physical Review B 65, 165401 (2002).
+[39] J. Schmidt, M. R. G. Marques, S. Botti, and M. A. L.
+Marques, npj Comput Mater 5 (2019).
+[40] B. Mortazavi,
+I. S. Novikov,
+E. V. Podryabinkin,
+S. Roche, T. Rabczuk, A. V. Shapeev, and X. Zhuang,
+Applied Materials Today 20, 100685 (2020).
+[41] J. Cai, X. Chu, K. Xu, H. Li, and J. Wei, Nanoscale Adv.
+2, 3115 (2020).
+[42] G. R. Schleder, C. M. Acosta, and A. Fazzio, ACS Ap-
+plied Materials & Interfaces 12, 20149 (2020).
+[43] B. Ryu, L. Wang, H. Pu, M. K. Y. Chan, and J. Chen,
+Chem. Soc. Rev. 51, 1899 (2022).
+[44] E. Canadell, M.-L. Doublet, and C. Iung, Orbital Ap-
+proach to the Electronic Structure of Solids (Oxford Uni-
+versity Press, 2012).
+[45] M. Elstner,
+D. Porezag,
+G. Jungnickel,
+J. Elsner,
+M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Phys-
+ical Review B 58, 7260 (1998).
+[46] S. Smidstrup, T. Markussen, P. Vancraeyveld, J. Wellen-
+dorff, J. Schneider, T. Gunst, B. Verstichel, D. Stradi,
+P. A. Khomyakov, U. G. Vej-Hansen, M.-E. Lee, S. T.
+Chill, F. Rasmussen, G. Penazzi, F. Corsetti, A. Ojan-
+per¨a, K. Jensen, M. L. N. Palsgaard, U. Martinez,
+A. Blom, M. Brandbyge, and K. Stokbro, Journal of
+Physics: Condensed Matter 32, 015901 (2019).
+[47] R. K. Harris, P. Hodgkinson, C. J. Pickard, J. R. Yates,
+and V. Zorin, Magnetic Resonance in Chemistry 45, S174
+(2007).
+[48] V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu,
+X. Ren, K. Reuter, and M. Scheffler, Computer Physics
+Communications 180, 2175 (2009).
+[49] M. van Setten, M. Giantomassi, E. Bousquet, M. Ver-
+straete, D. Hamann, X. Gonze, and G.-M. Rignanese,
+Computer Physics Communications 226, 39 (2018).
+[50] D. C. Liu and J. Nocedal, Mathematical Programming
+45, 503 (1989).
+[51] H. J. Monkhorst and J. D. Pack, Physical Review B 13,
+5188 (1976).
+[52] P. Koskinen and V. M¨akinen, Computational Materials
+Science 47, 237 (2009).
+[53] B. Sz˝ucs, Z. Hajnal, T. Frauenheim, C. Gonz´alez, J. Or-
+tega, R. P´erez, and F. Flores, Applied Surface Science
+212-213, 861 (2003).
+[54] B. Sz˝ucs, Z. Hajnal, R. Scholz, S. Sanna, and T. Frauen-
+heim, Applied Surface Science 234, 173 (2004).
+[55] J. P. Perdew and Y. Wang, Physical Review B 45, 13244
+(1992).
+[56] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re-
+view Letter 77, 3865 (1996).
+[57] B. Hammer, L. B. Hansen, and J. K. Nørskov, Physical
+Review B 59, 7413 (1999).
+[58] J. P. Perdew, P. Ziesche, and H. Eschrig, Electronic struc-
+ture of solids’ 91 (1991).
+[59] K. Burke, J. P. Perdew, and Y. Wang, Derivation of
+a generalized gradient approximation: The pw91 den-
+sity functional, in Electronic density functional theory
+(Springer, 1998) pp. 81–111.
+[60] Y. Matsushita, K. Nakamura, and A. Oshiyama, Physical
+Review B 84, 075205 (2011).
+[61] A. D. Becke, The Journal of Chemical Physics 98, 5648
+(1993).
+[62] J. P. Perdew, M. Ernzerhof, and K. Burke, The Journal
+of Chemical Physics 105, 9982 (1996).
+[63] R. R. Pela, M. Marques, and L. K. Teles, Journal of
+Physics: Condensed Matter 27, 505502 (2015).
+[64] J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal
+of Chemical Physics 118, 8207 (2003).
+[65] P. E. Bl¨ochl, Physical Review B 50, 17953 (1994).
+[66] C. Kittel, Introduction to solid state physics, 8th ed. (Wi-
+ley New York, 2005).
+[67] J. Paier, M. Marsman, and G. Kresse, The Journal of
+Chemical Physics 127, 024103 (2007).
+[68] S. Zhang and R. Zhang, Computer Physics Communica-
+tions 220, 403 (2017).
+[69] V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng,
+Computer Physics Communications 267, 108033 (2021).
+[70] N. W. Tschoegl, Australian Journal of Physics 11, 154
+(1958).
+[71] M. Ma´zdziarz, 2D Materials 6, 048001 (2019).
+[72] M. Jamal, S. Jalali Asadabadi, I. Ahmad, and H. Rahna-
+maye Aliabad, Computational Materials Science 95, 592
+(2014).
+[73] P. Koskinen and T. Korhonen, Nanoscale 7, 10140
+(2015).
+[74] P. Koskinen, H. H¨akkinen, B. Huber, B. von Issendorff,
+and M. Moseler, Physical Review Letter 98, 015701
+(2007).
+
+14
+[75] P.
+Koskinen,
+H.
+H¨akkinen,
+G.
+Seifert,
+S.
+Sanna,
+T. Frauenheim, and M. Moseler, New Journal of Physics
+8, 9 (2006).
+
+15
+APPENDIX
+TABLE V. Bond lengths d(�A) and Cohesive energies Ecoh(eV) for each lattice type corresponding to different DFT-attributes.
+1D
+Honeycomb
+Square
+Hexagonal
+3D
+DFT-Methods
+d
+Ecoh
+d
+Ecoh
+d
+Ecoh
+d
+Ecoh
+d
+Ecoh
+DFTB
+2.572
+1.691
+2.562
+2.450
+2.636
+2.804
+2.819
+2.967
+3.008
+3.891
+LDA-LCAO-M
+2.584
+1.513
+2.591
+2.012
+2.623
+2.475
+2.712
+2.761
+2.840
+3.547
+LDA-LCAO-H
+2.553
+1.563
+2.562
+2.105
+2.606
+2.563
+2.693
+2.858
+2.827
+3.660
+LDA-LCAO-U
+2.542
+1.587
+2.553
+2.126
+2.598
+2.590
+2.685
+2.887
+2.826
+3.672
+LDA-PW
+2.542
+1.591
+2.542
+2.138
+2.595
+2.586
+2.682
+2.881
+2.828
+3.638
+RPBE-LCAO-M
+2.732
+0.959
+2.760
+1.198
+2.764
+1.474
+2.853
+1.677
+2.982
+2.065
+RPBE-LCAO-H
+2.710
+0.989
+2.731
+1.251
+2.745
+1.531
+2.831
+1.738
+2.965
+2.130
+RPBE-LCAO-U
+2.691
+1.001
+2.723
+1.262
+2.736
+1.547
+2.824
+1.756
+2.963
+2.143
+RPBE-PW
+2.689
+0.992
+2.709
+1.248
+2.734
+1.523
+2.822
+1.732
+2.962
+2.100
+PW91-LCAO-M
+2.679
+1.145
+2.700
+1.470
+2.717
+1.806
+2.807
+2.026
+2.941
+2.529
+PW91-LCAO-H
+2.655
+1.171
+2.670
+1.522
+2.703
+1.858
+2.790
+2.083
+2.932
+2.586
+PW91-LCAO-U
+2.642
+1.186
+2.668
+1.536
+2.696
+1.876
+2.785
+2.103
+2.932
+2.598
+PW91-PW
+2.639
+1.185
+2.659
+1.534
+2.693
+1.862
+2.783
+2.089
+2.928
+2.560
+PBE-LCAO-M
+2.690
+1.126
+2.710
+1.441
+2.724
+1.771
+2.814
+1.994
+2.945
+2.501
+PBE-LCAO-H
+2.668
+1.155
+2.685
+1.497
+2.710
+1.826
+2.797
+2.053
+2.932
+2.558
+PBE-LCAO-U
+2.651
+1.170
+2.677
+1.510
+2.702
+1.844
+2.790
+2.073
+2.932
+2.571
+PBE-PW
+2.648
+1.167
+2.670
+1.506
+2.700
+1.829
+2.788
+2.058
+2.930
+2.533
+B3LYP-LCAO-M
+2.373
+3.734
+2.410
+5.164
+2.457
+6.029
+2.558
+6.586
+2.725
+8.340
+B3LYP-LCAO-H
+2.655
+1.067
+2.691
+1.426
+2.714
+1.772
+2.812
+1.977
+-
+-
+B3LYP-LCAO-U
+2.642
+1.100
+2.679
+1.461
+2.705
+1.816
+2.803
+2.025
+-
+-
+B3LYP-PW
+2.681
+0.944
+2.715
+1.211
+2.737
+1.470
+2.830
+1.659
+2.986
+1.963
+PBE0-LCAO-M
+2.322
+4.877
+2.358
+6.807
+2.409
+7.978
+2.512
+8.657
+-
+-
+PBE0-LCAO-H
+2.635
+1.092
+2.654
+1.523
+2.679
+1.970
+2.773
+2.219
+-
+-
+PBE0-LCAO-U
+2.626
+1.128
+2.642
+1.567
+2.670
+2.023
+2.764
+2.277
+-
+-
+PBE0-PW
+2.649
+0.963
+2.671
+1.288
+2.690
+1.640
+2.779
+1.879
+2.910
+2.444
+HSE03-LCAO-M
+2.694
+1.030
+2.715
+1.351
+2.729
+1.696
+2.825
+1.919
+2.725
+2.436
+HSE03-LCAO-H
+2.668
+1.044
+2.693
+1.385
+2.714
+1.728
+2.807
+1.949
+-
+-
+HSE03-LCAO-U
+2.663
+1.058
+2.687
+1.396
+2.710
+1.744
+2.801
+1.966
+-
+-
+HSE03-PW
+2.651
+1.049
+2.664
+1.392
+2.697
+1.742
+2.787
+1.971
+2.925
+2.484
+HSE06-LCAO-M
+2.697
+1.061
+2.716
+1.358
+2.733
+1.707
+2.829
+1.932
+2.954
+2.431
+HSE06-LCAO-H
+2.676
+1.075
+2.693
+1.391
+2.719
+1.738
+2.812
+1.961
+-
+-
+HSE06-LCAO-U
+2.666
+1.088
+2.686
+1.402
+2.709
+1.753
+2.803
+1.978
+-
+-
+HSE06-PW
+2.650
+1.075
+2.664
+1.396
+2.695
+1.750
+2.786
+1.982
+2.923
+2.479
+TABLE VI. Elastic constants for 1D (GPa nm2), 2D (GPa nm), and 3D (GPa) calculated by using different DFT-attributes.
+1D
+Honeycomb
+Square
+Hexagonal
+3D
+DFT-Methods
+C11
+C11
+C12
+C66
+C11
+C12
+C66
+C11
+C12
+C66
+C11
+C12
+C66
+DFTB
+88.2
+163.6
+63.8
+49.9
+57.8
+9.7
+-3.9
+42.7
+22.6
+10.1
+110.7
+102.4
+19.9
+LDA-LCAO-M
+24.5
+34.3
+14.9
+9.7
+80.5
+9.0
+-5.9
+77.9
+30.7
+23.6
+163.6
+133.0
+53.4
+LDA-LCAO-H
+25.2
+33.7
+17.0
+8.3
+79.5
+10.7
+-7.5
+79.1
+28.4
+25.3
+165.4
+131.0
+56.3
+LDA-LCAO-U
+25.8
+34.2
+17.4
+8.4
+80.6
+12.1
+-7.6
+85.3
+27.1
+29.1
+164.3
+131.4
+54.7
+LDA-PW
+24.7
+34.0
+18.3
+7.9
+79.4
+12.0
+-8.8
+79.2
+31.4
+23.9
+165.4
+131.1
+58.7
+RPBE-LCAO-M
+15.5
+19.0
+8.7
+5.1
+48.6
+4.6
+-2.9
+43.5
+13.8
+14.9
+103.4
+88.0
+35.9
+RPBE-LCAO-H
+15.3
+19.4
+9.0
+5.2
+47.8
+5.6
+-2.3
+48.6
+16.7
+16.0
+103.6
+83.9
+32.7
+RPBE-LCAO-U
+15.1
+19.6
+8.4
+5.6
+47.9
+6.5
+-2.7
+44.6
+21.8
+11.4
+103.4
+82.9
+31.3
+RPBE-PW
+16.0
+20.5
+9.4
+5.5
+48.2
+6.4
+-3.4
+49.0
+17.1
+16.0
+92.7
+72.0
+25.4
+PW91-LCAO-M
+18.8
+24.1
+10.7
+6.7
+57.3
+6.0
+-3.0
+56.2
+-9.0
+32.6
+133.3
+81.2
+16.7
+PW91-LCAO-H
+18.6
+24.5
+11.7
+6.6
+56.7
+7.4
+-3.4
+56.3
+21.8
+17.2
+116.4
+69.2
+19.7
+PW91-LCAO-U
+19.1
+24.2
+11.0
+6.6
+56.1
+8.1
+-3.6
+56.8
+21.1
+17.8
+117.7
+68.9
+19.0
+PW91-PW
+19.1
+24.7
+11.5
+6.6
+57.5
+8.2
+-4.1
+56.8
+21.3
+17.7
+109.8
+85.9
+29.7
+PBE-LCAO-M
+16.9
+25.5
+8.63
+8.4
+56.2
+5.5
+-3.1
+55.2
+20.1
+17.5
+114.1
+67.6
+34.2
+PBE-LCAO-H
+17.6
+23.0
+10.7
+6.2
+54.8
+6.8
+-3.6
+56.2
+18.8
+18.7
+113.2
+68.3
+20.5
+PBE-LCAO-U
+18.6
+23.2
+10.7
+6.2
+55.3
+7.7
+-3.7
+55.8
+20.7
+17.5
+115.2
+68.0
+20.0
+
+16
+TABLE VI. (Continued)
+1D
+Honeycomb
+Square
+Hexagonal
+3D
+DFT-Methods
+C11
+C11
+C12
+C66
+C11
+C12
+C66
+C11
+C12
+C66
+C11
+C12
+C66
+PBE-PW
+18.3
+23.4
+11.0
+6.2
+55.3
+7.7
+-4.3
+55.8
+20.4
+17.7
+107.7
+84.2
+31.0
+B3LYP-LCAO-M
+65.3
+97.4
+45.6
+25.9
+160.2
+52.2
+-31.9
+168.8
+85.3
+41.8
+-
+-
+-
+B3LYP-LCAO-H
+19.4
+23.3
+10.5
+6.4
+44.9
+17.0
+-3.6
+51.3
+19.2
+16.0
+-
+-
+-
+B3LYP-LCAO-U
+20.5
+24.2
+10.7
+6.7
+46.0
+18.2
+-3.4
+52.8
+20.6
+16.1
+-
+-
+-
+B3LYP-PW
+35.9
+20.8
+9.6
+5.6
+38.9
+15.5
+-1.8
+47.2
+17.6
+14.8
+-
+-
+-
+PBE0-LCAO-M
+80.4
+115.1
+58.9
+28.1
+205.4
+60.0
+-90.1
+203.9
+103.2
+50.3
+-
+-
+-
+PBE0-LCAO-H
+20.2
+25.1
+11.1
+7.0
+48.4
+23.8
+-8.0
+58.3
+20.2
+19.1
+-
+-
+-
+PBE0-LCAO-U
+20.8
+26.3
+14.8
+5.8
+45.2
+25.2
+-8.3
+59.9
+21.9
+19.0
+-
+-
+-
+PBE0-PW
+17.6
+22.5
+11.0
+5.7
+40.0
+22.2
+-5.3
+55.1
+19.6
+17.6
+138.4
+73.0
+-
+HSE03-LCAO-M
+17.7
+23.1
+10.2
+6.5
+53.9
+8.3
+-3.1
+54.0
+19.6
+17.2
+96.4
+84.5
+27.9
+HSE03-LCAO-H
+18.0
+23.4
+10.6
+6.4
+50.9
+10.2
+-3.3
+53.2
+20.5
+16.3
+-
+-
+-
+HSE03-LCAO-U
+17.4
+22.7
+10.9
+5.9
+50.3
+10.0
+-3.5
+53.8
+19.4
+17.2
+-
+-
+-
+HSE03-PW
+20.9
+22.3
+11.5
+5.4
+52.4
+9.1
+-4.5
+53.8
+21.2
+16.3
+96.2
+83.0
+13.7
+HSE06-LCAO-M
+17.4
+23.1
+10.3
+6.4
+52.1
+8.6
+-3.1
+52.8
+19.3
+16.8
+113.5
+95.6
+36.5
+HSE06-LCAO-H
+18.6
+23.2
+10.6
+6.3
+49.9
+11.0
+-3.2
+51.9
+19.5
+16.2
+-
+-
+-
+HSE06-LCAO-U
+17.1
+24.0
+9.9
+7.0
+49.1
+11.3
+-3.5
+53.3
+18.9
+17.2
+-
+-
+-
+HSE06-PW
+26.5
+21.9
+11.5
+5.2
+50.5
+11.1
+-5.3
+54.4
+20.3
+17.0
+94.2
+87.2
+14.4
+TABLE VII. Estimation of contribution of s, p, and d orbitals to the density of states by implementing different DFT-attributes
+1D
+Honeycomb
+Square
+Hexagonal
+3D
+DFT-Methods
+Ns
+Np
+Nd
+Ns
+Np
+Nd
+Ns
+Np
+Nd
+Ns
+Np
+Nd
+Ns
+Np
+Nd
+DFTB
+1.01
+0.19
+0.15
+0.52
+0.39
+0.12
+0.36
+0.45
+0.12
+0.34
+0.42
+0.13
+0.19
+0.46
+0.15
+LDA-LCAO-M
+0.97
+0.08
+0.53
+0.56
+0.14
+0.18
+0.39
+0.18
+0.20
+0.33
+0.16
+0.22
+0.20
+0.26
+0.14
+LDA-LCAO-H
+1.00
+0.04
+0.79
+0.57
+0.13
+0.26
+0.41
+0.18
+0.26
+0.34
+0.16
+0.27
+0.21
+0.22
+0.18
+LDA-LCAO-U
+0.99
+0.04
+0.81
+0.56
+0.13
+0.26
+0.40
+0.18
+0.27
+0.33
+0.16
+0.26
+0.21
+0.22
+0.18
+LDA-PW
+0.64
+0.39
+0.31
+0.17
+0.54
+0.08
+0.10
+0.50
+0.11
+0.08
+0.44
+0.09
+0.01
+0.41
+0.02
+RPBE-LCAO-M
+1.04
+0.08
+0.37
+0.67
+0.13
+0.13
+0.45
+0.17
+0.13
+0.40
+0.17
+0.18
+0.26
+0.26
+0.12
+RPBE-LCAO-H
+1.06
+0.04
+0.53
+0.67
+0.12
+0.18
+0.47
+0.17
+0.17
+0.41
+0.15
+0.22
+0.26
+0.22
+0.18
+RPBE-LCAO-U
+1.05
+0.04
+0.54
+0.67
+0.12
+0.18
+0.47
+0.17
+0.18
+0.40
+0.15
+0.22
+0.26
+0.22
+0.18
+RPBE-PW
+0.75
+0.33
+0.23
+0.28
+0.52
+0.07
+0.16
+0.49
+0.08
+0.14
+0.43
+0.08
+0.03
+0.49
+0.02
+PW91-LCAO-M
+1.01
+0.08
+0.28
+0.63
+0.13
+0.11
+0.43
+0.18
+0.11
+0.38
+0.17
+0.15
+0.24
+0.26
+0.11
+PW91-LCAO-H
+1.04
+0.04
+0.58
+0.63
+0.12
+0.20
+0.45
+0.17
+0.19
+0.39
+0.16
+0.23
+0.25
+0.23
+0.17
+PW91-LCAO-U
+1.03
+0.04
+0.59
+0.63
+0.12
+0.20
+0.45
+0.17
+0.19
+0.38
+0.16
+0.22
+0.25
+0.22
+0.17
+PW91-PW
+0.73
+0.33
+0.23
+0.25
+0.53
+0.07
+0.14
+0.50
+0.08
+0.12
+0.44
+0.08
+0.02
+0.48
+0.02
+PBE-LCAO-M
+1.02
+0.08
+0.31
+0.64
+0.13
+0.12
+0.44
+0.17
+0.12
+0.38
+0.17
+0.16
+0.24
+0.26
+0.12
+PBE-LCAO-H
+1.04
+0.04
+0.59
+0.65
+0.12
+0.20
+0.46
+0.17
+0.19
+0.39
+0.15
+0.23
+0.25
+0.22
+0.17
+PBE-LCAO-U
+1.04
+0.04
+0.60
+0.64
+0.12
+0.20
+0.45
+0.17
+0.19
+0.39
+0.15
+0.23
+0.25
+0.22
+0.18
+PBE-PW
+0.73
+0.33
+0.23
+0.25
+0.53
+0.07
+0.14
+0.50
+0.08
+0.12
+0.44
+0.08
+0.02
+0.49
+0.02
+B3LYP-LCAO-M
+0.62
+0.10
+0.01
+0.41
+0.14
+0.00
+0.29
+0.18
+0.00
+0.27
+0.15
+0.00
+0.19
+0.22
+0.01
+B3LYP-LCAO-H
+0.81
+0.03
+0.02
+0.55
+0.10
+0.01
+0.41
+0.15
+-0.01
+0.37
+0.13
+0.00
+-
+-
+-
+B3LYP-LCAO-U
+0.78
+0.03
+0.02
+0.55
+0.10
+0.01
+0.41
+0.15
+-0.01
+0.37
+0.14
+0.00
+-
+-
+-
+B3LYP-PW
+0.55
+0.26
+0.02
+0.21
+0.43
+0.01
+0.13
+0.40
+0.02
+0.12
+0.35
+0.02
+0.04
+0.39
+0.01
+PBE0-LCAO-M
+0.50
+0.10
+0.01
+0.37
+0.14
+0.00
+0.25
+0.19
+0.00
+0.24
+0.16
+0.00
+-
+-
+-
+PBE0-LCAO-H
+0.71
+0.03
+0.02
+0.51
+0.10
+0.01
+0.38
+0.15
+-0.01
+0.34
+0.13
+0.00
+-
+-
+-
+PBE0-LCAO-U
+0.70
+0.03
+0.02
+0.50
+0.10
+0.01
+0.37
+0.15
+-0.01
+0.34
+0.13
+0.00
+-
+-
+-
+PBE0-PW
+0.47
+0.26
+0.01
+0.18
+0.42
+0.01
+0.10
+0.39
+0.02
+0.10
+0.34
+0.01
+0.01
+0.38
+0.01
+HSE03-LCAO-M
+0.92
+0.07
+0.03
+0.58
+0.12
+0.03
+0.39
+0.16
+0.02
+0.35
+0.15
+0.04
+0.23
+0.25
+0.07
+HSE03-LCAO-H
+0.94
+0.04
+0.05
+0.57
+0.12
+0.05
+0.40
+0.16
+0.04
+0.35
+0.14
+0.06
+-
+-
+-
+HSE03-LCAO-U
+0.93
+0.04
+0.05
+0.57
+0.12
+0.05
+0.40
+0.16
+0.04
+0.35
+0.14
+0.06
+-
+-
+-
+HSE03-PW
+0.67
+0.30
+0.03
+0.22
+0.49
+0.01
+0.13
+0.45
+0.02
+0.11
+0.40
+0.02
+0.02
+0.46
+0.01
+HSE06-LCAO-M
+0.88
+0.07
+0.03
+0.55
+0.11
+0.03
+0.38
+0.15
+0.02
+0.34
+0.14
+0.04
+0.22
+0.24
+0.06
+HSE06-LCAO-H
+0.90
+0.03
+0.04
+0.55
+0.11
+0.04
+0.39
+0.15
+0.04
+0.34
+0.13
+0.05
+-
+-
+-
+HSE06-LCAO-U
+0.89
+0.03
+0.04
+0.55
+0.11
+0.05
+0.39
+0.15
+0.04
+0.34
+0.13
+0.05
+-
+-
+-
+HSE06-PW
+0.63
+0.29
+0.02
+0.21
+0.47
+0.01
+0.12
+0.43
+0.02
+0.11
+0.38
+0.02
+0.02
+0.45
+0.01
+

-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d2e43e242e82e9a21edb2013bfa5d1b31040b17036f7c12271d84a34632d3e3f
+size 1228283

-dAyT4oBgHgl3EQf3fmN/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:31e28e94a6e54e320298551e25c8c34e9a6fe55d6fa5d4104292ed27c856d46f
+size 3080237

-dAyT4oBgHgl3EQf3fmN/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b014f782a61584d447e791f9ed652f7923830dcdf169b18989c1e01fe60a28b6
+size 102790

-dFQT4oBgHgl3EQf6zaC/content/tmp_files/2301.13440v1.pdf.txt

@@ -0,0 +1,748 @@
+arXiv:2301.13440v1  [math.RT]  31 Jan 2023
+Real characters in nilpotent blocks
+Benjamin Sambale∗
+February 1, 2023
+Dedicated to Pham Huu Tiep on the occasion of his 60th birthday.
+Abstract
+We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally
+determined. We further conjecture that the Frobenius–Schur indicators of those characters can be
+computed for p = 2 in terms of the extended defect group. We derive this from a more general conjecture
+on the Frobenius–Schur indicator of projective indecomposable characters of 2-blocks with one simple
+module. This extends results of Murray on 2-blocks with cyclic and dihedral defect groups.
+Keywords: real characters; Frobenius–Schur indicators; nilpotent blocks
+AMS classification: 20C15, 20C20
+1 Introduction
+An important task in representation theory is to determine global invariants of a finite group G by means of
+local subgroups. Dade’s conjecture, for instance, predicts the number of irreducible characters χ ∈ Irr(G)
+such that the p-part χ(1)p is a given power of a prime p (see [23, Conjecture 9.25]). Since Gow’s work [7],
+there has been an increasing interest in counting real (i. e. real-valued) characters and more generally
+characters with a given field of values.
+The quaternion group Q8 testifies that a real irreducible character χ is not always afforded by a repre-
+sentation over the real numbers. The precise behavior is encoded by the Frobenius–Schur indicator (F-S
+indicator, for short)
+ǫ(χ) :=
+1
+|G|
+�
+g∈G
+χ(g2) =
+
+
+
+
+
+0
+if χ ̸= χ,
+1
+if χ is realized by a real representation,
+−1
+if χ is real, but not realized by a real representation.
+(1)
+A new interpretation of the F-S indicator in terms of superalgebras has been given recently in [13]. The case
+of the dihedral group D8 shows that ǫ(χ) is not determined by the character table of G. The computation
+∗Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Han-
+nover, Germany, [email protected]
+1
+
+of F-S indicators can be a surprisingly difficult task, which has not been fully completed for the simple
+groups of Lie type, for instance (see [25]). Problem 14 on Brauer’s famous list [2] asks for a group-theoretical
+interpretation of the number of χ ∈ Irr(G) with ǫ(χ) = 1.
+To obtain deeper insights, we fix a prime p and assume that χ lies in a p-block B of G with defect group
+D. By complex conjugation we obtain another block B of G. If B ̸= B, then clearly ǫ(χ) = 0 for all
+χ ∈ Irr(B). Hence, we assume that B is real, i. e. B = B. John Murray [18, 19] has computed the F-S
+indicators when D is a cyclic 2-group or a dihedral 2-group (including the Klein four-group). His results
+depend on the fusion system of B, on Erdmann’s classification of tame blocks and on the structure of the
+so-called extended defect group E of B (see Definition 7 below). For p > 2 and D cyclic, he obtained in
+[20] partial information on the F-S indicators in terms of the Brauer tree of B.
+The starting point of my investigation is the well-known fact that 2-blocks with cyclic defect groups are
+nilpotent. Assume that B is nilpotent and real. If B is the principal block, then G = Op′(G)D and
+Irr(B) = Irr(G/Op′(G)) = Irr(D). In this case the F-S indicators of B are determined by D alone. Thus,
+suppose that B is non-principal. By Broué–Puig [4], there exists a height-preserving bijection Irr(D) →
+Irr(B), λ �→ λ ∗ χ0 where χ0 ∈ Irr(B) is a fixed character of height 0 (see also [16, Definition 8.10.2]).
+However, this bijection does not in general preserve F-S indicators. For instance, the dihedral group D24
+has a nilpotent 2-block with defect group C4 and a nilpotent 3-block with defect group C3, although every
+character of D24 is real. Our main theorem asserts that the number of real characters in a nilpotent block
+is nevertheless locally determined. To state it, we introduce the extended inertial group
+NG(D, bD)∗ :=
+�
+g ∈ NG(D) : bg
+D ∈ {bD, bD}
+�
+where bD is a Brauer correspondent of B in DCG(D).
+Theorem A. Let B be a real, nilpotent p-block of a finite group G with defect group D. Let bD be a Brauer
+correspondent of B in DCG(D). Then the number of real characters in Irr(B) of height h coincides with
+the number of characters λ ∈ Irr(D) of degree ph such that λt = λ where
+NG(D, bD)∗/DCG(D) = ⟨tDCG(D)⟩.
+If p > 2, then all real characters in Irr(B) have the same F-S indicator.
+In contrast to arbitrary blocks, Theorem A implies that nilpotent real blocks have at least one real character
+(cf. [20, p. 92] and [8, Theorem 5.3]). If bD = bD, then B and D have the same number of real characters,
+because NG(D, bD) = DCG(D). This recovers a result of Murray [18, Lemma 2.2]. As another consequence,
+we will derive in Proposition 5 a real version of Eaton’s conjecture [5] for nilpotent blocks as put forward
+by Héthelyi–Horváth–Szabó [12].
+The F-S indicators of real characters in nilpotent blocks seem to lie somewhat deeper. We still conjecture
+that they are locally determined by a defect pair (see Definition 7) for p = 2 as follows.
+Conjecture B. Let B be a real, nilpotent, non-principal 2-block of a finite group G with defect pair (D, E).
+Then there exists a height preserving bijection Γ : Irr(D) → Irr(B) such that
+ǫ(Γ(λ)) =
+1
+|D|
+�
+e∈E\D
+λ(e2)
+(2)
+for all λ ∈ Irr(D).
+2
+
+The right hand side of (2) was introduced and studied by Gow [8, Lemma 2.1] more generally for any
+groups D ≤ E with |E : D| = 2. This invariant was later coined the Gow indicator by Murray [20, Eq.
+(2)]. For 2-blocks of defect 0, Conjecture B confirms the known fact that real characters of 2-defect 0
+have F-S indicator 1 (see [8, Theorem 5.1]). There is no such result for odd primes p. As a matter of fact,
+every real character has p-defect 0 whenever p does not divide |G|. In Theorem 10 we prove Conjecture B
+for abelian defect groups D. Then it also holds for all quasisimple groups G by work of An–Eaton [1].
+Murray’s results mentioned above, imply Conjecture B also for dihedral D.
+For p > 2, the common F-S indicator in the situation of Theorem A is not locally determined. For instance,
+G = Q8⋊C9 = SmallGroup(72, 3) has a non-principal real 3-block with D ∼= C9 and common F-S indicator
+−1, while its Brauer correspondent in NG(D) ∼= C18 has common F-S indicator 1. Nevertheless, for cyclic
+defect groups D we find another way to compute this F-S indicator in Theorem 3 below.
+Our second conjecture applies more generally to blocks with only one simple module.
+Conjecture C. Let B be a real, non-principal 2-block with defect pair (D, E) and a unique projective
+indecomposable character Φ. Then
+ǫ(Φ) = |{x ∈ E \ D : x2 = 1}|.
+Here ǫ(Φ) is defined by extending (1) linearly. If ǫ(Φ) = 0, then E does not split over D and Conjecture C
+holds (see Proposition 8 below). Conjecture C implies a stronger, but more technical statement on 2-blocks
+with a Brauer correspondent with one simple module (see Theorem 13 below). This allows us to prove the
+following.
+Theorem D. Conjecture C implies Conjecture B.
+We remark that our proof of Theorem D does not work block-by-block. For solvable groups we offer a
+purely group-theoretical version of Conjecture C at the end of Section 4.
+Theorem E. Conjectures B and C hold for all nilpotent 2-blocks of solvable groups.
+We have checked Conjectures B and C with GAP [6] in many examples using the libraries of small groups,
+perfect groups and primitive groups.
+2 Theorem A and its consequences
+Our notation follows closely Navarro’s book [22]. Let B be a p-block of a finite group G with defect group
+D. Recall that a B-subsection is a pair (u, b) where u ∈ D and b is a Brauer correspondent of B in CG(u).
+For χ ∈ Irr(B) and ϕ ∈ IBr(b) we denote the corresponding generalized decomposition number by du
+χϕ. If
+u = 1, we obtain the (ordinary) decomposition number dχϕ = d1
+χϕ. We put l(b) = |IBr(b)| as usual.
+Following [22, p. 114], we define a class function χ(u,b) by
+χ(u,b)(us) :=
+�
+ϕ∈IBr(b)
+du
+χϕϕ(s)
+3
+
+for s ∈ CG(u)0 and χ(u,b)(x) = 0 whenever x is outside the p-section of u. If R is a set of representatives
+for the G-conjugacy classes of B-subsections, then χ = �
+(u,b)∈R χ(u,b) by Brauer’s second main theorem
+(see [22, Problem 5.3]). Now suppose that B is nilpotent and λ ∈ Irr(D). By [16, Proposition 8.11.4], each
+Brauer correspondent b of B is nilpotent and in particular l(b) = 1. Broué–Puig [4] have shown that, if χ
+has height 0, then
+λ ∗ χ :=
+�
+(u,b)∈R
+λ(u)χ(u,b) ∈ Irr(B)
+and (λ ∗ χ)(1) = λ(1)χ(1). Note also that du
+λ∗χ,ϕ = λ(u)du
+χϕ.
+Proof of Theorem A. Let R be a set of representatives for the G-conjugacy classes of B-subsections
+(u, bu) ≤ (D, bB) (see [22, p. 219]). Since B is nilpotent, we have IBr(bu) = {ϕu} for all (u, bu) ∈ R.
+Since the Brauer correspondence is compatible with complex conjugation, (u, bu)t ≤ (D, bD)t = (D, bD)
+where NG(D, bD)∗/DCG(D) = ⟨tDCG(D)⟩. Thus, (u, bu)t is D-conjugate to some (u′, bu′) ∈ R.
+If p > 2, there exists a unique p-rational character χ0 ∈ Irr(B) of height 0, which must be real by
+uniqueness (see [4, Remark after Theorem 1.2]). If p = 2, there is a 2-rational real character χ0 ∈ Irr(B)
+of height 0 by [8, Theorem 5.1]. Then du
+χ0,ϕu = du
+χ0,ϕu ∈ Z and
+χ(u,bu)
+0
+= χ(u,bu)
+0
+= χ(u,bu)t
+0
+= χ(u′,bu′)
+0
+.
+Now let λ ∈ Irr(D). Then
+λ ∗ χ0 =
+�
+(u,bu)∈R
+λ(u)χ(u,bu)
+0
+=
+�
+(u,bu)∈R
+λ(u)χ(u′,bu′)
+0
+.
+Since the class functions χ(u,b)
+0
+have disjoint support, they are linearly independent. Therefore, λ ∗ χ0 is
+real if and only if λ(ut) = λ(u′) = λ(u) for all (u, bu) ∈ R. Since every conjugacy class of D is represented
+by some u with (u, bu) ∈ R, we conclude that λ ∗ χ0 is real if and only λt = λ. Moreover, if λ(1) = ph,
+then λ ∗ χ0 has height h. This proves the first claim.
+To prove the second claim, let p > 2 and IBr(B) = {ϕ}. Then the decomposition numbers dλ∗χ0,ϕ = λ(1)
+are powers of p; in particular they are odd. A theorem of Thompson and Willems (see [26, Theorem 2.8])
+states that all real characters χ with dχ,ϕ odd have the same F-S indicator. So in our situation all real
+characters in Irr(B) have the same F-S indicator.
+Since the automorphism group of a p-group is “almost always” a p-group (see [11]), the following conse-
+quence is of interest.
+Corollary 1. Let B be a real, nilpotent p-block with defect group D such that p and |Aut(D)| are odd.
+Then B has a unique real character.
+Proof. The hypothesis on Aut(D) implies that NG(D, bD)∗ = DCG(D). Hence by Theorem A, the number
+of real characters in Irr(B) is the number of real characters in D. Since p > 2, the trivial character is the
+only real character of D.
+4
+
+The next lemma is a consequence of Brauer’s second main theorem and the fact that |{g ∈ G : g2 = x}| =
+|{g ∈ CG(x) : g2 = x}| is locally determined for g, x ∈ G.
+Lemma 2 (Brauer). For every p-block B of G and every B-subsection (u, b) with ϕ ∈ IBr(b) we have
+�
+χ∈Irr(B)
+ǫ(χ)du
+χϕ =
+�
+ψ∈Irr(b)
+ǫ(ψ)du
+ψϕ =
+�
+ψ∈Irr(b)
+ǫ(ψ)ψ(u)
+ψ(1) dψϕ.
+If l(b) = 1, then
+�
+χ∈Irr(B)
+ǫ(χ)du
+χϕ =
+1
+ϕ(1)
+�
+ψ∈Irr(b)
+ǫ(ψ)ψ(u).
+Proof. The first equality is [3, Theorem 4A]. The second follows from u ∈ Z(CG(u)). If l(b) = 1, then
+ψ(1) = dψϕϕ(1) for ψ ∈ Irr(b) and the last claim follows.
+Recall that a canonical character of B is a character θ ∈ Irr(DCG(D)) lying in a Brauer correspondent of
+B such that D ≤ Ker(θ) (see [22, Theorem 9.12]). We define the extended stabilizer
+NG(D)∗
+θ :=
+�
+g ∈ NG(D) : θg ∈ {θ, θ}
+�
+.
+The following results adds some detail to the nilpotent case of [20, Theorem 1].
+Theorem 3. Let B be a real, nilpotent p-block with cyclic defect group D = ⟨u⟩ and p > 2. Let θ ∈
+Irr(CG(D)) be a canonical character of B and set T := NG(D)∗
+θ. Then one of the following holds:
+(1) θ ̸= θ. All characters in Irr(B) are real with F-S indicator ǫ(θT ).
+(2) θ = θ. The unique non-exceptional character χ0 ∈ Irr(B) is the only real character in Irr(B) and
+ǫ(χ0) = sgn(χ0(u))ǫ(θ) where sgn(χ0(u)) is the sign of χ0(u).
+Proof. Let bD be a Brauer correspondent of B in CG(D) containing θ. Then T = NG(D, bD)∗. If θ ̸= θ,
+then T inverts the elements of D since p > 2. Thus, Theorem A implies that all characters in Irr(B) are
+real. By [20, Theorem 1(v)], the common F-S indicator is the Gow indicator of θ with respect to T. This
+is easily seen to be ǫ(θT ) (see [20, after Eq. (2)]).
+Now assume that θ = θ. Here Theorem A implies that the unique p-rational character χ0 ∈ Irr(B) is the
+only real character. In particular, χ0 must be the unique non-exceptional character. Note that (u, bD) is
+a B-subsection and IBr(bD) = {ϕ}. Since χ0 is p-rational, du
+χ0ϕ = ±1. Since all Brauer correspondents of
+B in CG(u) are conjugate under NG(D), the generalized decomposition numbers are Galois conjugate, in
+particular du
+χ0ϕ does not depend on the choice of bD. Hence,
+χ0(u) = |NG(D) : NG(D)θ|du
+χ0ϕϕ(1)
+and du
+χ0ϕ = sgn(χ0(u)). Moreover, θ is the unique non-exceptional character of bD and θ(u) = θ(1). By
+Lemma 2, we obtain
+ǫ(χ0) = sgn(χ0(u))
+�
+χ∈Irr(B)
+ǫ(χ)du
+χϕ = sgn(χ0(u))
+ϕ(1)
+�
+ψ∈Irr(bD)
+ǫ(ψ)ψ(u) = sgn(χ0(u))ǫ(θ).
+5
+
+If B is a nilpotent block with canonical character θ ̸= θ, the common F-S indicator of the real characters
+in Irr(B) is not always ǫ(θT ) as in Theorem 3. A counterexample is given by a certain 3-block of G =
+SmallGroup(288, 924) with defect group D ∼= C3 × C3.
+We now restrict ourselves to 2-blocks. Héthelyi–Horváth–Szabó [12] introduced four conjectures, which
+are real versions of Brauer’s conjecture, Olsson’s conjecture and Eaton’s conjecture. We only state the
+strongest of them, which implies the remaining three. Let D(0) := D and D(k+1) := [D(k), D(k)] for k ≥ 0
+be the members of the derived series of D.
+Conjecture 4 (Héthelyi–Horváth–Szabó). Let B be a 2-block with defect group D. For every h ≥ 0, the
+number of real characters in Irr(B) of height ≤ h is bounded by the number of elements of D/D(h+1) which
+are real in NG(D)/D(h+1).
+A conjugacy class K of G is called real if K = K−1 := {x−1 : x ∈ K}. A conjugacy class K of a normal
+subgroup N ⊴ G is called real under G if there exists g ∈ G such that Kg = K−1.
+Proposition 5. Let B be a nilpotent 2-block with defect group D and Brauer correspondent bD in DCG(D).
+Then the number of real characters in Irr(B) of height ≤ h is bounded by the number of conjugacy classes
+of D/D(h+1) which are real under NG(D, bD)∗/D(h+1). In particular, Conjecture 4 holds for B.
+Proof. We may assume that B is real. As in the proof of Theorem A, we fix some 2-rational real character
+χ0 ∈ Irr(B) of height 0. Now λ ∗ χ0 has height ≤ h if and only if λ(1) ≤ ph for λ ∈ Irr(B). By [14,
+Theorem 5.12], the characters of degree ≤ ph in Irr(D) lie in Irr(D/D(h+1)). By Theorem A, λ ∗ χ0 is
+real if and only if λt = λ. By Brauer’s permutation lemma (see [23, Theorem 2.3]), the number of those
+characters λ coincides with the number of conjugacy classes K of D/D(h+1) such that Kt = K−1. Now
+Conjecture 4 follows from NG(D, bD)∗ ≤ NG(D).
+3 Extended defect groups
+We continue to assume that p = 2. As usual we choose a complete discrete valuation ring O such that
+F := O/J(O) is an algebraically closed field of characteristic 2. Let Cl(G) be the set of conjugacy classes
+of G. For K ∈ Cl(G) let K+ := �
+x∈K x ∈ Z(FG) be the class sum of K. We fix a 2-block B of FG with
+block idempotent 1B = �
+K∈Cl(G) aKK+ where aK ∈ F. The central character of B is defined by
+λB : Z(FG) → F,
+K+ �→
+�|K|χ(g)
+χ(1)
+�∗
+where g ∈ K, χ ∈ Irr(B) and ∗ denotes the canonical reduction O → F (see [22, Chapter 2]).
+Since λB(1B) = 1, there exists K ∈ Cl(G) such that aK ̸= 0 ̸= λB(K+). We call K a defect class of
+B. By [22, Corollary 3.8], K consists of elements of odd order. According to [22, Corollary 4.5], a Sylow
+2-subgroup D of CG(x) where x ∈ K is a defect group of B. For x ∈ K let
+CG(x)∗ := {g ∈ G : gxg−1 = x±1} ≤ G
+be the extended centralizer of x.
+6
+
+Proposition 6 (Gow, Murray). Every real 2-block B has a real defect class K. Let x ∈ K. Choose a
+Sylow 2-subgroup E of CG(x)∗ and put D := E ∩ CG(x). Then the G-conjugacy class of the pair (D, E)
+does not depend on the choice of K or x.
+Proof. For the principal block (which is always real since it contains the trivial character), K = {1} is
+a real defect class and E = D is a Sylow 2-subgroup of G. Hence, the uniqueness follows from Sylow’s
+theorem. Now suppose that B is non-principal. The existence of K was first shown in [8, Theorem 5.5].
+Let L be another real defect class of B and choose y ∈ L. By [9, Corollary 2.2], we may assume after
+conjugation that E is also a Sylow 2-subgroup of CG(y)∗. Let Dx := E ∩ CG(x) and Dy := E ∩ CG(y).
+We may assume that |E : Dx| = 2 = |E : Dy| (cf. the remark after the proof).
+We now introduce some notation in order to apply [17, Proposition 14]. Let Σ = ⟨σ⟩ ∼= C2. We consider
+FG as an F[G × Σ]-module where G acts by conjugation and gσ = g−1 for g ∈ G (observe that these
+actions indeed commute). For H ≤ G × Σ let
+TrG×Σ
+H
+: (FG)H → (FG)G×Σ, α �→
+�
+x∈R
+αx
+be the relative trace with respect to H, where R denotes a set of representatives of the right cosets of H
+in G × Σ. By [17, Proposition 14], we have 1B ∈ TrG×Σ
+Ex
+(FG) where Ex := Dx⟨exσ⟩ for some ex ∈ E \ Dx.
+By the same result we also obtain that Dy⟨eyσ⟩ with ey ∈ E \ Dy is G-conjugate to Ex. This implies that
+Dy is conjugate to Dx inside NG(E). In particular, (Dx, E) and (Dy, E) are G-conjugate as desired.
+Definition 7. In the situation of Proposition 6 we call E an extended defect group and (D, E) a defect
+pair of B.
+We stress that real 2-blocks can have non-real defect classes and non-real blocks can have real defect classes
+(see [10, Theorem 3.5]).
+It is easy to show that non-principal real 2-blocks cannot have maximal defect (see [22, Problem 3.8]).
+In particular, the trivial class cannot be a defect class and consequently, |E : D| = 2 in those cases.
+For non-real blocks we define the extended defect group by E := D for convenience. Every given pair of
+2-groups D ≤ E with |E : D| = 2 occurs as a defect pair of a real (nilpotent) block. To see this, let Q ∼= C3
+and G = Q ⋊ E with CE(Q) = D. Then G has a unique non-principal block with defect pair (D, E).
+We recall from [14, p. 49] that
+�
+χ∈Irr(G)
+ǫ(χ)χ(g) = |{x ∈ G : x2 = g}|
+(3)
+for all g ∈ G. The following proposition provides some interesting properties of defect pairs.
+Proposition 8 (Gow, Murray). Let B be a real 2-block with defect pair (D, E). Let bD be a Brauer
+correspondent of B in DCG(D). Then the following holds:
+(i) NG(D, bD)∗ = NG(D, bD)E. In particular, bD is real if and only if E = DCE(D).
+(ii) For u ∈ D, we have �
+χ∈Irr(B) ǫ(χ)χ(u) ≥ 0 with strict inequality if and only if u is G-conjugate to
+e2 for some e ∈ E \ D. In particular, E splits over D if and only if �
+χ∈Irr(B) ǫ(χ)χ(1) > 0.
+7
+
+(iii) E/D′ splits over D/D′ if and only if all height zero characters in Irr(B) have non-negative F-S
+indicator.
+Proof.
+(i) See [19, Lemma 1.8] and [18, Theorem 1.4].
+(ii) See [19, Lemma 1.3].
+(iii) See [8, Theorem 5.6].
+The next proposition extends [18, Lemma 1.3].
+Corollary 9. Suppose that B is a 2-block with defect pair (D, E) where D is abelian. Then E splits over
+D if and only if all characters in Irr(B) have non-negative F-S indicator.
+Proof. If B is non-real, then E = D splits over D and all characters in Irr(B) have F-S indicator 0. Hence,
+let B = B. By Kessar–Malle [15], all characters in Irr(B) have height 0. Hence, the claim follows from
+Proposition 8(iii).
+Theorem 10. Let B be a real, nilpotent 2-block with defect pair (D, E) where D is abelian. If E splits over
+D, then all real characters in Irr(B) have F-S indicator 1. Otherwise exactly half of the real characters
+have F-S indicator 1. In either case, Conjecture B holds for B.
+Proof. If E splits over D, then all real characters in Irr(B) have F-S indicator 1 by Corollary 9. Otherwise
+we have �
+χ∈Irr(B) ǫ(χ) = 0 by Proposition 8(ii), because all characters in Irr(B) have the same degree.
+Hence, exactly half of the real characters have F-S indicator 1. Using Theorem A we can determine the
+number of characters for each F-S indicator. For the last claim, we may therefore replace B by the unique
+non-principal block of G = Q ⋊ E where Q ∼= C3 and CE(Q) = D (mentioned above). In this case
+Conjecture B follows from Gow [8, Lemma 2.2] or Theorem E.
+Example 11. Let B be a real block with defect group D ∼= C4 × C2. Then B is nilpotent since Aut(D)
+is a 2-group and D is abelian. Moreover |Irr(B)| = 8. The F-S indicators depend not only on E, but also
+on the way D embeds into E. The following cases can occur (here M16 denotes the modular group and
+[16, 3] refers to the small group library):
+F-S indicators
+E
++ + + + + + ++
+D8 × C2
++ + + + − − −−
+Q8 × C2, C4 ⋊ C4 with Φ(D) = E′
++ + + + 0 0 0 0
+D, D × C2, D8 ∗ C4, [16, 3]
++ + − − 0 0 0 0
+C2
+4, C8 × C2, M16, C4 ⋊ C4 with Φ(D) ̸= E′
+The F-S indicator ǫ(Φ) appearing in Conjecture C has an interesting interpretation as follows. Let Ω :=
+{g ∈ G : g2 = 1}. The conjugation action of G on Ω turns FΩ into an FG-module, called the involution
+module.
+8
+
+Lemma 12 (Murray). Let B be a real 2-block and ϕ ∈ IBr(B). Then ǫ(Φϕ) is the multiplicity of ϕ as a
+constituent of the Brauer character of FΩ.
+Proof. See [18, Lemma 2.6].
+Next we develop a local version of Conjecture C. Let B be a real 2-block with defect pair (D, E) and B-
+subsection (u, b). If E = DCE(u), then b is real and (CD(u), CE(u)) is a defect pair of b by [19, Lemma 2.6]
+applied to the subpair (⟨u⟩, b). Conversely, if b is real, we may assume that (CD(u), CE(u)) is a defect
+pair of b by [19, Theorem 2.7]. If b is non-real, we may assume that (CD(u), CD(u)) = (CD(u), CE(u)) is
+a defect pair of b.
+Theorem 13. Let B be 2-block of a finite group G with defect pair (D, E). Suppose that Conjecture C
+holds for all 2-blocks of sections of G. Let (u, b) be a B-subsection with defect pair (CD(u), CE(u)) such
+that IBr(b) = {ϕ}. Then
+�
+χ∈Irr(B)
+ǫ(χ)du
+χϕ =
+�
+|{x ∈ D : x2 = u}|
+if B is the principal block,
+|{x ∈ E \ D : x2 = u}|
+otherwise.
+Proof. If B is not real, then B is non-principal and E = D. It follows that ǫ(χ) = 0 for all χ ∈ Irr(B) and
+|{x ∈ E \ D : x2 = u}| = 0.
+Hence, we may assume that B is real. By Lemma 2, we have
+�
+χ∈Irr(B)
+ǫ(χ)du
+χϕ =
+�
+ψ∈Irr(b)
+ǫ(ψ)du
+ψϕ =
+1
+ϕ(1)
+�
+ψ∈Irr(b)
+ǫ(ψ)ψ(u).
+(4)
+Suppose that B is the principal block. Then b is the principal block of CG(u) by Brauer’s third main
+theorem (see [22, Theorem 6.7]). The hypothesis l(b) = 1 implies that ϕ = 1CG(u) and CG(u) has a normal
+2-complement N (see [22, Corollary 6.13]). It follows that Irr(b) = Irr(CG(u)/N) = Irr(CD(u)) and
+�
+ψ∈Irr(b)
+ǫ(ψ)du
+ψϕ =
+�
+λ∈Irr(CD(u))
+ǫ(λ)λ(u) = |{x ∈ CD(u) : x2 = u}|
+by (3). Since every x ∈ D with x2 = u lies in CD(u), we are done in this case.
+Now let B be a non-principal real 2-block. If b is not real, then (4) shows that �
+χ∈Irr(B) ǫ(χ)du
+χϕ = 0. On
+the other hand, we have CE(u) = CD(u) ≤ D and |{x ∈ E \ D : x2 = u}| = 0. Hence, we may assume
+that b is real. Since every x ∈ E with x2 = u lies in CE(u), we may assume that u ∈ Z(G) by (4).
+Then χ(u) = du
+χϕϕ(1) for all χ ∈ Irr(B). If u2 /∈ Ker(χ), then χ(u) /∈ R and ǫ(χ) = 0. Thus, it suffices to
+sum over χ with du
+χϕ = ±dχϕ. Let Z := ⟨u⟩ ≤ Z(G) and G := G/Z. Let ˆB be the unique (real) block of
+9
+
+G dominated by B. By [19, Lemma 1.7], (D, E) is a defect pair for ˆB. Then, using [14, Lemma 4.7] and
+Conjecture C for B and ˆB, we obtain
+�
+χ∈Irr(B)
+ǫ(χ)du
+χϕ =
+�
+χ∈Irr(B)
+ǫ(χ)(dχϕ + du
+χϕ) −
+�
+χ∈Irr(B)
+ǫ(χ)dχϕ
+= 2
+�
+χ∈Irr( ˆB)
+ǫ(χ)dχϕ −
+�
+χ∈Irr(B)
+ǫ(χ)dχϕ
+= 2|{x ∈ E \ D : x2 = 1}| − |{x ∈ E \ D : x2 = 1}|
+=
+�
+λ∈Irr(E)
+ǫ(λ)(λ(1) + λ(u)) −
+�
+λ∈Irr(D)
+ǫ(λ)(λ(1) + λ(u))
+−
+�
+λ∈Irr(E)
+ǫ(λ)λ(1) +
+�
+λ∈Irr(D)
+ǫ(λ)λ(1)
+=
+�
+λ∈Irr(E)
+ǫ(λ)λ(u) −
+�
+λ∈Irr(D)
+ǫ(λ)λ(u) = |{x ∈ E \ D : x2 = u}|.
+4 Theorems D and E
+The following result implies Theorem D.
+Theorem 14. Suppose that B is a real, nilpotent, non-principal 2-block fulfilling the statement of Theorem 13.
+Then Conjecture B holds for B.
+Proof. Let (D, E) be defect pair of B. By Gow [8, Theorem 5.1], there exists a 2-rational character
+χ0 ∈ Irr(B) of height 0 and ǫ(χ0) = 1. Let
+Γ : Irr(D) → Irr(B),
+λ �→ λ ∗ χ0
+be the Broué–Puig bijection. Let (u1, b1), . . . , (uk, bk) be representatives for the conjugacy classes of B-
+subsections. Since B is nilpotent, we may assume that u1, . . . , uk ∈ D represent the conjugacy classes of
+D. Let IBr(bi) = {ϕi} for i = 1, . . . , k. Since χ0 is 2-rational, we have σi := du
+χ0,ϕi ∈ {±1} for i = 1, . . . , k.
+Hence, the generalized decomposition matrix of B has the form
+Q = (λ(ui)σi : λ ∈ Irr(D), i = 1, . . . , k)
+(see [16, Section 8.10]). Let v := (ǫ(Γ(λ)) : λ ∈ Irr(D)) and w := (w1, . . . , wk) where wi := |{x ∈ E \ D :
+x2 = ui}|. Then Theorem 13 reads as vQ = w.
+Let di := |CD(ui)| and d = (d1, . . . , dk). Then the second orthogonality relation yields QtQ = diag(d)
+where Qt denotes the transpose of Q. It follows that Q−1 = diag(d)−1Q
+t and
+v = w diag(d)−1Q
+t = w diag(d)−1Qt,
+10
+
+because v = v. Since wi = |{x ∈ E \ D : x2 = uy
+i }| for every y ∈ D, we obtain �k
+i=1 wi|D : CD(ui)| =
+|E \ D| = |D|. In particular,
+1 = ǫ(χ0) =
+k
+�
+i=1
+wiσi
+|CD(ui)| ≤
+k
+�
+i=1
+wi|σi|
+|CD(ui)| = 1.
+Therefore, σi = 1 or wi = 0 for each i. This means that the signs σi have no impact on the solution of the
+linear system xQ = w. Hence, we may assume that Q = (λ(ui)) is just the character table of D. Since Q
+has full rank, v is the only solution of xQ = w. Setting µ(λ) :=
+1
+|D|
+�
+e∈E\D λ(e2), it suffices to show that
+(µ(λ) : λ ∈ Irr(D)) is another solution of xQ = w. Indeed,
+�
+λ∈Irr(D)
+λ(ui)
+|D|
+�
+e∈E\D
+λ(e2) =
+1
+|D|
+�
+e∈E\D
+�
+λ∈Irr(D)
+λ(ui)λ(e2)
+=
+1
+|D|
+�
+e∈E\D
+e2=u−1
+i
+|D : CD(ui)||CD(ui)| = wi
+for i = 1, . . . , k.
+Theorem E. Conjectures B and C hold for all nilpotent 2-blocks of solvable groups.
+Proof. Let B be a real, nilpotent, non-principal 2-block of a solvable group G with defect pair (D, E).
+We first prove Conjecture C for B. Since all sections of G are solvable and all blocks dominated by B-
+subsections are nilpotent, Conjecture C holds for those blocks as well. Hence, the hypothesis of Theorem 13
+is fulfilled for B. Now by Theorem 14, Conjecture B holds for B.
+Let N := O2′(G) and let θ ∈ Irr(N) such that the block {θ} is covered by B. Since B is non-principal,
+θ ̸= 1N and therefore θ ̸= θ as N has odd order. Since B also lies over θ, it follow that Gθ < G. Let b
+be the Fong–Reynolds correspondent of B in the extended stabilizer G∗
+θ. By [22, Theorem 9.14] and [20,
+p. 94], the Clifford correspondence Irr(b) → Irr(B), ψ �→ ψG preserves decomposition numbers and F-S
+indicators. Thus, we need to show that b has defect pair (D, E). Let β be the Fong–Reynolds correspondent
+of B in Gθ. By [22, Theorem 10.20], β is the unique block over θ. In particular, the block idempotents
+1β = 1θ are the same (we identify θ with the block {θ}). Since b is also the unique block of G∗
+θ over θ, we
+have 1b = 1θ + 1θ = �
+x∈N αxx for some αx ∈ F. Let S be a set of representatives for the cosets G/G∗
+θ.
+Then
+1B =
+�
+s∈S
+(1θ + 1θ)s =
+�
+s∈S
+1s
+b =
+�
+g∈N
+��
+s∈S
+αgs−1
+�
+g.
+Hence, there exists a real defect class K of B such that αgs−1 ̸= 0 for some g ∈ K and s ∈ S. Of course
+we can assume that g = gs−1. Then 1b does not vanish on g. By [22, Theorem 9.1], the central characters
+λB, λb and λθ agree on N. It follows that K is also a real defect class of b. Hence, we may assume that
+(D, E) is a defect pair of b.
+It remains to consider G = G∗
+θ and B = b. Then D is a Sylow 2-subgroup of Gθ by [22, Theorem 10.20]
+and E is a Sylow 2-subgroup of G. Since |G : Gθ| = 2, it follows that Gθ ⊴ G and N = O2′(Gθ). By
+11
+
+[21, Lemma 1 and 2], β is nilpotent and Gθ is 2-nilpotent, i. e. Gθ = N ⋊ D and G = N ⋊ E. Let
+�Φ := �
+χ∈Irr(B) χ(1)χ = ϕ(1)Φ where IBr(B) = {ϕ}. We need to show that
+ǫ(�Φ) = ϕ(1)|{x ∈ E \ D : x2 = 1}|.
+Note that χN = χ(1)
+2θ(1)(θ + θ). By Frobenius reciprocity, it follows that �Φ = 2θ(1)θG and
+�ΦN = |G : N|θ(1)(θ + θ).
+Since Φ vanishes on elements of even order, �Φ vanishes outside N. Since �ΦGθ is a sum of non-real characters
+in β, we have
+ǫ(�Φ) =
+1
+|G|
+�
+g∈Gθ
+�Φ(g2) + 1
+|G|
+�
+g∈G\Gθ
+�Φ(g2) =
+1
+|G|
+�
+g∈G\Gθ
+�Φ(g2).
+Every g ∈ G \ Gθ = NE \ ND with g2 ∈ N is N-conjugate to a unique element of the form xy where
+x ∈ E \ D is an involution and y ∈ CN(x) (Sylow’s theorem). Setting ∆ := {x ∈ E \ D : x2 = 1}, we
+obtain
+ǫ(�Φ) = θ(1)
+|N|
+�
+x∈∆
+|N : CN(x)|
+�
+y∈CN (x)
+(θ(y) + θ(y)) = 2θ(1)
+�
+x∈∆
+1
+|CN(x)|
+�
+y∈CN(x)
+θ(y).
+(5)
+For x ∈ ∆ let Hx := N⟨x⟩. Again by Sylow’s theorem, the N-orbit of x is the set of involutions in Hx.
+From θx = θ we see that θHx is an irreducible character of 2-defect 0. By [8, Theorem 5.1], we have
+ǫ(θHx) = 1. Now applying the same argument as before, it follows that
+1 = ǫ(θHx) =
+1
+|N|
+�
+g∈Hx\N
+θHx(g2) =
+2
+|CN(x)|
+�
+y∈CN(x)
+θ(y).
+Combined with (5), this yields ǫ(�Φ) = 2θ(1)|∆|. By Green’s theorem (see [22, Theorem 8.11]), ϕN = θ + θ
+and ǫ(�Φ) = ϕ(1)|∆| as desired.
+For non-principal blocks B of solvable groups with l(B) = 1 it is not true in general that Gθ is 2-
+nilpotent in the situation of Theorem E. For example, a (non-real) 2-block of a triple cover of A4 × A4
+has a unique simple module. Extending this group by an automorphism of order 2, we obtain the group
+G = SmallGroup(864, 3988), which fulfills the assumptions with D ∼= C4
+2, N ∼= C3 and |G : NE| = 9.
+In order to prove Conjecture C for arbitrary 2-blocks of solvable groups, we may follow the steps in the
+proof above and invoke a result on fully ramified Brauer characters [24, Theorem 2.1]. The claim then
+boils down to a purely group-theoretical statement: Let B be a real, non-principal 2-block of a solvable
+group G with defect pair (D, E) and l(B) = 1. Let G := G/O2′(G). Then
+|{x ∈ G \ Gθ : x2 = 1}| = |{x ∈ E \ D : x2 = 1}|
+�
+|G : EN|.
+Unfortunately, I am unable to prove this.
+Acknowledgment
+I thank Gabriel Navarro for providing some arguments for Theorem E from his paper [21]. John Mur-
+ray and three anonymous referees have made many valuable comments, which improved the quality of the
+manuscript. The work is supported by the German Research Foundation (SA 2864/3-1 and SA 2864/4-1).
+12
+
+References
+[1] J. An and C. W. Eaton, Nilpotent Blocks of Quasisimple Groups for the Prime Two, Algebr. Represent.
+Theory 16 (2013), 1–28.
+[2] R. Brauer, Representations of finite groups, in: Lectures on Modern Mathematics, Vol. I, 133–175,
+Wiley, New York, 1963.
+[3] R. Brauer, Some applications of the theory of blocks of characters of finite groups. III, J. Algebra 3
+(1966), 225–255.
+[4] M. Broué and L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.
+[5] C. W. Eaton, Generalisations of conjectures of Brauer and Olsson, Arch. Math. (Basel) 81 (2003),
+621–626.
+[6] The
+GAP
+Group,
+GAP
+–
+Groups,
+Algorithms,
+and
+Programming,
+Version
+4.12.0;
+2022,
+(http://www.gap-system.org).
+[7] R. Gow, Real-valued characters and the Schur index, J. Algebra 40 (1976), 258–270.
+[8] R. Gow, Real-valued and 2-rational group characters, J. Algebra 61 (1979), 388–413.
+[9] R. Gow, Real 2-blocks of characters of finite groups, Osaka J. Math. 25 (1988), 135–147.
+[10] R. Gow and J. Murray, Real 2-regular classes and 2-blocks, J. Algebra 230 (2000), 455–473.
+[11] G. T. Helleloid and U. Martin, The automorphism group of a finite p-group is almost always a p-group,
+J. Algebra 312 (2007), 294–329.
+[12] L. Héthelyi, E. Horváth and E. Szabó, Real characters in blocks, Osaka J. Math. 49 (2012), 613–623.
+[13] T. Ichikawa and Y. Tachikawa, The Super Frobenius–Schur Indicator and Finite Group Gauge Theories
+on Pin− Surfaces, to appear in Commun. Math. Phys., DOI: 10.1007/s00220-022-04601-9.
+[14] I. M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006.
+[15] R. Kessar and G. Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann. of Math. (2)
+178 (2013), 321–384.
+[16] M. Linckelmann, The block theory of finite group algebras. Vol. II, London Mathematical Society
+Student Texts, Vol. 92, Cambridge University Press, Cambridge, 2018.
+[17] J. Murray, Strongly real 2-blocks and the Frobenius-Schur indicator, Osaka J. Math. 43 (2006), 201–
+213.
+[18] J. Murray, Components of the involution module in blocks with cyclic or Klein-four defect group, J.
+Group Theory 11 (2008), 43–62.
+[19] J. Murray, Real subpairs and Frobenius-Schur indicators of characters in 2-blocks, J. Algebra 322
+(2009), 489–513.
+[20] J. Murray, Frobenius-Schur indicators of characters in blocks with cyclic defect, J. Algebra 533 (2019),
+90–105.
+13
+
+[21] G. Navarro, Nilpotent characters, Pacific J. Math. 169 (1995), 343–351.
+[22] G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series,
+Vol. 250, Cambridge University Press, Cambridge, 1998.
+[23] G. Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathemat-
+ics, Vol. 175, Cambridge University Press, Cambridge, 2018.
+[24] G. Navarro, B. Späth and P. H. Tiep, On fully ramified Brauer characters, Adv. Math. 257 (2014),
+248–265.
+[25] S. Trefethen and C. R. Vinroot, A computational approach to the Frobenius-Schur indicators of finite
+exceptional groups, Internat. J. Algebra Comput. 30 (2020), 141–166.
+[26] W. Willems, Duality and forms in representation theory, in: Representation theory of finite groups
+and finite-dimensional algebras (Bielefeld, 1991), 509–520, Progr. Math., Vol. 95, Birkhäuser, Basel,
+1991.
+14
+

.gitattributes

@@ -6209,3 +6209,88 @@ xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf filter=lfs diff=lfs merge=lfs -tex
 79FAT4oBgHgl3EQfoh2y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 99AzT4oBgHgl3EQfSvs8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf filter=lfs diff=lfs merge=lfs -text
+OdFLT4oBgHgl3EQfPC8X/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf filter=lfs diff=lfs merge=lfs -text
+ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf filter=lfs diff=lfs merge=lfs -text
+q9FPT4oBgHgl3EQf9TVT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+f9AzT4oBgHgl3EQfMfsT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+xtFJT4oBgHgl3EQfhCzP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+I9AyT4oBgHgl3EQfsPkA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+p9FPT4oBgHgl3EQf8DXy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf filter=lfs diff=lfs merge=lfs -text
+KNFRT4oBgHgl3EQfDTe1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+gNAyT4oBgHgl3EQfj_gw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+HNFJT4oBgHgl3EQfuC2G/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+LdE4T4oBgHgl3EQfig1d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf filter=lfs diff=lfs merge=lfs -text
+aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf filter=lfs diff=lfs merge=lfs -text
+gtE0T4oBgHgl3EQf6QKw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PdE0T4oBgHgl3EQfTwBe/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf filter=lfs diff=lfs merge=lfs -text
+7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf filter=lfs diff=lfs merge=lfs -text
+ktFPT4oBgHgl3EQf2TWb/content/2301.13186v1.pdf filter=lfs diff=lfs merge=lfs -text
+sNE2T4oBgHgl3EQffAfS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8tFST4oBgHgl3EQfaTh3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf filter=lfs diff=lfs merge=lfs -text
+7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf filter=lfs diff=lfs merge=lfs -text
+eNAyT4oBgHgl3EQfwvnx/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+r9FJT4oBgHgl3EQfbyya/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ptE3T4oBgHgl3EQf8As1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+7NE1T4oBgHgl3EQfTgOa/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+SNFGT4oBgHgl3EQfgygC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+4dAyT4oBgHgl3EQfpPgc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf filter=lfs diff=lfs merge=lfs -text
+eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf filter=lfs diff=lfs merge=lfs -text
+kb_47/content/kb_47.pdf filter=lfs diff=lfs merge=lfs -text
+b9E4T4oBgHgl3EQfPgyg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf filter=lfs diff=lfs merge=lfs -text
+c9FPT4oBgHgl3EQfyDV1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf filter=lfs diff=lfs merge=lfs -text
+jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf filter=lfs diff=lfs merge=lfs -text
+btE1T4oBgHgl3EQfKwNx/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf filter=lfs diff=lfs merge=lfs -text
+-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf filter=lfs diff=lfs merge=lfs -text
+kb_47/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf filter=lfs diff=lfs merge=lfs -text
+u9AzT4oBgHgl3EQfP_sX/content/2301.01191v1.pdf filter=lfs diff=lfs merge=lfs -text
+09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf filter=lfs diff=lfs merge=lfs -text
+K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf filter=lfs diff=lfs merge=lfs -text
+-dAyT4oBgHgl3EQf3fmN/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PNFRT4oBgHgl3EQfIjco/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf filter=lfs diff=lfs merge=lfs -text
+4dE0T4oBgHgl3EQfvQGV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+7dAyT4oBgHgl3EQfcvf_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf filter=lfs diff=lfs merge=lfs -text
+GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf filter=lfs diff=lfs merge=lfs -text
+bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf filter=lfs diff=lfs merge=lfs -text
+Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf filter=lfs diff=lfs merge=lfs -text
+3NE0T4oBgHgl3EQfeABh/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf filter=lfs diff=lfs merge=lfs -text
+9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf filter=lfs diff=lfs merge=lfs -text
+btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf filter=lfs diff=lfs merge=lfs -text
+gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf filter=lfs diff=lfs merge=lfs -text
+hdE4T4oBgHgl3EQfSAzu/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+YdA0T4oBgHgl3EQfFv8F/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf filter=lfs diff=lfs merge=lfs -text
+o9E3T4oBgHgl3EQfLgkW/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf filter=lfs diff=lfs merge=lfs -text
+stFJT4oBgHgl3EQfcixn/content/2301.11544v1.pdf filter=lfs diff=lfs merge=lfs -text
+7tE2T4oBgHgl3EQf7whk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf filter=lfs diff=lfs merge=lfs -text
+_tE1T4oBgHgl3EQfDAJg/content/2301.02871v1.pdf filter=lfs diff=lfs merge=lfs -text
+LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf filter=lfs diff=lfs merge=lfs -text
+stFJT4oBgHgl3EQfcixn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf filter=lfs diff=lfs merge=lfs -text
+K9E2T4oBgHgl3EQfAwYv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf filter=lfs diff=lfs merge=lfs -text
+bNE2T4oBgHgl3EQfwAiL/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+9dFQT4oBgHgl3EQf5zZD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+99AyT4oBgHgl3EQfqfgS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf filter=lfs diff=lfs merge=lfs -text
+PNE4T4oBgHgl3EQf-A73/content/2301.05361v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tFQT4oBgHgl3EQfHDWI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+_tE1T4oBgHgl3EQfDAJg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+q9E1T4oBgHgl3EQf2wUi/content/2301.03481v1.pdf filter=lfs diff=lfs merge=lfs -text
+TNE2T4oBgHgl3EQfWwfK/content/2301.03838v1.pdf filter=lfs diff=lfs merge=lfs -text
+9NAyT4oBgHgl3EQfQ_Zv/content/2301.00057v1.pdf filter=lfs diff=lfs merge=lfs -text

09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b695af2616285d98273841018a18aa78094c4b40106abf6bf4f7a3f24cd300d4
+size 943062

2tA0T4oBgHgl3EQfM_-6/content/tmp_files/2301.02141v1.pdf.txt

@@ -0,0 +1,955 @@
+arXiv:2301.02141v1  [math.NT]  5 Jan 2023
+A refinement of Lang’s formula for the sum of powers of integers
+Jos´e Luis Cereceda
+Collado Villalba, 28400 (Madrid), Spain
+[email protected]
+Abstract
+In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integers Sk(n) =
+1k + 2k + · · · + nk involving simultaneously the Stirling numbers of the first and second kind. In
+this note, we first recall and then slightly refine Lang’s formula for Sk(n). As it turns out, the
+modified Lang’s formula constitutes a special case of a general relationship discovered by Merca
+between the power sums, the elementary symmetric functions, and the complete homogeneous
+symmetric functions.
+1
+Introduction
+For integers n ≥ 1 and k ≥ 0, let Sk(n) denote the sum of k-th powers of the first n positive integers
+1k + 2k + · · · + nk. In a 2011 technical note [8], W. Lang derived the following explicit formula for
+Sk(n) (in our notation):
+Sk(n) =
+min (k,n−1)
+�
+m=0
+(−1)m(n − m)
+�
+n + 1
+n + 1 − m
+��n + k − m
+n
+�
+,
+(1)
+see [8, Equation (10)], where
+�k
+j
+�
+and
+�k
+j
+�
+are the (unsigned) Stirling numbers of the first and second
+kind, respectively.
+For completeness and for its intrinsic interest, in Section 2 of the present note we outline the
+proof of the formula (1) as given by Lang. Then, in Section 3, we slightly refine the formula (1). The
+refinement made essentially amounts to the removal of n from the factor (n − m). In Section 4, we
+show that the modified Lang’s formula arises as a direct consequence of the Newton-Girard identities
+involving the power sums Sk(n) and the elementary symmetric functions with natural arguments.
+Moreover, in Section 5, we point out that, actually, the modified Lang’s formula constitutes a
+special case of a general relationship discovered by Merca (see [10, Lemma 2.1]) between the power
+sums, the elementary symmetric functions, and the complete homogeneous symmetric functions.
+2
+Proof of Lang’s formula
+Following Lang’s own derivation [8], next we give a simplified proof sketch of the formula (1). We
+start with the ordinary generating function of Sk(n), i.e.
+Gn(x) =
+∞
+�
+k=0
+(1k + 2k + · · · + nk)xk =
+n
+�
+j=1
+1
+1 − jx.
+This generating function can be rewritten in the form
+Gn(x) =
+Pn(x)
+�n
+j=1(1 − jx),
+(2)
+1
+
+where Pn(x) is the following polynomial in x of degree n − 1 with coefficients Pn,r:
+Pn(x) =
+n
+�
+j=1
+n
+�
+l=1
+l̸=j
+(1 − lx) =
+n−1
+�
+r=0
+Pn,rxr.
+(3)
+Hence, noting that
+1
+�n
+j=1(1−jx) = �∞
+m=0
+�n+m
+n
+�
+xm, from (2) and (3) it follows that
+Sk(n) =
+min (k,n−1)
+�
+m=0
+Pn,m
+�n + k − m
+n
+�
+.
+(4)
+Now, as pointed out by Lang [8], the elementary symmetric functions σm(1, 2, . . . , n) enter the
+scene because we have that
+n
+�
+j=1
+(1 − jx) =
+n
+�
+m=0
+(−1)mσm(1, 2, . . . , n)xm,
+(5)
+with σ0 = 1. In view of (3) and (5), it is clear that, by symmetry, Pn(x) must be of the form
+Pn(x) =
+n−1
+�
+m=0
+Cn,m(−1)mσm(1, 2, . . . , n)xm,
+for certain positive integer coefficients Cn,m. Indeed, it can be seen that
+Pn,0 = n,
+Pn,1 = (n − 1)(−1)(1 + 2 + · · · + n) = (n − 1)(−1)σ1(1, 2, . . . , n),
+Pn,2 = (n − 2)(1 · 2 + 1 · 3 + · · · + (n − 1)n) = (n − 2)σ2(1, 2, . . . , n),
+and, in general,
+Pn,m = n
+�n−1
+m
+�
+�n
+m
+� (−1)mσm(1, 2, . . . , n) = (n − m)(−1)mσm(1, 2, . . . , n),
+so that Cn,m = n − m, for m = 0, 1, . . . , n − 1.
+Therefore, recalling (4), and invoking the well-known relationship σm(1, 2, . . . , n) =
+�
+n+1
+n+1−m
+�
+(see, e.g., [7, Equation (2.6)]), we get (1).
+3
+A refinement of Lang’s formula
+Having considered Lang’s original formula for the sum of powers of integers, we show that this
+formula can be simplified somewhat. To see this, we write (1) in the equivalent form
+Sk(n) = n
+min (k,n)
+�
+m=0
+(−1)m
+�
+n + 1
+n + 1 − m
+��n + k − m
+n
+�
++
+min (k,n)
+�
+m=1
+(−1)m−1 m
+�
+n + 1
+n + 1 − m
+��n + k − m
+n
+�
+,
+2
+
+where the second summation on the right-hand side is zero when k = 0 or, in other words, it applies
+for the case that k ≥ 1. Regarding the first summation, it turns out that
+min (k,n)
+�
+m=0
+(−1)m
+�
+n + 1
+n + 1 − m
+��n + k − m
+n
+�
+= δk,0,
+(6)
+where δk,0 is the Kronecker’s delta. This is so because
+
+�
+i≥0
+(−1)i
+� n + 1
+n + 1 − i
+�
+xi
+
+
+
+�
+j≥0
+�n + j
+n
+�
+xj
+
+ = 1.
+Consequently, Lang’s original formula (1) can be reduced to
+Sk(n) = n δk,0 +
+min (k,n)
+�
+m=1
+(−1)m−1 m
+�
+n + 1
+n + 1 − m
+��n + k − m
+n
+�
+,
+(7)
+which holds for any integers n ≥ 1 and k ≥ 0, and where, as noted above, the summation on the
+right-hand side is zero when k = 0. Moreover, for the general case where k ≥ 1, the formula (7)
+can in turn be expressed without loss of generality as
+Sk(n) =
+k
+�
+m=1
+(−1)m−1 m
+��
+n + 1
+n + 1 − m
+��n + k − m
+n
+�
+,
+k ≥ 1,
+(8)
+assuming the natural convention that
+�
+n+1
+n+1−m
+�
+= σm(1, 2, . . . , n) = 0 whenever m > n.
+4
+Connection with the Newton-Girard identities
+As we shall presently see, the modified Lang’s formula for Sk(n) in eq. (8) can be readily ob-
+tained from the Newton-Girard identities (cf. Exercise 2 of [3]).
+Let {x1, x2, . . . , xn} denote a
+(possibly infinite) set of variables and let σm(x1, x2, . . . , xn) denote the corresponding elementary
+symmetric function. Generally speaking, the Newton-Girard identities are, within the ring of sym-
+metric functions, the connection formulas between the generating sets {σm(x1, x2, . . . , xn)}k
+m=1 and
+{pm(x1, x2, . . . , xn)}k
+m=1, where k stands for any fixed positive integer and the pm’s stand for the
+power sums pm(x1, x2, . . . , xn) = xm
+1 + xm
+2 + · · · + xm
+n .
+For our purposes here, we focus on the case where xi = i, ∀i. Also, to abbreviate the notation,
+in what follows we write σm(1, 2, . . . , n) in the shortened form σm(n). Then, for any given positive
+integer m, the Newton-Girard identities can be formulated as follows (see, e.g., [5, Equation (5)]
+and [14, Theorem 1.2])
+m−1
+�
+j=1
+σm−j(n)Sj(n) + Sm(n) + mσm(n) = 0,
+m ≥ 1,
+(9)
+where σj(n) = (−1)jσj(n), and where the summation on the left-hand side is zero when m = 1.
+Thus, letting successively m = 1, 2, 3, . . . , k in (9) yields the following system of k equations in the
+3
+
+unknowns S1(n), S2(n), . . . , Sk(n):
+S1(n) = −σ1(n),
+σ1(n)S1(n) + S2(n) = −2σ2(n),
+σ2(n)S1(n) + σ1(n)S2(n) + S3(n) = −3σ3(n),
+...
+σk−1(n)S1(n) + σk−2(n)S2(n) + · · · + σ1(n)Sk−1(n) + Sk(n) = −kσk(n),
+which can be expressed in matrix form as
+
+
+
+
+
+
+
+
+
+1
+0
+0
+· · ·
+0
+σ1(n)
+1
+0
+...
+0
+σ2(n)
+σ1(n)
+1
+...
+0
+...
+...
+...
+...
+0
+σk−1(n)
+σk−2(n)
+· · ·
+σ1(n)
+1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+S1(n)
+S2(n)
+S3(n)
+...
+Sk(n)
+
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+−σ1(n)
+−2σ2(n)
+−3σ3(n)
+...
+−kσk(n)
+
+
+
+
+
+
+
+
+
+.
+On the other hand, it is easily seen that the orthogonality relation in eq. (6) is equivalent to the
+matrix identity
+
+
+
+
+
+
+
+
+
+1
+0
+0
+· · ·
+0
+σ1(n)
+1
+0
+...
+0
+σ2(n)
+σ1(n)
+1
+...
+0
+...
+...
+...
+...
+0
+σk−1(n)
+σk−2(n)
+· · ·
+σ1(n)
+1
+
+
+
+
+
+
+
+
+
+−1
+=
+
+
+
+
+
+
+
+
+
+1
+0
+0
+· · ·
+0
+h1(n)
+1
+0
+...
+0
+h2(n)
+h1(n)
+1
+...
+0
+...
+...
+...
+...
+0
+hk−1(n)
+hk−2(n)
+· · ·
+h1(n)
+1
+
+
+
+
+
+
+
+
+
+,
+where hk(n) =
+�n+k
+n
+�
+and h0(n) = 1. Hence, it follows that
+
+
+
+
+
+
+
+
+
+S1(n)
+S2(n)
+S3(n)
+...
+Sk(n)
+
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+1
+0
+0
+· · ·
+0
+h1(n)
+1
+0
+...
+0
+h2(n)
+h1(n)
+1
+...
+0
+...
+...
+...
+...
+0
+hk−1(n)
+hk−2(n)
+· · ·
+h1(n)
+1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−σ1(n)
+−2σ2(n)
+−3σ3(n)
+...
+−kσk(n)
+
+
+
+
+
+
+
+
+
+.
+Finally, solving for Sk(n), we get (8).
+We conclude this section with the following two remarks.
+Remark 1. The Newton-Girard identities (9) can equally be written as the recurrence relation
+Sm(n) = (−1)m−1mσm(n) −
+m−1
+�
+j=1
+(−1)jσj(n)Sm−j(n),
+m ≥ 1,
+giving Sm(n) in terms of σ1(n), σ2(n), . . . , σm(n) and the earlier power sums Sj(n), j = 1, 2, . . . , m−
+1. This recurrence may be compared with the following one appearing in [3, Remark 3]:
+Sm(n) = m!
+�n + m
+m + 1
+�
+−
+m−1
+�
+j=1
+σj(m − 1)Sm−j(n),
+m ≥ 1.
+4
+
+Remark 2. It should be mentioned that the formula for Sk(n) in eq. (8) was (re)discovered by
+Merca in [9, Theorem 1] by manipulating the formal power series for the Stirling numbers.
+5
+Generalized Lang’s formula
+The proof given in the preceding section of the formula (8) naturally generalizes to arbitrary
+elementary symmetric functions σm(x1, x2, . . . , xn), complete homogenous symmetric functions
+hm(x1, x2, . . . , xn), and associated power sums pm(x1, x2, . . . , xn). Indeed, as shown by Merca (see
+[10, Lemma 2.1]), the power sum pk(x1, x2, . . . , xn) can be expressed in terms of the σm(x1, x2, . . . , xn)
+and hk−m(x1, x2, . . . , xn) as
+pk(x1, x2, . . . , xn) =
+k
+�
+m=1
+(−1)m−1mσm(x1, x2, . . . , xn)hk−m(x1, x2, . . . , xn),
+(10)
+which becomes the formula (8) when xi = i, ∀i. Next, we describe some other applications of the
+formula (10).
+Consider first the case in which xi = 1, ∀i. Then, recalling that σm(1, 1, . . . , 1) =
+� n
+m
+�
+and
+hm(1, 1, . . . , 1) =
+�n+m−1
+m
+�
+, from (10) we obtain the identity
+k
+�
+m=1
+(���1)m−1 m
+�n
+m
+��n + k − m − 1
+k − m
+�
+= n,
+which holds for any integers k, n ≥ 1.
+On the other hand, for integers 1 ≤ r ≤ n, it turns
+out that the r-Stirling numbers of the first kind are the elementary symmetric functions of the
+numbers r, r + 1, . . . , n, that is,
+�
+n+1
+n+1−m
+�
+r = σm(r, r + 1, . . . , n); and the r-Stirling numbers of
+the second kind are the complete symmetric functions of the numbers r, r + 1, . . . , n, that is,
+�n+m
+n
+�
+r = hm(r, r + 1, . . . , n) (see [2, Section 5]). Therefore, from (10), we find that
+rk + (r + 1)k + · · · + nk =
+k
+�
+m=1
+(−1)m−1 m
+�
+n + 1
+n + 1 − m
+�
+r
+�n + k − m
+n
+�
+r
+,
+where
+�
+n+1
+n+1−m
+�
+r = σm(r, r + 1, . . . , n) = 0 whenever m > n + 1 − r. In particular, for r = 1, this
+equation reduces to (8). A further generalization of (8) in terms of the r-Whitney numbers of both
+kinds and the Bernoulli polynomials can be found in [11].
+As another application of eq. (10), we can evaluate the sum of even powers of the first n positive
+integers, S2k(n) = 12k + 22k + · · · + n2k, by using the fact that (see [12])
+u(n + 1, n + 1 − m) = (−1)mσm(12, 22, . . . , n2),
+and
+U(n + m, n) = hm(12, 22, . . . , n2),
+where u(n, k) [respectively, U(n, k)] are the central factorial numbers of the first [respectively,
+second] kind with even indices. Therefore, we have [12, Theorem 1.1]
+12k + 22k + · · · + n2k = −
+k
+�
+m=1
+m u(n + 1, n + 1 − m)U(n + k − m, n).
+5
+
+Likewise, noting that (see [12])
+v(n, n − m) = (−1)mσm(12, 32, . . . , (2n − 1)2),
+and
+V (n − 1 + m, n − 1) = hm(12, 32, . . . , (2n − 1)2),
+where v(n, k) [respectively, V (n, k)] are the central factorial numbers of the first [respectively,
+second] kind with odd indices, we can evaluate the sum of even powers of the first n odd integers,
+12k + 32k + · · · + (2n − 1)2k, as follows
+12k + 32k + · · · + (2n − 1)2k = −
+k
+�
+m=1
+m v(n, n − m)V (n − 1 + k − m, n − 1).
+Incidentally, it is to be noted that the above power sum can alternatively be expressed in the form
+(see [6, 4])
+12k + 32k + · · · + (2n − 1)2k = n
+k
+�
+m=1
+dk,mN m,
+where N = (2n − 1)(2n + 1), and where the dk,m are certain (non-zero) rational coefficients.
+Our last application concerns the so-called Legendre-Stirling (LS) numbers of the first and
+second kind, which, following [1], we denote by Ps(j)
+n
+and PS(j)
+n , respectively. Furthermore, we
+assume that n and j are non-negative integers fulfilling 0 ≤ j ≤ n. Table 1 (2) displays the first
+few LS numbers of the first (second) kind. The LS numbers of the first kind are the elementary
+symmetric functions of the numbers 2, 6, . . . , n(n + 1), i.e
+Ps(n+1−k)
+n+1
+= (−1)kσk(2, 6, . . . , n(n + 1)),
+whereas the LS numbers of the second kind are the complete homogeneous symmetric functions of
+the numbers 2, 6, . . . , n(n + 1), i.e
+PS(n)
+n+k = hk(2, 6, . . . , n(n + 1)).
+Equivalently, we can write the above two expressions as
+Ps(n+1−k)
+n+1
+= (−1)k2kσk(T1, T2, . . . , Tn),
+n\ j
+0
+1
+2
+3
+4
+5
+6
+7
+0
+1
+1
+0
+1
+2
+0
+−2
+1
+3
+0
+12
+−8
+1
+4
+0
+−144
+108
+−20
+1
+5
+0
+2880
+−2304
+508
+−40
+1
+6
+0
+−86400
+72000
+−17544
+1708
+−70
+1
+7
+0
+3628800
+−3110400
+808848
+−89280
+4648
+−112
+1
+Table 1: The LS numbers of the first kind, Ps(j)
+n , up to n = 7.
+6
+
+n\ j
+0
+1
+2
+3
+4
+5
+6
+7
+0
+1
+1
+0
+1
+2
+0
+2
+1
+3
+0
+4
+8
+1
+4
+0
+8
+52
+20
+1
+5
+0
+16
+320
+292
+40
+1
+6
+0
+32
+1936
+3824
+1092
+70
+1
+7
+0
+64
+11648
+47824
+25664
+3192
+112
+1
+Table 2: The LS numbers of the second kind, PS(j)
+n , up to n = 7.
+and
+PS(n)
+n+k = 2khk(T1, T2, . . . , Tn),
+where Tn = 1
+2n(n + 1) is the n-th triangular number. Therefore, we conclude from (10) that
+T k
+1 + T k
+2 + · · · + T k
+n = − 1
+2k
+k
+�
+m=1
+m Ps(n+1−m)
+n+1
+PS(n)
+n+k−m.
+(11)
+In particular, for k = 1, we have
+T1 + T2 + · · · + Tn =
+�n + 2
+3
+�
+= −1
+2Ps(n)
+n+1.
+In addition, we note that the sum of k-th powers of the first n triangular numbers can also be
+expressed by
+T k
+1 + T k
+2 + · · · + T k
+n = 1
+2k
+k
+�
+j=0
+�k
+j
+�
+Sk+j(n) = 1
+2k
+k
+�
+j=0
+�k
+j
+�Bk+j+1(n + 1) − Bk+j+1(1)
+k + j + 1
+,
+(12)
+where the Bk(n) are the Bernoulli polynomials. Moreover, Merca showed that, see [13, Corollary
+1.1] (in our notation)
+−
+k
+�
+m=1
+m Ps(n+1−m)
+n+1
+PS(n)
+n+k−m =
+(−1)k
+(k + 1)
+�2k+2
+k+1
+� +
+k
+�
+j=0
+�k
+j
+�Bk+j+1(n + 1)
+k + j + 1
+.
+(13)
+Hence, combining (11) and (13), and taking into account (12), we obtain the identity
+k
+�
+j=0
+(−1)j
+�k
+j
+� Bk+j+1
+k + j + 1 =
+1
+(k + 1)
+�2k+2
+k+1
+�,
+k ≥ 1,
+where the Bk are the Bernoulli numbers.
+7
+
+6
+Conclusion
+In this note, we have brought to light an outstanding (though largely unnoticed) contribution of
+W. Lang to the subject of sums of powers of integers, namely, his formula for Sk(n) stated in
+eq. (1). We have shown that Lang’s original formula (1) can be slightly refined so that the integer
+variable n can be effectively removed from the factor (n − m), as can be seen by looking at formula
+(7). Furthermore, we have shown that the modified Lang’s formula for Sk(n) in eq. (8) follows
+straightforwardly from the Newton-Girard identities formulated in eq. (9). Finally, to broaden the
+scope of the present note, we have pointed out several extensions of the formula (8) achieved by
+Merca [10, 11, 12, 13].
+References
+[1] G. E. Andrews, W. Gawronski, and L. L. Littlejohn, The Legendre-Stirling numbers, Discrete
+Math., 311(14):1255–1272 (2011).
+[2] A. Z. Broder, The r-Stirling numbers. Discrete Math., 49(3):241–259 (1984).
+[3] J. L. Cereceda, Sums of powers of integers and Stirling numbers, Resonance, 27(5):769–784
+(2022).
+[4] J. L. Cereceda, Explicit polynomial for sums of powers of odd integers, Int. Math. Forum,
+9(30):1441–1446 (2014).
+[5] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci
+sequences, Fibonacci Quart., 37(2):135–140 (1999).
+[6] S. Guo and Y. Shen, On sums of powers of odd integers, J. Math. Res. Appl., 33(6):666–672
+(2013).
+[7] D. E. Knuth, Two notes on notation, Amer. Math. Monthly, 99(5):403–422 (1992).
+[8] W. Lang, A196837: Ordinary generating functions for sums of powers of the first n positive
+integers, online note (2011), available at http://oeis.org/A196837/a196837.pdf
+[9] M. Merca, An alternative to Faulhaber’s formula, Amer. Math. Monthly, 122(6):599–601
+(2015).
+[10] M. Merca, New convolutions for complete and elementary symmetric functions, Integral Trans-
+forms Spec. Funct., 27(12):965–973 (2016).
+[11] M. Merca, A new connection between r-Whitney numbers and Bernoulli polynomials, Integral
+Transforms Spec. Funct., 25(12):937–942 (2014).
+[12] M. Merca, Connections between central factorial numbers and Bernoulli polynomials, Period.
+Math. Hungar., 73(2):259–264 (2016).
+[13] M. Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J. Num-
+ber Theory, 151:223–229 (2015).
+[14] M.
+Moss´e,
+Newton’s
+identities,
+online
+note
+(2019),
+available
+at
+https://web.stanford.edu/~marykw/classes/CS250_W19/Netwons_Identities.pdf
+8
+

2tA0T4oBgHgl3EQfM_-6/content/tmp_files/load_file.txt

@@ -0,0 +1,361 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf,len=360
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='02141v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='NT]  5 Jan 2023  A refinement of Lang’s formula for the sum of powers of integers  Jos´e Luis Cereceda  Collado Villalba, 28400 (Madrid), Spain  jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='cereceda@movistar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='es  Abstract  In 2011, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Lang derived a novel, explicit formula for the sum of powers of integers Sk(n) =  1k + 2k + · · · + nk involving simultaneously the Stirling numbers of the first and second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' In  this note, we first recall and then slightly refine Lang’s formula for Sk(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' As it turns out, the  modified Lang’s formula constitutes a special case of a general relationship discovered by Merca  between the power sums, the elementary symmetric functions, and the complete homogeneous  symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  1  Introduction  For integers n ≥ 1 and k ≥ 0, let Sk(n) denote the sum of k-th powers of the first n positive integers  1k + 2k + · · · + nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' In a 2011 technical note [8], W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Lang derived the following explicit formula for  Sk(n) (in our notation):  Sk(n) =  min (k,n−1)  �  m=0  (−1)m(n − m)  �  n + 1  n + 1 − m  ��n + k − m  n  �  ,  (1)  see [8, Equation (10)], where  �k  j  �  and  �k  j  �  are the (unsigned) Stirling numbers of the first and second  kind, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  For completeness and for its intrinsic interest, in Section 2 of the present note we outline the  proof of the formula (1) as given by Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Then, in Section 3, we slightly refine the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' The  refinement made essentially amounts to the removal of n from the factor (n − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' In Section 4, we  show that the modified Lang’s formula arises as a direct consequence of the Newton-Girard identities  involving the power sums Sk(n) and the elementary symmetric functions with natural arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Moreover, in Section 5, we point out that, actually, the modified Lang’s formula constitutes a  special case of a general relationship discovered by Merca (see [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='1]) between the power  sums, the elementary symmetric functions, and the complete homogeneous symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  2  Proof of Lang’s formula  Following Lang’s own derivation [8], next we give a simplified proof sketch of the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' We  start with the ordinary generating function of Sk(n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Gn(x) =  ∞  �  k=0  (1k + 2k + · · · + nk)xk =  n  �  j=1  1  1 − jx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  This generating function can be rewritten in the form  Gn(x) =  Pn(x)  �n  j=1(1 − jx),  (2)  1  where Pn(x) is the following polynomial in x of degree n − 1 with coefficients Pn,r:  Pn(x) =  n  �  j=1  n  �  l=1  l̸=j  (1 − lx) =  n−1  �  r=0  Pn,rxr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  (3)  Hence, noting that  1  �n  j=1(1−jx) = �∞  m=0  �n+m  n  �  xm, from (2) and (3) it follows that  Sk(n) =  min (k,n−1)  �  m=0  Pn,m  �n + k − m  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  (4)  Now, as pointed out by Lang [8], the elementary symmetric functions σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) enter the  scene because we have that  n  �  j=1  (1 − jx) =  n  �  m=0  (−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n)xm,  (5)  with σ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' In view of (3) and (5), it is clear that, by symmetry, Pn(x) must be of the form  Pn(x) =  n−1  �  m=0  Cn,m(−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n)xm,  for certain positive integer coefficients Cn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Indeed, it can be seen that  Pn,0 = n,  Pn,1 = (n − 1)(−1)(1 + 2 + · · · + n) = (n − 1)(−1)σ1(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n),  Pn,2 = (n − 2)(1 · 2 + 1 · 3 + · · · + (n − 1)n) = (n − 2)σ2(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n),  and, in general,  Pn,m = n  �n−1  m  �  �n  m  � (−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) = (n − m)(−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n),  so that Cn,m = n − m, for m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Therefore, recalling (4), and invoking the well-known relationship σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) =  �  n+1  n+1−m  �  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', [7, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='6)]), we get (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  3  A refinement of Lang’s formula  Having considered Lang’s original formula for the sum of powers of integers, we show that this  formula can be simplified somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' To see this, we write (1) in the equivalent form  Sk(n) = n  min (k,n)  �  m=0  (−1)m  �  n + 1  n + 1 − m  ��n + k − m  n  �  +  min (k,n)  �  m=1  (−1)m−1 m  �  n + 1  n + 1 − m  ��n + k − m  n  �  ,  2  where the second summation on the right-hand side is zero when k = 0 or, in other words, it applies  for the case that k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Regarding the first summation, it turns out that  min (k,n)  �  m=0  (−1)m  �  n + 1  n + 1 − m  ��n + k − m  n  �  = δk,0,  (6)  where δk,0 is the Kronecker’s delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' This is so because  \uf8eb  \uf8ed�  i≥0  (−1)i  � n + 1  n + 1 − i  �  xi  \uf8f6  \uf8f8  \uf8eb  \uf8ed�  j≥0  �n + j  n  �  xj  \uf8f6  \uf8f8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Consequently, Lang’s original formula (1) can be reduced to  Sk(n) = n δk,0 +  min (k,n)  �  m=1  (−1)m−1 m  �  n + 1  n + 1 − m  ��n + k − m  n  �  ,  (7)  which holds for any integers n ≥ 1 and k ≥ 0, and where, as noted above, the summation on the  right-hand side is zero when k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Moreover, for the general case where k ≥ 1, the formula (7)  can in turn be expressed without loss of generality as  Sk(n) =  k  �  m=1  (−1)m−1 m  �  n + 1  n + 1 − m  ��n + k − m  n  �  ,  k ≥ 1,  (8)  assuming the natural convention that  �  n+1  n+1−m  �  = σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) = 0 whenever m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  4  Connection with the Newton-Girard identities  As we shall presently see, the modified Lang’s formula for Sk(n) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (8) can be readily ob-  tained from the Newton-Girard identities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Exercise 2 of [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Let {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn} denote a  (possibly infinite) set of variables and let σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn) denote the corresponding elementary  symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Generally speaking, the Newton-Girard identities are, within the ring of sym-  metric functions, the connection formulas between the generating sets {σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn)}k  m=1 and  {pm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn)}k  m=1, where k stands for any fixed positive integer and the pm’s stand for the  power sums pm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn) = xm  1 + xm  2 + · · · + xm  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  For our purposes here, we focus on the case where xi = i, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Also, to abbreviate the notation,  in what follows we write σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) in the shortened form σm(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Then, for any given positive  integer m, the Newton-Girard identities can be formulated as follows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', [5, Equation (5)]  and [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='2])  m−1  �  j=1  σm−j(n)Sj(n) + Sm(n) + mσm(n) = 0,  m ≥ 1,  (9)  where σj(n) = (−1)jσj(n), and where the summation on the left-hand side is zero when m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Thus, letting successively m = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , k in (9) yields the following system of k equations in the  3  unknowns S1(n), S2(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , Sk(n):  S1(n) = −σ1(n),  σ1(n)S1(n) + S2(n) = −2σ2(n),  σ2(n)S1(n) + σ1(n)S2(n) + S3(n) = −3σ3(n),  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  σk−1(n)S1(n) + σk−2(n)S2(n) + · · · + σ1(n)Sk−1(n) + Sk(n) = −kσk(n),  which can be expressed in matrix form as  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  0  0  · ·  0  σ1(n)  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  σ2(n)  σ1(n)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  σk−1(n)  σk−2(n)  · ·  σ1(n)  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  S1(n)  S2(n)  S3(n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Sk(n)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −σ1(n)  −2σ2(n)  −3σ3(n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  −kσk(n)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  On the other hand, it is easily seen that the orthogonality relation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (6) is equivalent to the  matrix identity  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  0  0  · ·  0  σ1(n)  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  σ2(n)  σ1(n)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  σk−1(n)  σk−2(n)  · ·  σ1(n)  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  −1  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  0  0  · ·  0  h1(n)  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  h2(n)  h1(n)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  hk−1(n)  hk−2(n)  · ·  h1(n)  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  where hk(n) =  �n+k  n  �  and h0(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Hence, it follows that  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  S1(n)  S2(n)  S3(n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Sk(n)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  0  0  · ·  0  h1(n)  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  h2(n)  h1(n)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  0  hk−1(n)  hk−2(n)  · ·  h1(n)  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −σ1(n)  −2σ2(n)  −3σ3(n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  −kσk(n)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Finally, solving for Sk(n), we get (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  We conclude this section with the following two remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' The Newton-Girard identities (9) can equally be written as the recurrence relation  Sm(n) = (−1)m−1mσm(n) −  m−1  �  j=1  (−1)jσj(n)Sm−j(n),  m ≥ 1,  giving Sm(n) in terms of σ1(n), σ2(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , σm(n) and the earlier power sums Sj(n), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , m−  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' This recurrence may be compared with the following one appearing in [3, Remark 3]:  Sm(n) = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  �n + m  m + 1  �  −  m−1  �  j=1  σj(m − 1)Sm−j(n),  m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  4  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' It should be mentioned that the formula for Sk(n) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (8) was (re)discovered by  Merca in [9, Theorem 1] by manipulating the formal power series for the Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  5  Generalized Lang’s formula  The proof given in the preceding section of the formula (8) naturally generalizes to arbitrary  elementary symmetric functions σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn), complete homogenous symmetric functions  hm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn), and associated power sums pm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Indeed, as shown by Merca (see  [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='1]), the power sum pk(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn) can be expressed in terms of the σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn)  and hk−m(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn) as  pk(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn) =  k  �  m=1  (−1)m−1mσm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn)hk−m(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , xn),  (10)  which becomes the formula (8) when xi = i, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Next, we describe some other applications of the  formula (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Consider first the case in which xi = 1, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Then, recalling that σm(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , 1) =  � n  m  �  and  hm(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , 1) =  �n+m−1  m  �  , from (10) we obtain the identity  k  �  m=1  (−1)m−1 m  �n  m  ��n + k − m − 1  k − m  �  = n,  which holds for any integers k, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  On the other hand, for integers 1 ≤ r ≤ n, it turns  out that the r-Stirling numbers of the first kind are the elementary symmetric functions of the  numbers r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n, that is,  �  n+1  n+1−m  �  r = σm(r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' and the r-Stirling numbers of  the second kind are the complete symmetric functions of the numbers r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n, that is,  �n+m  n  �  r = hm(r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) (see [2, Section 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Therefore, from (10), we find that  rk + (r + 1)k + · · · + nk =  k  �  m=1  (−1)m−1 m  �  n + 1  n + 1 − m  �  r  �n + k − m  n  �  r  ,  where  �  n+1  n+1−m  �  r = σm(r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n) = 0 whenever m > n + 1 − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' In particular, for r = 1, this  equation reduces to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' A further generalization of (8) in terms of the r-Whitney numbers of both  kinds and the Bernoulli polynomials can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  As another application of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (10), we can evaluate the sum of even powers of the first n positive  integers, S2k(n) = 12k + 22k + · · · + n2k, by using the fact that (see [12])  u(n + 1, n + 1 − m) = (−1)mσm(12, 22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n2),  and  U(n + m, n) = hm(12, 22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n2),  where u(n, k) [respectively, U(n, k)] are the central factorial numbers of the first [respectively,  second] kind with even indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Therefore, we have [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='1]  12k + 22k + · · · + n2k = −  k  �  m=1  m u(n + 1, n + 1 − m)U(n + k − m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  5  Likewise, noting that (see [12])  v(n, n − m) = (−1)mσm(12, 32, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , (2n − 1)2),  and  V (n − 1 + m, n − 1) = hm(12, 32, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , (2n − 1)2),  where v(n, k) [respectively, V (n, k)] are the central factorial numbers of the first [respectively,  second] kind with odd indices, we can evaluate the sum of even powers of the first n odd integers,  12k + 32k + · · · + (2n − 1)2k, as follows  12k + 32k + · · · + (2n − 1)2k = −  k  �  m=1  m v(n, n − m)V (n − 1 + k − m, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Incidentally, it is to be noted that the above power sum can alternatively be expressed in the form  (see [6, 4])  12k + 32k + · · · + (2n − 1)2k = n  k  �  m=1  dk,mN m,  where N = (2n − 1)(2n + 1), and where the dk,m are certain (non-zero) rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Our last application concerns the so-called Legendre-Stirling (LS) numbers of the first and  second kind, which, following [1], we denote by Ps(j)  n  and PS(j)  n , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Furthermore, we  assume that n and j are non-negative integers fulfilling 0 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Table 1 (2) displays the first  few LS numbers of the first (second) kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' The LS numbers of the first kind are the elementary  symmetric functions of the numbers 2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n(n + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='e  Ps(n+1−k)  n+1  = (−1)kσk(2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n(n + 1)),  whereas the LS numbers of the second kind are the complete homogeneous symmetric functions of  the numbers 2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n(n + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='e  PS(n)  n+k = hk(2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , n(n + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Equivalently, we can write the above two expressions as  Ps(n+1−k)  n+1  = (−1)k2kσk(T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , Tn),  n\\ j  0  1  2  3  4  5  6  7  0  1  1  0  1  2  0  −2  1  3  0  12  −8  1  4  0  −144  108  −20  1  5  0  2880  −2304  508  −40  1  6  0  −86400  72000  −17544  1708  −70  1  7  0  3628800  −3110400  808848  −89280  4648  −112  1  Table 1: The LS numbers of the first kind, Ps(j)  n , up to n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  6  n\\ j  0  1  2  3  4  5  6  7  0  1  1  0  1  2  0  2  1  3  0  4  8  1  4  0  8  52  20  1  5  0  16  320  292  40  1  6  0  32  1936  3824  1092  70  1  7  0  64  11648  47824  25664  3192  112  1  Table 2: The LS numbers of the second kind, PS(j)  n , up to n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  and  PS(n)  n+k = 2khk(T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' , Tn),  where Tn = 1  2n(n + 1) is the n-th triangular number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Therefore, we conclude from (10) that  T k  1 + T k  2 + · · · + T k  n = − 1  2k  k  �  m=1  m Ps(n+1−m)  n+1  PS(n)  n+k−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  (11)  In particular, for k = 1, we have  T1 + T2 + · · · + Tn =  �n + 2  3  �  = −1  2Ps(n)  n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  In addition, we note that the sum of k-th powers of the first n triangular numbers can also be  expressed by  T k  1 + T k  2 + · · · + T k  n = 1  2k  k  �  j=0  �k  j  �  Sk+j(n) = 1  2k  k  �  j=0  �k  j  �Bk+j+1(n + 1) − Bk+j+1(1)  k + j + 1  ,  (12)  where the Bk(n) are the Bernoulli polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Moreover, Merca showed that, see [13, Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='1] (in our notation)  −  k  �  m=1  m Ps(n+1−m)  n+1  PS(n)  n+k−m =  (−1)k  (k + 1)  �2k+2  k+1  � +  k  �  j=0  �k  j  �Bk+j+1(n + 1)  k + j + 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  (13)  Hence, combining (11) and (13), and taking into account (12), we obtain the identity  k  �  j=0  (−1)j  �k  j  � Bk+j+1  k + j + 1 =  1  (k + 1)  �2k+2  k+1  �,  k ≥ 1,  where the Bk are the Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  7  6  Conclusion  In this note, we have brought to light an outstanding (though largely unnoticed) contribution of  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Lang to the subject of sums of powers of integers, namely, his formula for Sk(n) stated in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' We have shown that Lang’s original formula (1) can be slightly refined so that the integer  variable n can be effectively removed from the factor (n − m), as can be seen by looking at formula  (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Furthermore, we have shown that the modified Lang’s formula for Sk(n) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (8) follows  straightforwardly from the Newton-Girard identities formulated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Finally, to broaden the  scope of the present note, we have pointed out several extensions of the formula (8) achieved by  Merca [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Andrews, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Gawronski, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Littlejohn, The Legendre-Stirling numbers, Discrete  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 311(14):1255–1272 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Broder, The r-Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 49(3):241–259 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Cereceda, Sums of powers of integers and Stirling numbers, Resonance, 27(5):769–784  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Cereceda, Explicit polynomial for sums of powers of odd integers, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Forum,  9(30):1441–1446 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci  sequences, Fibonacci Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 37(2):135–140 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Guo and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Shen, On sums of powers of odd integers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 33(6):666–672  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Knuth, Two notes on notation, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Monthly, 99(5):403–422 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Lang, A196837: Ordinary generating functions for sums of powers of the first n positive  integers, online note (2011), available at http://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='org/A196837/a196837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='pdf  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Merca, An alternative to Faulhaber’s formula, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Monthly, 122(6):599–601  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Merca, New convolutions for complete and elementary symmetric functions, Integral Trans-  forms Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 27(12):965–973 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Merca, A new connection between r-Whitney numbers and Bernoulli polynomials, Integral  Transforms Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 25(12):937–942 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Merca, Connections between central factorial numbers and Bernoulli polynomials, Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=', 73(2):259–264 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content=' Num-  ber Theory, 151:223–229 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='  Moss´e,  Newton’s  identities,  online  note  (2019),  available  at  https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='edu/~marykw/classes/CS250_W19/Netwons_Identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}
+page_content='pdf  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'}

2tE0T4oBgHgl3EQfuwEX/content/tmp_files/2301.02608v1.pdf.txt

@@ -0,0 +1,1639 @@
+A CAD System for Colorectal Cancer from WSI: A
+Clinically Validated Interpretable ML-based Prototype
+Pedro C. Netoa,b,1, Diana Montezumac,f,d,1, Sara P. Oliveiraa,b,1, Domingos
+Oliveirac, Jo˜ao Fragae, Ana Monteiroc, Jo˜ao Monteiroc, Liliana Ribeiroc,
+Sofia Gon¸calvesc, Stefan Reinhardg, Inti Zlobecg, Isabel M. Pintoc, Jaime S.
+Cardosoa,b
+aInstitute for Systems and Computer Engineering, Technology and Science (INESC
+TEC), R. Dr. Roberto Frias, Porto, 4200-465, Porto, Portugal
+bFaculty of Engineering, University of Porto (FEUP), R. Dr. Roberto
+Frias, Porto, 4200-465, Porto, Portugal
+cIMP Diagnostics, Praca do Bom Sucesso, 61, sala
+809, Porto, 4150-146, Porto, Portugal
+dCancer Biology and Epigenetics Group, IPO-Porto, R. Dr. Ant´onio Bernardino de
+Almeida 865, Porto, 4200-072, Porto, Portugal
+eDepartment of Pathology, IPO-Porto, R. Dr. Ant´onio Bernardino de Almeida
+865, Porto, 4200-072, Porto, Portugal
+fSchool of Medicine and Biomedical Sciences, University of Porto (ICBAS), R. Jorge de
+Viterbo Ferreira 228, Porto, 4050-313, Porto, Portugal
+gInstitute of Pathology, University of Bern, Uni Bern, Murtenstrasse
+31, Bern, 3008, Bern, Switzerland
+Abstract
+The integration of Artificial Intelligence (AI) and Digital Pathology has
+been increasing over the past years. Nowadays, applications of deep learn-
+ing (DL) methods to diagnose cancer from whole-slide images (WSI) are,
+more than ever, a reality within different research groups. Nonetheless, the
+development of these systems was limited by a myriad of constraints re-
+garding the lack of training samples, the scaling difficulties, the opaqueness
+of DL methods, and, more importantly, the lack of clinical validation. As
+such, we propose a system designed specifically for the diagnosis of colorectal
+samples. The construction of such a system consisted of four stages: (1) a
+careful data collection and annotation process, which resulted in one of the
+largest WSI colorectal samples datasets; (2) the design of an interpretable
+1These authors contributed equally.
+Preprint submitted to .
+January 9, 2023
+arXiv:2301.02608v1  [eess.IV]  6 Jan 2023
+
+mixed-supervision scheme to leverage the domain knowledge introduced by
+pathologists through spatial annotations; (3) the development of an effective
+sampling approach based on the expected severeness of each tile, which de-
+creased the computation cost by a factor of almost 6x; (4) the creation of
+a prototype that integrates the full set of features of the model to be eval-
+uated in clinical practice. During these stages, the proposed method was
+evaluated in four separate test sets, two of them are external and completely
+independent. On the largest of those sets, the proposed approach achieved
+an accuracy of 93.44%. DL for colorectal samples is a few steps closer to stop
+being research exclusive and to become fully integrated in clinical practice.
+Keywords:
+Clinical Prototype, Colorectal Cancer, Interpretable Artificial
+Intelligence, Deep Learning, Whole-Slide Images
+1. Introduction
+Colorectal cancer (CRC) incidence and mortality are increasing, and it
+is estimated that they will keep growing at least until 2040 [1], according to
+estimations of the International Agency for Research on Cancer. Nowadays,
+it is the third most incident (10.7% of all cancer diagnoses) and the second
+most deadly type of cancer [1]. Due to the effect of lifestyle, genetics, envi-
+ronmental factors and an increase in life expectancy, the current increase in
+world wealth and the adoption of western lifestyles further advocates for an
+increase in the capabilities to perform more CRC evaluations for potential
+diagnosis [2, 3]. Despite the pessimist predictions for an increase in the in-
+cidence, CRC is preventable and curable when detected in its earlier stages.
+Thus, effective screening through medical examination, imaging techniques
+and colonoscopy are of utmost importance [4, 5].
+Despite the CRC detection capabilities shown by imaging techniques,
+the diagnosis of cancer is always based on the pathologist’s evaluation of
+biopsies/surgical specimen samples. The stratification of neoplasia develop-
+ment stages consists of non-neoplastic (NNeo), low-grade dysplasia (LGD),
+high-grade dysplasia (HGD, including intramucosal carcinomas), and inva-
+sive carcinomas, from the initial to the latest stage of cancer progression,
+respectively. In spite of the inherent subjectivity of the dysplasia grading
+system [6], recent guidelines from the European Society of Gastrointestinal
+Endoscopy (ESGE), as well as those from the US multi-society task force on
+CRC, consistently recommend shorter surveillance intervals for patients with
+2
+
+polyps with high-grade dysplasia, regardless of their dimension [5, 7]. Hence,
+grading dysplasia is still routinely performed by pathologists worldwide when
+assessing colorectal tissue samples.
+Private datasets of digitised slides are becoming widely available, in the
+form of whole-slide images (WSI), with an increase in the adoption of digital
+workflows [8, 9, 10]. Despite the burden of the additional scanning step, WSI
+eases the revision of old cases, data sharing and quick peer-review [11, 12].
+It has also created several research opportunities within the computer vision
+domain, especially due to the complexity of the problem and the high dimen-
+sions of WSI [13, 14, 15, 16]. As such, robust and high-performance systems
+can be valuable assets to the digital workflow of a laboratory, especially if
+they are transparent and interpretable [11, 12]. However, some limitations
+still affect the applicability of such solutions in practice [17].
+The majority of the works on CRC diagnosis direct their focus towards the
+classification of cropped regions of interest, or small tiles, instead of tackling
+the challenging task of diagnosing the entire WSI [18, 19, 17, 20]. Notwith-
+standing, some authors already presented methods to assess the grading of
+the complete slide of colorectal samples. In 2020, Iizuka et al. [21] used a re-
+current neural network (RNN) to aggregate the predictions of individual tiles
+processed by an Inception-v3 network into non-neoplastic, adenoma (AD)
+and adenocarcinoma (ADC). Due to the large dimensions of WSI related to
+their pyramidal format (with several magnification levels) [22], usually over
+50,000 × 50,000 pixels, it is usual to use a scheme consisting of a grid of
+tiles. This scheme permits the acceleration of the processing steps since the
+tiles are small enough to fit in the memory of the graphics processing units
+(GPU), popular units for the training of deep learning (DL). Wei et al. [23]
+studied the usage of an ensemble of five distinct ResNet networks, in order to
+distinguish the types of CRC adenomas H&E stained slides. Song et al. [24]
+experimented with a modified DeepLab-v2 network for tile classification, and
+proposed pixel probability thresholding to detect CRC adenomas. Both Xu et
+al. [25] and Wang et al. [26, 27] looked into the performance of the Inception-
+v3 architecture to detect CRC, with the latter also retrieving a cluster-based
+slide classification and a map of predictions. The MuSTMIL [28] method,
+classifieds five colon-tissue findings: normal glands, hyperplastic polyps, low-
+grade dysplasias, high-grade dysplasias and carcinomas. This classification
+originates from a multitask architecture that leverages several levels of mag-
+nification of a slide. Ho et al. [29] extended the experiments with multitask
+learning, but instead of leveraging the magnification, its model aims to jointly
+3
+
+segment glands, detect tumour areas and sort the slides into low-risk (benign,
+inflammation or reactive changes) and high-risk (adenocarcinoma or dyspla-
+sia) categories. The architecture of this model is considerably more complex,
+with regard to the number of parameters, and is known as Faster-RCNN with
+a ResNet-101 backbone network for the segmentation task. Further to this
+task, a gradient-boosted decision tree completes the pipeline that results in
+the final grade.
+As stated in the previous paragraph, recent state-of-the-art computer-
+aided diagnosis systems are based on deep learning approaches. These sys-
+tems rely on large volumes of data to learn how to perform a particular task.
+Increasing the complexity of the task often demands an increase in the data
+available. Collecting this data is expensive and tedious due to the annotation
+complexity and the need for expert knowledge. Despite recent publications
+that present approaches using large volumes of data to train CAD systems,
+the majority do not publicly release the data used. To reverse this trend,
+the novel and completely anonymised dataset introduced in this document
+will have the majority of the available slides publicly released. This dataset
+contains, approximately, 10500 high-quality slides. The available slides origi-
+nate one of the largest colorectal samples (CRS) datasets to be made publicly
+available. This high volume of data, in addition to the massive resolution
+of the images, creates a significant bottleneck of deep learning approaches
+that extract patches from the whole slide. Hence, we introduce an efficient
+sampling approach that is performed once without sacrificing prediction per-
+formance. The proposed sampling leverages knowledge learnt from the data
+to create a proxy that reduces the rate of important information discarded,
+when compared to random sampling. Due to the cost of annotating the en-
+tire dataset, it is only annotated for a portion at the pixel level, while the
+remaining portion is labelled for a portion at the slide level. To leverage these
+two levels of supervision, we propose a mixed supervision training scheme.
+One other increasingly relevant issue with current research is the lack
+of external validation. It is not uncommon to observe models that perform
+extremely well on a test set collected from the same data distribution used
+to train, but fail on samples from other laboratories or scanners. Hence,
+we validate our proposed model in two different external datasets that vary
+in quality, country of origin and laboratory. While the results on a similar
+dataset show the performance of the model if implemented in the institution
+that collects that data, the test on external samples indicates its capabilities
+to be deployed in other scenarios.
+4
+
+In order to bring this CAD system into production, and to infer its ca-
+pabilities within clinical practice, we developed a prototype that has been
+used by pathologists in clinical practice. We further collected information on
+the misdiagnoses and the pathologists’ feedback. Since it is expected that
+the model indicates some rationale behind its predictions, we developed a
+visual approach to explain the decision and guide pathologists’ focus toward
+more aggressive areas. This mechanism increased the acceptance of the al-
+gorithm in clinical practice, and its usefulness. When collected from routine
+the slides can be digitised with duplicated tissue areas, known as fragments,
+which might be of lower quality.
+Hence, the workflow for the automatic
+diagnostic also included an automatic fragment detection and counting sys-
+tem [30]. Moreover, it was possible to utilize this prototype to do the second
+round of labelling based on the test set prediction given by the model. This
+second round led to the correction of certain labels (that the model pre-
+dicted correctly and were mislabeled by the expert pathologist) and insights
+regarding the areas where the model has to be improved.
+To summarise, in this paper we propose a novel dataset with more than
+thirteen million tiles, a sampling approach to reduce the difficulty of using
+large datasets, a deep learning model that is trained with mixed supervision,
+evaluated on two external datasets, and incorporated in a prototype that
+provides a simple integration in clinical practice and visual explanations of
+the model’s predictions. This way, we come a step closer to making CAD
+tools a reality for colorectal diagnosis.
+2. Methods
+In this section, after defining the problem at hand, we introduce the pro-
+posed dataset used to train, validate and test the model, the external datasets
+to evaluate the generalisation capabilities of the model and the pre-processing
+pipeline. Afterwards, we describe in detail the methodology followed to cre-
+ate the deep learning model and to design the experiments. Finally, we also
+detail the clinical assessment and evaluation of the model.
+2.1. Problem definition
+Digitised colorectal cancer histological samples have large dimensions,
+which are far larger than the dimensions of traditional images used in medi-
+cal or general computer vision problems. Labelling such images is expensive
+and highly dependent on the availability of expert knowledge. It limits the
+5
+
+availability of whole slide images, and, in scenarios where these are available,
+meaningful annotations are usually lacking. On the other hand, it is easier
+to label the dataset at the slide level. The inclusion of detailed spatial an-
+notations on approximately 10% of the dataset has been shown to positively
+impact the performance of deep learning algorithms [17, 31].
+Figure 1: Problem definition as a fully supervised task (on top), and as a weakly-supervised
+task (bottom).
+To fully leverage the potential of spatial and slide labels, we propose a
+deep learning pipeline, based on previous approaches [17, 31], using mixed
+supervision. Each slide, S is composed of a set of tiles Ts,n, where s represents
+the index of the slide and n ∈ {1, · · · , ns} the tile number. Furthermore,
+there is an inherent order in the grading used to classify the input into one of
+the C(k) classes, which represents a variation in severity. For fully supervised
+learning, only strongly annotated slides are useful, and for those, the label of
+each tile Cs,n is known. The remaining slides are deprived of these detailed
+labels, hence, they can only be leveraged by training algorithms with weakly
+supervision. To be used by these algorithms, the weakly annotated slides
+have only a single label for the entire bag (set) of tiles, as seen in Figure 1.
+6
+
+Following the order of the labels and the clinical knowledge, we assume that
+the predicted slide label Cs is the most severe case of the tile labels:
+Cs = maxn{Cs,n}.
+In other words, if there is at least one tile classified as containing high-
+grade dysplasia, then the entire slide that contains the tile is classified ac-
+cordingly. On the other end of the spectrum, if the worst tile is classified as
+non-neoplastic, then it is assumed that there is no dysplasia in the entire set
+of tiles. This is a generalisation of multiple-instance learning (MIL) to an
+ordinal classification problem, as proposed by Oliveira et al. [17].
+2.2. Datasets
+The spectrum of large-scale CRC/CRS datasets is slowly increasing due to
+the contributions of several researchers. Two datasets that have been recently
+introduced in the literature are the CRS1K [17] and CRS4K [31] datasets.
+Since the latter is an extension of the former with roughly four times more
+slides, it will be the baseline dataset for the remaining of this document.
+Moreover, we further extend these with the CRS10K dataset, which contains
+9.26x and 2.36x more slides than CRS1K and CRS4K, respectively. Similarly,
+the number of tiles is multiplied by a factor of 12.2 and 2.58 (Table 1). This
+volume of slides is translated into an increase in the flexibility to design
+experiments and infer the robustness of the model. Thus, the inclusion of a
+test set separated from the validation set is now facilitated.
+The set is composed of colorectal biopsies and polypectomies (excluding
+surgical specimens). Following the same annotation process as the previ-
+ous datasets, CRS10K slides are labelled according to three main categories:
+non-neoplastic (NNeo), low-grade lesions (LG), and high-grade lesions (HG).
+The first, contains normal colorectal mucosa, hyperplasia and non-specific
+inflammation. LG lesions categorise conventional adenomas with low-grade
+dysplasia. Finally, HG lesions are composed of adenomas with high-grade
+dysplasia (including intra-mucosal carcinomas) and invasive adenocarcino-
+mas. In order to avoid diversions from the main goal, slides with suspicion
+of known history of inflammatory bowel disease/infection, serrated lesions or
+other polyp types were not included in the dataset.
+The slides, retrieved from an archive of previous cases, were digitised with
+Leica GT450 WSI scanners, at 40× magnification. The cases were initially
+seen and classified (labelled) by one of three pathologists. The pathologist
+revised and classified the slides, and then compared them with the initial
+report diagnosis (which served as a second-grader). If there was a match
+7
+
+between both, no further steps were taken.
+In discordant cases, a third
+pathologist served as a tie-breaker. Roughly 9% of the dataset (967 slides and
+over a million tiles) were manually annotated by a pathologist and rechecked
+by the other, in turn, using the Sedeen Viewer software [32]. For complex
+cases, or when the agreement for a joint decision could not be reached, a
+third pathologist reevaluated the annotation.
+The CRS10K dataset was divided into train, validation and test sets.
+The first includes all the strongly annotated slides and other slides randomly
+selected.
+Whereas the second is composed of only non-annotated slides.
+Finally, the test set was selected from the new data added to extend the
+previous datasets. Thus, it is completely separated from the training and
+validation sets of previous works. The test set, will be publicly available, so
+that future research can directly compare their results and use that set as a
+benchmark.
+Table 1: Dataset summary, with the number of slides (annotated samples are detailed in
+parentheses) and tiles distributed by class for all the datasets used in this study.
+NNeo
+LG
+HG
+Total
+# slides
+300 (6)
+552 (35)
+281 (59)
+1133 (100)
+CRS1K dataset [17]
+# annotated tiles
+49,640
+77,946
+83,649
+211,235
+# non-annotated tiles
+-
+-
+-
+1,111,361
+# slides
+663 (12)
+2394 (207)
+1376 (181)
+4433 (400)
+CRS4K dataset [31]
+# annotated tiles
+145,898
+196,116
+163,603
+505,617
+# non-annotated tiles
+-
+-
+-
+5,265,362
+# slides
+1740 (12)
+5387 (534)
+3369 (421)
+10,496 (967)
+CRS10K dataset
+# annotated tiles
+338,979
+371,587
+341,268
+1,051,834
+# non-annotated tiles
+-
+-
+-
+13,571,871
+CRS Prototype
+# slides
+28
+44
+28
+100
+# non-annotated tiles
+-
+-
+-
+244,160
+PAIP [33]
+# slides
+-
+-
+100
+100
+# non-annotated tiles
+-
+-
+-
+97,392
+TCGA [34]
+# slides
+1
+1
+230
+232
+# non-annotated tiles
+-
+-
+-
+1,568,584
+Furthermore, as detailed in the following sections, this work comprises the
+development of a fully-functional prototype to be used in clinical practice.
+Leveraging this prototype, it was possible to further collect a new set with
+100 slides. It differs from the CRS10K dataset, in the sense that they were
+not carefully selected from the archives. Instead, these cases were actively
+collected from the current year’s routine exams. We argue that this might
+8
+
+better reflect the real-world data distribution. Hence, we introduce this set as
+a distinct dataset to evaluate the robustness of the presented methodology.
+Differently from the datasets discussed below, the CRS Prototype dataset
+has a more balanced distribution of the slide labels. Although it is useful
+in practice, the usage of the fragment counting and selection algorithm for
+the evaluation could potentiate the propagation of errors from one system
+to another. Thus, in this evaluation, we did not use the fragment selection
+algorithm, and as shown in Table 1, the number of tiles per slide doubles
+when compared to CRS10K, which had its fragments carefully selected.
+To evaluate the domain generalisation of the proposed approach, two
+external datasets were used. We evaluate the proposed approaches on two
+external datasets publicly available. The first dataset is composed of samples
+of the TCGA-COAD [35] and TCGA-READ [36] collections from The Can-
+cer Imaging Archive [34], which are composed in general by resection samples
+(in contrast to our dataset, composed only of biopsies and polypectomies).
+Samples containing pen markers, large air bubbles over tissue, tissue folds,
+and other artefacts affecting large areas of the slide were excluded. The fi-
+nal selection includes 232 whole-slide images reviewed and validated by the
+same pathologists that reviewed the in-house datasets. 230 of those sam-
+ples were diagnosed as high-grade lesions, whereas the remaining two have
+been diagnosed as low-grade and non-neoplastic. For this dataset, the spe-
+cific model of the scanner used to digitise the images is unknown, but the
+file type (”.svs”) matches the file type of the training data. The second ex-
+ternal dataset used to evaluate the model contains 100H&E slides from the
+Pathology AI Platform [33] colorectal cohort, which contains all the cases
+with a more superficial sampling of the lesion, for a better comparison with
+our datasets. All the whole slide images in this dataset were digitised with
+an Aperio AT2 at 20X magnification. Finally, the pathologists’ team fol-
+lowed the same guidelines to review and validate all the WSI, which were all
+classified as high-grade lesions. It is interesting to note that while the PAIP
+contains significantly fewer tiles per slide, around 973, than the CRS10K
+dataset, around 1293, the TCGA dataset shows the largest amount of tissue
+per slide with an average of 6761 tiles as seen in Table 1.
+2.3. Data pre-processing
+H&E slides are composed of two distinct elements, white background
+and colourful tissue. Since the former is not meaningful for the diagnostic,
+9
+
+the pre-processing of these slides incorporates an automatic tissue segmenta-
+tion with Otsu’s thresholding [37] on the saturation (S) channel of the HSV
+colour space, resulting in a separation between the tissue regions and the
+background. The result of this step, which receives as input a 32× down-
+sampled slide, is the mask used for the following steps.
+Leveraging this
+previous output, tiles with a dimension of 512 × 512 pixels (Figure 2) were
+extracted from the original slide (without any downsampling) at its maxi-
+mum magnification (40×), if they did not include any portion of background
+(i.e. a 100% tissue threshold was used). Following previous experiments in
+the literature, our empirical assessment, and the confirmation that smaller
+tiles would significantly increase the number of tiles and the complexity of
+the task, 512 × 512 was chosen as the tile size. Moreover, it is believed that
+512 × 512 is the smallest tile size that still incorporates enough information
+to make a good diagnostic with the possibility of visually explaining the de-
+cision [17]. The selected threshold of 100% further reduces the number of
+tiles by not including the tissue at the edges and decreases the complexity
+of the task, since the model does not see the background at any moment.
+Due to tissue variations in different images, there is also a different number
+of tiles extracted per image.
+2.4. Methodology
+The massive size of images, which translates to thousands of tiles per
+image, allied to a large number of samples in the CRS10K dataset, bottle-
+necks the training of weakly-supervised models based on multiple instance
+learning (MIL). Hence, in this document, we propose a mix-supervision ap-
+proach with self-contained tile sampling to diagnose colorectal cancer samples
+from whole-slide images. This subsection comprises the methodology, which
+includes supervised training, sampling and weakly-supervised learning.
+2.4.1. Supervised Training
+As mentioned in previous sections, spatial annotations are rare in large
+quantities. However, these include domain information, given by the expert
+annotator, concerning the most meaningful areas and what are the most and
+less severe tiles. Thus, they can facilitate the initial optimisation of a deep
+neural network. As shown in the literature, there has been some research on
+the impact of starting the training with a few iterations of fully-supervised
+training [17, 38]. We further explore this in three different ways. First, we
+have 967 annotated slides resulting in more than one million annotated tiles
+10
+
+Figure 2: Examples of regions with and a sample tile with 512 × 512 pixels (40× mag-
+nification), representing each class: non-neoplastic (on top), low-grade dysplasia (on the
+middle) and high-grade dysplasia (on the bottom).
+for supervised training. Secondly, attending to the size of our dataset and
+the need for a stronger initial supervised training, the models are trained
+for 50 epochs, and their performance was monitored over specific checkpoint
+epochs. Finally, we explore this pre-trained model as the main tool to sample
+useful tiles for the weakly-supervised task.
+11
+
+Figure 3: Overall scheme of the proposed methodology containing the mix-supervision
+framework that is responsible for diagnosing colorectal samples from WSI.
+2.4.2. Tile Sampling
+Our scenario presents a particularly difficult condition for scaling the
+training data. First, let’s consider the structure of the data, which consists
+of, on average, more than one thousand tiles per slide. Within this set of
+tiles, some tiles provide meaningful value for the prediction, and others do
+not add extra information. In other words, for the CRS10K dataset, the
+extensive, lengthy, time and energy-consuming process of going through 13
+million tiles every epoch can be avoided, and as result, these models can be
+trained for more epochs. Nowadays, there is an increasing concern regarding
+energy and electricity consumption. Thus, these concerns, together with the
+sustainability goals, further support the importance of more efficient training
+processes.
+Let T be the original set of tiles, and Ts be the original set of tiles from
+the slide s, the former is composed by a union of the latter of all the slides
+(Eq. 1). We propose to map T to a smaller set of tiles M without affecting
+the overall performance and behaviour of the trained algorithm.
+12
+
+85
+annotated WsI
+annotatedtiles (512x512px)
+tiles classifier
+non-annotated WSI
+tiles set (512x512 px)
+tiles classifier
+tiles ranking
+tiles sampling
+(inference)
+(top 200)
+orw grad
+T
+sampled tiles
+tiles classifier
+tiles ranking
+tiles classifier
+slide diagnosis
+(inference)
+(training with top 5)
+(top tile prediction)T =
+S�
+s=1
+Ts
+(1)
+The model trained in a fully supervised task, previously described, pro-
+vides a good estimation of the utility of each tile.
+Hence, we utilise the
+function (Φ) learned by the model to compute the predicted severity of each
+tile. As will be shown below, the weakly-supervised method utilises only the
+five most severe tiles per slide to train in each epoch. As such, we select M
+tiles per slide (M=200 in our experimental setup) utilising a Top-k function
+(with k set to 200) to be retained for the weakly-supervised training. As in-
+dicated by the results presented in the following sections, the value of M was
+selected in accordance with a trade-off between information lost and training
+time. This is formalised in Eq. 2.
+Ms = Top-k(Φ(Ts))
+(2)
+For instance, in the CRS10K dataset, the total number of tiles after
+sampling would be at most 2,099,200, which represents a reduction of 6.46×
+when compared to the total number of slides. Despite this upper bound on
+the number of tiles, there are WSI samples that contain less than M tiles, and
+as such, they remain unsampled and the actual total number of tiles after
+sampling is potentially lower. During the evaluation and test time, there is
+no sampling.
+We conducted extensive studies on the performance of our methodology,
+without sampling, with sampling on the training data, and with sampling
+on training and validation. The results on the CRS4K dataset validate our
+proposal. The number of selected tiles considers a trade-off between compu-
+tational cost and information potentially lost, and for that reason, it is the
+success of empirical optimization.
+2.4.3. Weakly-Supervised Learning
+The weakly-supervised learning approach designed for our methodology
+follows the same principles of recent work [31]. It is divided into two distinct
+stages, tile severity analysis and training. The former utilises the pre-trained
+model to evaluate the severity of every tile in a set of tiles. In the first epoch,
+T , the set of all the tiles in the complete dataset is used. This is possible
+since the model used to assess the severity in this epoch is the same one used
+for sampling. Hence, both tasks are integrated with the initial epoch. The
+13
+
+following epochs utilise the sampled tile set M instead of the original set.
+This overall structure is represented in Figure 3.
+The link between both stages is guided by a slide-wise tile ranking ap-
+proach based on the expected severity. For tile Ts,n, the expected severity is
+defined as
+E( ˆCs,n) =
+K
+�
+i=1
+i × p
+�
+ˆCs,n = C(i)�
+(3)
+where ˆCs,n is a random variable on the set of possible class labels {C(1), · · · , C(K)}
+and p
+�
+ˆCs,n = C(i)�
+are the K output values of the neural network. After
+this analysis, the five most severe tiles are selected for training. The number
+of selected tiles was chosen in accordance with previous studies [31]. These
+five tiles per slide are used to train the proposed model for one more epoch.
+Each epoch is composed of both stages, which means that the tiles used for
+training vary across epochs. The slide label is used as the ground truth of
+all five tiles of that same slide used for network optimisation. For validation
+and evaluation, only the most severe tile is used for diagnostics. Although
+it might lead to an increase in false positives, it shall significantly reduce
+false negatives. Furthermore, we argue that increasing the variability and
+quantity of data available leads to a better balance between the reduction of
+these two types of errors.
+2.5. Confidence Interval
+In order to quantify the uncertainty of a result, it is common to compute
+the 95 percent confidence interval. In this way, two different models can
+be easily understood and compared based on the overlap of their confidence
+intervals. The standard approach to calculating these intervals requires sev-
+eral runs of a single experiment. As we increase the number of runs, our
+interval becomes narrower. However, this procedure is impractical for the
+computationally intensive experiments presented in this document. Hence,
+we use an independent test set to approximate the confidence interval as a
+Gaussian function [39]. To do so, we compute the standard error (SE) of an
+evaluation metric m, which is dependent on the number of samples (n), as
+seen in Equation 4.
+SE =
+�
+1
+n × m × (1 − m)
+(4)
+14
+
+For the SE computation to be mathematically correct, the metric m must
+originate from a set of Bernoulli trials. In other words, if each prediction is
+considered a Bernoulli trial, then the metric should classify them as correct
+or incorrect. The number of correct samples is then given by a Binomial
+distribution X ∼ (n, p), where p is the probability of correctly predicting a
+label, and n is the number of samples. For instance, the accuracy is a metric
+that fits all these constraints.
+Following the definition and the properties of a Normal distribution, we
+compute the number of standard deviations (z), known as a standard score,
+that can be translated to the desired confidence (c) set to 95% of the area
+under a normal distribution. This is a well-studied value, which is approx-
+imately z ≈ 1.96.
+This value z is then used to calculate the confidence
+interval, calculated as the product of z and SE as seen in Equation 5.
+M ± z ∗
+�
+1
+n × m × (1 − m)
+(5)
+2.6. Experimental setup
+For our experimental setup, we divide our data into training and valida-
+tion sets. Besides, we further evaluate the performance of the former in our
+test set composed of slides never seen by any of the methods presented or
+in the literature. Following the split of these three sets, we have 8587, 1009
+and 900 stratified non-overlapping samples in the training, validation and
+test set, respectively.
+In an attempt to also contribute to reproducible research, the training of
+all the versions of the proposed algorithm uses the deterministic constraints
+available on Pytorch. The usage of deterministic constraints implies a trade-
+off between performance, either in terms of algorithmic efficiency or on its
+predictive power, and the complement with reproducible research guidelines.
+As such, due to the current progress in the field, we have chosen to comply
+with the reproducible research guidelines.
+All the trained backbone networks were ResNet-34 [40]. Pytorch was used
+to train these networks with the Adaptive Moment Estimation (Adam) [41]
+optimiser, a learning rate of 6 × 10−6 and a weight decay of 3 × 10−4. The
+training batch size was set to 32 for both fully and weakly supervised train-
+ing, while the test and inference batch size was 256. The performance of the
+model was evaluated on the validation set used for model selection in terms
+of the best accuracies and quadratic weighted kappa (QWK). The training
+15
+
+was accelerated by an Nvidia Tesla V100 (32GB) GPU for 50 epochs of both
+weakly and fully supervised learning. In addition to the proposed method-
+ology, we extended our experiments to include the aggregation approach
+proposed by Neto et al. [31] on top of our best-performing method.
+2.7. Label correction
+The complex process of labelling thousands of whole-slide images with
+colorectal cancer diagnostic grades is a task of increased difficulty. It should
+also be noted and taken into account that grading colorectal dysplasia is hur-
+dled by considerable subjectivity, so it is to be expected that some borderline
+cases will be classified by some pathologists as low-grade and others as high-
+grade. Moreover, as the number of cases increases, it becomes increasingly
+difficult to maintain perfect criteria and avoid mislabelling. For this reason,
+we have extended the analysis of the model’s performance to understand its
+errors and its capability to detect mislabelled slides.
+After training the proposed model, it was evaluated on the test data. Fol-
+lowing this evaluation, we identified the misclassified slides and conducted
+a second round of labelling. These cases were all blindly reviewed by two
+pathologists, and discordant cases from the initial ground truth were dis-
+cussed and classified by both pathologists (and in case of doubt/complexity,
+a third pathologist was also consulted). We tried to maintain similar criteria
+between the graders and always followed the same guidelines. These new
+labels were used to rectify the performance of all the algorithms evaluated in
+the test set. We argue that the information regarding the strength/confidence
+of predictions of a model used as a second opinion is of utter importance. A
+correct integration of this feature can be shown as extremely insightful for
+the pathologists using the developed tool.
+2.8. Prototype and Interpretability Assessment
+The proposed algorithm was integrated into a fully functional prototype
+to enable its use and validation in a real clinical workflow.
+This system
+was developed as a server-side web application that can be accessed by any
+pathologist in the lab. The system supports the evaluation of either a sin-
+gle slide or a batch of slides simultaneously and in real time. It also caches
+the most recent results, allowing re-evaluation without the need to re-upload
+slides. In addition to displaying the slide diagnosis, and confidence level for
+each class, a visual explanation map is also retrieved, to draw the patholo-
+gist’s attention to key tissue areas within each slide (all seen in Figure 4).
+16
+
+The opaqueness of the map can be set to different thresholds, allowing the
+pathologist to control its overlay over the tissue. An example of the zoomed
+version of a slide with lower overlay of the map is shown in Figure 5.
+Figure 4: Main view of the CAD system prototype for CRS: Slide identification, confidence
+per class, diagnostic, mask overlay controller, results download as csv and slide search are
+some of the features visible. Slide identification is anonymised.
+Furthermore, the prototype also allows user feedback where the user can
+accept/reject a result and provide a justification (Figure 6), an important
+feature for software updates, research development and possible active learn-
+ing frameworks that can be developed in the future. These results can be
+downloaded with the corrected labels to allow for further retraining of the
+model.
+There are several advantages to developing such a system as a server-
+side web application. First, it does not require any specific installation or
+dedicated local storage in the user’s device. Secondly, it can be accessed at
+the same time by several pathologists from different locations, allowing for
+a quick review of a case by multiple pathologists without data transference.
+Moreover, the lack of local storage of clinical data increases the privacy of
+patient data, which can only be accessed through a highly encrypted virtual
+private network (VPN). Finally, all the processing can be moved to an effi-
+cient graphics processing unit (GPU), thus reducing the processing time by
+17
+
+ CADPath
++
+1
+人
+CADPath
+ Search
+Search
+Download Results
+00000000001-A-
+OOOO
+01-001_xxx_000
+000001.svs
+processed
+Processed
+00000000002-A-
+Case: 00000000003
+02-002_xxx_000
+Specimen: A
+_000002.svs
+Block: 03
+ processed
+Slide: 003
+Mask
+00000000003-A
+03-003_xxx_000
+_000003.svs
+Model Results
+processed
+High grade
+100%
+Low grade
+00000000004-A-
+Normal
+04-004_xxx_000
+_000004.svs
+processed
+High Grade
+00000000005-A-
+Results approved
+05-005_xxx_000
+_000005.svs
+processed
+00000000006-A.
+06-006_xxx_000
+_000006.svs
+Upload new slidesFigure 5: Zoomed view of a slide from the CAD system prototype, with the predictions
+map with a small overlay threshold. Slide identification is anonymised.
+several orders of magnitude. Similar behaviour on a local machine would
+require the installation of dedicated GPUs in the pathologists’ personal de-
+vices. This platform is the first Pathology prototype for colorectal diagnosis
+developed in Portugal, and, as far as we know, one of the pioneers in the
+world. Its design was also carefully thought to be aligned with the needs of
+the pathologists.
+3. Results
+In this section, we present the results of the proposed method. The results
+are organised to first demonstrate the effectiveness of sampling, followed by
+an evaluation of the model in the two internal datasets (CRS10K and the
+prototype dataset), and finalise with the results on external datasets. We also
+discuss the advantages and disadvantages of the proposed approach, perform
+an analysis of the results from a clinical perspective, provide pathologists’
+feedback on the use of the prototype, and finally discuss future directions.
+18
+
+ CADPath
++
+人☆
+CADPath
+Search
+Search
+Download Results
+processed
+00000000006-A
+06-006_xxx_000
+Upload new slides
++0:00000000Figure 6: CAD system prototype report tool: the user can report if the result is either
+correct, wrong or inconclusive and leave a comment for each case individually.
+Slide
+identification is anonymised.
+3.1. On the effectiveness of sampling
+To find the most suitable threshold for sampling the tiles used in the
+weakly supervised training, we evaluated the percentage of relevant tiles that
+would be left out of the selection, if the original set was reduced to 75, 100,
+150 or 200 tiles, over the first five inference epochs.
+A tile is considered
+relevant if it shares the same label as the slide, or if it would take part in
+the learning process in the weakly-supervised stage. As it is possible to see
+in Figure 7, if we set the maximum number of tiles to 200 after the second
+loop of inference, we would discard only 3.5% of the potentially informative
+tiles, in the worst-case scenario. On the other side of the spectrum, a more
+radical sampling of only 50 tiles would lead to a cut of up to 8%.
+Moreover, to assess the impact of this sampling on the model’s perfor-
+mance, we also evaluated the accuracy and the QWK with and without
+sampling the top 200 tiles after the first inference iteration (Table 2). This
+evaluation considered sampling applied only to the training tile set, and to
+both the training and validation tile sets. As can be noticed, the performance
+is not degraded and the model is trained in a much faster way. In fact, using
+the setup previously mentioned, the first epoch of inference, with the full set
+19
+
+CADPath
+x
+人☆
+CADPath
+ Search
+ Download Results
+ownload
+Feedback
+Approval State:
+00000000001-A-
+01-001_xxx_000
+000001.svs
+Expected:
+(Low Grade 
+ Normal
+High Grade
+processed
+Processed
+Comments
+00000000002-A-
+Case: 00000000002
+02-002_xxx_000
+Specimen: A
+000002.svS
+Block: 02
+processed
+Slide: 002
+Mask
+00000000003-A-
+03-003_xxx_000
+000003.SvS
+Model Results
+processed
+80%
+Low grade
+20%
+00000000004-A-
+ Save changes 
+Normal
+04-004_xxx_000
+000004.svs
+High Grade
+processed
+00000000005-A.
+Results rejected!
+05-005_xxx_000
+000005.SVS
+processed
+O
+00000000006-A-
+06-006_xxx_000
+000006.svs
+Upload new slidesFigure 7: Tile sampling impact on information loss: percentage of tiles not selected due
+to sampling with different thresholds, over the first four inference epochs.
+of tiles takes 28h to be completed, while from the second loop the training
+time decreases to only 5h per epoch. Without sampling, training the model
+for 50 epochs would take around 50 days, whereas with sampling it takes
+around 10.
+3.2. CRS10K and Prototype
+CRS10K test set and the prototype dataset were collected through dif-
+ferent procedures. The first followed the same data collection process as the
+complete dataset, whereas the second originated from routine samples. Thus,
+the evaluation of both these sets is done separately.
+The first experiment was conducted on the CRS10K test set.
+As ex-
+pected, the steep increase in the number of training samples led to a signif-
+icantly better algorithm in this test set. Initially, the model trained on the
+CRS10K correctly predicted the class of 819 out of 900 samples. For the
+20
+
+8
+Info
+Samplingof 20o tiles
+sampling
+Sampling of 1oo tiles
+selected
+Sampling of 75 tiles
+Sampling of 50 tiles
+01
+due
+tiles
+not selected
+of
+total number
+4
+3
+tiles
+2
+1
+0
+1
+2
+3
+4
+5
+EpochsTable 2: Model performance comparison with and without tile sampling of the top 200
+tiles from the first inference iteration. Compared the best epoch of the initial five epochs
+and of the initial ten epochs. Validation is represented as Val.
+Best Accuracy at
+Best QWK at
+Sampling
+5th epoch
+10th epoch
+5th epoch
+10th epoch
+No
+84.94% ± 2.20
+86.42% ± 2.11
+0.809 ± 0.024
+0.829 ± 0.023
+Train
+85.43% ± 2.18
+86.82% ± 2.08
+0.817 ± 0.024
+0.828 ± 0.023
+Train and Val.
+86.12% ± 2.13
+86.92% ± 2.08
+0.824 ± 0.023
+0.829 ± 0.023
+Table 3: Model performance evaluation on the CRS10K test set. The binary accuracy is
+calculated as NNeo vs all. Accuracy is represented as (ACC). In bold are the best results
+per column.
+Method
+ACC
+Binary ACC
+Sensitivity
+iMIL4Path
+91.33% ± 1.84
+97.00% ± 1.11
+0.997 ± 0.004
+Ours (CRS4K)
+89.44% ± 2.01
+96.11% ± 1.26
+0.997 ± 0.004
+Ours (CRS10K) wo/ Agg
+93.44% ± 1.62
+97.78% ± 0.96
+0.996 ± 0.005
+Ours (CRS10K) w/ Agg
+90.67% ± 1.90
+97.55% ± 1.01
+0.985 ± 0.009
+wrong 81 cases, the pathologists performed a blind review of these cases and
+found that the algorithm was indeed correct in 22 of them. This led to a
+correction in the labels of the test set, and the appropriate adjustment of
+the metrics. In Table 3, the performance of the different algorithms is pre-
+sented. CRS10K outperforms the other approaches by a reasonable margin.
+We further applied the aggregation proposed by Neto et al. [31] to the best
+performing method, but without gains in performance. Despite being trained
+on the same dataset iMIL4Path and the proposed methodology trained on
+CRS4K, utilise different splits for training and validation, as well as different
+optimisation techniques due to the deterministic approach.
+In addition to examining quantitative metrics, such as the accuracy of
+the model, we extended our study to include an analysis of the confidence in
+the model when it correctly predicts a class and when it makes an incorrect
+prediction. To this end, we recorded the confidence of the model for the
+predicted class and divided it into the set of correct and incorrect predictions.
+These were then used to fit a kernel density estimator (KDE). Figure 8 shows
+the density estimation of the confidence values for the three different models.
+It is worth noting that, when correct, the model trained on the CRS10K,
+21
+
+Figure 8: Kernel density estimation of the confidences of correct and incorrect predic-
+tions performed on the three-class classification problem by three distinct models on the
+CRS10K test set. The plots represent, from left to right, the proposed method trained on
+CRS10K, the proposed method trained on CRS4K and iMIL4Path.
+returns higher confidence levels as shown by the shift of its mean towards
+values close to one. On the other hand, the confidence values of its incorrect
+predictions decrease significantly, and although it does not present the lowest
+values, it shows the largest gap between correct and incorrect means.
+Table 4: Model performance evaluation on the prototype test set. Accuracy is represented
+as (ACC). The binary accuracy is calculated as NNeo vs all. In bold are the best results
+per column.
+Method
+ACC
+Binary ACC
+Sensitivity
+iMIL4Path
+89.00% ± 6.13
+96.00% ± 3.84
+1.000 ± 0.000
+Ours (CRS4K)
+85.00% ± 6.99
+93.00% ± 5.00
+1.000 ± 0.000
+Ours (CRS10K) wo/ Agg
+89.00% ± 6.13
+98.00% ± 2.74
+0.986 ± 0.026
+Ours (CRS10K) w/ Agg
+85.00% ± 6.99
+98.00% ± 2.74
+0.986 ± 0.026
+When tested on the prototype data (n=100), the importance of a higher
+volume of data remains visible (Table 4). Nonetheless, the performance of
+iMIL4Path [31] approach is comparable to the proposed approach trained on
+CRS10K. It is worth noting that the latter achieves better performance on
+the binary accuracy at the cost of a decrease in sensibility. In other words,
+the capability to detect negatives increases significantly. Due to the smaller
+22
+
+CRS10KTest Set-Ours(CRS10K)
+CRS10KTestSet-Ours(CRS4K)
+CRS10K Test Set - iMIL4Path
+8
+8
+8
+Correct
+Correct
+Correct
+Mean = 0.961
+Mean = 0.953
+.
+Mean = 0.956
+7
+Incorrect
+7
+Incorrect
+7
+Incorrect
+Mean = 0.769
+Mean = 0.762
+Mean = 0.773
+6
+6
+6
+5
+5
+5
+/p(c)
+/p(c)
+........
+3
+3
+3
+2
+2 
+2
+1
+1
+1
+0
+0
+0
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Confidencec
+Confidencec
+Confidence cset of slides, the confidence interval is much wider, as such, the performance
+on the CRS10K test set is a good indication of how these values would shift
+if more data was added. Similar performance drops were linked with the
+introduction of aggregation.
+Figure 9: Kernel density estimation of the confidences of correct and incorrect predictions
+performed on the three-class classification problem by three distinct models on the proto-
+type set. The plots represent, from left to right, the proposed method trained on CRS10K,
+the proposed method trained on CRS4K and iMIL4Path.
+Despite similar results, the confidence of the model in its predictions is
+distinct in all three approaches, as seen in Figure 9. The proposed approach
+when trained on the CRS10K dataset has a larger density on values close
+to one when the predictions are correct, and the mean confidence of those
+predictions is, once more, higher than the other approaches. However, espe-
+cially when compared to the proposed approach trained on the CRS4K, the
+confidence of wrong predictions is also higher. It can be a result of a larger
+set of wrong predictions available on the latter approach. Nonetheless, the
+steep increase in the density of values closer to one further indicates that
+there is room to explore other effects of extending the number of training
+samples, besides benefits in quantitative metrics.
+3.3. Domain Generalisation Evaluation
+To ensure the generalisation of the proposed approach across external
+datasets, we have evaluated their performance on TCGA and PAIP. More-
+23
+
+Prototype-Ours(CRS10K)
+Prototype-Ours(CRS4K)
+Prototype - iMIL4Path
+8
+8
+8
+Correct
+Correct
+Correct
+Mean = 0.943
+Mean = 0.923
+Mean = 0.916
+7
+Incorrect
+7
+Incorrect
+7
+Incorrect
+....
+Mean = 0.84
+Mean = 0.757
+Mean =0.818
+9
+6
+6
+5
+5
+5 
+p(c)
+/ p(c)
+3
+3
+3
+2
+2 
+2
+1
+1
+1
+0
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Confidencec
+Confidencec
+Confidence cover, we conducted a similar analysis of both of these datasets, as the one
+done for the internal datasets.
+Table 5: Model performance evaluation on the PAIP test set. The binary accuracy is
+calculated as NNeo vs all. Accuracy is represented as (ACC). In bold are the best results
+per column.
+Method
+ACC
+Binary ACC
+Sensitivity
+iMIL4Path
+99.00% ± 1.95
+100.00% ± 0.00
+1.000 ± 0.000
+Ours (CRS4K)
+69.00% ± 9.06
+100.00% ± 0.00
+1.000 ± 0.000
+Ours (CRS10K) wo/ Agg
+100.00% ± 0.00
+100.00% ± 0.00
+1.000 ± 0.000
+Ours (CRS10K) w/ Agg
+52.00 ± 9.79
+100.00% ± 0.00
+1.000 ± 0.000
+From the two datasets, PAIP is arguably the closest to CRS10K. It con-
+tains similar tissue, despite its colour differences. The performances of the
+proposed approaches were expected to match the performance of iMIL4Path
+in this dataset. However, it did not happen for the version trained on the
+CRS4K dataset, as seen in Table 5. A viable explanation concerns potential
+overfitting to the training data potentiated by an increase in the number of
+epochs of fully and weakly supervised training, a slight decrease in the tile
+variability in the latter approach, and a smaller number of samples when
+compared to the version trained on CRS10K. This version, trained on the
+larger set, mitigates the problems of the other method due to a significant
+increase in the training samples. Moreover, it is worth noting that in all
+three approaches, the errors corresponded only to a divergence between low
+and high-grade cases, with no non-neoplastic cases being classified as high-
+grade or vice-versa. As in previous sets, the version trained on the CRS10K
+dataset outperforms the remaining approaches. Using aggregation in this
+dataset leads to a discriminative power to distinguish between high- and
+low-grade lesions that is close to random.
+In two of the three approaches, the number of incorrect samples is one
+or zero, as such, there is no density estimation for wrong samples in their
+confidence plot as seen in Figure 10. Yet, it is visible the shift towards higher
+values of confidence in the proposed approach trained on the CRS10K when
+compared to the method of iMIL4Path.
+The version trained on CRS4K
+shows very little separability between the confidence of correct and incorrect
+predictions.
+The TCGA dataset has established itself as the most challenging for the
+proposed approaches. Besides the expected differences in colour and other
+24
+
+Figure 10: Kernel density estimation of the confidences of correct and incorrect predictions
+performed on the three-class classification problem by three distinct models on the PAIP
+dataset. The plots represent, from left to right, the proposed method trained on CRS10K,
+the proposed method trained on CRS4K and iMIL4Path.
+Table 6: Model performance evaluation on the TCGA test set. The binary accuracy is
+calculated as NNeo vs all. Accuracy is represented as (ACC). In bold are the best results
+per column.
+Method
+ACC
+Binary ACC
+Sensitivity
+iMIL4Path
+71.55% ± 5.80
+80.60% ± 5.05
+0.805 ± 0.051
+Ours (CRS4K) wo/ Agg
+70.69% ± 5.86
+98.71% ± 1.45
+0.991 ± 0.012
+Ours (CRS10K) wo/ Agg
+84.91% ± 4.61
+99.13% ± 1.19
+0.996 ± 0.008
+Ours (CRS10K) w/ Agg
+69.83% ± 5.91
+97.41% ± 2.04
+0.983 ± 0.017
+elements, this dataset is mostly composed of resection samples, which are not
+present in the training dataset. As such, this presents itself as an excellent
+dataset to assess the capability of the model to handle these different types of
+samples. Both iMIL4Path and the proposed method trained on CRS4K have
+shown substantial problems in correctly classifying TCGA slides, as shown
+in Table 6. Despite having a lower performance on the general accuracy,
+the binary accuracy shows that our proposed method trained on CRS4K has
+much lower misclassification errors regarding the classification of high-grade
+samples as normal, demonstrating higher robustness of the new training ap-
+proach against errors with a gap of two classes. As with other datasets, the
+proposed approach trained on CRS10K shows better results, this time by a
+25
+
+PAIP -Ours(CRS10K)
+PAIP - Ours(CRS4K)
+PAIP - iMIL4Path
+8
+8
+8
+Correct
+Correct
+Correct
+Mean = 0.99
+Mean = 0.843
+Mean = 0.964
+7
+7
+Incorrect
+7
+Mean =0.835
+9
+6
+6
+5
+5
+5
+/ p(c)
+p(c)
+3
+3
+3
+2
+2
+2
+1
+1
+1
+0
+:
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Confidencec
+Confidencec
+Confidence csignificant margin with no overlapping between the confidence intervals.
+Figure 11: Kernel density estimation of the confidences of correct and incorrect predictions
+performed on the three-class classification problem by three distinct models on the TCGA
+dataset. The plots represent, from left to right, the proposed method trained on CRS10K,
+the proposed method trained on CRS4K and iMIL4Path.
+Inspecting the predictions’ confidence for the three models indicates a be-
+haviour in line with the accuracy-based performance (Figure 11). Moreover,
+a confidence shift of wrong predictions’ confidence towards smaller values is
+clearly visible in the plot corresponding to the model trained on CRS10K.
+The shown gap of 0.2 between the confidence of correct and wrong predic-
+tions, indicates that it is possible to quantify the uncertainty of the model
+and avoid the majority of the wrong predictions. In other words, when the
+uncertainty is above a learnt threshold, then the model refuses to make any
+prediction. It is extremely useful in models designed as a second opinion
+system.
+3.4. Prototype usability in clinical practice
+As it is currently designed, the algorithm works preferentially as a “second
+opinion”, allowing the assessment of difficult and troublesome cases, without
+the immediate need for the intervention of a second pathologist. Due to its
+“user-friendly” nature and very practical interface, the cases can be easily
+introduced into the system and results are rapidly shown and easily accessed.
+Also, by not only providing results but presenting visualisation maps (cor-
+responding to each diagnostic class), the pathologist is able to compare his
+26
+
+TCGA-Ours(CRS10K)
+TCGA-Ours(CRS4K)
+TCGA- iMIL4Path
+8
+8
+8
+Correct
+Correct
+Correct
+Mean=0.965
+Mean=0.956
+.
+Mean=0.932
+7
+Incorrect
+7
+Incorrect
+7
+Incorrect
+Mean = 0.764
+Mean = 0.876
+Mean = 0.815
+6
+6
+6
+5
+5
+5
+p(c)
+p(c)
+3
+3
+3
+2
+2 
+2
+1
+1
+1
+0
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+Confidencec
+Confidencec
+Confidencecown remarks to those of the algorithm itself, towards a future “AI-assisted
+diagnosis”. Another relevant aspect is the fact that the prototype allows for
+user feedback (agreeing or not with the model’s proposed result), which can
+be further integrated into further updates of the software. Also interesting,
+is the possibility of using the prototype as a triage system on a pathologist’s
+daily workflow (by running front, before the pathologist checks the cases).
+Signalling the cases that would need to be more urgently observed (namely
+high-risk lesions) would allow the pathologists to prioritise their workflow.
+Further, by providing a previous assessment of the cases, it would also con-
+tribute to enhancing the pathologists’ efficiency. Although it is possible to
+use the model as it is upfront, it would classify some samples incorrectly
+(since it was not trained on the full spectrum of colorectal pathology). As
+such, the uncertainty quantification based on the provided confidence given
+in the user interface could also be extremely useful. Presently, there is no rec-
+ommendation for dual independent diagnosis of colorectal biopsies (contrary
+to gastric biopsies, where, in cases in which surgical treatment is considered,
+it is recommended to obtain a pre-treatment diagnostic second opinion [42]),
+but, in case that in the future this also becomes a requirement, a tool such as
+CADPath.AI prototype could assist in this task. This has increased impor-
+tance due to the worldwide shortage of pathologists and so, such CAD tools
+can really make a difference in patient care (in similarity, for example, with
+Google Health’s research, using deep learning to screen diabetic retinopa-
+thy in low/middle-income countries, in which the system showed real-time
+retinopathy detection capability similar to retina specialists, alleviating the
+significant manpower constrictions in this setting [43]). Lastly, we also an-
+ticipate that this prototype, and similar tools, can be used in a teaching
+environment since its easy use and explainable capability (through the visu-
+alisation maps) allows for easy understanding of the given classifications and
+having a web-based interface allows for easy sharing.
+3.5. Future work
+The proposed algorithm still has potential for improvement.
+We aim
+to include the recognition of serrated lesions, to distinguish normal mucosa
+from significant inflammatory alterations/diseases, to stratify high-risk le-
+sions into high-grade dysplasia and invasive carcinomas and to identify other
+neoplasia subtypes. Further, we would like to leverage the model to be able
+to evaluate also surgical specimens. Another relevant step will be the merge
+of our dataset and external ones for training, besides only testing it on ex-
+27
+
+ternal samples. This will enhance its generalisation capabilities and provide
+a more robust system. Lastly, we intend to measure the “user experience”
+and feedback from the pathologists, by its gradual implementation in general
+laboratory routine work.
+The following goals comprise a more extensive evaluation of the model
+across more scanner brands and labs.
+We also want to promote certain
+behaviours that would allow for more direct and integrated uncertainty esti-
+mation. We have also been looking towards aggregation methods, but, since
+in the majority of them there is an increased risk of false negatives, we have
+work to do in that research direction.
+4. Discussion
+In this document, we have redesigned the previous methodology on MIL
+for colorectal cancer diagnosis. First, we extended and leveraged the mixed
+supervision approach to design a sampling strategy, which utilises the knowl-
+edge from the full supervision training as a proxy to tile utility. Secondly, we
+studied the confidence that the model shows in its predictions when they are
+correct and when they are incorrect. Additionally, this confidence is shown to
+be a potential resource to quantify uncertainty and avoid wrong predictions
+on low-certainty scenarios. This is entirely integrated within a web-based
+prototype to aid pathologists in their routine work.
+The proposed methodology was evaluated on several datasets, including
+two external sets. Through this evaluation, it was possible to infer that the
+performance of the proposed methodology benefits from a larger dataset and
+surpasses the performance of previous state-of-the-art models. As such, and
+given the excelling results that originated from the increase in the dataset,
+we are also publicly releasing the majority of the CRS10K dataset, one of
+the largest publicly available colorectal datasets composed of H&E images in
+the literature, including the test set for the benchmark of distinct approaches
+across the literature.
+Finally, we have clearly defined a set of potential future directions to be
+explored, either for better model design, the development of useful prototypes
+or even the integration of uncertainty in the predictions.
+References
+[1] International Agency for Research on Cancer (IARC), Global cancer
+observatory, https://gco.iarc.fr/ (2022).
+28
+
+[2] H. Brody, Colorectal cancer, Nature 521 (2015) S1.
+doi:10.1038/
+521S1a.
+[3] D. Holmes, A disease of growth, Nature 521 (2015) S2–S3. doi:10.
+1038/521S2a.
+[4] Digestive Cancers Europe (DiCE), Colorectal screening in europe,
+https://bit.ly/3rFxSEL.
+[5] C. Hassan, G. Antonelli, J.-M. Dumonceau, J. Regula, M. Bretthauer,
+S. Chaussade, E. Dekker, M. Ferlitsch, A. Gimeno-Garcia, R. Jover,
+M. Kalager, M. Pellis´e, C. Pox, L. Ricciardiello, M. Rutter, L. M.
+Helsingen, A. Bleijenberg, C. Senore, J. E. van Hooft, M. Dinis-Ribeiro,
+E. Quintero, Post-polypectomy colonoscopy surveillance: European so-
+ciety of gastrointestinal endoscopy guideline - update 2020, Endoscopy
+52 (8) (2020) 687–700. doi:10.1055/a-1185-3109.
+[6] D. Mahajan, E. Downs-Kelly, X. Liu, R. Pai, D. Patil, L. Rybicki,
+A. Bennett, T. Plesec, O. Cummings, D. Rex, J. Goldblum, Repro-
+ducibility of the villous component and high-grade dysplasia in col-
+orectal adenomas <1 cm:
+Implications for endoscopic surveillance,
+American Journal of Surgical Pathology 37 (3) (2013) 427–433. doi:
+10.1097/PAS.0b013e31826cf50f.
+[7] S. Gupta, D. Lieberman, J. C. Anderson, C. A. Burke, J. A. Dominitz,
+T. Kaltenbach, D. J. Robertson, A. Shaukat, S. Syngal, D. K. Rex,
+Recommendations for follow-up after colonoscopy and polypectomy: A
+consensus update by the us multi-society task force on colorectal cancer,
+Gastrointestinal Endoscopy (2020). doi:10.1016/j.gie.2020.01.014.
+[8] C. Eloy, J. Vale, M. Curado, A. Pol´onia, S. Campelos, A. Caramelo,
+R. Sousa, M. Sobrinho-Sim˜oes, Digital pathology workflow imple-
+mentation at ipatimup, Diagnostics 11 (11) (2021).
+doi:10.3390/
+diagnostics11112111.
+URL https://www.mdpi.com/2075-4418/11/11/2111
+[9] F. Fraggetta, A. Caputo, R. Guglielmino, M. G. Pellegrino, G. Runza,
+V. L’Imperio, A survival guide for the rapid transition to a fully digital
+workflow: The “caltagirone example”, Diagnostics 11 (10) (2021). doi:
+29
+
+10.3390/diagnostics11101916.
+URL https://www.mdpi.com/2075-4418/11/10/1916
+[10] D. Montezuma, A. Monteiro, J. Fraga, L. Ribeiro, S. Gon¸calves,
+A. Tavares, J. Monteiro, I. Macedo-Pinto, Digital pathology implemen-
+tation in private practice: Specific challenges and opportunities, Diag-
+nostics 12 (2) (2022) 529.
+[11] A. Madabhushi, G. Lee, Image analysis and machine learning in digital
+pathology: challenges and opportunities, Medical Image Analysis 33
+(2016) 170–175. doi:10.1016/j.media.2016.06.037.
+[12] E. A. Rakha, M. Toss, S. Shiino, P. Gamble, R. Jaroensri, C. H. Mermel,
+P.-H. C. Chen, Current and future applications of artificial intelligence in
+pathology: a clinical perspective, Journal of Clinical Pathology (2020).
+doi:10.1136/jclinpath-2020-206908.
+[13] M. Veta, P. J. van Diest, S. M. Willems, H. Wang, A. Madabhushi,
+A. Cruz-Roa, F. Gonzalez, A. B. Larsen, J. S. Vestergaard, A. B. Dahl,
+D. C. Cire¸san, J. Schmidhuber, A. Giusti, L. M. Gambardella, F. B.
+Tek, T. Walter, C.-W. Wang, S. Kondo, B. J. Matuszewski, F. Precioso,
+V. Snell, J. Kittler, T. E. de Campos, A. M. Khan, N. M. Rajpoot,
+E. Arkoumani, M. M. Lacle, M. A. Viergever, J. P. Pluim, Assessment of
+algorithms for mitosis detection in breast cancer histopathology images,
+Medical Image Analysis 20 (1) (2015) 237–248. doi:10.1016/j.media.
+2014.11.010.
+[14] G. Campanella, M. G. Hanna, L. Geneslaw, A. Miraflor, V. Werneck
+Krauss Silva, K. J. Busam, E. Brogi, V. E. Reuter, D. S. Klimstra, T. J.
+Fuchs, Clinical-grade computational pathology using weakly supervised
+deep learning on whole slide images, Nat. Med. 25 (8) (2019) 1301–1309.
+doi:10.1038/s41591-019-0508-1.
+[15] S. P. Oliveira, J. Ribeiro Pinto, T. Gon¸calves, R. Canas-Marques, M.-
+J. Cardoso, H. P. Oliveira, J. S. Cardoso, Weakly-supervised classifi-
+cation of HER2 expression in breast cancer haematoxylin and eosin
+stained slides, Applied Sciences 10 (14) (2020) 4728.
+doi:10.3390/
+app10144728.
+30
+
+[16] T. Albuquerque, A. Moreira, J. S. Cardoso, Deep ordinal focus assess-
+ment for whole slide images, in: Proceedings of the IEEE/CVF Inter-
+national Conference on Computer Vision, 2021, pp. 657–663.
+[17] S. P. Oliveira, P. C. Neto, J. Fraga, D. Montezuma, A. Monteiro,
+J. Monteiro, L. Ribeiro, S. Gon¸calves, I. M. Pinto, J. S. Cardoso, CAD
+systems for colorectal cancer from WSI are still not ready for clini-
+cal acceptance, Scientific Reports 11 (1) (2021) 14358. doi:10.1038/
+s41598-021-93746-z.
+[18] N. Thakur, H. Yoon, Y. Chong, Current trends of artificial intelligence
+for colorectal cancer pathology image analysis: a systematic review,
+Cancers 12 (7) (2020). doi:10.3390/cancers12071884.
+[19] Y. Wang, X. He, H. Nie, J. Zhou, P. Cao, C. Ou, Application of artificial
+intelligence to the diagnosis and therapy of colorectal cancer, Am. J.
+Cancer Res. 10 (11) (2020) 3575–3598.
+[20] A. Davri, E. Birbas, T. Kanavos, G. Ntritsos, N. Giannakeas, A. T.
+Tzallas, A. Batistatou, Deep learning on histopathological images for
+colorectal cancer diagnosis: A systematic review, Diagnostics 12 (4)
+(2022) 837.
+[21] O. Iizuka, F. Kanavati, K. Kato, M. Rambeau, K. Arihiro, M. Tsuneki,
+Deep learning models for histopathological classification of gastric and
+colonic epithelial tumours, Scientific Rep. 10 (2020).
+doi:10.1038/
+s41598-020-58467-9.
+[22] H. Tizhoosh, L. Pantanowitz, Artificial intelligence and digital pathol-
+ogy:
+challenges and opportunities, J. Pathol. Inform 9 (1) (2018).
+doi:10.4103/jpi.jpi\_53\_18.
+[23] J. W. Wei, A. A. Suriawinata, L. J. Vaickus, B. Ren, X. Liu, M. Lisovsky,
+N. Tomita, B. Abdollahi, A. S. Kim, D. C. Snover, J. A. Baron, E. L.
+Barry, S. Hassanpour, Evaluation of a deep neural network for auto-
+mated classification of colorectal polyps on histopathologic slides, JAMA
+Network Open 3 (4) (2020).
+doi:10.1001/jamanetworkopen.2020.
+3398.
+[24] Z. Song, C. Yu, S. Zou, W. Wang, Y. Huang, X. Ding, J. Liu, L. Shao,
+J. Yuan, X. Gou, W. Jin, Z. Wang, X. Chen, H. Chen, C. Liu, G. Xu,
+31
+
+Z. Sun, C. Ku, Y. Zhang, X. Dong, S. Wang, W. Xu, N. Lv, H. Shi,
+Automatic deep learning-based colorectal adenoma detection system and
+its similarities with pathologists, BMJ Open 10 (9) (2020). doi:10.
+1136/bmjopen-2019-036423.
+[25] L. Xu, B. Walker, P.-I. Liang, Y. Tong, C. Xu, Y. Su, A. Karsan, Col-
+orectal cancer detection based on deep learning, J. Pathol. Inf. 11 (1)
+(2020). doi:10.4103/jpi.jpi\_68\_19.
+[26] K.-S. Wang, G. Yu, C. Xu, X.-H. Meng, J. Zhou, C. Zheng, Z. Deng,
+L. Shang, R. Liu, S. Su, et al., Accurate diagnosis of colorectal can-
+cer based on histopathology images using artificial intelligence, BMC
+medicine 19 (1) (2021) 1–12. doi:10.1186/s12916-021-01942-5.
+[27] G. Yu, K. Sun, C. Xu, X.-H. Shi, C. Wu, T. Xie, R.-Q. Meng, X.-H.
+Meng, K.-S. Wang, H.-M. Xiao, et al., Accurate recognition of colorec-
+tal cancer with semi-supervised deep learning on pathological images,
+Nature communications 12 (1) (2021) 1–13.
+[28] N. Marini, S. Ot´alora, F. Ciompi, G. Silvello, S. Marchesin, S. Vatrano,
+G. Buttafuoco, M. Atzori, H. M¨uller, Multi-scale task multiple instance
+learning for the classification of digital pathology images with global
+annotations, in: M. Atzori, N. Burlutskiy, F. Ciompi, Z. Li, F. Minhas,
+H. M¨uller, T. Peng, N. Rajpoot, B. Torben-Nielsen, J. van der Laak,
+M. Veta, Y. Yuan, I. Zlobec (Eds.), Proceedings of the MICCAI Work-
+shop on Computational Pathology, Vol. 156 of Proceedings of Machine
+Learning Research, PMLR, 2021, pp. 170–181.
+[29] C. Ho, Z. Zhao, X. F. Chen, J. Sauer, S. A. Saraf, R. Jialdasani,
+K. Taghipour, A. Sathe, L.-Y. Khor, K.-H. Lim, et al., A promising
+deep learning-assistive algorithm for histopathological screening of col-
+orectal cancer, Scientific Reports 12 (1) (2022) 1–9.
+[30] T. Albuquerque, A. Moreira, B. Barros, D. Montezuma, S. P. Oliveira,
+P. C. Neto, J. Monteiro, L. Ribeiro, S. Gon¸calves, A. Monteiro, I. M.
+Pinto, J. S. Cardoso, Quality control in digital pathology: Automatic
+fragment detection and counting, in: 2022 44th Annual International
+Conference of the IEEE Engineering in Medicine & Biology Society
+(EMBC), 2022, pp. 588–593. doi:10.1109/EMBC48229.2022.9871208.
+32
+
+[31] P. C. Neto, S. P. Oliveira, D. Montezuma, J. Fraga, A. Monteiro,
+L. Ribeiro, S. Gon¸calves, I. M. Pinto, J. S. Cardoso, imil4path: A
+semi-supervised interpretable approach for colorectal whole-slide images,
+Cancers 14 (10) (2022) 2489.
+[32] Pathcore, Sedeen viewer, https://pathcore.com/sedeen (2020).
+[33] P. A. Platform, Paip, http://www.wisepaip.org, last accessed on
+20/01/22 (2020).
+[34] K. Clark, B. Vendt, K. Smith, J. Freymann, J. Kirby, P. Koppel,
+S. Moore, S. Phillips, D. Maffitt, M. Pringle, L. Tarbox, F. Prior, The
+cancer imaging archive (TCIA): Maintaining and operating a public in-
+formation repository, Journal of Digital Imaging 26 (2013) 1045–1057.
+doi:10.1007/s10278-013-9622-7.
+[35] S. Kirk, Y. Lee, C. A. Sadow, S. Levine, C. Roche, E. Bonaccio, J. Fil-
+iippini, Radiology data from the cancer genome atlas colon adenocarci-
+noma [TCGA-COAD] collection. (2016). doi:10.7937/K9/TCIA.2016.
+HJJHBOXZ.
+[36] S. Kirk, Y. Lee, C. A. Sadow, S. Levine, Radiology data from the cancer
+genome atlas rectum adenocarcinoma [TCGA-READ] collection. (2016).
+doi:10.7937/K9/TCIA.2016.F7PPNPNU.
+[37] N. Otsu, A threshold selection method from gray-level histograms, IEEE
+Transactions on Systems, Man, and Cybernetics 9 (1) (1979) 62–66.
+doi:10.1109/TSMC.1979.4310076.
+[38] J. Boˇziˇc, D. Tabernik, D. Skoˇcaj, Mixed supervision for surface-defect
+detection: From weakly to fully supervised learning, Computers in In-
+dustry 129 (2021) 103459.
+[39] S. Raschka, Model evaluation, model selection, and algorithm selection
+in machine learning, arXiv preprint arXiv:1811.12808 (2018).
+[40] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image
+recognition, in: Proceedings of the IEEE conference on computer vision
+and pattern recognition, 2016, pp. 770–778. doi:10.1109/CVPR.2016.
+90.
+33
+
+[41] D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, in:
+ICLR (Poster), 2015.
+[42] W. C. of Tumours Editorial Board, WHO classification of tumours of
+the digestive system, no. Ed. 5, World Health Organization, 2019.
+[43] P. Ruamviboonsuk, R. Tiwari, R. Sayres, V. Nganthavee, K. Hemarat,
+A. Kongprayoon, R. Raman, B. Levinstein, Y. Liu, M. Schaekermann,
+et al., Real-time diabetic retinopathy screening by deep learning in a
+multisite national screening programme: a prospective interventional
+cohort study, The Lancet Digital Health 4 (4) (2022) e235–e244.
+34
+

3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:44edec5b3f481e72d9b56d0fc5b31803c065aaf3faee3765a64d63b089c0879a
+size 347933

3NE0T4oBgHgl3EQfeABh/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:14624e64cdd30113541af8f70d07edd0a0dcaf0abbd95296d05afe130a9907ef
+size 2293805

3NE0T4oBgHgl3EQfeABh/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f0a5d69744d79c569f32eb517c2aa5a10f794e2ad3cb7bc38d2acf6ab3bbd618
+size 77683

3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:53cc2003fd204352489201d57931cf871fa65b07c427461100a5b6593e39fbdc
+size 2425722

3tFQT4oBgHgl3EQfHDWI/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9966a9144a046a56e8921f8b39f978c527a73306f6a7954da3e990304edc070f
+size 5832749

3tFQT4oBgHgl3EQfHDWI/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:76a088ee89284b4c4181a14d348a3ea15bd0f08245dd35547f80f79082762b1b
+size 191278

4dAyT4oBgHgl3EQfpPgc/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a98d90550a166ad0ed06d6d8060718d7c46b7fccf6c97867ce92a56d57487482
+size 5308461

4dE0T4oBgHgl3EQfvQGV/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5f730d428de41eded28e91ac7c54e034fa2a3ff133a9fec074f17dbb59a882f4
+size 458797

4dE0T4oBgHgl3EQfvQGV/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:cb0ef2120cc7ce0b2d91dad1a8bec2a41a884c9775898dd663aa34929dcafc92
+size 17939

7NE0T4oBgHgl3EQffQAA/content/tmp_files/2301.02400v1.pdf.txt

@@ -0,0 +1,2236 @@
+arXiv:2301.02400v1  [cs.IT]  6 Jan 2023
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+with Flexible Array Size and Large Set Size
+Gobinda Ghosh1, Sudhan Majhi2* and Shubhabrata Paul1
+1Mathematics, IIT Patna, Bihta, Patna, 801103, Bihar, India.
+2*Electrical Communication Engineering, IISc Bangalore, CV
+Raman Rd, Bengaluru, 560012, Karnataka, India.
+*Corresponding author(s). E-mail(s): [email protected];
+Contributing authors: gobinda [email protected];
+[email protected];
+Abstract
+In this paper, we propose a direct construction of optimal two-
+dimensional
+Z-complementary
+array
+code
+sets
+(2D-ZCACS)
+using
+multivariable functions (MVFs). In contrast to earlier works, the
+proposed construction allows for a flexible array size and a large
+set size. Additionally, the proposed design can be transformed into
+a one-dimensional Z-complementary code set (1D-ZCCS). Many of
+the 1D-ZCCS described in the literature appeared to be special
+cases of this proposed construction. At last, we compare our work
+with the current state of the art and then draw our conclusions.
+Keywords: Two dimensional complete complementary codes (2D-CCC),
+multivariable function (MVF), two dimensional Z- complementary array code
+set (2D-ZCACS).
+1 Introduction
+For an asynchronous two dimensional multi-carrier code-division multiple
+access (2D-MC-CDMA) system, the ideal 2D correlation properties of two
+dimensional complete complementary codes (2D-CCCs)[1] can be properly uti-
+lized to obtain interference-free performance [2]. Similar to one dimensional
+complete complementary code (1D-CCC)[3–5], one of the most significant
+1
+
+Springer Nature 2021 LATEX template
+2
+A Direct Construction of Optimal 2D-ZCACS
+drawbacks of 2D-CCC is that the set size is restricted [6]. Motivated by
+the scarcity of 2D-CCC with flexible set sizes, Zeng et al. proposed 2D Z-
+complementary array code sets (2D-ZCACSs) in [6, 7]. For a 2D − (K, Z1 ×
+Z2)−ZCACSL1×L2
+M
+, K, Z1×Z2, L1×L2 and M denote the set size, two dimen-
+sional zero-correlation zone (2D-ZCZ) width, array size and the number of
+constituent arrays, respectively. In [6, 7], authors obtained ternary 2D-ZCACSs
+by inserting some zeros into the existing binary 2D-ZCACSs. In 2021, Pai
+et al. presented a new construction method of 2D binary Z-complementary
+array pairs (2D-ZCAP) [8]. Recently, Das et al. in [9] proposed a construction
+of 2D-ZCACS by using Z-paraunitary (ZPU) matrices. All these construc-
+tions of 2D-ZCACS depend heavily on initial sequences and matrices which
+increase hardware storage. For the first time in the literature, Roy et al. in
+[10] proposed a direct construction of 2D-ZCACS based on MVF. The array
+size of the proposed 2D-ZCACS is of the form L1 × L2, where L1 = 2m,
+L2 = 2pm1
+1 pm2
+2
+. . . pmk
+k , m ≥ 1, mi ≥ 2 and the set size is of the form 2p2
+1p2
+2 . . . p2
+k
+where pi is a prime number. Therefore the array size and the set size is
+restricted to some even numbers.
+Existing array and set size limitations through direct construction in the
+literature motivates us to search multivariable function (MVF) for more flex-
+ible array and set sizes. Our proposed construction provides 2D-ZCACS with
+parameter 2D − (R1R2M1M2, N1 × N2) − ZCACSR1N1×R2N2
+M1M2
+where M1 =
+�a
+i=1 pki
+i , M2 = �b
+j=1 qtj
+j , pi is any prime or 1, qj is prime, a, b, ki, tj ≥ 1, R1
+and R2 are positive integer, such that R1 ≥ 1 and R2 ≥ 2, N1 = �a
+i=1 pmi
+i ,
+N2 = �b
+j=1 qnj
+j , mi, nj ≥ 1. The set size in our proposed 2D-ZCACS construc-
+tion, R1R2M1M2, is more adaptable than the set size of 2D-ZCACS given in
+[10]. Unlike [10], the proposed 2D-ZCACS can be reduced to 1D-ZCCS [11–18]
+also. As a result, many existing optimal 1D-ZCCSs have become special cases
+of the proposed construction [16–18]. The proposed construction also derived a
+new set of optimal 1D-ZCCS that had not previously been presented by direct
+method.
+The rest of the paper is organized as follows. Section 2 discusses construc-
+tion related definitions and lemmas. Section 3 contains the construction of
+2D-ZCACS and the comparison with the existing state-of-the-art. Finally, in
+Section 4, the conclusions are drawn.
+2 Notations and definitions
+The following notations will be followed throughout this paper: ωn
+=
+exp
+�
+2π√−1/n
+�
+, An = {0, 1, . . ., n− 1} ⊂ Z, where n is a positive integer and
+Z is the ring of integer.
+2.1 Two Dimensional Array
+Definition 1 ([9]) Let A =
+�
+ag,i
+�
+and B =
+�
+bg,i
+�
+be complex-valued arrays of size
+l1 × l2 where 0 ≤ g < l1, 0 ≤ i < l2. The two dimensional aperiodic cross correlation
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+3
+function (2D-ACCF) of arrays A and B at shift (τ1, τ2) is defined as
+C (A, B) (τ1, τ2) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+�l1−1−τ1
+g=0
+�l2−1−τ2
+i=0
+ag,ib∗
+g+τ1,i+τ2, if
+0 ≤ τ1 < l1,
+0 ≤ τ2 < l2;
+�l1−1−τ1
+g=0
+�l2−1+τ2
+i=0
+ag,i−τ2b∗
+g+τ1,i, if
+0 ≤ τ1 < l1,
+−l2 < τ2 < 0;
+�l1−1+τ1
+g=0
+�l2−1−τ2
+i=0
+ag−τ1,ib∗
+g,i+τ2, if −l1 < τ1 < 0,
+0 ≤ τ2 < l2;
+�l1−1+τ1
+g=0
+�l2−1+τ2
+i=0
+ag−τ1,i−τ2b∗
+g,i, if −l1 < τ1 < 0,
+−l2 < τ2 < 0.
+Here, (.)∗ denotes the complex conjugate. If A = B, then C (A, B) (τ1, τ2)
+is called the two dimensional aperiodic auto correlation function (2D-AACF)
+of A and referred to as C (A) (τ1, τ2).
+When l1 = 1, the complex-valued arrays A and B are reduced to one
+dimensional complex-valued sequences A = (aj)l2−1
+j=0 and B = (bj)l2−1
+j=0 with
+the corresponding one dimensional aperiodic cross correlation function (1D-
+ACCF) given by
+C(A, B)(τ2) =
+
+
+
+
+
+
+
+�l2−1−τ2
+i=0
+aib∗
+i+τ2,
+0 ≤ τ2 < l2,
+�l2+τ2−1
+i=0
+ai−τ2b∗
+i ,
+−l2 < τ2 < 0,
+0,
+otherwise.
+(1)
+Definition 2 [19],[9] For a set of s sets of arrays A =
+�
+Ak | k = 0, 1, . . . , s − 1},
+each set Ak =
+�
+Ak
+0, Ak
+1, . . . , Ak
+s−1
+�
+is composed of s arrays of size is l1 × l2. The
+set A is said to be 2D-CCC with parameters (s, s, l1, l2) if the following holds
+C
+�
+Ak, Ak′�
+(τ1, τ2) =
+s−1
+�
+i=0
+C
+�
+Ak
+i , Ak′
+i
+�
+(τ1, τ2)
+=
+
+
+
+
+
+
+
+sl1l2,
+(τ1, τ2) = (0, 0), k = k′;
+0,
+(τ1, τ2) ̸= (0, 0), k = k′;
+0,
+k ̸= k′.
+(2)
+Definition 3 [10],[9] Let z1, z2, l1, l2 are positive integers and z1 ≤ l1, z2 ≤ l2.
+Consider the sets of ˆs set of arrays A =
+�
+Ak | k = 0, 1, . . . , ˆs − 1}, where each set
+Ak =
+�
+Ak
+0, . . . , Ak
+s−1
+�
+is composed of s arrays of size l1 × l2. The set A is said to
+
+Springer Nature 2021 LATEX template
+4
+A Direct Construction of Optimal 2D-ZCACS
+be 2D − (ˆs, z1 × z2) − ZCACSl1×l2
+s
+if the following holds
+C
+�
+Ak, Ak′�
+(τ1, τ2) =
+s−1
+�
+i=0
+C
+�
+Ak
+i , Ak′
+i
+�
+(τ1, τ2)
+=
+
+
+
+
+
+
+
+sl1l2,
+(τ1, τ2) = (0, 0), k = k′;
+0,
+(τ1, τ2) ̸= (0, 0),|τ1| < z1,|τ2| < z2, k = k′;
+0,
+|τ1| < z1,|τ2| < z2, k ̸= k′.
+(3)
+When z1 = l1, z2 = l2, ˆs = s the 2D-ZCACS becomes 2D-CCC[19, 20] with
+parameter (s, l1, l2). It should be noted that for l1 = 1, each array Ak
+i becomes
+l2-length sequence. Therefore, 2D-ZCACS can be reduced to a conventional
+1D-(ˆs, z2) − ZCCSl2
+s [21], [22],[23], where, ˆs, s, z2, l2 represents no. of set, set
+size, ZCZ width and sequence length respectively.
+Lemma 1 [9] For a 2D − (ˆs, z1 × z2) − ZCACSl1×l2
+s
+, the following inequality holds
+ˆsz1z2 ≤ s (l1 + z1 − 1) (l2 + z2 − 1) .
+(4)
+We called 2D-ZCACS is optimal if the following equality holds
+ˆs = s
+� l1
+z1
+�� l2
+z2
+�
+,
+(5)
+where ⌊.⌋ denotes the floor function.
+2.2 Multivariable Function
+Let a, b, mi, and nj be positive integers for 1 ≤ i ≤ a and 1 ≤ j ≤ b. Let pi be
+any prime or 1, and qj be a prime number. A multivariable function (MVF)
+can be defined as
+f : Am1
+p1 × Am2
+p2 × · · · × Ama
+pa × An1
+q1 × An2
+q2 × · · · × Anb
+qb → Z.
+Let c, d ≥ 0 be integers such that 0 ≤ c < r and 0 ≤ d < s where r =
+pm1
+1 pm2
+2
+. . . pma
+a
+and s = qn1
+1 qn2
+2 . . . qnb
+b . Then c and d can be written as
+c = c1 + c2pm1
+1
++ · · · + capm1
+1 pm2
+2
+. . . pma−1
+a−1 ,
+d = d1 + d2qn1
+1 + · · · + dbqn1
+1 qn2
+2 . . . qnb−1
+b−1 ,
+(6)
+where, 0 ≤ ci < pmi
+i
+and 0 ≤ dj < qnj
+j . Let Ci = (ci,1, ci,2, . . . , ci,mi) ∈ Ami
+pi ,
+be the vector representation of ci with base pi, i.e., ci = �mi
+k=1 ci,kpk−1
+i
+and
+Dj = (dj,1, dj,2, . . . , dj,nj) ∈ Anj
+qj be the vector representation of dj with base
+qj, i.e., dj = �nj
+l=1 dj,lql−1
+j
+where 0 ≤ ci,k < pi, and 0 ≤ dj,l < qj. We define
+vectors associated with c and d as
+φ(c) = (C1, C2, . . . , Ca) ∈ Am1
+p1 × Am2
+p2 × · · · × Ama
+pa ,
+φ(d) = (D1, D2, . . . , Db) ∈ An1
+q1 × An2
+q2 × · · · × Anb
+qb ,
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+5
+respectively. We also define an array associated with f as
+ψλ(f) =
+
+
+
+
+
+
+
+ωf0,0
+λ
+ωf0,1
+λ
+· · ·
+ωf0,r−1
+λ
+ωf1,0
+λ
+ωf1,1
+λ
+· · ·
+ωf1,r−1
+λ
+...
+...
+...
+...
+ωfs−1,0
+λ
+ωfs−1,1
+λ
+· · · ωfs−1,r−1
+λ
+
+
+
+
+
+
+
+,
+(7)
+where fc,d = f
+�
+φ(c), φ(d)
+�
+and λ is a positive integer.
+Lemma 2 ([24]) Let t and t′ be two non-negative integers, where t ̸= t′, and p is a
+prime number. Then
+p−1
+�
+j=0
+ω(t−t′)j
+p
+= 0.
+(8)
+Let us consider the set C as
+C =
+�
+Am1
+p1 × Am2
+p2 × · · · × Ama
+pa
+�
+×
+�
+An1
+q1 × An2
+q2 × · · · × Anb
+qb
+�
+.
+(9)
+Let 0 ≤ γ < pm1
+1 pm2
+2
+. . . pma
+a
+and 0 ≤ µ < qn1
+1 qn2
+2 . . . qnb
+b
+be positive integers
+such that
+γ = γ1 +
+a
+�
+i=2
+γi
+
+
+i−1
+�
+i1=1
+p
+mi1
+i1
+
+ ,
+µ = µ1 +
+b
+�
+j=2
+µj
+
+
+j−1
+�
+j1=1
+q
+nj1
+j1
+
+ ,
+(10)
+where 0 ≤ γi < pmi
+i
+and 0 ≤ µj < qnj
+j . Let γi = (γi,1, γi,2, . . . , γi,mi) ∈
+Ami
+pi be the vector representation of γi with base pi, i.e., γi = �mi
+k=1 γi,kpk−1
+i
+,
+where 0 ≤ γi,k < pi. Similarly µj = (µj,1, µj,2, . . . , µj,nj) ∈ Anj
+qj be the vector
+representation of µj with base qj i.e., µj = �nj
+l=1 µj,lql−1
+j
+where 0 ≤ µj,l < qj.
+Let
+φ(γ) = (γ1, γ2, . . . , γa) ∈ Am1
+p1 ×Am2
+p2 × · · · × Ama
+pa ,
+(11)
+be the vector associated with γ and
+φ(µ) = (µ1, µ2, . . . , µb) ∈ An1
+q1×An2
+q2 × · · · × Anb
+qb ,
+(12)
+be the vector associated with µ. Let πi and σj be any permutations of the
+set {1, 2, . . ., mi} and {1, 2, . . ., nj}, respectively. Let us also define the MVF
+
+Springer Nature 2021 LATEX template
+6
+A Direct Construction of Optimal 2D-ZCACS
+f : C → Z, as
+f(φ(γ), φ(µ))
+= f (γ1, γ2, . . . , γa, µ1, µ2, . . . , µb)
+=
+a
+�
+i=1
+λ
+pi
+mi−1
+�
+e=1
+γi,πi(e)γi,πi(e+1) +
+a
+�
+i=1
+mi
+�
+e=1
+di,eγi,e +
+b
+�
+j=1
+λ
+qj
+nj−1
+�
+o=1
+µj,σj(o)µj,σj(o+1)
++
+b
+�
+j=1
+nj
+�
+o=1
+cj,oµj,o,
+(13)
+where di,e, cj,o ∈ {0, 1, . . ., λ − 1} and λ = l.c.m.(p1, . . . , pa, q1, . . . , qb). Let us
+define the set Θ and T as
+Θ = {θ : θ = (r1, r2, . . . , ra, s1, s2, . . . , sb)},
+T = {t : t = (x1, x2, . . . , xa, y1, y2, . . . , yb)},
+where 0 ≤ ri, xi < pki
+i
+and 0 ≤ sj, yj < qrj
+j
+and ki, rj are positive integers.
+Now, we define a function aθ
+t: C →Z, as
+aθ
+t
+�
+φ(γ), φ(µ)
+�
+= aθ
+t (γ1, γ2, . . . , γa, µ1, µ2, . . . , µb)
+=f
+�
+φ(γ), φ(µ)
+�
++
+a
+�
+i=1
+λ
+pi
+γi,πi(1)ri +
+b
+�
+j=1
+λ
+qj
+µj,σj(1)sj +
+a
+�
+i=1
+λ
+pi
+γi,πi(mi)xi
++
+b
+�
+j=1
+λ
+qj
+µj,σj(nj)yj + dθ,
+(14)
+where 0 ≤ dθ < λ, γi,πi(1), γi,πi(mi) denote πi(1)−th and πi(mi)−th element
+of γi respectively. Similarly, µj,σj(1), µj,σj(nj) denote σj(1)−th and σj(nj)−th
+element of µj respectively. For simplicity, we denote aθ
+t
+�
+φ(γ), φ(µ)
+�
+by (aθ
+t)γ,µ
+and f
+�
+φ(γ), φ(µ)
+�
+by fγ,µ.
+Lemma 3 ([20]) We define the ordered set of arrays At = {ψλ
+�
+aθ
+t
+�
+: θ ∈ Θ}.
+Then the set {At : t ∈ T } forms a 2D-CCC with parameter (α, α, m, n), where, α =
+�a
+i=1 pki
+i
+�b
+j=1 qrj
+j , m = �a
+i=1 pmi
+i
+, n = �b
+j=1 qnj
+j
+and ki, mi, nj, rj are non-negative
+integers.
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+7
+3 Proposed construction of 2D-ZCACS
+Let a′, b′ be positive integers for 1 ≤ i′ ≤ a′ and 1 ≤ j′ ≤ b′, p′
+i′ be any
+prime or 1, and q′
+j′ be prime number. Let γ′, µ′ are positive integers such that
+0 ≤ γ′ <
+��a
+i=1 pmi
+i
+� ��a′
+i′=1 p′
+i′
+�
+and 0 ≤ µ′ <
+��b
+j=1 qnj
+j
+� ��b′
+j′=1 q′
+j′
+�
+. Then
+γ′, µ′ can be written as
+γ′ =γ1+
+a
+�
+i=2
+γi
+
+
+i−1
+�
+i1=1
+p
+mi1
+i1
+
++
+
+
+γ′
+1 +
+a′
+�
+i′=2
+γ′
+i′
+
+
+i′−1
+�
+i1=1
+p′
+i1
+
+
+
+
+ m,
+µ′ =µ1+
+b
+�
+j=2
+µj
+
+
+j−1
+�
+j1=1
+q
+nj1
+j1
+
++
+
+
+µ′
+1 +
+b′
+�
+j′=2
+µ′
+j′
+
+
+j′−1
+�
+j1=1
+q′
+j1
+
+
+
+
+ n,
+(15)
+where m = �a
+i=1 pmi
+i , n = �b
+j=1 qnj
+j , 0 ≤ γi < pmi
+i , 0 ≤ µj < qnj
+j , 0 ≤ γ′
+i′ < p′
+i′
+and 0 ≤ µ′
+j′ < q′
+j′. We denote the vectors associated with γ′ and µ′ are
+φ(γ′) =
+�
+γ1, . . . , γa, γ′
+1, . . . , γ′
+a
+�
+∈ Am1
+p1 × . . . × Ama
+pa × Ap′
+1 × . . . × Ap′
+a′ ,
+φ(µ′) =
+�
+µ1, . . . , µb, µ′
+1, . . . , µ′
+b
+�
+∈ An1
+q1 × . . . × Anb
+qb × Aq′
+1 × . . . × Aq′
+b′ ,
+(16)
+respectively, where γi ∈ Ami
+pi , µj ∈ Anj
+qj
+are the vectors associated with
+γi and µj
+respectively i.e., γi
+=
+(γi,1, γi,2, . . . , γi,mi)
+∈
+Ami
+pi , µj
+=
+(µj,1, µj,2, . . . , µj,nj) ∈ Anj
+qj , γi = �mi
+k=1 γi,kpk−1
+i
+, µj = �nj
+l=1 µi,lql−1
+j
+, 0 ≤
+γi,k < pi and 0 ≤ µj,l < qj. Let us consider the set D as
+D = Am1
+p1 ×. . .×Ama
+pa ×Ap′
+1 ×. . .×Ap′
+a′ ×An1
+q1 ×. . .×Anb
+qb ×Aq′
+1 ×. . .×Aq′
+b′ . (17)
+Let f be the function as defined (13). We define the MVF M c,d : D → Z as
+M c,d �
+φ(γ′), φ(µ′)
+�
+= M c,d �
+γ1, . . . , γa, γ′
+1, . . . , γ′
+a′, µ1, . . . , µb, µ′
+1, . . . , µ′
+b′
+�
+= δ
+λf (γ1, . . . , γa, µ1, . . . , µb)+
+a′
+�
+i′=1
+ci′ δ
+p′
+i′ γ′
+i′ +
+b′
+�
+j′=1
+dj′ δ
+q′
+j′ µ′
+j′,
+(18)
+where
+0
+≤
+ci′
+<
+p′
+i′,
+0
+≤
+dj′
+<
+q′
+j′,
+c
+=
+(c1, c2, . . . , ca′)
+and
+d
+=
+(d1, d2, . . . , db′).
+For
+simplicity,
+now
+on-wards
+we
+denote
+M c,d(γ1, . . . , γa, γ′
+1, . . . , γ′
+a′, µ1, . . . , µb, µ′
+1, . . . , µ′
+b′) by M c,d. Consider the set
+Θ and T as
+Θ = {θ : θ = (r1, r2, . . . , ra, s1, s2, . . . , sb)},
+T = {t : t = (x1, x2, . . . , xa, y1, y2, . . . , yb)},
+
+Springer Nature 2021 LATEX template
+8
+A Direct Construction of Optimal 2D-ZCACS
+where 0 ≤ ri, xi < pki
+i
+and 0 ≤ sj, yj < qrj
+j
+and ki, rj are positive integers. Let
+us define MVF, bθ,c,d
+t
+: D → Z, as
+bθ,c,d
+t
+=M c,d +
+a
+�
+i=1
+δ
+pi
+γi,πi(1)ri +
+b
+�
+j=1
+δ
+qj
+µj,σj(1)sj +
+a
+�
+i=1
+δ
+pi
+γi,πi(mi)xi
++
+b
+�
+j=1
+δ
+qj
+µj,σj(nj)yj + δ
+λdθ,
+(19)
+where 0 ≤ dθ < λ. By (14), (18) and (19) we have
+bθ,c,d
+t
+= δ
+λaθ
+t +
+a′
+�
+i′=1
+ci′ δ
+p′
+i′ γ′
+i′ +
+b′
+�
+j′=1
+dj′ δ
+q′
+j′ µ′
+j′.
+(20)
+We define the ordered set of arrays as
+Ωc,d
+t
+= {ψδ(bθ,c,d
+t
+) : θ ∈ Θ}.
+(21)
+where δ = l.c.m(λ, p′
+1, p′
+2, . . . , p′
+a′, q′
+1, q′
+2, . . . , q′
+b′).
+Theorem 1 Let m = �a
+i=1 pmi
+i
+, n = �b
+j=1 qnj
+j , c = (c1, . . . , ca′), d = (d1, . . . , db′).
+Then the set S = {Ωc,d
+t
+: t ∈ T, 0 ≤ ci′ < p′
+i′, 0 ≤ dj′ < q′
+j′} forms a 2D − (α1, z1 ×
+z2) − ZCACSl1×l2
+α
+, where, α1 =
+��a′
+i′=1 p′
+i′
+� ��b′
+j′=1 q′
+j′
+�
+α, l1 = m
+��a′
+i′=1 p′
+i′
+�
+,
+l2 = n
+��b′
+j′=1 q′
+j′
+�
+, z1 = m ,z2 = n, α = (�a
+i=1 pki
+i )(�b
+j=1 qrj
+j ), ki, rj, mi, nj ≥ 1.
+Proof Let ˆγ, ˆµ are positive integers such that 0 ≤ ˆγ < l1 and 0 ≤ ˆµ < l2. Then ˆγ, ˆµ
+can be written as
+ˆγ = γ1+
+a
+�
+i=2
+γi
+
+
+i−1
+�
+i1=1
+p
+mi1
+i1
+
++
+
+
+γ′
+1 +
+a′
+�
+i′=2
+γ′
+i′
+
+
+i′−1
+�
+i1=1
+p′
+i1
+
+
+
+
+ m,
+ˆµ = µ1+
+b
+�
+j=2
+µj
+
+
+j−1
+�
+j1=1
+qnj1
+j1
+
++
+
+
+
+µ′
+1 +
+b′
+�
+j′=2
+µ′
+j′
+
+
+
+j′−1
+�
+j1=1
+q′
+j1
+
+
+
+
+
+
+ n,
+where 0 ≤ γi < pmi
+i
+, 0 ≤ µj < qnj
+j , 0 ≤ γ′
+i′ < p′
+i′ and 0 ≤ µ′
+j′ < q′
+j′. The proof will
+be split into following cases
+Case 1. (τ1 = 0, τ2 = 0)
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+9
+The ACCF between Ωc,d
+t
+and Ωc′,d′
+t′
+at τ1 = 0 and τ2 = 0 can be expressed as
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, 0)
+=
+�
+θ∈Θ
+C(ψδ((bθ,c,d
+t
+)), ψδ((bθ,c′,d′
+t′
+)))(0, 0)
+=
+�
+θ∈Θ
+l1−1
+�
+ˆγ=0
+l2−1
+�
+ˆµ=0
+ω
+(bθ,c,d
+t
+)ˆγ,ˆ
+µ−(bθ,c′,d′
+t′
+)ˆγ,ˆ
+µ
+δ
+=
+�
+θ∈Θ
+m−1
+�
+γ=0
+n−1
+�
+µ=0
+p′
+1−1
+�
+γ′
+1=0
+. . .
+p′
+a′ −1
+�
+γ′
+a′=0
+q′
+1−1
+�
+µ1=0
+. . .
+q′
+b′ −1
+�
+µ′
+b′ =0
+ωD
+δ ,
+(22)
+where D = δ
+λ
+�
+(aθ
+t )γ,µ − (aθ
+t′)γ,µ
+�
++�a′
+i′=1
+δ
+p′
+i′ (ci′ −c′
+i′)γi′ +�b′
+j′=1
+δ
+q′
+j′ (dj′ −d′
+j′)µj′.
+After splitting (22), we get
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, 0)
+=
+
+�
+θ∈Θ
+m−1
+�
+γ=0
+n−1
+�
+µ=0
+ω
+δ
+λ
+�
+(aθ
+t )γ,µ−(aθ
+t′ )γ,µ
+�
+δ
+
+ EF
+=
+
+�
+θ∈Θ
+m−1
+�
+γ=0
+n−1
+�
+µ=0
+ω
+�
+(aθ
+t )γ,µ−(aθ
+t′ )γ,µ
+�
+λ
+
+ EF
+= C(At, At′
+)(0, 0)EF,
+(23)
+where
+E =
+a′
+�
+i′=1
+
+
+
+p′
+i′ −1
+�
+γ′
+i′ =0
+ω
+(ci′−c′
+i′)γ′
+i′
+p′
+i′
+
+
+ ,
+F =
+b′
+�
+j′=1
+
+
+
+
+q′
+j′ −1
+�
+µ′
+j′ =0
+ω
+(dj′ −d′
+j′ )µ′
+j′
+q′
+j′
+
+
+
+ .
+(24)
+Subcase (i): (t ̸= t′)
+By lemma 2 we know, the set {At : t ∈ T } forms a 2D-CCC. Hence By lemma 2, we
+have
+C(At, At′
+)(0, 0) = 0.
+(25)
+Hence by (23) and (25) we have
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, 0) = 0.
+(26)
+Subcase (ii): (t = t′)
+By lemma 2, we know
+C(At, At′
+)(0, 0) =
+
+
+a
+�
+i=1
+pmi+ki
+i
+
+
+
+
+b
+�
+j=1
+qnj+rj
+j
+
+ .
+(27)
+
+Springer Nature 2021 LATEX template
+10
+A Direct Construction of Optimal 2D-ZCACS
+Let M =
+��a
+i=1 pmi+ki
+i
+� ��b
+j=1 qnj+rj
+j
+�
+hence by Lemma 2, (23), (24), (27), we
+have the following
+C(Ωc,d
+t
+, Ωc′,d′
+t
+)(0, 0) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+M
+��a′
+i′=1 p′
+i′
+� ��b′
+j′=1 q′
+j′
+�
+c = c′, d = d′
+0,
+c ̸= c′, d = d′
+0,
+c = c′, d ̸= d′
+0,
+c ̸= c′, d ̸= d′.
+(28)
+Case 2. (0 < τ1 < �a
+i=1 pmi
+i
+, 0 < τ2 < �b
+j=1 qnj
+j )
+Let σ, ρ are positive integers such that 0 ≤ σ < m′ and 0 ≤ ρ < n′ where m′ =
+�a′
+i′=1 p′
+i′, n′ = �b′
+j′=1 q′
+j′. Then σ and ρ can be written as
+σ = σ1 + σ2p′
+1 + . . . + σa′
+
+
+a′−1
+�
+i′=1
+p′
+i′
+
+ ,
+ρ = ρ1 + ρ2q′
+1 + . . . + ρb′
+
+
+b′−1
+�
+j′=1
+q′
+j′
+
+ ,
+(29)
+respectively where 0 ≤ σi′ < p′
+i′ and 0 ≤ ρj′ < q′
+j′ . We define vectors associated
+with σ and ρ to be
+φ(σ) = (σ1, . . . , σa′) ∈ Ap′
+1 × . . . × Ap′
+a′ ,
+φ(ρ) = (ρ1, . . . , ρb′) ∈ Aq′
+1 × . . . × Aq′
+b′ ,
+(30)
+respectively. The ACCF between Ωc,d
+t
+and Ωc′,d′
+t′
+for 0 < τ1 < �a
+i=1 pmi
+i
+and 0 <
+τ2 < �b
+j=1 qnj
+j , can be derived as
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, τ2) =C(At, At′
+)(τ1, τ2)DE+C(At, At′
+)(τ1−
+a
+�
+i=1
+pmi
+i
+, τ2)D′E+
+C(At, At′
+)(τ1, τ2 −
+b
+�
+j=1
+qnj
+j )DE′ + C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+, τ2 −
+b
+�
+j=1
+qnj
+j )D′E′,
+(31)
+where
+D =
+m′−1
+�
+σ=0
+
+
+a′
+�
+i′=1
+ω
+(ci′−c′
+i′ )(σi′ )
+p′
+i′
+
+ ,
+(32)
+E =
+n′−1
+�
+ρ=0
+
+
+b′
+�
+j′=1
+ω
+(dj′ −d′
+j′)(ρj′ )
+q′
+j′
+
+ ,
+(33)
+D′ =
+m′−2
+�
+σ=0
+
+
+a′
+�
+i′=1
+ω(ci′(σi′ )−c′
+i′ (σ+1)i′)
+p′
+i′
+
+ ,
+(34)
+E′ =
+n′−2
+�
+ρ=0
+
+
+b′
+�
+j′=1
+ω
+�
+dj′ (ρj′ )−d′
+j′(ρ+1)j′
+�
+q′
+j′
+
+ ,
+(35)
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+11
+and (σ + 1)i′ , (ρ + 1)j′ denotes the i′-th and j′-th components of φ (σ + 1) and
+φ (ρ + 1) respectively. By Lemma 2, for 0 < τ1 < �a
+i=1 pmi
+i
+and 0 < τ2 < �b
+j=1 qnj
+j ,
+we have
+C(At, At′
+)(τ1, τ2) = 0,
+(36)
+C(At, At′
+)(τ1−
+a
+�
+i=1
+pmi
+i
+, τ2) = 0,
+(37)
+C(At, At′
+)(τ1, τ2 −
+b
+�
+j=1
+qnj
+j ) = 0,
+(38)
+C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+, τ2 −
+b
+�
+j=1
+qnj
+j ) = 0.
+(39)
+By (31), (36), (37), (38), (39) we have
+C(Ωc,d
+t
+, Ωc
+′ ,d
+′
+t′
+)(τ1, τ2) = 0.
+(40)
+Case 3. (0 < τ1 < �a
+i=1 pmi
+i
+, − �b
+j=1 qnj
+j
+< τ2 < 0)
+The ACCF between Ωc,d
+t
+and Ωc′,d′
+t′
+for 0 < τ1 < �a
+i=1 pmi
+i
+and − �b
+j=1 qnj
+j
+< τ2 <
+0, can be derived as
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, τ2)
+=C(At, At′
+)(τ1, τ2)DE+C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+, τ2)D′E
++ C(At, At′
+)(τ1,
+b
+�
+j=1
+qnj
+j
++ τ2)DE′′ + C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+,
+b
+�
+j=1
+qnj
+j
++ τ2)D′E′′,
+(41)
+where
+E′′ =
+n′−2
+�
+ρ=0
+
+
+b′
+�
+j′=1
+ω
+�
+dj′ (ρ+1)j′ −d′
+j′ (ρj′ )
+�
+q′
+j′
+
+ .
+(42)
+By Lemma 2, for 0 < τ1 < �a
+i=1 pmi
+i
+and − �b
+j=1 qnj
+j
+< τ2 < 0, we have
+C(At, At′
+)(τ1,
+b
+�
+j=1
+qnj
+j
++ τ2) = 0,
+(43)
+C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+,
+b
+�
+j=1
+qnj
+j
++ τ2) = 0.
+(44)
+By (41) , (43) and (44) we have
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, τ2) = 0.
+(45)
+Case 4. (0 < τ1 < �a
+i=1 pmi
+i
+, τ2 = 0)
+
+Springer Nature 2021 LATEX template
+12
+A Direct Construction of Optimal 2D-ZCACS
+The ACCF between Ωc,d
+t
+and Ωc′,d′
+t′
+for 0 < τ1 < �a
+i=1 pmi
+i
+and τ2 = 0 , can be
+derived as
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, 0) =C(At, At′
+)(τ1, 0)DE+ C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+, 0)D′E.
+(46)
+By Lemma 2, for 0 < τ1 < �a
+i=1 pmi
+i
+, we have
+C(At, At′
+)(τ1, 0) = 0.
+C(At, At′
+)(τ1 −
+a
+�
+i=1
+pmi
+i
+, 0) = 0,
+(47)
+by (46) and (47) we have
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, 0) = 0.
+(48)
+Case 5.
+(τ1 = 0, 0 < τ2 < �b
+j=1 qnj
+j )
+The ACCF between Ωc,d
+t
+and Ωc′,d′
+t′
+for τ1 = 0 and 0 < τ2 < �b
+j=1 qnj
+j , can be
+derived as
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, τ2) =C(At, At′
+)(0, τ2)DE+ C(At, At′
+)(0, τ2 −
+b
+�
+j=1
+qnj
+j )DE′.
+(49)
+By Lemma 2, for 0 < τ2 < �b
+j=1 qnj
+j , we have
+C(At, At′
+)(0, τ2) = 0,
+C(At, At′
+)(0, τ2 −
+b
+�
+j=1
+qnj
+j ) = 0.
+(50)
+By (49) and (50) we have
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, τ2) = 0.
+(51)
+Case 6. (τ1 = 0, − �b
+j=1 qnj
+j
+< τ2 < 0)
+Similarly the ACCF between Ωc,d
+t
+and Ωc′,d′
+t′
+for τ1 = 0 and − �b
+j=1 qnj
+j
+< τ2 < 0 is
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, τ2) =C(At, At′
+)(0, τ2)DE+ C(At, At′
+)(0, τ2 +
+b
+�
+j=1
+qnj
+j )DE′′.
+(52)
+By Lemma 2, for − �b
+j=1 qnj
+j
+< τ2 < 0, we have
+C(At, At′
+)(0, τ2 +
+b
+�
+j=1
+qnj
+j ) = 0.
+(53)
+Hence by (50), (52) and (53) we have
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(0, τ2) = 0.
+(54)
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+13
+Combining all the cases we have
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, τ2) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+M
+��a′
+i′=1 p′
+i′
+� ��b′
+j′=1 q′
+j′
+�
+,
+(c, d, t) = (c′, d′, t′)
+(τ1, τ2) = (0, 0),
+0,
+(c, d, t) ̸= (c′, d′, t′)
+(τ1, τ2) = (0, 0),
+0,
+0 ≤ τ1 < �a
+i=1 pmi
+i
+,
+(τ1, τ2) ̸= (0, 0).
+(55)
+Similarly it can be shown
+C(Ωc,d
+t
+, Ωc′,d′
+t′
+)(τ1, τ2) = 0, −
+a
+�
+i=1
+pmi
+i
+< τ1 < 0.
+(56)
+Hence from (55), (56) we derive our conclusion.
+□
+Example 1 Suppose that a = 1, b = 1, a′ = 1, b′ = 1, p1 = 2, m1 = 2, k1 = 1, q1 = 3,
+n1 = 2, r1 = 1, p′
+1 = 3, q′
+1 = 2. Let δ = 6, λ = 6, γ1 = (γ11, γ12) ∈ A2
+2 = {0, 1}2
+be the vector associated with γ1 where 0 ≤ γ1 ≤ 3, i.e., γ1 = γ11 + 2γ12 and
+µ1 = (µ11, µ12) ∈ A2
+3 = {0, 1, 2}2 be the vector associated with µ1 where 0 ≤ µ1 ≤ 8,
+i.e., µ1 = µ11+3µ12 and 0 ≤ γ′
+1 ≤ 2, 0 ≤ µ′
+1 ≤ 1. We define the MVF f : A2
+2×A2
+3 → Z
+as
+f (γ1, µ1)=3γ1,2γ1,1+γ1,1+2γ1,2+2µ1,2µ1,1+2µ1,1+µ1,2.
+Consider the MVF, Mc,d : A2
+2 × A3 × A2
+3 × A2 → Z as
+Mc,d �
+γ1, γ′
+1, µ1, µ′
+1
+�
+= f(γ1, µ1) + 2c1γ′
+1 + 3d1µ′
+1
+= 3γ1,2γ1,1 + γ1,1 + 2γ1,2 + 2µ1,2µ1,1 + 2µ1,1 + µ1,2 + 2c1γ′
+1 + 3d1µ′
+1,
+(57)
+where 0 ≤ c1 < p′
+1 = 2, 0 ≤ d1 < q′
+1 = 3, c = c1 ∈ {0, 1}, and d = d1 ∈ {0, 1, 2}. We
+have
+Θ = {θ : θ = (r1, s1) : 0 ≤ r1 ≤ 1, 0 ≤ s1 ≤ 2},
+T = {t : t = (x1, y1) : 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ 2}.
+(58)
+Let dθ = 0, now from (19) we have
+bθ,c,d
+t
+= Mc,d + 3γ1,2r1 + 2µ1,2s1 + 3γ1,1x1 + 2µ1,2y1,
+(59)
+and
+Ωc,d
+t
+=
+�
+ψ6(bθ,c,d
+t
+) : θ = (r1, s1) ∈ {0, 1} × {0, 1, 2}
+�
+.
+(60)
+Therefore, the set
+S = {Ωc,d
+t
+: t ∈ T, 0 ≤ c1 ≤ 1, 0 ≤ d1 ≤ 2},
+(61)
+forms an optimal 2D − (36, 4 × 9) − ZCACS12×18
+6
+over Z6.
+
+Springer Nature 2021 LATEX template
+14
+A Direct Construction of Optimal 2D-ZCACS
+Table 1 Comparison with Previous Works
+Source
+No. of set
+Array Size
+Condition
+Based on
+[7]
+K = K′r
+L′
+1×(L′
+2 + r + 1)
+r ≥ 0
+2D − ZCACS of
+set size K′ and
+array size L′
+1×L′
+2
+[8]
+1
+2m × 2nL
+m, n ≥ 0
+ZCP of length L
+[9]
+K
+K × K
+K divides set size
+BH matrices
+[10]
+2 �ki
+i=1 p2
+i
+2m × �ki
+i=1 pmi
+i
+ki, mi ≥ 1, pi’s are prime
+MVF
+Thm 2
+rsα
+rm × sn
+α = (�a
+i=1 pki
+i )(�b
+j=1 q
+rj
+j ),
+m=�a
+i=1pmi
+i
+, n=�b
+j=1q
+nj
+j ,
+r, s, α ≥ 1, pi, qjareprimes
+MVF
+Remark 1 In Theorem 1, if we take a = 1, p1 = 1, a′ = 1, p′
+1 = 1, b = 1, q1 =
+2, b′ = l, r1 ≥ 2, we have optimal 1D-ZCCS with parameter (�l
+i=1 q′
+i2r1, 2n1) −
+ZCCS
+�l
+i=1 q′
+i2n1
+2r1
+, which is exactly the same result as in [18]. Also if we take l = 1, then
+we have optimal 1D-ZCCS of the form (q′
+12r1, 2n1) − ZCCSq′
+12n1
+2r1
+, which is exactly
+the same result in [17]. Therefore the optimal 1D-ZCCS given by [17, 18] appears as
+a special case of the proposed construction
+Remark 2 In Theorem 1, if a = 1, p1 = 1, a′ = 1, p′
+1 = 1, b = 1, q1 = 2, b′ = l, r1 = 1,
+we have 1d-ZCCS with parameter (2 �l
+i=1 q′
+i, 2n1) − ZCCS
+�l
+i=1 q′
+i2n1
+2
+, which is just
+a collection of 2 �l
+i=1 q′
+i ZCPs with sequence length �l
+i=1 q′
+i2n1 and ZCZ width 2n1.
+Hence our work produces collections of ZCPs[15] as well.
+Remark 3 In Theorem 1, if we take a = 1, p1 = 1, a′ = 1, p′
+1 = 1, b = 1, q1 = 2,
+b′ = r, q′
+1 = q′
+2 = . . . = q′r = 2, n1 = m − r and r1 = s + 1 then we have 1D-ZCCS
+with parameter (2s+r+1, 2m−r)−ZCCS2m
+2s+1, which is exactly the same result in [16].
+Hence, the ZCCS in [16] appears as a special case of our proposed construction.
+Remark 4 The 2D-ZCACS given by the proposed construction satisfies the equality
+given in (5). Therefore the 2D-ZCACS obtained by the proposed construction is
+optimal.
+Remark 5 If we take a = 1, a′ = 1, p1 = 1 and p′
+1 = 1, in Theorem 1, we have optimal
+1D-ZCCS with parameter
+���b′
+j′=1 q′
+j′
+� �b
+j=1 qrj
+j , n
+�
+− ZCCS
+n
+��b′
+j′=1 q′
+j′
+�
+�b
+j=1 q
+rj
+j
+where,
+n = �b
+j=1 qnj
+j . Hence, we have optimal 1D-ZCCS of length nm where, n, m > 1 and
+m = �b′
+j′=1 q′
+j′. Therefore our construction produces optimal 1D-ZCCS with a new
+length which is not present in the literature by direct method.
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+15
+Remark
+6 The
+set
+size
+of
+our
+proposed
+2D-ZCACS
+is
+��a′
+i′=1 p′
+i′
+� ��b′
+j′=1 q′
+j′
+� �a
+i=1 pki
+i
+�b
+j=1 qrj
+j
+where,
+ki, tj
+≥
+1.
+If
+we
+take
+a = 1, p1 = 1, a′ = 1, p′
+1 = 1, r1 = r2 = . . . = rb = 2, b′ = 1, and q′
+1 = 2 then we
+have set size 2 �b
+j=1 q2
+j which is the set size of the 2D-ZCACS in [10]. Therefore, we
+have flexible number of set sizes compared to [10].
+3.1 Comparison with Previous Works
+Table I compares the proposed work with indirect constructions from [7–9] and
+direct construction from [10]. The constructions in [7–9] heavily rely on initial
+sequences, increasing hardware storage. The construction in [10] is direct, but
+set size and array sizes are limited to some even numbers. Our construction
+doesn’t require initial matrices or sequences and produces flexible parameters.
+4 Conclusion
+In this paper, 2D-ZCACSs are designed by using MVF. The proposed design
+does not depend on initial sequences or matrices, so it is direct. Our proposed
+design produces flexible array size and set size compared to existing works.
+Also, our proposed construction can be reduced to 1D-ZCCS. As a result,
+many 1D-ZCCSs become special cases of our work. Finally, we compare our
+work to the existing state-of-the-art and show that it’s more versatile.
+References
+[1] Farkas, P., Turcs´any, M.: Two-dimensional orthogonal complete com-
+plementary codes. In: Joint IEEE 1st Workshop on Mobile Future and
+Symposium on Trends in Communications (sympoTIC), pp. 21–24 (2003)
+[2] Turcs´any, M., Farkaˇs, P.: New 2d-mc-ds-ss-cdma techniques based on two-
+dimensional orthogonal complete complementary codes. in Multi-Carrier
+Spread-Spectrum, Berlin, Germany: Springer, 49–56 (2004)
+[3] Chen, C.-Y., Wang, C.-H., Chao, C.-C.: Complete complementary codes
+and generalized reed-muller codes. IEEE Commun. Lett. 12(11), 849–851
+(2008)
+[4] Das, S., Majhi, S., Liu, Z.: A novel class of complete complementary codes
+and their applications for apu matrices. IEEE Sig. Process. Lett. 25(9),
+1300–1304 (2018)
+[5] Liu, Z., Guan, Y.L., Parampalli, U.: New complete complementary codes
+for peak-to-mean power control in multi-carrier cdma. IEEE Trans.
+Commun. 62(3), 1105–1113 (2014)
+
+Springer Nature 2021 LATEX template
+16
+A Direct Construction of Optimal 2D-ZCACS
+[6] Xeng, F., Zhang, Z., Ge, L.: Theoretical limit on two dimensional gener-
+alized complementary orthogonal sequence set with zero correlation zone
+in ultra wideband communications. International Workshop on UWBST
+& IWUWBS, 197–201 (2004)
+[7] Zeng, F., Zhang, Z., Ge, L.: Construction of two-dimensional comple-
+mentary orthogonal sequences with ZCZ and their lower bound. IET
+(2005)
+[8] Pai,
+C.-Y.,
+Ni,
+Y.-T.,
+Chen,
+C.-Y.:
+Two-dimensional
+binary
+Z-
+complementary array pairs. IEEE Trans. Inf. Theory 67(6), 3892–3904
+(2021)
+[9] Das, S., Majhi, S.: Two-dimensional Z-complementary array code sets
+based on matrices of generating polynomials. IEEE Trans. Signal Process.
+68, 5519–5532 (2020)
+[10] Roy, A., Majhi, S.: Construction of inter-group complementary code set
+and 2D Z-complementary array code set based on multivariable functions
+(2021). https://doi.org/10.48550/arXiv.2109.00970
+[11] Shen, B., Meng, H., Yang, Y., Zhou, Z.: New constructions of z-
+complementary code sets and mutually orthogonal complementary
+sequence sets. Des. Codes Cryptogr., 1–19 (2022)
+[12] Sarkar, P., Roy, A., Majhi, S.: Construction of z-complementary code sets
+with non-power-of-two lengths based on generalized boolean functions.
+IEEE Commun. Lett. 24(8), 1607–1611 (2020)
+[13] Sarkar, P., Majhi, S.: A direct construction of optimal zccs with maximum
+column sequence pmepr two for mc-cdma system. IEEE Commun. Lett
+25(2), 337–341 (2020)
+[14] Wu, S.-W., S¸ahin, A., Huang, Z.-M., Chen, C.-Y.: Z-complementary code
+sets with flexible lengths from generalized boolean functions. IEEE Access
+9, 4642–4652 (2020)
+[15] Kumar, P., Sarkar, P., Majhi, S., Paul, S.: A direct construction of even
+length zcps with large zcz ratio. Cryptogr. Commun., 1–10 (2022)
+[16] Sarkar, P., Majhi, S., Liu, Z.: Optimal Z-complementary code set from
+generalized Reed-Muller codes. IEEE Trans. Commun. 67(3), 1783–1796
+(2018)
+[17] Sarkar, P., Majhi, S., Liu, Z.: Pseudo-Boolean functions for optimal Z-
+complementary code sets with flexible lengths. IEEE Signal Process. Lett.
+28, 1350–1354 (2021)
+
+Springer Nature 2021 LATEX template
+A Direct Construction of Optimal 2D-ZCACS
+17
+[18] Ghosh, G., Majhi, S., Sarkar, P., Upadhaya, A.K.: Direct construction of
+optimal Z-complementary code sets with even lengths by using generalized
+boolean functions. IEEE Signal Process. Lett. 29, 872–876 (2022)
+[19] Pai, C.-Y., Liu, Z., Zhao, Y.-Q., Huang, Z.-M., Chen, C.-Y.: Design-
+ing two-dimensional complete complementary codes for omnidirectional
+transmission in massive mimo systems. In: International Symposium on
+Information Theory (ISIT), pp. 2285–2290 (2022). IEEE
+[20] Ghosh, G., Majhi, S., Upadhyay, A.K.: A direct construction of 2D-CCC
+with arbitrary array size and flexible set size using multivariable function
+(2022). https://doi.org/10.48550/arXiv.2207.13395
+[21] Wu, S.-W., Chen, C.-Y.: Optimal Z-complementary sequence sets with
+good peak-to-average power-ratio property. IEEE Signal Process. Lett.
+25(10), 1500–1504 (2018)
+[22] Yu, T., Adhikary, A.R., Wang, Y., Yang, Y.: New class of optimal Z-
+complementary code sets. IEEE Trans. Signal Process. 29, 1477–1481
+(2022)
+[23] Shen,
+B.,
+Yang,
+Y.,
+Fan,
+P.,
+Zhou,
+Z.:
+New
+z-
+complementary/complementary sequence sets with non-power-of-two
+length and low papr. Cryptogr. Commun. 14(4), 817–832 (2022)
+[24] Vaidyanathan, P.: Ramanujan sums in the context of signal process-
+ing���part i: Fundamentals. IEEE Trans. Signal Process. 62(16), 4145–
+4157 (2014)
+
+

7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:25195e416ca02bf6e54fdf64a8da957ecf23ce821b1d3c4b7ec9d4960e77e789
+size 193776

7NE1T4oBgHgl3EQfTgOa/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9677a5744679db229a3c6f6e8feb3484fd0d953696f509d7c23fc33eb302c6a9
+size 1769517

7NE1T4oBgHgl3EQfTgOa/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3c99fa220eec1b6ec02f5a6f546cc39307755d2479d1ea8fe19b63440cb1d04f
+size 73770

7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9e4bbf14b29cb6c69027850cd8177ed73b3c75f57cde996b953503a483a34d42
+size 1038570

7dAyT4oBgHgl3EQfcvf_/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:46aaedc39643a3935c0acbe5ec93bc600f5303db66a769bb42254451f51c58a5
+size 3538989

7dAyT4oBgHgl3EQfcvf_/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4e5caf710b94e7e717b4d6a7ef39d03cc389ec24d3473e774f6e305890a3f305
+size 154934

7tE2T4oBgHgl3EQf7whk/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1840ff0cf247ec59d7b8a13afe8394a523525caab92bb4304d62c7581ae87041
+size 2031661

8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fc3143438268239655c5c080276647dbb1a530bfdc8730e9e5edf5cacec39a79
+size 2542622

8tFST4oBgHgl3EQfaTh3/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7b08492e11b4988f3eb45ab26329827f09ed99bf887748c0ab40c2635f76a430
+size 3866669

8tFST4oBgHgl3EQfaTh3/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:751a8fe33ac92cc1ab56bb7843b2a31f7164559bf9124ce78d1f5918fe2406a7
+size 135947

99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:49e50df0b4cfc15f7f2a7745aed106a68343f2dbab2e2485e072c148e3f4cfdd
+size 1761809

99AyT4oBgHgl3EQfqfgS/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:85b57d5419628f66ceb2db578324133dfd812826880e670d17e32ee003142c29
+size 4587565

99AyT4oBgHgl3EQfqfgS/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:76221a79ea8ddbc14c684225b2459b6ef2f3a01e8b28edbb32c8e86d5d7bf970
+size 169574

9NAyT4oBgHgl3EQfQ_Zv/content/2301.00057v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9a52f23f1d936ba35c19df5fe1ad3d8d6be76eba9afcd095abf52bafbef53407
+size 476678

9NAyT4oBgHgl3EQfQ_Zv/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f2ff7b4e5eb4686090a6d867f5aea9ac39ba1d0e695b1183f8a53ddf81717724
+size 140386

9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:68db25aff4d2427f42d5d67b4d172d2a0621d24e2c1e11361ea801facdeff276
+size 208947

9dFQT4oBgHgl3EQf5zZD/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ba63630d9875ff436065035bfd43fa331c51e276671a2455fbf1420e9ccbb197
+size 3211309

9dFQT4oBgHgl3EQf5zZD/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c61314c21670e5f567e50cc19fe70e26f10349a17d3fadb8001bb49af7b6dd47
+size 119903

BNE2T4oBgHgl3EQf8gli/content/tmp_files/2301.04219v1.pdf.txt

@@ -0,0 +1,741 @@
+arXiv:2301.04219v1  [math.CO]  10 Jan 2023
+EXTENSIONS OF A FAMILY FOR SUNFLOWERS
+JUNICHIRO FUKUYAMA
+Abstract. This paper refines the original construction of the recent proof
+of the sunflower conjecture to prove the same general bound [ck log(k + 1)]m
+on the cardinality of a family of m-cardinality sets without a sunflower of k
+elements. Our proof uses a structural claim on an extension of a family that
+has been previously developed.
+1. Motivation, Terminology and Related Facts
+The sunflower conjecture states that a family F of sets each of cardinality at
+most m includes a k-sunflower if |F| > cm
+k for some ck ∈ R>0 depending only on
+k, where k-sunflower stands for a family of k different sets with common pair-wise
+intersections. It had been open since the sunflower lemma was presented in 1960
+[1], until it was recently proven [2] with the following statement confirmed.
+Theorem 1.1. There exists c ∈ R>0 such that for every k, m ∈ Z>0, a family F of
+sets each of cardinality at most m includes a k-sunflower if |F| > [ck log(k + 1)]m.
+□
+The base of the obtained bound [ck log(k + 1)]m is asymptotically close to the
+lower bound k − 1. Our investigation on finding such a near-optimal bound had
+continued from the previous work [3]. The paper attempts to explore some combi-
+natorial structure involving sunflowers, to prove that a uniform family F includes
+three mutually disjoint sets, not just a 3-sunflower, if it satisfies the Γ
+�
+cm
+1
+2 +ǫ�
+-
+condition for any given ǫ ∈ (0, 1) and c depending on ǫ only. Here the Γ (b)-condition
+of F (b ∈ R>0) means |{U : U ∈ F, S ⊂ U}| < b−|S||F| for all nonempty sets S.
+The original construction [4] of the work [2] proves the most noted three-petal
+case of the conjecture, referring to Theorem 1.2 given below that derives the exten-
+sion generator theorem presented in [3]. The goal of this paper is to further refine1
+the original construction to prove the same [ck log(k + 1)]m bound. We will find
+such proof at the end of the next section.
+The rest of this section describes the similar terminology and related facts. De-
+note the universal set by X, its cardinality by n, and a sufficiently small positive
+2010 Mathematics Subject Classification. 05D05: Extremal Set Theory (Primary).
+Key words and phrases. sunflower lemma, sunflower conjecture, ∆-system.
+1Extra information on this paper and the references [2, 3, 4] is available at Penn State Sites.
+Web address: https://sites.psu.edu/sunflowerconjecture/2023/01/10/index-page/
+1
+
+2
+JUNICHIRO FUKUYAMA
+number depending on no other variables by ǫ ∈ (0, 1). In addition,
+i, j, m, p, r ∈ Z≥0,
+[b] = [1, b] ∩ Z,
+F ⊂ 2X,
+�X′
+m
+�
+= {U : U ⊂ X′, |U| = m} ,
+for X′ ⊂ X,
+and
+F[S] = {U : U ∈ F, S ⊂ U} ,
+for S ⊂ X,
+A set means a subset of X, and one in
+�X
+m
+�
+is called m-set.
+Weight X by some w : 2X → R≥0, which induces the norm ∥ · ∥ of a family
+defined by ∥F∥ = �
+U∈F w(U) for any F. Denote set/family subtraction by −,
+while we use the symbol \ for the different notion described below.
+Use ← to
+express substitution into a variable. For simplicity, a real interval may denote the
+integral interval of the same range, e.g., use (1, t] instead of (1, t] ∩ Z if it is clear
+by context. Obvious floor/ceiling functions will be ommited throughout the paper.
+Now let F be a family of m-sets, i.e., F ⊂
+�X
+m
+�
+. We say
+κ (F) =
+�n
+m
+�
+− ln |F|.
+is the sparsity of F. The family satisfies the Γ (b)-condition on ∥ · ∥ (b ∈ R>0) if
+∥G∥ = ∥G ∩ F∥,
+for all G ⊂ 2X,
+and
+∥F[S]∥ < b−|S|∥F∥,
+for every nonempty set S ⊂ X.
+As used above, the norm ∥ · ∥ can be omitted if it is induced by the unit weight,
+i.e.,
+w : V �→
+� 1,
+if V ∈ F,
+0,
+otherwise.
+The following theorem is proven2 in [3].
+Theorem 1.2. Let X be weighted to induce the norm ∥ · ∥. For every sufficiently
+small ǫ ∈ (0, 1), and F ⊂
+�X
+m
+�
+satisfying the Γ
+� 4γn
+l
+�
+-condition on ∥ · ∥ for some
+l ∈ [n], m ∈ [l], and γ ∈
+�
+ǫ−2, lm−1�
+, there are
+��n
+l
+�
+(1 − ǫ)
+�
+sets Y ∈
+�X
+l
+�
+such
+that
+�
+1 −
+� 2
+ǫγ
+� � l
+m
+�
+� n
+m
+� ∥F∥ <
+����
+�Y
+m
+����� <
+�
+1 +
+� 2
+ǫγ
+� � l
+m
+�
+� n
+m
+� ∥F∥ .
+□
+With Theorem 1.2, we can prove the aforementioned extention generator theorem
+that is about the l-extension of F, i.e.,
+Ext (F, l) =
+�
+T : T ∈
+�X
+l
+�
+, and ∃U ∈ F, U ⊂ T
+�
+,
+for l ∈ [n] − [m].
+It is not difficult to see
+(1.1)
+κ [Ext (F, l)] ≤ κ (F) ,
+as in [3], where it is also shown:
+Lemma 1.3. For F ⊂
+�X
+m
+�
+such that m ≤ n/2,
+κ
+�� X
+2m
+�
+− Ext (F, 2m)
+�
+≥ 2κ
+��X
+m
+�
+− F
+�
+.
+□
+2In [3], the theorem uses so called Γ2 (b, 1)-condition on ∥ · ∥. It is straightforward to check it
+means the Γ (b)-condition on ∥ · ∥ here.
+
+EXTENSIONS OF A FAMILY FOR SUNFLOWERS
+3
+Further denote
+Gp = G × G × · · · × G
+�
+��
+�
+p
+,
+X = (X1, X2, . . . , Xp) ∈
+�
+2X�p ,
+Rank (X) = p,
+and
+Union (X) =
+p�
+j=1
+Xj.
+Suppose m divides n and p = m. If Union (X) = X, and all Xi are mutually
+disjoint n/m-sets, then such an X is an m-split of X with Xi called strips. Its
+subsplit X′ of rank r ∈ [m], or r-subsplit of X, is the tuple of some r strips of X
+preserving the order.
+A set S is on X′ if S ⊂ Union
+�
+X′�
+, and |Xi ∩ S| ∈ {0, 1} for every strip Xi of
+X′. Denote
+- by 2X′ the family of all sets on X′,
+- by
+�X′
+p
+�
+the family of p-sets on X′,
+- by X \ X′ the subsplit of rank m − m′ consisting of the strips in X but not in
+X′,
+- and by X′ \ B for a set B, abusing the symbol \, the subsplit of X consisting of
+the strips each disjoint with B.
+For notational convenience, allow Rank
+�
+X′�
+= 0 for which X′ = (∅), Union
+�
+X′�
+=
+∅, and
+�X′
+p
+�
+= {∅}. We have:
+Lemma 1.4. For any nonempty family F ⊂
+�X
+m
+�
+such that m divides n = |X|,
+there exists an m-split X of X such that
+����F ∩
+�X
+m
+����� ≥
+� n
+m
+�m |F|
+� n
+m
+� > |F| exp (−m) .
+□
+The lemma proven in [2] poses a special case of the general statement presented in
+[3].
+2. Proof of Theorem 1.1
+We prove Theorem 1.1 with Theorem 1.2 in the two subsections below. Given F
+and k, we will find a subfamily ˆF ⊂ F with a property that implies the existence
+of a k-sunflower in itself.
+2.1. Formulation and Construction. Letting
+h = exp
+�1
+ǫ
+�
+,
+and
+c = exp (h) ,
+assume WLOG that
+- k ≥ 3,
+- n = |X| is larger than ckm and divisible by m. Otherwise add some extra elements
+to X.
+- F ⊂
+�X
+m
+�
+for an m-split X of X by Lemma 1.4, satisfying the Γ (cck ln k)-condition
+and |F| > (cck ln k)m.
+- |F| < (km)m and m > cc ln k, otherwise F includes a k-sunflower by the sunflower
+lemma.
+
+4
+JUNICHIRO FUKUYAMA
+FindCores
+Input:
+i) the family F ⊂
+�X
+m
+�
+.
+Outputs:
+i) C ⊂
+�X
+r0
+�
+for some r0 ∈ [0, m].
+ii) ˆF ⊂ �
+C∈C F[C] such that | ˆF| ≥ 3−m−1|F|.
+1. F′ ← F;
+ˆF ← ∅;
+C ← ∅;
+2. for r = m down to 0 do:
+2-1. repeat:
+a) find an r-set C such that |F′[C]| ≥ f(r) putting TC ← F′[C];
+b) if found then:
+F′ ← F′ − TC;
+ˆF ← ˆF ∪ TC;
+C ← C ∪ {C};
+else exit Loop 2-1;
+2-2. if | ˆF| ≥ 3−m+r−1|F| then return
+�
+r, C, ˆF
+�
+;
+Figure 1. Algorithm FindCores
+Let
+i ∈ [k],
+r ∈ [0, m],
+b = ck ln k,
+δ =
+ǫ
+k ln k,
+F′, Fi ⊂ F,
+C ∈ 2X,
+and
+Yi ∈ 2X.
+A tuple Z = (C, Y1; F1, Y2; F2, · · · , Yk; Fk) is said to be a partial sunflower of
+rank r over F′ if there exists an r-subsplit X∗ of X satisfying the four conditions:
+Z-i) C ∈ �m/c
+u=0
+�X∗
+u
+�
+.
+Z-ii) Yi are mutually disjoint k subsets of Union (X∗ \ C) such that
+|Yi ∩ X†| = δ|X†| for each strip X† of X∗ \ C.
+Z-iii) The k families Fi are each nonempty included in
+F′[C] ∩
+�X − Union (X∗ \ C) ∪ Yi
+m
+�
+,
+and are identical if Rank (X∗ \ C) = 0.
+Z-iv) |Fi| < 2|Fi′| for i ∈ [k] and i′ ∈ [k] − {i}.
+We say that such an Fi occurs on Z and in Z with the core C. Also Z and
+Fi are on X∗. A family Z of Z on one or more X∗ is a partial sunflower family
+(PSF) of rank r over F′, if each two Fi occurring on two different Z are mutually
+disjoint, i.e., the universal disjoint property of Z is met. Denote
+F (Z) :=
+�
+Z∈Z
+i∈[k]
+Fi of Z,
+for any PSF Z abusing the symbol F.
+In the rest of our proof, we construct a nonempty PSF of rank m over F. This
+means a k-sunflower in F proving Theorem 1.1.
+With
+f : Z≥0 → R≥0,
+x �→ ǫ3m(chk)−x
+k
+|F|,
+
+EXTENSIONS OF A FAMILY FOR SUNFLOWERS
+5
+obtain the families C and ˆF by the algorithm FindCores described in Fig. 1. It
+is straightforward to see that the two outputs correctly satisfy the properties i)-ii).
+In addition:
+A) | ˆF[U]| < f(|U|) for all U ∈ �m
+r′=r0+1
+�X
+r′
+�
+.
+B) We will construct partial sunflowers over ˆF with cores C in C. The families TC
+Step 2-1 finds for r = r0 are mutually disjoint each with |TC| ≥ f(r0). By the
+Γ (cck ln k)-condition of F and cc ln k < m,
+k−1ǫ3m(chk ln k)−|C||F| = f (r0) ≤ |TC| ≤ |F[C]| < (cck ln k)−|C||F|,
+⇒
+r0 = |C| < ln k − 3m ln ǫ
+(c − h) ln c
+< m
+2c
+�ln k
+m + 1
+�
+< m
+c .
+Define a statement on the obtained objects.
+Proposition Πr for r ∈ [r0, m]: there exists a PSF Z of rank r over ˆF such that
+|F(Z)| > ǫ2r−2r0| ˆF|.
+□
+Such a Z is said to be r-normal. By definition, Z is the union of PSFs ZX∗ on
+r-subsplits X∗ satisfying the universal disjoint property.
+Our final goal of finding a nonempty PSF of rank m over F would be met if Πm.
+The proposition Πr0 holds since
+Z =
+
+
+
+
+C, ∅; TC, ∅; TC, · · · , ∅; TC
+�
+��
+�
+k
+
+ : C ∈ ˆC
+
+
+
+is an r0-normal PSF such that F (Z) = ˆF by B) where TC are the ones mentioned
+there. So it suffices to show
+(2.1)
+Πr ⇒ Πr+1,
+for every r ∈ [r0, m),
+to have proof of a k-sunflower in F.
+2.2. Proof of (2.1). We start showing (2.1) as the only remaining task. Assume
+Πr for a particular r ∈ [r0, m), so we have an r-normal PSF Z that is the union of
+ZX∗ on some r-subsplits X∗ by definition. We confirm Πr+1 in four steps.
+Step 1. Reconstruct Z into another PSF Z′. Obtain such a Z′ by the algorithm
+Reconstruct described in Fig. 2. It is a PSF of rank r over F (Z) satisfying the
+two conditions:
+C) |F (Z′) | > 2−1ǫ|F (Z) |.
+D) For each Z ∈ Z′ on an r-subsplit X∗, there exists an r+1-subsplit X′ containing
+X∗ such that each Fi on Z meets
+|Fi[S]| < 1
+b |Fi|,
+∀S ∈
+�X′ \ X∗
+1
+�
+.
+□
+We see their truth by the notes below. Such a Z ∈ Z′ is said to be on the split
+pair
+�
+X∗, X′�
+. In Steps 2 and 3, we will construct our desired r + 1-normal PSF
+Z′′ from Z′ confirming Πr+1.
+Justification of Z′ being a PSF with C) and D).
+- F (ZX′) of an X′ disregarded by Step 2-3 is negligible as their union will be
+smaller than 2−m� m
+r+1
+�
+<
+� 3
+2
+�−m <
+� 3
+2
+�−cc ln k < ǫ3/k of F (Z).
+
+6
+JUNICHIRO FUKUYAMA
+Reconstruct
+Input:
+an r-normal PSF Z for some r ∈ [r0, m).
+Output:
+a PSF Z′ of rank r over F (Z) satisfying C) and D).
+1. X ← the family of all r-subsplits X∗;
+/* The given Z is the union of PSFs ZX∗ on some X∗ by definition. */
+2. for each r + 1-subsplit X′ do:
+2-1. X∗ ← the family of all X∗ ∈ X that are r-subsplits of X′;
+2-2. ZX′ ← �
+X∗∈X∗ ZX∗;
+2-3. if |F (ZX′) | < 3−m|F (Z) | then go to Step 2 for the next X′ else X ← X − X∗;
+2-4. for each Fi in ZX′ do F′
+i ← Fi;
+2-5. for each Fi in ZX′ and on X∗, and sets B ∈
+�X∗
+r
+�
+and U ∈
+� X′
+r+1
+�
+[B] such that
+|Fi[U]| >
+�
+c
+√
+hk ln k
+�−r−1
+|Fi[B]| do F′
+i ← F′
+i − Fi[U];
+2-6. for each Z ∈ ZX′ do:
+a) if |F′
+i| < ǫ|Fi| for some Fi on Z then delete Z from ZX′;
+b) else normalize Z for the condition Z-iv) as follows:
+b)-i) for each Fi on Z do:
+γi ← |F′
+i|−1 mini′∈[k] |F′
+i′|;
+F′
+i ← any subfamily of F′
+i of cardinality min (|F′
+i|, ⌊2γi|F′
+i|⌋);
+b)-ii) replace all Fi by the F′
+i to reconstruct Z;
+3. return the union of all ZX′ found in Loop 2 as Z′;
+Figure 2. Algorithm Reconstruct
+- |F(ZX′)[U]| ≤ | ˆF[U]| < f (r + 1) for all U ∈
+� X
+r+1
+�
+by A). So,
+|F (ZX′) [U]|
+|F (ZX′) |
+<
+f (r + 1)
+3−mǫ2r−2r0| ˆF|
+<
+�
+chk ln k
+�−r−1
+k
+,
+before Step 2-4.
+- Fi[B] are mutually disjoint for all different Fi occurring in ZX′ each on an X∗,
+and B ∈
+�X∗
+r
+�
+right before Step 2-5, by the universal disjoint property of Z. (By
+the rule Z-iii), Fi on a single Z are identified when r = r0.) In addition, Fi[U]
+are mutually disjoint for all Fi and U ∈
+� X′
+r+1
+�
+meeting
+�
+X∗∈X∗, Fi in ZX′ and on X∗
+B∈(X∗
+r ), U∈(X′
+m′)[B]
+Fi[U] = F(ZX′).
+- By the above two, Step 2-5 may reduce
+V =
+�
+Fi in ZX′
+F′
+i
+by less than its ǫ3/k, leaving only Fi, B, and U such that
+|F′
+i[U]| < (cb)−r−1|F′
+i[B]|,
+⇒
+|F′
+i[B]| > (cb)r+1.
+- By Z-iv) of Z, Step 2-6-a) may only reduce less than 2ǫ3 of V.
+- The process of normalization is well-defined by Step 2-6-b) due to |F′
+i| > (cb)r+1
+before it. It could further reduce V into its ǫ/2 or larger.
+
+EXTENSIONS OF A FAMILY FOR SUNFLOWERS
+7
+- The condition Z-iv) of the obtained Z′ follows the above as well as the two
+properties C) and D).
+Step 2. For each Fi occurring in Z′, construct a family Yi of Y ∈
+�
+X†
+δ|X†|
+�
+such
+that Fi ∩
+�X−X†∪Y
+m
+�
+is sufficiently large, where X† = Union(X′ \ X∗). Consider
+each Z ∈ Z′ on (X∗, X′) with the unique strip X† of X′ \ X∗, and an Fi on Z.
+Weight X† by 2X† → Z≥0, W �→ |Fi[W]| inducing the norm ∥ · ∥ as in Section 1.
+The family H =
+�X†
+1
+�
+satisfies the Γ(b)-condition on ∥ ·∥ by D). Apply Theorem 1.2
+to H. There exists Yi ⊂
+� X†
+δ|X†|
+�
+such that
+|Yi| >
+� |X†|
+δ|X†|
+�
+[1 − exp (−h)] ,
+(2.2)
+δ [1 − exp (−h)] |Fi| <
+��FY
+i
+�� < δ [1 + exp (−h)] |Fi|,
+for every Y ∈ Yi,
+where
+FY
+i ⊂ Fi ∩
+�X − X† ∪ Y
+m
+�
+.
+Step 3. Find Z′′ by (2.2). Now consider the same Z with the k families Fi and
+sets Yi. Put
+δ′ = 2δ ln k,
+and
+Y′
+i = Ext (Yi, δ′|X†|) ,
+for each Fi to see
+|Y′
+i| >
+� |X†|
+δ′|X†|
+�
+[1 − exp (−h ln k)] >
+� |X†|
+δ′|X†|
+� �
+1 − ǫ
+k
+�
+,
+by Lemma 1.3, (1.1) and (2.2): to the Yi, repeatedly apply the lemma ⌈log2 ln k⌉
+times doubling the second parameter of Ext. Then κ
+��
+X†
+δ′|X†|
+�
+− Y′
+i
+�
+> h ln k.
+Hence, there exist more than
+� |X†|
+δ′|X†|
+��|X†| − δ′|X†|
+δ′|X†|
+�
+· · ·
+�|X†| − (k − 1)δ′|X†|
+δ′|X†|
+�
+(1 − ǫ)
+tuples (Y ′
+1, Y ′
+2, . . . , Y ′
+k) ∈
+�
+X†
+δ′|X†|
+�k such that each Y ′
+i is in Y′
+i, disjoint with the other
+k − 1.
+For such a (Y ′
+1, Y ′
+2, . . . , Y ′
+k), find a δ|X†|-set Y †
+i ∈ Yi included in each Y ′
+i . Add
+the tuple
+�
+C, Y1 ∪ Y †
+1 ; F
+Y †
+1
+1 , Y2 ∪ Y †
+2 ; F
+Y †
+2
+2 , . . . , Yk ∪ Y †
+k ; F
+Y †
+k
+k
+�
+to Z′′, where the set C is that of Z. By construction, it satisfies the conditions Z-i)
+to iii) with F′ ← F (Z′).
+Subtract �k
+i=1 F
+Y †
+1
+i
+from Fi. Repeat the above ǫ−1/2 times including Step 2 for
+the current Z. Then denote an element of Z′′ by Z′, and family F
+Y †
+i
+i
+by F′
+i. Such
+a Z′ and F′
+i are produced from Z and Fi, and we assume it for the four objects
+anywhere below.
+Finally, normalize each Z′ for Z-iv) the same way as Step 2-6-b)-i) of Recon-
+struct with the γi given there. It possibly reduces F′
+i into its 1 − exp (−h/2) or
+larger.
+Perform the process for all Z ∈ Z′ to complete our construction of Z′′.
+
+8
+JUNICHIRO FUKUYAMA
+Step 4. Confirm Πr+1 to finish the proof. For the r + 1-normality of Z′′, it can
+be checked by straightforward recursive arguments with (2.2) that
+1
+2ǫ1/2|Fi| < ∆|Fi| < 2ǫ1/2|Fi|,
+where |Fi| expresses the value after Step 1, and ∆|Fi| the difference between |Fi|
+and its final value. This means Step 2 can use the Γ
+�
+b
+�
+1 − 2ǫ1/2��
+-condition on ∥ ·
+∥B instead of the Γ (b)-condition throughout the construction, constantly achieving
+(2.2) for a Z. In addition, the normalization of each Z′ always keeps more than
+half of its �k
+i=1 F′
+i.
+Hence, the recursive loop for every Z terminates without an exception defining
+our Z′′ with |F (Z′′) | > ǫ2/3|F (Z′) | > ǫ2|F (Z) | by C). As it is a PSF with the
+universal disjoint property by construction, we confirm the proposition Πr+1 to
+complete our proof of (2.1).
+We now have Theorem 1.1.
+References
+1. Erd¨os, P., Rado, R. : Intersection theorems for systems of sets. Journal of the London Math-
+ematical Society, Second Series, 35 (1), pp. 85 - 90 (1960)
+2. Fukuyama, J. : The sunflower conjecture proven. arXiv:2212.13609 [math.CO] (2022)
+3. Fukuyama,
+J.:
+Improved
+bound
+on
+sets
+including
+no
+sunflower
+with
+three
+petals.
+arXiv:1809.10318v3 [math.CO] (2021)
+4. Fukuyama, J. : The case k = 3 of the sunflower conjecture. Private work available at Penn
+State Sites.
+Web address:
+https://sites.psu.edu/sunflowerconjecture/files/2023/01/Proof-of-the-3-petal-
+sunflower-conjecture.pdf (2023)
+Department of Computer Science and Engineering, The Pennsylvania State Univer-
+sity, PA 16802, USA
+E-mail address: [email protected]
+

BNE2T4oBgHgl3EQf8gli/content/tmp_files/load_file.txt

@@ -0,0 +1,271 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf,len=270
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='04219v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='CO]  10 Jan 2023  EXTENSIONS OF A FAMILY FOR SUNFLOWERS  JUNICHIRO FUKUYAMA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' This paper refines the original construction of the recent proof  of the sunflower conjecture to prove the same general bound [ck log(k + 1)]m  on the cardinality of a family of m-cardinality sets without a sunflower of k  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Our proof uses a structural claim on an extension of a family that  has been previously developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Motivation, Terminology and Related Facts  The sunflower conjecture states that a family F of sets each of cardinality at  most m includes a k-sunflower if |F| > cm  k for some ck ∈ R>0 depending only on  k, where k-sunflower stands for a family of k different sets with common pair-wise  intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' It had been open since the sunflower lemma was presented in 1960  [1], until it was recently proven [2] with the following statement confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' There exists c ∈ R>0 such that for every k, m ∈ Z>0, a family F of  sets each of cardinality at most m includes a k-sunflower if |F| > [ck log(k + 1)]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  □  The base of the obtained bound [ck log(k + 1)]m is asymptotically close to the  lower bound k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Our investigation on finding such a near-optimal bound had  continued from the previous work [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' The paper attempts to explore some combi-  natorial structure involving sunflowers, to prove that a uniform family F includes  three mutually disjoint sets, not just a 3-sunflower, if it satisfies the Γ  �  cm  1  2 +ǫ� condition for any given ǫ ∈ (0, 1) and c depending on ǫ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Here the Γ (b)-condition  of F (b ∈ R>0) means |{U : U ∈ F, S ⊂ U}| < b−|S||F| for all nonempty sets S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  The original construction [4] of the work [2] proves the most noted three-petal  case of the conjecture, referring to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2 given below that derives the exten-  sion generator theorem presented in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' The goal of this paper is to further refine1  the original construction to prove the same [ck log(k + 1)]m bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' We will find  such proof at the end of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  The rest of this section describes the similar terminology and related facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' De-  note the universal set by X, its cardinality by n, and a sufficiently small positive  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' 05D05: Extremal Set Theory (Primary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' sunflower lemma, sunflower conjecture, ∆-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  1Extra information on this paper and the references [2, 3, 4] is available at Penn State Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Web address: https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='edu/sunflowerconjecture/2023/01/10/index-page/  1  2  JUNICHIRO FUKUYAMA  number depending on no other variables by ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' In addition,  i, j, m, p, r ∈ Z≥0,  [b] = [1, b] ∩ Z,  F ⊂ 2X,  �X′  m  �  = {U : U ⊂ X′, |U| = m} ,  for X′ ⊂ X,  and  F[S] = {U : U ∈ F, S ⊂ U} ,  for S ⊂ X,  A set means a subset of X, and one in  �X  m  �  is called m-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Weight X by some w : 2X → R≥0, which induces the norm ∥ · ∥ of a family  defined by ∥F∥ = �  U∈F w(U) for any F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Denote set/family subtraction by −,  while we use the symbol \\ for the different notion described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Use ← to  express substitution into a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' For simplicity, a real interval may denote the  integral interval of the same range, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=', use (1, t] instead of (1, t] ∩ Z if it is clear  by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Obvious floor/ceiling functions will be ommited throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Now let F be a family of m-sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=', F ⊂  �X  m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' We say  κ (F) =  �n  m  �  − ln |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  is the sparsity of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' The family satisfies the Γ (b)-condition on ∥ · ∥ (b ∈ R>0) if  ∥G∥ = ∥G ∩ F∥,  for all G ⊂ 2X,  and  ∥F[S]∥ < b−|S|∥F∥,  for every nonempty set S ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  As used above, the norm ∥ · ∥ can be omitted if it is induced by the unit weight,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=',  w : V �→  � 1,  if V ∈ F,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  The following theorem is proven2 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Let X be weighted to induce the norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' For every sufficiently  small ǫ ∈ (0, 1), and F ⊂  �X  m  �  satisfying the Γ  � 4γn  l  �  condition on ∥ · ∥ for some  l ∈ [n], m ∈ [l], and γ ∈  �  ǫ−2, lm−1�  , there are  ��n  l  �  (1 − ǫ)  �  sets Y ∈  �X  l  �  such  that  �  1 −  � 2  ǫγ  � � l  m  �  � n  m  � ∥F∥ <  ����  �Y  m  ����� <  �  1 +  � 2  ǫγ  � � l  m  �  � n  m  � ∥F∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  □  With Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2, we can prove the aforementioned extention generator theorem  that is about the l-extension of F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=',  Ext (F, l) =  �  T : T ∈  �X  l  �  , and ∃U ∈ F, U ⊂ T  �  ,  for l ∈ [n] − [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  It is not difficult to see  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1)  κ [Ext (F, l)] ≤ κ (F) ,  as in [3], where it is also shown:  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' For F ⊂  �X  m  �  such that m ≤ n/2,  κ  �� X  2m  �  − Ext (F, 2m)  �  ≥ 2κ  ��X  m  �  − F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  □  2In [3], the theorem uses so called Γ2 (b, 1)-condition on ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' It is straightforward to check it  means the Γ (b)-condition on ∥ · ∥ here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  EXTENSIONS OF A FAMILY FOR SUNFLOWERS  3  Further denote  Gp = G × G × · · · × G  �  ��  �  p  ,  X = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' , Xp) ∈  �  2X�p ,  Rank (X) = p,  and  Union (X) =  p�  j=1  Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Suppose m divides n and p = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' If Union (X) = X, and all Xi are mutually  disjoint n/m-sets, then such an X is an m-split of X with Xi called strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Its  subsplit X′ of rank r ∈ [m], or r-subsplit of X, is the tuple of some r strips of X  preserving the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  A set S is on X′ if S ⊂ Union  �  X′�  , and |Xi ∩ S| ∈ {0, 1} for every strip Xi of  X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Denote  by 2X′ the family of all sets on X′,  by  �X′  p  �  the family of p-sets on X′,  by X \\ X′ the subsplit of rank m − m′ consisting of the strips in X but not in  X′,  and by X′ \\ B for a set B, abusing the symbol \\, the subsplit of X consisting of  the strips each disjoint with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  For notational convenience, allow Rank  �  X′�  = 0 for which X′ = (∅), Union  �  X′�  =  ∅, and  �X′  p  �  = {∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' We have:  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' For any nonempty family F ⊂  �X  m  �  such that m divides n = |X|,  there exists an m-split X of X such that  ����F ∩  �X  m  ����� ≥  � n  m  �m |F|  � n  m  � > |F| exp (−m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  □  The lemma proven in [2] poses a special case of the general statement presented in  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1  We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2 in the two subsections below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Given F  and k, we will find a subfamily ˆF ⊂ F with a property that implies the existence  of a k-sunflower in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Formulation and Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Letting  h = exp  �1  ǫ  �  ,  and  c = exp (h) ,  assume WLOG that  k ≥ 3,  n = |X| is larger than ckm and divisible by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Otherwise add some extra elements  to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  F ⊂  �X  m  �  for an m-split X of X by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='4, satisfying the Γ (cck ln k)-condition  and |F| > (cck ln k)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  |F| < (km)m and m > cc ln k, otherwise F includes a k-sunflower by the sunflower  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  4  JUNICHIRO FUKUYAMA  FindCores  Input:  i) the family F ⊂  �X  m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Outputs:  i) C ⊂  �X  r0  �  for some r0 ∈ [0, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  ii) ˆF ⊂ �  C∈C F[C] such that | ˆF| ≥ 3−m−1|F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' F′ ← F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  ˆF ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  C ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' for r = m down to 0 do:  2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' repeat:  a) find an r-set C such that |F′[C]| ≥ f(r) putting TC ← F′[C];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  b) if found then:  F′ ← F′ − TC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  ˆF ← ˆF ∪ TC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  C ← C ∪ {C};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  else exit Loop 2-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' if | ˆF| ≥ 3−m+r−1|F| then return  �  r, C, ˆF  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Algorithm FindCores  Let  i ∈ [k],  r ∈ [0, m],  b = ck ln k,  δ =  ǫ  k ln k,  F′, Fi ⊂ F,  C ∈ 2X,  and  Yi ∈ 2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  A tuple Z = (C, Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' F1, Y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' F2, · · · , Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Fk) is said to be a partial sunflower of  rank r over F′ if there exists an r-subsplit X∗ of X satisfying the four conditions:  Z-i) C ∈ �m/c  u=0  �X∗  u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Z-ii) Yi are mutually disjoint k subsets of Union (X∗ \\ C) such that  |Yi ∩ X†| = δ|X†| for each strip X† of X∗ \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Z-iii) The k families Fi are each nonempty included in  F′[C] ∩  �X − Union (X∗ \\ C) ∪ Yi  m  �  ,  and are identical if Rank (X∗ \\ C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Z-iv) |Fi| < 2|Fi′| for i ∈ [k] and i′ ∈ [k] − {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  We say that such an Fi occurs on Z and in Z with the core C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Also Z and  Fi are on X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' A family Z of Z on one or more X∗ is a partial sunflower family  (PSF) of rank r over F′, if each two Fi occurring on two different Z are mutually  disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=', the universal disjoint property of Z is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Denote  F (Z) :=  �  Z∈Z  i∈[k]  Fi of Z,  for any PSF Z abusing the symbol F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  In the rest of our proof, we construct a nonempty PSF of rank m over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' This  means a k-sunflower in F proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  With  f : Z≥0 → R≥0,  x �→ ǫ3m(chk)−x  k  |F|,  EXTENSIONS OF A FAMILY FOR SUNFLOWERS  5  obtain the families C and ˆF by the algorithm FindCores described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' It  is straightforward to see that the two outputs correctly satisfy the properties i)-ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  In addition:  A) | ˆF[U]| < f(|U|) for all U ∈ �m  r′=r0+1  �X  r′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  B) We will construct partial sunflowers over ˆF with cores C in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' The families TC  Step 2-1 finds for r = r0 are mutually disjoint each with |TC| ≥ f(r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' By the  Γ (cck ln k)-condition of F and cc ln k < m,  k−1ǫ3m(chk ln k)−|C||F| = f (r0) ≤ |TC| ≤ |F[C]| < (cck ln k)−|C||F|,  ⇒  r0 = |C| < ln k − 3m ln ǫ  (c − h) ln c  < m  2c  �ln k  m + 1  �  < m  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Define a statement on the obtained objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Proposition Πr for r ∈ [r0, m]: there exists a PSF Z of rank r over ˆF such that  |F(Z)| > ǫ2r−2r0| ˆF|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  □  Such a Z is said to be r-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' By definition, Z is the union of PSFs ZX∗ on  r-subsplits X∗ satisfying the universal disjoint property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Our final goal of finding a nonempty PSF of rank m over F would be met if Πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  The proposition Πr0 holds since  Z =  \uf8f1  \uf8f2  \uf8f3  \uf8eb  \uf8edC, ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' TC, ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' TC, · · · , ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' TC  �  ��  �  k  \uf8f6  \uf8f8 : C ∈ ˆC  \uf8fc  \uf8fd  \uf8fe  is an r0-normal PSF such that F (Z) = ˆF by B) where TC are the ones mentioned  there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' So it suffices to show  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1)  Πr ⇒ Πr+1,  for every r ∈ [r0, m),  to have proof of a k-sunflower in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' We start showing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1) as the only remaining task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Assume  Πr for a particular r ∈ [r0, m), so we have an r-normal PSF Z that is the union of  ZX∗ on some r-subsplits X∗ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' We confirm Πr+1 in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Reconstruct Z into another PSF Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Obtain such a Z′ by the algorithm  Reconstruct described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' It is a PSF of rank r over F (Z) satisfying the  two conditions:  C) |F (Z′) | > 2−1ǫ|F (Z) |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  D) For each Z ∈ Z′ on an r-subsplit X∗, there exists an r+1-subsplit X′ containing  X∗ such that each Fi on Z meets  |Fi[S]| < 1  b |Fi|,  ∀S ∈  �X′ \\ X∗  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  □  We see their truth by the notes below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Such a Z ∈ Z′ is said to be on the split  pair  �  X∗, X′�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' In Steps 2 and 3, we will construct our desired r + 1-normal PSF  Z′′ from Z′ confirming Πr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Justification of Z′ being a PSF with C) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  F (ZX′) of an X′ disregarded by Step 2-3 is negligible as their union will be  smaller than 2−m� m  r+1  �  <  � 3  2  �−m <  � 3  2  �−cc ln k < ǫ3/k of F (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  6  JUNICHIRO FUKUYAMA  Reconstruct  Input:  an r-normal PSF Z for some r ∈ [r0, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Output:  a PSF Z′ of rank r over F (Z) satisfying C) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' X ← the family of all r-subsplits X∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  /* The given Z is the union of PSFs ZX∗ on some X∗ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' */  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' for each r + 1-subsplit X′ do:  2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' X∗ ← the family of all X∗ ∈ X that are r-subsplits of X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' ZX′ ← �  X∗∈X∗ ZX∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' if |F (ZX′) | < 3−m|F (Z) | then go to Step 2 for the next X′ else X ← X − X∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' for each Fi in ZX′ do F′  i ← Fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' for each Fi in ZX′ and on X∗, and sets B ∈  �X∗  r  �  and U ∈  � X′  r+1  �  [B] such that  |Fi[U]| >  �  c  √  hk ln k  �−r−1  |Fi[B]| do F′  i ← F′  i − Fi[U];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' for each Z ∈ ZX′ do:  a) if |F′  i| < ǫ|Fi| for some Fi on Z then delete Z from ZX′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  b) else normalize Z for the condition Z-iv) as follows:  b)-i) for each Fi on Z do:  γi ← |F′  i|−1 mini′∈[k] |F′  i′|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  F′  i ← any subfamily of F′  i of cardinality min (|F′  i|, ⌊2γi|F′  i|⌋);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  b)-ii) replace all Fi by the F′  i to reconstruct Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' return the union of all ZX′ found in Loop 2 as Z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Algorithm Reconstruct  |F(ZX′)[U]| ≤ | ˆF[U]| < f (r + 1) for all U ∈  � X  r+1  �  by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' So,  |F (ZX′) [U]|  |F (ZX′) |  <  f (r + 1)  3−mǫ2r−2r0| ˆF|  <  �  chk ln k  �−r−1  k  ,  before Step 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Fi[B] are mutually disjoint for all different Fi occurring in ZX′ each on an X∗,  and B ∈  �X∗  r  �  right before Step 2-5, by the universal disjoint property of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' (By  the rule Z-iii), Fi on a single Z are identified when r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=') In addition, Fi[U]  are mutually disjoint for all Fi and U ∈  � X′  r+1  �  meeting  �  X∗∈X∗, Fi in ZX′ and on X∗  B∈(X∗  r ), U∈(X′  m′)[B]  Fi[U] = F(ZX′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  By the above two, Step 2-5 may reduce  V =  �  Fi in ZX′  F′  i  by less than its ǫ3/k, leaving only Fi, B, and U such that  |F′  i[U]| < (cb)−r−1|F′  i[B]|,  ⇒  |F′  i[B]| > (cb)r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  By Z-iv) of Z, Step 2-6-a) may only reduce less than 2ǫ3 of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  The process of normalization is well-defined by Step 2-6-b) due to |F′  i| > (cb)r+1  before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' It could further reduce V into its ǫ/2 or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  EXTENSIONS OF A FAMILY FOR SUNFLOWERS  7  The condition Z-iv) of the obtained Z′ follows the above as well as the two  properties C) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' For each Fi occurring in Z′, construct a family Yi of Y ∈  �  X†  δ|X†|  �  such  that Fi ∩  �X−X†∪Y  m  �  is sufficiently large, where X† = Union(X′ \\ X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Consider  each Z ∈ Z′ on (X∗, X′) with the unique strip X† of X′ \\ X∗, and an Fi on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Weight X† by 2X† → Z≥0, W �→ |Fi[W]| inducing the norm ∥ · ∥ as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  The family H =  �X†  1  �  satisfies the Γ(b)-condition on ∥ ·∥ by D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2  to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' There exists Yi ⊂  � X†  δ|X†|  �  such that  |Yi| >  � |X†|  δ|X†|  �  [1 − exp (−h)] ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2)  δ [1 − exp (−h)] |Fi| <  ��FY  i  �� < δ [1 + exp (−h)] |Fi|,  for every Y ∈ Yi,  where  FY  i ⊂ Fi ∩  �X − X† ∪ Y  m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Find Z′′ by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Now consider the same Z with the k families Fi and  sets Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Put  δ′ = 2δ ln k,  and  Y′  i = Ext (Yi, δ′|X†|) ,  for each Fi to see  |Y′  i| >  � |X†|  δ′|X†|  �  [1 − exp (−h ln k)] >  � |X†|  δ′|X†|  � �  1 − ǫ  k  �  ,  by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='3, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2): to the Yi, repeatedly apply the lemma ⌈log2 ln k⌉  times doubling the second parameter of Ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Then κ  ��  X†  δ′|X†|  �  − Y′  i  �  > h ln k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Hence, there exist more than  � |X†|  δ′|X†|  ��|X†| − δ′|X†|  δ′|X†|  �  · ·  �|X†| − (k − 1)δ′|X†|  δ′|X†|  �  (1 − ǫ)  tuples (Y ′  1, Y ′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' , Y ′  k) ∈  �  X†  δ′|X†|  �k such that each Y ′  i is in Y′  i, disjoint with the other  k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  For such a (Y ′  1, Y ′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' , Y ′  k), find a δ|X†|-set Y †  i ∈ Yi included in each Y ′  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Add  the tuple  �  C, Y1 ∪ Y †  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' F  Y †  1  1 , Y2 ∪ Y †  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' F  Y †  2  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' , Yk ∪ Y †  k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' F  Y †  k  k  �  to Z′′, where the set C is that of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' By construction, it satisfies the conditions Z-i)  to iii) with F′ ← F (Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Subtract �k  i=1 F  Y †  1  i  from Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Repeat the above ǫ−1/2 times including Step 2 for  the current Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Then denote an element of Z′′ by Z′, and family F  Y †  i  i  by F′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Such  a Z′ and F′  i are produced from Z and Fi, and we assume it for the four objects  anywhere below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Finally, normalize each Z′ for Z-iv) the same way as Step 2-6-b)-i) of Recon-  struct with the γi given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' It possibly reduces F′  i into its 1 − exp (−h/2) or  larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Perform the process for all Z ∈ Z′ to complete our construction of Z′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  8  JUNICHIRO FUKUYAMA  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Confirm Πr+1 to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' For the r + 1-normality of Z′′, it can  be checked by straightforward recursive arguments with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2) that  1  2ǫ1/2|Fi| < ∆|Fi| < 2ǫ1/2|Fi|,  where |Fi| expresses the value after Step 1, and ∆|Fi| the difference between |Fi|  and its final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' This means Step 2 can use the Γ  �  b  �  1 − 2ǫ1/2��  condition on ∥ ·  ∥B instead of the Γ (b)-condition throughout the construction, constantly achieving  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='2) for a Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' In addition, the normalization of each Z′ always keeps more than  half of its �k  i=1 F′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Hence, the recursive loop for every Z terminates without an exception defining  our Z′′ with |F (Z′′) | > ǫ2/3|F (Z′) | > ǫ2|F (Z) | by C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' As it is a PSF with the  universal disjoint property by construction, we confirm the proposition Πr+1 to  complete our proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  We now have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Erd¨os, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=', Rado, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' : Intersection theorems for systems of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Journal of the London Math-  ematical Society, Second Series, 35 (1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' 85 - 90 (1960)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Fukuyama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' : The sunflower conjecture proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='13609 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='CO] (2022)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Fukuyama,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=':  Improved  bound  on  sets  including  no  sunflower  with  three  petals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='10318v3 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='CO] (2021)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Fukuyama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' : The case k = 3 of the sunflower conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content=' Private work available at Penn  State Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='  Web address:  https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='edu/sunflowerconjecture/files/2023/01/Proof-of-the-3-petal-  sunflower-conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='pdf (2023)  Department of Computer Science and Engineering, The Pennsylvania State Univer-  sity, PA 16802, USA  E-mail address: jxf140@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'}

BNFAT4oBgHgl3EQfsB5-/content/tmp_files/2301.08656v1.pdf.txt

@@ -0,0 +1,1052 @@
+Quantum Control of Trapped Polyatomic Molecules for eEDM Searches
+Lo¨ıc Anderegg,1, 2, ∗ Nathaniel B. Vilas,1, 2 Christian Hallas,1, 2 Paige
+Robichaud,1, 2 Arian Jadbabaie,3 John M. Doyle,1, 2 and Nicholas R. Hutzler3, †
+1Department of Physics, Harvard University, Cambridge, MA 02138, USA
+2Harvard-MIT Center for Ultracold Atoms, Cambridge, MA 02138, USA
+3Division of Physics, Mathematics, and Astronomy,
+California Institute of Technology, Pasadena, CA 91125, USA
+(Dated: January 23, 2023)
+Ultracold polyatomic molecules are promising candidates for experiments in quantum science,
+quantum sensing, ultracold chemistry, and precision measurements of physics beyond the Standard
+Model. A key, yet unrealized, requirement of these experiments is the ability to achieve full quantum
+control over the complex internal structure of the molecules. Here, we establish coherent control of
+individual quantum states in a polyatomic molecule, calcium monohydroxide (CaOH), and use these
+techniques to demonstrate a method for searching for the electron electric dipole moment (eEDM).
+Optically trapped, ultracold CaOH molecules are prepared in a single quantum state, polarized in
+an electric field, and coherently transferred into an eEDM sensitive state where an electron spin
+precession measurement is performed. To extend the coherence time of the measurement, we utilize
+eEDM sensitive states with tunable, near-zero magnetic field sensitivity. The spin precession coher-
+ence time is limited by AC Stark shifts and uncontrolled magnetic fields. These results establish a
+path for eEDM searches with trapped polyatomic molecules, towards orders-of-magnitude improved
+experimental sensitivity to time-reversal-violating physics.
+The rich structure of polyatomic molecules makes them
+an appealing platform for experiments in quantum sci-
+ence [1–4], ultracold chemistry [5], and precision mea-
+surements [6–10]. Key to this structure is the presence
+of near-degenerate states of opposite parity, which allow
+the molecules to be easily polarized in the laboratory
+frame with the application of a small electric field. Such
+states are a novel resource, generic among polyatomic
+molecules while rare in diatomics, that may be useful
+for applications such as analog simulation of quantum
+magnetism models [1, 2] or for realizing switchable inter-
+actions and long-lived qubit states for quantum comput-
+ing [4]. Additionally, the parity-doublet states in trapped
+polyatomic molecules are expected to be an invaluable
+tool for systematic error rejection in precision measure-
+ments of physics beyond the Standard Model (BSM) [6].
+To date, several species of polyatomic molecules have
+been laser cooled and/or trapped at ultracold temper-
+atures [11–17].
+One powerful avenue for tabletop BSM searches is
+probing for the electric dipole moment of the electron
+(eEDM) [18–22], de, which violates time-reversal (T)
+symmetry and is predicted by many BSM theories to
+be orders of magnitude larger than the Standard Model
+prediction [19, 20]. Current state-of-the-art eEDM ex-
+periments are broadly sensitive to T-violating physics at
+energies much greater than 1 TeV [23–28]. All such ex-
+periments use Ramsey spectroscopy to measure an en-
+ergy shift due to the interaction of the electron with the
+large electric field present inside a polarized molecule [24–
+27, 29]. Molecular beam experiments have achieved high
+statistical sensitivity by measuring a large number of
+molecules over a ≈ 1 ms coherence time [24, 25], while
+molecular ion-based experiments have used long Ram-
+sey interrogation times (≈ 1 s) though with lower num-
+bers [26, 27, 29].
+Measurements with trapped neutral
+polyatomic molecules can potentially combine the best
+features of each approach to achieve orders-of-magnitude
+improved statistical sensitivity [6].
+In this Report, we demonstrate full quantum control
+over the internal states of a trapped polyatomic molecule
+in a vibrational bending mode with high polarizability
+in small electric fields. The method starts with prepar-
+ing ultracold, optically trapped molecules in a single hy-
+perfine level, after which a static electric field is applied
+to polarize the molecules.
+The strength of the polar-
+izing electric field is tuned to obtain near-zero g-factor
+spin states, which have strongly suppressed sensitivity
+to magnetic field noise while retaining eEDM sensitivity.
+Microwave pulses are applied to create a coherent super-
+position of these zero g-factor spin states that precesses
+under the influence of an external magnetic field. The
+precession phase is then read out by a combination of
+microwave pulses and optical cycling.
+We observe spin precession over a range of electric and
+magnetic fields and characterize the current limitations
+to the coherence time of the measurement. With readily
+attainable experimental parameters, coherence times on
+the order of the state lifetime (>100 ms) could be realisti-
+cally achieved. We therefore realize the key components
+of an eEDM measurement in this system. Although the
+light mass of CaOH precludes a competitive eEDM mea-
+surement [30], the protocol demonstrated here is directly
+transferable to heavier laser-cooled alkaline earth mono-
+hydroxides with identical internal level structures, such
+as SrOH, YbOH, and RaOH, which have significantly en-
+arXiv:2301.08656v1  [physics.atom-ph]  20 Jan 2023
+
+2
+FIG. 1.
+(a) A geometric picture of the bending molecule at the zero g-factor crossing, showing the electron spin (⃗S) has a finite
+projection on the molecule axis (ˆn), giving eEDM sensitivity. However, the electron spin (⃗S) is orthogonal to the magnetic field
+( ⃗B), resulting in suppressed magnetic field sensitivity. (b) The magnetic sensitivity (upper plot) and eEDM sensitivity (lower
+plot) for a pair of zero g-factor states (N = 1, J = 1/2+, F = 1, MF = ±1) are shown as a function of the applied electric
+field. (c) Experimental sequence to prepare the eEDM sensitive state. First, the molecules are pumped into a single quantum
+state (N = 1, J = 1/2−, F = 0) with a combination of microwave drives and optical pumping (I). Next, a microwave π-pulse
+drives the molecules into the N = 2, J = 3/2−, F = 2, MF = 0 state (II). Lastly, the eEDM measurement state is prepared as
+a coherent superposition of the N = 1, J = 1/2−, F = 1 MF = ±1 states with a microwave π-pulse (III). The states which are
+optically detectable with the detection light are shown in black, while those not addressed by the detection light are in grey.
+hanced sensitivity to the eEDM [6, 11, 12, 30, 31].
+In eEDM measurements with polarized molecules, the
+electron spin ⃗S precesses under the influence of an ex-
+ternal magnetic field BZ and the internal electric field of
+the molecule, Eeff, which can be large due to relativistic
+effects. Time evolution is described by the Hamiltonian
+H = gSµBBZ ⃗S · ˆZ − deEeff⃗S · ˆn
+= gSµBBZMS − deEeffΣ.
+(1)
+Here, gS ≈ 2 is the electron spin g-factor, µB is the
+Bohr magneton, BZ points along the lab ˆZ axis, and
+the internal field Eeff points along the molecule’s inter-
+nuclear axis ˆn.
+We define the quantities MS = ⃗S · ˆZ
+and Σ = ⃗S · ˆn to describe the electron’s magnetic sen-
+sitivity and EDM sensitivity, respectively. The effect of
+the eEDM can be isolated by switching the orientation of
+the applied magnetic field or, alternatively, by switching
+internal states to change the sign of MS or Σ. Perform-
+ing both switches is a powerful technique for suppressing
+systematic errors [25, 26].
+Current EDM bounds rely on specific states in di-
+atomic molecules that have an unusually small g-factor,
+reducing sensitivity to stray magnetic fields [24, 26].
+However, CaOH, like other laser-coolable molecules with
+structure amenable to eEDM searches [6, 31–33], has
+a single valence electron, which results in large mag-
+netic g-factors. In this work, we engineer reduced mag-
+netic sensitivity by using an applied electric field EZ to
+tune MS to a zero-crossing, while maintaining signifi-
+cant eEDM sensitivity Σ. This technique is generic to
+polyatomic molecules with parity-doublets. Details of a
+specific M = ±1 pair of zero g-factor states are shown
+in Figure 1 (a)-(b), with further information in the Sup-
+plemental Material. Sensitivity to transverse magnetic
+fields is also suppressed in these zero g-factor states (see
+Supplemental Material).
+The
+experiment
+begins
+with
+laser-cooled
+CaOH
+molecules loaded from a magneto-optical trap [14] into
+an optical dipole trap (ODT) formed by a 1064 nm laser
+beam with a 25 µm waist size, as described in previous
+work [15]. The ODT is linearly polarized and its polar-
+ization vector ⃗ϵODT defines the ˆZ axis, along which we
+also apply magnetic and electric fields, ⃗B = BZ ˆZ and
+⃗E = EZ ˆZ, respectively, as depicted in Figure 1(a). We
+first non-destructively image the molecules in the ODT
+for 10 ms as normalization against variation in the num-
+ber of trapped molecules. The molecules are then opti-
+cally pumped into the N = 1− levels of the �
+X2Σ+(010)
+vibrational bending mode [15] (Figure 1(c)), and the trap
+depth is adiabatically lowered by 3.5× to reduce the effect
+of AC Stark shifts from the trap light and to lower the
+temperature of the molecules to 34 µK. Any molecules
+that were not pumped into N = 1− levels of the bending
+
+(c)
+A21(010)2-
+(a)
+(b)
+1/2
+0,1+
+μb(Ms) (MHz/G)
+0.3
+0.2
+623 nm
+2
+M=
+E,B,EoDT
+ 0. 0
+(Z)= . 0
+1
+M=+1
+3/2
+-0.2
+2-
+n
+Ca
+40 GHz
+(010)+3zX
+0.6
+Sensitivity (22)
+Relative EDM
+0.4
+M=-1
+0.2 
+0+
+1/2
+<Ms) = 3.2 = 0
+1+
+0.0 
+-0.2 
+21
+M=+1
+3/2
+0.4
+0-
+0.6.
+_1/2
+1-
+40
+50
+60
+70
+80
++1
+N
+J
+MF
+F
+E (Vlcm)
+(I)
+(II)
+(III)3
+FIG. 2. (a) Spin precession of the eEDM sensitive state in the presence of a bias magnetic field. (b) Magnetic field sensitivity
+of the eEDM state in CaOH as a function of electric field. The field sensitivity is determined by measuring the spin precession
+frequency at different electric fields with an applied magnetic field of BZ = 110 mG. Error bars are smaller than the markers.
+The solid curve is the calculated magnetic field sensitivity in the presence of trap shifts using known molecular parameters, as
+described in the Supplemental Material.
+mode are heated out of the trap with a pulse of resonant
+laser light.
+Following transfer to the �
+X2Σ+(010)(N = 1−) state,
+the molecular population is initially spread across twelve
+hyperfine Zeeman sublevels in the spin-rotation compo-
+nents J = 1/2 and J = 3/2. To prepare the molecules in
+a single hyperfine state, we use a combination of optical
+pumping and microwave pulses, as shown in Figure 1(c).
+We first apply microwaves from the (N = 1, J = 3/2−)
+state up to the (N = 2, J = 3/2−) state.
+As this
+transition is parity-forbidden, we apply a small electric
+field EZ = 7.5 V/cm to slightly mix the parity of the
+N = 1 levels and provide transition strength. From the
+N = 2 state, we drive an optical transition to the excited
+�A2Π(010)κ2Σ(−), J = 1/2+ state. This state predomi-
+nately decays to both F = 0 (the target state) and F = 1
+states in the N = 1, J = 1/2− manifold. After 3 ms of
+optical pumping, the microwaves are switched to drive
+the accumulated N = 1, J = 1/2−, F = 1 population to
+the same N = 2, J = 3/2− state in �
+X(010), where they
+are excited by the optical light and pumped into the tar-
+get F = 0 state. Once this optical pumping sequence
+is complete, we adiabatically ramp the electric field to
+EZ =150 V/cm to significantly mix parity, then drive
+population up to the N = 2, J = 3/2−, F = 2, M = 0
+state with a microwave π-pulse (Figure 1(c)(II)). We
+clean out any remaining population in the N = 1 state
+with a depletion laser that resonantly drives population
+to undetected rotational levels.
+To perform spin precession in the eEDM sensitive
+state, we first adiabatically ramp the electric field to a
+value EZ, then turn on a small bias magnetic field BZ.
+We measure the electron spin precession frequency using
+a procedure analogous to Ramsey spectroscopy [24, 25].
+The molecules are prepared by driving a π-pulse (2.5
+µs), with microwaves linearly polarized along the lab ˆX
+axis, into the “bright” superposition state |B⟩ = (|M =
+1⟩ + |M = −1⟩)/
+√
+2 within the N = 1, J = 1/2+, F =
+1, M = ±1 eEDM sensitive manifold (Figure 1(c)). The
+state begins to oscillate between the bright state and the
+“dark” state |D⟩ = (|M = 1⟩ − |M = −1⟩)/
+√
+2 at a
+rate ωSP = µeffBZ, where the effective magnetic moment
+µeff = µBgeff = gSµB(⟨MS⟩M=1 − ⟨MS⟩M=−1) is tuned
+via the applied electric field EZ (Figure 1(b)). The con-
+tribution from the deEeff term in eqn. 1 is negligible in
+CaOH, but could be measured in heavier molecules with
+much larger Eeff. After a given time, a second π-pulse
+is applied to stop spin precession and transfer the bright
+state to the optically detectable N = 2, J = 3/2− level.
+Once the electric field is ramped down, the population
+remaining in the eEDM manifold, which has the oppo-
+site parity, is not optically detectable. We then image
+the ODT again and take the ratio of the first and sec-
+ond images (Figure 2(a)). At long spin precession times
+(> 10 ms), losses from background gas collisions (∼1 sec),
+blackbody excitation (∼1 sec), and the spontaneous life-
+time of the bending mode (∼0.7 sec) lead to an overall
+loss of signal, as characterized in Ref. [15]. This effect
+is mitigated with a fixed duration between the first and
+second images, making the loss independent of the pre-
+cession time.
+To map out the location of the zero g-factor cross-
+ing, we perform spin precession measurements at a fixed
+magnetic field BZ = 110 mG for different electric fields
+(Figure 2(b)). The spin precession frequency corresponds
+to an effective g-factor at that electric field.
+We find
+that the zero g-factor crossing within the N = 1, J =
+1/2+, F = 1, M = ±1 eEDM manifold occurs at an elec-
+tric field of 59.6 V/cm, in agreement with theory cal-
+culations described in the Supplemental Material.
+We
+note that there is another zero g-factor crossing for the
+N = 1, J = 3/2+, F = 1 manifold at ≈ 64 V/cm, which
+
+(a)
+(b)
+(au)
+0.6
+0.5
+ remaining
+(MHz/G)
+0.55
+0
+0.05
+0.5
+eff
+Fraction
+0
+0.45
+-0.05
+-0.5
+58
+59
+60
+61
+0.4
+40
+50
+60
+70
+500
+1000
+80
+0
+Time (μs)
+E- (V/cm)4
+FIG. 3. Coherence time of the spin precession signal. (a) Measured coherence times τ versus BZ at different electric fields
+(red and blue markers, corresponding to different magnetic field sensitivity). The coherence time scales as 1/BZ due to AC
+Stark shift broadening, then plateaus at a limit set by the magnetic field instability δB. This limit increases as the g-factor
+approaches zero. Solid and dashed curves are fit to the data. The ambient magnetic field noise determined from the fit is
+δB = 4+2
+−1 mG, while the fitted decoherence time due to light shifts is τ = (1/BZ) × 80+20
+−10 ms×mG. (b) The spin precession
+coherence time at BZ = 15 mG is extended by 16× by approching the zero g-factor point.
+has a smaller eEDM sensitivity but the opposite slope
+of geff vs.
+EZ, thereby providing a powerful resource
+to reject systematic errors related to imperfect field re-
+versals (see Supplemental Material). We emphasize that
+while the location of these crossings is dependent on the
+structure of a specific molecule, their existence is generic
+in polyatomic molecules, which naturally have parity-
+doublet structure [6].
+A critical component of the spin precession measure-
+ment is the coherence time, which sets the sensitivity
+of an eEDM search.
+Figure 3(a) shows the measured
+coherence time of our system at different applied fields
+BZ and EZ. We characterize two dominant limitations
+that wash out oscillations at long times. Variations in
+the spin precession frequency can be linearly expanded
+as δωSP = µeff(δBZ) + (δµeff)BZ.
+The first term de-
+scribes magnetic field noise and drift of the applied bias
+field, given by δBZ. The second term describes noise and
+drifts in the g-factor, δgeff, which can arise from insta-
+bility in the applied electric field, EZ, or from AC Stark
+shifts (described below). Drifts in the bias electric field
+EZ are found to be negligible in our apparatus.
+Decoherence due to magnetic field noise, δBZ, is inde-
+pendent of the applied magnetic field but is proportional
+to µeff, and can be mitigated by operating near the zero g-
+factor crossing. As shown in Fig. 3(b), at an electric field
+of 90 V/cm, corresponding to a large magnetic moment
+of µeff = 1.0 MHz/G, we realize a magnetic field noise-
+limited coherence time of 0.5 ms at BZ ≈ 15 mG. At an
+electric field of 61.5 V/cm, corresponding to µeff = 0.06
+MHz/G, much closer to the zero g-factor location, we
+find a coherence time of 4 ms at the same BZ.
+At higher magnetic fields, the primary limitation to
+the coherence time is due to AC stark shifts from the
+optical trapping light (Fig. 4). The intense Z-polarized
+ODT light leads to a shift in the electric field at which the
+zero g-factor crossing occurs. Due to the finite temper-
+ature of the molecules within the trap, they will explore
+different intensities of trap light and hence have differ-
+ent values of geff. The spread δgeff causes variation of
+ωSP, which leads to decoherence. In contrast to the mag-
+netic field noise term, this effect is independent of the
+electric field EZ but decreases monotonically with BZ,
+which scales the frequency sensitivity to g-factor vari-
+ations, δωSP = BZδµeff.
+The insensitivity of g-factor
+broadening to the exact value of geff is demonstrated in
+Fig. 4(c). Decoherence due to AC Stark shifts can be
+reduced by cooling the molecules to lower temperatures
+or by decreasing BZ.
+The bias magnetic field can be
+reduced arbitrarily far until either transverse magnetic
+fields or magnetic field noise become dominant.
+From
+the decoherence rates measured in this work, it is ex-
+pected that AC Stark shift-limited coherence times ∼1 s
+could be achieved at bias fields of BZ ∼ 100 µG.
+From the above discussion, it is expected that the
+longest achievable coherence times will occur for very
+small g-factors, geff ≈ 0, and very small bias fields,
+BZ ≈ 0. Minimizing BZ requires reducing the effects of
+
+(a)
+(b)
+0.55
+0.06 MHz/G
+Ambient Magnetic Field Noise
+(ne)
+10
+1.0 MHz/G
+raction
+0.5
+5
+Trap Light Shift
+0.45
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+(sw
+Time (μus)
+Abient Magnetic Field Noise
+0.5
+0.5
+0.1
+0
+0.06 MHz/G
+g
+1.0 MHz/G
+0.45
+5
+10
+50
+100
+500
+0
+100
+200
+300
+400
+500
+600
+(mG)
+Time (μs)5
+FIG. 4. Effect of trap light on coherence time. (a) The trap
+light shifts the location of the zero crossing in µeff. As a result,
+molecules at a finite temperature explore different magnetic
+field sensitivities µeff. (b) Dependence of the spin precession
+frequency (scaled by the trap depth U0) on position within
+the trap.
+At lower magnetic fields, the relative change in
+spin precession frequency is reduced. (c) Two spin precession
+curves taken at the same magnetic field (BZ = 210 mG) but
+at different electric fields, showing that the AC Stark shift
+limitation is independent of the effective g-factor, since AC
+Stark shifts dominate the coherence time for large bias fields.
+both magnetic field noise and transverse magnetic fields
+to well below the level of the bias field energy shifts. We
+cancel the transverse magnetic fields to below 1 mG by
+maximizing the spin precession period under the influ-
+ence of transverse B fields only, and actively monitor
+and feedback on the magnetic field along each axis to
+minimize noise and drifts in BZ. Note that the stainless
+steel vacuum chamber has no magnetic shielding, lead-
+ing to high levels of magnetic field noise which would not
+be present in an apparatus designed for an eEDM search.
+Even under these conditions, we achieve a coherence time
+of 30 ms at an electric field of 60.3 V/cm (corresponding
+to µeff = 0.02 MHz/G) and a bias field of BZ ≈ 2 mG,
+(see Supplemental Material). However, at such a low bias
+field, the molecules are sensitive to 60 Hz magnetic field
+noise present in the unshielded apparatus, which is on
+the same order as the bias field. Since the experiment is
+phase stable with respect to the AC line frequency, this
+60 Hz magnetic field fluctuation causes a time-dependent
+spin precession frequency. Nevertheless, our prototype
+experiment confirms that long coherence times are possi-
+ble. Any future eEDM experiment would have magnetic
+shielding that would greatly suppress nefarious magnetic
+fields from the environment. Such shielding could readily
+enable coherence times exceeding that of the ∼ 0.5 s life-
+time of the bending modes of similar linear polyatomic
+molecules with larger eEDM sensitivity [15].
+In summary, we have realized coherent control of opti-
+cally trapped polyatomic molecules and demonstrated a
+realistic experimental roadmap for future eEDM mea-
+surements.
+By leveraging the unique features of the
+quantum levels in polyatomic molecules, we achieve a
+coherence time of 30 ms for paramagnetic molecules in a
+stainless steel chamber with no magnetic shielding. With
+common shielding techniques employed in past EDM ex-
+periments, there is a clear path to reducing stray fields
+and extending coherence times to > 100 ms.
+At such
+a level, the dominant limitation becomes the finite life-
+time of the bending mode [15]. Even longer coherence
+times are possible with the right choice of parity dou-
+blet states, as found in symmetric or asymmetric top
+molecules [6, 13, 34, 35].
+Following
+our
+established
+roadmap
+with
+heavier
+trapped polyatomic molecules has the potential to
+provide orders-of-magnitude improvements to current
+bounds on T-violating physics.
+Using a recent study
+of the �
+X(010) state in YbOH [36], we have identified
+similar N = 1 zero g-factor states for eEDM measure-
+ments with significantly improved sensitivity. In addi-
+tion to the g-factor tuning demonstrated in this work,
+polyatomic molecules provide the ability to reverse the
+sign of Σ without reversing MS - a crucial feature of re-
+cent experiments that have greatly improved the limit
+on the eEDM [25, 27]. For example, in the N = 1 mani-
+fold of CaOH, there is another zero g-factor crossing at a
+nearby electric field value, with 69% smaller values of Σ
+and opposite sign. Since the ratio of eEDM sensitivity to
+g-factor vs. EZ slope differs between these two crossings,
+measurements at both points could be used to suppress
+systematics due to non-reversing fields coupling to the
+electric field dependence of the g-factor [25].
+This work provides a first experimental demonstration
+of the advantages of the rich level structure of polyatomic
+molecules for precision measurements.
+While we have
+focused here on spin precession with T-reversed states
+(M = ±1), many levels of interest can be favorably en-
+gineered for precision measurement experiments.
+In a
+recent proposal [9], parity-doublets, magnetically tuned
+to degeneracy in optically trapped polyatomic molecules,
+were shown to be advantageous for searches for parity vi-
+olating physics. In another recent work [7], a microwave
+clock between rovibrational states in SrOH was proposed
+as a sensitive probe of ultra-light dark matter, utilizing
+transitions tuned to electric and/or magnetic insensitiv-
+ity. In these proposals, and now experimentally demon-
+strated in our work, coherent control and state engineer-
+ing in polyatomic molecules can mitigate systematic er-
+rors and enable robust searches for new physics.
+
+(a)
+(b)
+0.04
+10
+μeff (MHz/G)
+0.02
+100 mG
+I=1/2 
+I=0
+0.
+5
+50 mG
+-0.02
+10 mG
+0
+-0.04
+59
+60
+61
+-wo
+0
+Wo
+Ez (V/cm)
+Position in trap
+(c)
+0.58
+0.01 MHz/G, 210 mG
+(a.u.)
+0.06 MHz/G, 210 mG
+0.55
+Fraction remaining (
+0.52
+0.49
+0.46
+0.43
+0.4
+0
+200
+400
+600
+800
+1000
+Time (μs)6
+This work was supported by the AFOSR and the NSF.
+LA acknowledges support from the HQI, NBV from the
+DoD NDSEG fellowship program, and PR from the NSF
+GRFP. NRH and AJ acknowledge support from NSF CA-
+REER (PHY-1847550), The Gordon and Betty Moore
+Foundation (GBMF7947), and the Alfred P. Sloan Foun-
+dation (G-2019-12502). AJ acknowledges helpful discus-
+sions with Chi Zhang and Phelan Yu.
+∗ [email protected]
+† [email protected]
+[1] M. L. Wall, K. Maeda, and L. D. Carr, Simulating quan-
+tum magnets with symmetric top molecules, Ann. Phys.
+(Berlin) 525, 845 (2013).
+[2] M. Wall, K. Maeda, and L. D. Carr, Realizing un-
+conventional quantum magnetism with symmetric top
+molecules, New J. Phys. 17, 025001 (2015).
+[3] Q. Wei, S. Kais, B. Friedrich, and D. Herschbach, Entan-
+glement of polar symmetric top molecules as candidate
+qubits, J. Chem Phys 135, 154102 (2011).
+[4] P. Yu, L. W. Cheuk, I. Kozyryev, and J. M. Doyle, A
+scalable quantum computing platform using symmetric-
+top molecules, New J. Phys. 21, 093049 (2019).
+[5] L. D. Augustoviˇcov´a and J. L. Bohn, Ultracold collisions
+of polyatomic molecules: CaOH, New J. Phys. 21, 103022
+(2019).
+[6] I. Kozyryev and N. R. Hutzler, Precision measurement of
+time-reversal symmetry violation with laser-cooled poly-
+atomic molecules, Phys. Rev. Lett. 119, 133002 (2017).
+[7] I. Kozyryev, Z. Lasner, and J. M. Doyle, Enhanced sen-
+sitivity to ultralight bosonic dark matter in the spectra
+of the linear radical SrOH, Phys. Rev. A 103, 043313
+(2021).
+[8] N. R. Hutzler, Polyatomic molecules as quantum sensors
+for fundamental physics, Quantum Science and Technol-
+ogy 5, 044011 (2020).
+[9] E. B. Norrgard, D. S. Barker, S. Eckel, J. A. Fedchak,
+N. N. Klimov, and J. Scherschligt, Nuclear-spin depen-
+dent parity violation in optically trapped polyatomic
+molecules, Communications Physics 2, 1 (2019).
+[10] Y.
+Hao,
+P.
+Navr´atil,
+E.
+B.
+Norrgard,
+M.
+Iliaˇs,
+E. Eliav, R. G. E. Timmermans, V. V. Flambaum, and
+A. Borschevsky, Nuclear spin-dependent parity-violating
+effects in light polyatomic molecules, Phys. Rev. A 102,
+052828 (2020).
+[11] I. Kozyryev, L. Baum, K. Matsuda, B. L. Augenbraun,
+L. Anderegg, A. P. Sedlack, and J. M. Doyle, Sisyphus
+laser cooling of a polyatomic molecule, Phys. Rev. Lett.
+118, 173201 (2017).
+[12] B. L. Augenbraun, Z. D. Lasner, A. Frenett, H. Sawaoka,
+C. Miller, T. C. Steimle, and J. M. Doyle, Laser-
+cooled polyatomic molecules for improved electron elec-
+tric dipole moment searches, New J. Phys. 22, 022003
+(2020).
+[13] D. Mitra, N. B. Vilas, C. Hallas, L. Anderegg, B. L.
+Augenbraun, L. Baum, C. Miller, S. Raval, and J. M.
+Doyle, Direct laser cooling of a symmetric top molecule,
+Science 369, 1366 (2020).
+[14] N. B. Vilas, C. Hallas, L. Anderegg, P. Robichaud,
+A. Winnicki, D. Mitra, and J. M. Doyle, Magneto-
+optical trapping and sub-Doppler cooling of a polyatomic
+molecule, Nature 606, 70 (2022).
+[15] C. Hallas, N. B. Vilas, L. Anderegg, P. Robichaud,
+A. Winnicki, C. Zhang, L. Cheng, and J. M. Doyle, Opti-
+cal trapping of a polyatomic molecule in an ℓ-type parity
+doublet state, arXiv:2208.13762 (2022).
+[16] M. Zeppenfeld, B. G. Englert, R. Gl¨ockner, A. Prehn,
+M. Mielenz, C. Sommer, L. D. van Buuren, M. Motsch,
+and G. Rempe, Sisyphus cooling of electrically trapped
+polyatomic molecules, Nature 491, 570 (2012).
+[17] A. Prehn, M. Ibr¨ugger, R. Gl¨ockner, G. Rempe, and
+M. Zeppenfeld, Optoelectrical cooling of polar molecules
+to submillikelvin temperatures, Phys. Rev. Lett. 116,
+063005 (2016).
+[18] D. DeMille, J. M. Doyle, and A. O. Sushkov, Probing
+the frontiers of particle physics with tabletop-scale ex-
+periments, Science 357, 990 (2017).
+[19] T. E. Chupp, P. Fierlinger, M. J. Ramsey-Musolf, and
+J. T. Singh, Electric dipole moments of atoms, molecules,
+nuclei, and particles, Rev. Mod. Phys. 91, 015001 (2019).
+[20] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,
+A. Derevianko, and C. W. Clark, Search for new physics
+with atoms and molecules, Rev. Mod. Phys. 90, 025008
+(2018).
+[21] C. Cesarotti, Q. Lu, Y. Nakai, A. Parikh, and M. Reece,
+Interpreting the electron EDM constraint, Journal of
+High Energy Physics 2019, 59 (2019).
+[22] M. Pospelov and A. Ritz, Electric dipole moments as
+probes of new physics, Annals of Physics 318, 119 (2005).
+[23] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer,
+M. R. Tarbutt, and E. A. Hinds, Improved measurement
+of the shape of the electron, Nature 473, 493 (2011).
+[24] J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle,
+G. Gabrielse, Y. V. Gurevich, P. W. Hess, N. R. Hut-
+zler, E. Kirilov, I. Kozyryev, et al., Order of magnitude
+smaller limit on the electric dipole moment of the elec-
+tron, Science 343, 269 (2014).
+[25] A. Collaboration et al., Improved limit on the electric
+dipole moment of the electron., Nature 562, 355 (2018).
+[26] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel,
+T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, and E. A. Cornell,
+Precision measurement of the electron’s electric dipole
+moment using trapped molecular ions, Phys. Rev. Lett.
+119, 153001 (2017).
+[27] T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross,
+Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park,
+A. Wang, J. Ye, and E. A. Cornell, A new bound on
+the electron’s electric dipole moment, arXiv:2212.11841
+(2022).
+[28] R. Alarcon, J. Alexander, V. Anastassopoulos, T. Aoki,
+R. Baartman, S. Baeßler, L. Bartoszek, D. H. Beck,
+F. Bedeschi, R. Berger, et al., Electric dipole moments
+and the search for new physics, arXiv:2203.08103 (2022).
+[29] Y. Zhou, Y. Shagam, W. B. Cairncross, K. B. Ng, T. S.
+Roussy, T. Grogan, K. Boyce, A. Vigil, M. Pettine,
+T. Zelevinsky, J. Ye, and E. A. Cornell, Second-Scale
+Coherence Measured at the Quantum Projection Noise
+Limit with Hundreds of Molecular Ions, Phys. Rev. Lett.
+124, 053201 (2020).
+[30] K. Gaul and R. Berger, Ab initio study of parity
+and time-reversal violation in laser-coolable triatomic
+molecules, Phys. Rev. A 101, 012508 (2020).
+[31] T. A. Isaev, A. V. Zaitsevskii, and E. Eliav, Laser-
+
+7
+coolable polyatomic molecules with heavy nuclei, J. Phys.
+B 50, 225101 (2017).
+[32] T. A. Isaev and R. Berger, Polyatomic candidates for
+cooling of molecules with lasers from simple theoretical
+concepts, Phys. Rev. Lett. 116, 063006 (2016).
+[33] I. Kozyryev, L. Baum, K. Matsuda, and J. M. Doyle, Pro-
+posal for laser cooling of complex polyatomic molecules,
+ChemPhysChem 17, 3641 (2016).
+[34] B. L. Augenbraun, J. M. Doyle, T. Zelevinsky, and
+I. Kozyryev, Molecular asymmetry and optical cycling:
+Laser cooling asymmetric top molecules, Phys. Rev. X
+10, 031022 (2020).
+[35] B. L. Augenbraun, Z. D. Lasner, A. Frenett, H. Sawaoka,
+A. T. Le, J. M. Doyle, and T. C. Steimle, Observa-
+tion and laser spectroscopy of ytterbium monomethoxide,
+YbOCH3, Phys. Rev. A 103, 022814 (2021).
+[36] A. Jadbabaie, Y. Takahashi, N. H. Pilgram, C. J.
+Conn,
+C. Zhang, and N. R. Hutzler, Characteriz-
+ing the fundamental bending vibration of a linear
+polyatomic molecule for symmetry violation searches,
+arXiv:2301.04124 (2022).
+
+8
+Supplemental Material for “Quantum Control of Trapped Polyatomic Molecules for
+eEDM Searches”
+ZERO g-FACTOR STATES
+Origin
+In 2Σ electronic states of linear polyatomic molecules, the spin-rotation interaction, γ ⃗N · ⃗S, couples the molecular
+rotation N and the electron spin S to form the total angular momentum J. These states are well described in the
+Hund’s case (b) coupled basis. An applied electric field EZ will interact with the molecular-frame electric dipole
+moment µE, connecting states with opposite parity, ∆MF = 0, and ∆J ≤ 1. When µEEZ ≫ γ, N and S are
+uncoupled and well described by their lab frame projections MN and MS. However, in the intermediate field regime
+with µEEZ ∼ γ, the molecular eigenstates are mixed in both the Hund’s case (b) coupled basis and the decoupled
+basis. MF remains a good quantum number in the absence of transverse fields. In this regime, MF ̸= 0 states with
+⟨MS⟩ = 0 can arise at specific field values. These states have no first order electron spin magnetic sensitivity, and,
+unlike MF = 0 clock states, have large eEDM sensitivity near BZ = 0. We refer to these states as zero g-factor
+states [6].
+Zero g-factor states arise from avoided level crossings as free field states are mixed by the electric field. One of the
+crossing states has ⟨MS⟩ < 0, the other state has ⟨MS⟩ > 0, and both have mixed MN. The spin-rotation interaction
+couples the states and lifts the crossing degeneracy, resulting in eigenstates that are superpositions of electron spin
+up and down with ⟨MS⟩ = 0, while retaining non-zero molecular orientation with ⟨ˆn⟩ = ⟨MNℓ⟩ ̸= 0. The lab frame
+projection of ˆn ensures that the eEDM interaction in the molecule frame does not rotationally average away.
+Zero g-factor states are generically present in the Stark tuning of polyatomic molecules. The reduction of symmetry
+in a polyatomic molecule allows for rotation about the internuclear axis, resulting in closely spaced doublets of opposite
+parity. When these doublets are mixed by an applied electric field, they split into 2N +1 groups of levels representing
+the values of the molecular orientation ⟨MNℓ⟩. For each N manifold with parity doubling, avoided level crossings
+generically occur between an MNℓ = ±1 Stark manifold and an MNℓ = 0 Stark manifold.
+In diatomic molecules without parity-doubling, the existence of zero g-factor states requires an inverted spin rotation
+structure (γ < 0), such that the two J states are tuned closer to each other by an electric field. For example, the
+YbF molecule (γ = −13.4 MHz [37, 38]) has zero g-factor states at E ≈ 866 V/cm in the N = 1 manifold, while
+CaF does not. However, since |γ|/B ≪ 1 for most 2Σ diatomic molecules, the electric fields that mix spin-rotation
+states are much less than those that polarize the molecule. Therefore, zero g-factor states occur when the molecule
+has negligible lab-frame polarization, limiting eEDM sensitivity. For example, the aforementioned states in YbF have
+|⟨Σ⟩| ≈ 0.006, which is ∼3% the value of Σ in the zero g-factor states used in this work.
+Characterization
+To locate zero g-factor crossings and calculate eEDM sensitivities, we model the �
+X(010) level structure using an
+effective Hamiltonian approach [40–42]:
+Heff = HRot + HSR + Hℓ + HHyp + HZeeman + HStark + HODT
+(2a)
+HRot = B
+�
+⃗N 2 − ℓ2�
+(2b)
+HSR = γ
+�
+⃗N · ⃗S − NzSz
+�
+(2c)
+Hℓ = −qℓ
+�
+N 2
++e−i2φ + N 2
+−ei2φ�
+(2d)
+HHyp = bF ⃗I · ⃗S + c
+3
+�
+3IzSz − ⃗I · ⃗S
+�
+(2e)
+HZeeman = gSµBBZSZ
+(2f)
+HStark = −µZEZ
+(2g)
+HODT = −⃗d · ⃗EODT
+(2h)
+
+9
+Here, we use a similar Hamilton as Ref. [7].
+HRot is the rotational energy; HSR is the spin-rotation interaction
+accurate for low-N bending mode levels, with z defined in the molecule frame; Hℓ is the ℓ-type doubling Hamiltonian,
+with ± defined in the molecule frame, φ as the nuclear bending coordinate, and using the same phase convention as
+Ref. [43]; HHyp is the hyperfine Fermi-contact and dipolar spin interactions, defined in the molecule frame; HZeeman
+describes the interaction of the electron spin magnetic moment with the lab-frame magnetic field; HStark is the
+interaction of the Z-component of molecule-frame electric dipole moment µE with the lab frame DC electric field,
+EZ; and HODT is the interaction of the molecular dipole moment operator ⃗d with the electric field of the ODT laser,
+⃗EODT = E0/2(ˆϵODTe−iωt + c.c.).
+To evaluate the molecule frame matrix elements, we follow the techniques outlined in Refs. [40, 41] to transform into
+the lab frame. The field-free Hamiltonian parameters are taken from Ref. [44], except for the hyperfine parameters,
+which were determined by the observed line positions to be bF = 2.45 MHz and c = 2.6 MHz, similar to those of the
+�
+X(000) state [45]. We use the same dipole moment, |µ| = 1.47 D, as the �
+X(000) state, determined in Ref. [46]. Matrix
+elements of HODT are calculated following Ref. [47] using the 1064nm dynamic polarizabilities reported in Ref. [15].
+For the calculations discussed below and in the main text, the ODT is polarized along the laboratory Z axis and
+the molecules sit at a fixed trap depth of 160 µK (corresponding to the average trap intensity seen by the molecules in
+the experiment). As detailed in the main text, when the trapping light is aligned with EZ, it acts like a weak electric
+field, shifting the zero g-factor crossing by ∼ 1 V/cm from the field-free value. If the trapping light polarization is
+rotated relative to EZ, tensor light shifts can couple states with ∆MF = ±2 or ±1 (the linearity of the light ensures
+there are no ∆MF = ±1 vector shifts) [47]. The effects of this coupling are similar to those of transverse magnetic
+fields, which we discuss below.
+In the current work, we ignore nuclear and rotational Zeeman effects. Specifically, the magnetic sensitivity of CaOH
+receives small contributions from nuclear spin of the H atom and the rotational magnetic moment of both the electrons
+and the nuclear framework. While they have not yet been fully characterized, all of these effects will contribute at the
+10−3µB level or less. These additional g-factors do not depend strongly on the applied electric field, and result in a
+small shift of the zero g-factor crossing location. Future work characterizing rotational magnetic moments of �
+X(010)
+states of laser-coolable metal hydroxides can enable more accurate predictions of zero g-factor field values.
+In CaOH, each rotational state N supports multiple M = ±1 pairs of zero g-factor states. The states at finite
+electric field can be labeled in terms of their adiabatically correlated zero-field quantum numbers |N, Jp, F, M⟩. In the
+presence of trap shifts, the zero g-factor states for N = 1 occur at E = 59.6 V/cm for |J = 1/2+, F = 1, M = ±1⟩ and
+at E = 64.1 V/cm for |J = 3/2+, F = 1, M = ±1⟩. The J = 1/2, M = 1 state is a superposition of 47% MNℓ = −1,
+50% MNℓ = 0, and 3% MNℓ = 1, while the J = 3/2, M = 1 state is 43% MNℓ = −1, 48% MN = 0, and 9% MNℓ = 1.
+Both states are weak-electric-field seekers, yet the opposite molecule frame orientation of the spin results in differences
+(a)
+(b)
+FIG. S1. Electric field tuning of N = 1 zero g-factor states near BZ = 0 in the absence of trap shifts. Blue lines denote
+MF = +1 states and red lines MF = −1. Solid traces denote the J = 1/2 state pair and dashed traces denote the J = 3/2
+pair. The dotted vertical lines mark the electric field value of the zero g-factor crossing without trap shifts, ≈60.5 V/cm for
+J = 1/2 and ≈64.4 V/cm for J = 3/2. Grayed out traces are other states in the N = 1 manifold. (a) The g-factor gSµB⟨MS⟩
+as a function of the applied electric field. (b) eEDM sensitivity ⟨Σ⟩ as a function of the applied electric field. A consequence of
+the Hund’s case (b) coupling scheme is that Σ asymptotes to a maximum magnitude of S/(N(N + 1)) = 1/4 for fields where
+the parity doublets are fully mixed but rotational mixing is negligible [39]. For fields where J is not fully mixed, some states
+can exhibit |Σ| > 1/4.
+
+10
+(a)
+(b)
+(d)
+(c)
+FIG. S2. Full electric and magnetic characterization of zero g-factor states in the N = 1 manifold of CaOH, without trap shifts.
+(a, b) 2D plots of the effective g-factor difference between two M = ±1 states, defined by geff = gSµB (⟨MS⟩M=+1 − ⟨MS⟩M=−1).
+The plotted g-factor is normalized by gSµB. The black line represents the contour where the M = ±1 levels are nominally
+degenerate. (c, d) 2D plots of the eEDM sensitivity, ⟨Σ⟩M=+1 − ⟨Σ⟩M=−1. The black line represents the geff = 0 contour.
+in the value of Σ and the g-factor slope. For CaOH, the magnetic sensitivity and eEDM sensitivity of N = 1 zero
+g-factor states are shown in Fig. S1.
+By diagonalizing Heff over a grid of (EZ, BZ) values, we can obtain 2D plots of g-factors and eEDM sensitivities
+shown in Fig. S2. For generality, we consider the molecular structure in the absence of trap shifts. Using the Z-
+symmetry of the Hamiltonian, we separately diagonalize each MF block to avoid degeneracies at BZ = 0. Continuous
+2D surfaces for eigenvalues and eigenvectors are obtained by ordering eigenstates at each value of (E, B) according to
+their adiabatically correlated free field state. The application of an external magnetic field parallel to the electric field
+results in ⟨MS⟩ ̸= 0 for an individual zero g-factor state, but the differential value between a zero g-factor pair can
+still have ∆⟨MS⟩ = 0. This differential value means the superposition of a zero g-factor pair can maintain magnetic
+insensitivity and EDM sensitivity over a range of fields, for example up to ∼5 G for the J = 1/2, N = 1 pair.
+The procedure we use here for identifying zero g-factor states can be generically extended to searching for favorable
+transitions between states with differing eEDM sensitivities, similar to what has been already demonstrated in a
+recent proposal to search for ultra-light dark matter using SrOH [7]. In addition, there are also fields of BZ ≈ 10 − 20
+G and EZ ≈ 0 where opposite parity states are tuned to near degeneracy. This is the field regime that has been
+proposed for precision measurements of parity-violation in optically trapped polyatomic molecules [9].
+We note that zero g-factor pairs also occur in N = 2−. The crossings occur around 400 − 500 V/cm for states
+
+11
+MF = 0-
+MF = 0+
+MF = 1
+MF = -1
+SXBX
+~540 kHz
+~980 kHz
+geffBZ
+SXBX
+SXBX
+SXBX
+(a)
+(b)
+FIG. S3. (a) Stark shifts for N = 1 in CaOH. The J = 1/2+ zero g-factor states are shown with a solid green line, while the
+J = 3/2+ zero g-factor states are indicated with a dashed green line. All other levels are grayed out. A vertical dotted line
+indicates the location of the J = 1/2+ zero g-factor crossing. (b) A zoomed in level diagram of the J = 1/2+ zero g-factor
+hyperfine manifold. The bias field splitting geffBZ is not to scale. Transverse field couplings are shown with double sided
+arrows, with blue (red) indicating negative (positive) SX matrix element.
+correlated with the negative parity manifold.
+Since many interactions increase in magnitude with larger N, the
+overall electric field scale of the intermediate regime increases. Additionally, the robustness of zero g-factor states
+also improves, with some pairs able to maintain ∆⟨MS⟩ = 0 for magnetic fields up to 40 G. These N = 2 pairs also
+have non-zero eEDM sensitivity for a wide range of magnetic field values.
+TRANSVERSE MAGNETIC FIELDS
+Transverse Field Sensitivity
+We now expand our discussion to include the effect of transverse magnetic fields. Their effects can by modeled by
+adding BXSX and BY SY terms to the effective Hamiltonian, which have the selection rule ∆MF = ±1. For this
+discussion, we focus on the level structure of the N = 1, J = 1/2+ manifold in CaOH near the zero g-factor crossing
+at 60.5 V/cm in the absence of trap shifts, shown in Figure S3. We note if there were no nuclear spin I, the two zero
+g-factor states would be MJ = ±1/2 states separated by ∆M = 1. In such a case these degenerate states would be
+directly sensitive to transverse fields at first order, thereby reducing the g-factor suppression.
+Due to the hyperfine structure from the nuclear spin of the H atom in CaOH, the degenerate MF = ±1 states in a
+zero g-factor pair are coupled by second order transverse field interactions. These interactions are mediated via the
+MF = 0± states, where ± denotes the upper or lower states. Using a Schrieffer–Wolff (aka Van-Vleck) transformation,
+we can express the effective Hamiltonian matrix for second order coupling between the MF = ±1 states. We write
+the states as |MF ⟩, and for convenience we take the transverse field to point along X:
+
+12
+H+1,−1 = −(gSµBBX)2
+�⟨−1|SX|0+⟩⟨0+|SX| + 1⟩
+∆E0+
++ ⟨−1|SX|0−⟩⟨0−|SX| + 1⟩
+∆E0−
+�
+(3)
+Here, ∆E0± is the energy difference of the MF = 0± levels from the MF = ±1 levels. Our model provides the following
+values: ⟨0−|SX|+1⟩ = ⟨0−|SX|−1⟩ = −0.18, ⟨0+|SX|+1⟩ = −0.16, and ⟨0+|SX|−1⟩ = 0.16. The difference in sign is
+a result of Clebsh-Gordon coefficient phases, and only the relative phase is relevant. We also have ∆E0+ = 0.98 MHz
+and ∆E0− = −0.54 MHz. The combination of phases precludes the possibility of destructive interference. With these
+parameters and defining g⊥ = H+1,−1/BX, then eqn. 3 evaluates to (gSµBBX)2(0.086/MHz) ≈ (0.68 MHz/G2)B2
+X.
+Our model estimates the transverse sensitivity at BX ∼ 1 mG to be g⊥µB ∼ 7 × 10−4 MHz/G, of the same order as
+the neglected nuclear and rotational Zeeman terms. The suppressed transverse field sensitivity bounds the magnitude
+of BZ, which must be large enough to define a quantization axis for the spin, geffBZ ≫ g⊥B⊥.
+Cancellation of transverse magnetic fields
+When transverse magnetic fields are dominant, the electron will be quantized along the transverse axis and there
+is minimal spin precession by the bias BZ field. The transverse coupling results in eigenstates given by (|MF =
+1⟩±eiφ|MF = −1⟩)/
+√
+2, where the phase φ is set by the direction of ⃗B in the transverse plane. If φ = 0 or π, only one
+of these states is bright to the ˆX-polarized state preparation microwaves, which means the initial state is stationary
+under the transverse fields. For all other orientations, the transverse field causes spin precession with varying contrast,
+depending on the specific value of φ.
+We are able to use transverse spin precesion to measure and zero transverse fields to the mG level. We do so by
+operating with minimal bias field BZ ≈ 0 and operating EZ near the zero g-factor crossing, such that geffBZ < g⊥B⊥.
+We then apply a small transverse magnetic field to perform transverse spin precession.
+Here, the dynamics are
+dominated by the transverse fields rather than the Z fields.
+We obtain field zeros by iteratively minimizing the
+precession frequency by tuning the bias fields BX and BY .
+IMPERFECT FIELD REVERSAL
+We briefly present a systematic effect involving non-reversing fields in eEDM measurements with zero g-factor
+states and discuss methods for its mitigation. The electric field dependence of geff can mimic an eEDM signal when
+combined with other systematic effects, very much like in 3∆1 molecules [25, 26]. When the sign of EZ is switched, a
+non-reversing electric field ENR will cause a g-factor difference of gNR = (dgeff/dEZ)ENR. This will give an additional
+spin precession signal gNRBZ. By perfectly reversing BZ as well, this precession signal can be distinguished from a
+true EDM signal. However, if there is also a non-reversing magnetic field BNR, there will still be a residual EDM signal
+given by (dg/dE)ENRBNR. Using the measured slope of ∼0.03 (MHz/G)/(V/cm), and using conservative estimates
+of ENR ∼ 1 mV/cm and BNR ∼ 1 µG, we obtain an estimate precession frequency of ∼30 µHz. While this is an order
+of magnitude smaller than the statistical error for the current best eEDM measurement measurement [48], it is still
+desirable to devise methods to reduce the effect further.
+Performing eEDM measurements at different zero g-factor states can help suppress systematic errors resulting from
+the above mechanism. For example, the N = 1, J = 3/2 zero crossing has a different magnitude for Σ, which can be
+used to distinguish a true eEDM from a systematic effect. Both N = 1 crossings are only separated by ∼4 V/cm.
+Furthermore, the zero g-factor states in N = 2− can also be used for systematic checks, as they additionally offer
+different geff vs EZ slopes as well as different Σ values.
+The N = 2− states can be populated directly by the
+photon-cycling used to pump into the bending mode.
+SPIN PRECESSION NEAR ZERO G-FACTOR
+As discussed in the main text, the longest achievable coherence times occur at at combination of low effective g-
+factors (which suppress δBZ decoherence) and low magnetic bias fields (which suppress δµeff decoherence). These low
+g-factors and bias fields only very weakly enforce a quantization axis along Z, enhancing the potential for transverse
+magnetic fields B⊥ to contribute. Such fields have the effect of (a) reducing the spin precession contrast and (b)
+
+13
+FIG. S4. Spin precession at EZ = 60.3 V/cm and BZ = 2 mG. The fit includes a 60 Hz time-varying magnetic field whose
+amplitude and phase are measured with a magnetometer. The coherence time fits to 30 ms.
+altering the observed precession frequency. To avoid these effects, the condition geffBZ > g⊥B⊥ must therefore be
+satisfied. To achieve this, we zero the transverse magnetic fields by intentionally taking spin precession data at BZ ≈ 0
+and geff ≈ 0 while varying the transverse fields BX and BY . By minimizing the spin precession frequency as a function
+of the transverse fields, we reduce B⊥ to approximately 1 mG. In addition, long-term drifts in the dc magnetic field
+along all three axes are compensated by actively feeding back on the magnetic field as measured with a fluxgate
+magnetometer. Under these conditions, at an electric field of 60.3 V/cm (corresponding to µeff = 0.02 MHz/G) and
+a bias field of BZ ≈ 2 mG, we achieve a coherence time of 30 ms (Fig. S4).
+At these very low bias fields, the molecules are also sensitive to 60 Hz magnetic field noise present in the unshielded
+apparatus, whose amplitude is on the same order as BZ. Since the experiment is phase stable with respect to the AC
+line frequency, this 60 Hz magnetic field fluctuation causes a time-dependent spin precession frequency. A fluxgate
+magnetometer is used to measure the amplitude and phase of this 60 Hz field, which are then used as fixed parameters
+in the fit shown in Figure S4.
+[37] B. E. Sauer, J. Wang, and E. A. Hinds, Laser-rf double resonance spectroscopy of 174YbF in the X2Σ+ state: Spin-rotation,
+hyperfine interactions, and the electric dipole moment, J. Chem. Phys. 105, 7412 (1996).
+[38] C. S. Dickinson, J. A. Coxon, N. R. Walker, and M. C. L. Gerry, Fourier transform microwave spectroscopy of the 2Σ+
+ground states of YbX (X=F, Cl, Br): Characterization of hyperfine effects and determination of the molecular geometries,
+J. Chem. Phys. 115, 6979 (2001).
+[39] A. Petrov and A. Zakharova, Sensitivity of the YbOH molecule to P,T-odd effects in an external electric field, Phys. Rev.
+A 105, L050801 (2022).
+[40] J. M. Brown and A. Carrington, Rotational spectroscopy of diatomic molecules (Cambridge University Press, 2003).
+[41] E. Hirota, High-Resolution Spectroscopy of Transient Molecules, Springer Series in Chemical Physics, Vol. 40 (Springer
+Berlin Heidelberg, Berlin, Heidelberg, 1985).
+[42] A. Merer and J. Allegretti, Rotational energies of linear polyatomic molecules in vibrationally degenerate levels of electronic
+2Σ and 3Σ states, Canadian Journal of Physics 49, 2859 (1971).
+[43] J. M. Brown, The rotational dependence of the Renner-Teller interaction: a new term in the effective Hamiltonian for
+linear triatomic molecules in Π electronic states, Mol. Phys. 101, 3419 (2003).
+[44] M. Li and J. A. Coxon, High-resolution analysis of the fundamental bending vibrations in the ˜A2Π and ˜
+X2Σ+ states of
+caoh and caod: Deperturbation of Renner-Teller, spin-orbit and K-type resonance interactions, J. Chem. Phys. 102, 2663
+(1995).
+[45] C. Scurlock, D. Fletcher, and T. Steimle, Hyperfine structure in the (0,0,0) ˜
+X2Σ+ state of CaOH observed by pump/probe
+microwave-optical double resonance, J. Mol. Spectrosc. 159, 350 (1993).
+[46] T. Steimle, D. Fletcher, K. Jung, and C. Scurlock, A supersonic molecular beam optical stark study of CaOH and SrOH,
+J. Chem. Phys. 96, 2556 (1992).
+[47] L. Caldwell and M. R. Tarbutt, Sideband cooling of molecules in optical traps, Phys. Rev. Res. 2, 013251 (2020).
+[48] Z. Lasner, Order-of-magnitude-tighter bound on the electron electric dipole moment, Ph.D. thesis, Yale University (2019).
+
+Q
+60.3 V/cm
+0.44
+Spin Precession with 60 Hz Modulation
+Fraction (au)
+0.42
+0.4
+0.38
+0
+10
+20
+30
+40
+50
+60
+70
+Time (ms)

C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:83bb576b7fd1b1fdc39c5a8669c7e89ade832960507420b2e91457d43888260c
+size 421763

C9E2T4oBgHgl3EQf9QlY/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2ab0031dd62b4e3e231e1d7297ee5152f0df10d5fe21817026086a97c2c0569f
+size 146323