Tootfinder

At this point, C# should introduce multiple return types as a syntactic part of the language. Bending generics to this extent seems like an abuse of the original intent.

Random Walks and the Meeting Time for Trees
Andrew Beveridge, Ben Bridenbaugh, Ari Holcombe Pomerance
https://arxiv.org/abs/2508.02804 https://arxiv.org/pd…

Random Walks and the Meeting Time for Trees
Consider a random walk on a tree $G=(V,E)$. For $v,w \in V$, let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at $v$ to reach $w$, and let $π_v = \mathrm{deg}(v)/2|E|$ denote the stationary distribution for the random walk. We characterize the extremal tree structures for the meeting time $T_{\mathrm{meet}}(G) = \max_{w \in V} \sum_{v \in V} π_v H(v,w)$. For fixed order $n$ and diameter $d$, the meeting time is maximized by the b…

The linear minimal 4-chart with three crossings
Teruo Nagase, Akiko Shima
https://arxiv.org/abs/2509.04114 https://arxiv.org/pdf/2509.04114

The linear minimal 4-chart with three crossings
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $Γ$ be a chart, and we denote by $Cross(Γ)$ the set of all the crossings of $Γ$, and we denote by $Γ_m$ the union of all the edges of label $m$. For a 4-chart $Γ$, if the closure of each connected component of t…

On the pro-\'etale cohomology of quotient stacks of Drinfeld spaces
Zecheng Yi
https://arxiv.org/abs/2510.02699 https://arxiv.org/pdf/2510.02699…

On the pro-étale cohomology of quotient stacks of Drinfeld spaces
Let $\mathcal{H}^{n-1}_{K}$ denote the $(n-1)$-dimensional Drinfeld space over a $p$-adic field $K$. We give an explicit description of the $\ell$-adic and $p$-adic pro-étale cohomology of quotient stacks $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(\mathcal{O}_K)]$ and $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(K)]$, which are moduli stacks of special formal $\mathcal{O}_D$-modules. The computation makes use of the isomorphism between the Lubin-Tate tower and the Drinfeld tower due to Falting…

Modeling Carreau fluid flows through a very thin porous medium
Mar\'ia Anguiano (ECL, ICJ, MMCS), Matthieu Bonnivard (ECL, ICJ, MMCS), Francisco J. Su\'arez-Grau
https://arxiv.org/abs/2508.03214

Modeling Carreau fluid flows through a very thin porous medium
This study investigates three-dimensional, steady-state, and non-Newtonian flows within a very thin porous medium (VTPM). The medium is modeled as a domain confined between two parallel plates and perforated by solid cylinders that connect the plates and are distributed periodically in perpendicular directions. We denote the order of magnitude of the thickness of the domain by $ε$ and define the period and order of magnitude of the cylinders' diameter by $ε$^l, where 0 < l < 1 is fixed. In ot…

Spaces of triangularizable matrices (II): Finite fields with odd characteristic
Cl\'ement de Seguins Pazzis
https://arxiv.org/abs/2509.03716 https://ar…

Spaces of triangularizable matrices (II): Finite fields with odd characteristic
Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed. The structure of the spaces of maximal dimension was also elucidated provided $\mathbb{F}$ is infinite and not quadratically closed. In this sequel, we extend this re…

Central limit theorem and Cram\'{e}r-type moderate deviations for Milstein scheme
Peng Chen, Hui Jiang, Jing Wang
https://arxiv.org/abs/2510.03053 https://

Central limit theorem and Cramér-type moderate deviations for Milstein scheme
In this paper, we investigate the Milstein numerical scheme with step size $η$ for a stochastic differential equation driven by multiplicative Brownian motion. Under some appropriate coefficient conditions, the continuous-time system and its discrete Milstein scheme approximation each possess unique invariant measures, which we denote by $π$ and $π_η$ respectively. We first establish a central limit theorem for the empirical measure $Π_η$, a statistical consistent estimator of $π_η$. Su…

A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs
Hau-Wen Huang
https://arxiv.org/abs/2508.02032 https://arxiv.org/…

A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs
The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ are a family of discrete orthogonal polynomials involving two real parameters $r$ and $s$. Let $L,L^*$ denote the corresponding Leonard pair. Assume that $r\not=0$ and $r+s=0$. We show that $L,(L^*+\frac{r-d}{2})^{2}$ is a Leonard pair. According to the theory of Leonard pairs, the polynomials $\{u_i(x)\}_{i=0}^d$ are not only the dual Hahn polynomials but also the Racah polynomials with respect to the same inner product.

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

New conjectures on the inertia of graphs
Saieed Akbari, Clive Elphick, Hitesh Kumar, Shivaramakrishna Pragada, Quanyu Tang
https://arxiv.org/abs/2508.01163 https://

New conjectures on the inertia of graphs
Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgašev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the …

The locus of plane curves in the moduli stack of curves
Aaron Landesman
https://arxiv.org/abs/2509.25432 https://arxiv.org/pdf/2509.25432

The locus of plane curves in the moduli stack of curves
Let $d \geq 4$ and let $U_d$ denote the locus of smooth curves in the Hilbert scheme of degree $d$ plane curves. If the members of $U_d$ have genus $g$, let $\mathscr{M}_g$ denote the moduli stack of genus $g$ curves. We show that the natural map $[U_d/\operatorname{PGL}_3] \to \mathscr{M}_g$ is a locally closed embedding.

Spectral extremal problems for the $(p,Q)$-spectral radius of hypergraphs
Jian Zheng, Honghai Li, Li Su
https://arxiv.org/abs/2510.02776 https://arxiv.org/…

Spectral extremal problems for the $(p,Q)$-spectral radius of hypergraphs
Let $Q$ be an $s$-vertex $r$-uniform hypergraph, and let $H$ be an $n$-vertex $r$-uniform hypergraph. Denote by $\mathcal{N}(Q,H)$ the number of isomorphic copies of $Q$ in $H$. For a hereditary family $\mathcal{P}$ of $r$-uniform hypergraphs, define $$π(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\binom{n}{s}^{-1}\max\{\mathcal{N}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}.$$ For $p\geq1$, the $(p,Q)$-spectral radius of $H$ is defined as $$λ^{(p)}(Q,H):=\max_{\|\mathbf{x}\
…

The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW$^*$-algebra
Gerardo M. Escolano, Antonio M. Peralta, Armando R. Villena
https://arxiv.org/abs/2509.03213

The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW$^*$-algebra
Let $\mathcal{P} (\mathfrak{J})$ denote the lattice of projections of a JBW$^*$-algebra $\mathfrak{J}$, and let $X$ be a Banach space. A bounded finitely additive $X$-valued measure on $\mathcal{P}(\mathfrak{J})$ is a mapping $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ satisfying: $(a)$ $μ(p +q) = μ(p) + μ(q)$, whenever $p \circ q = 0$ in $\mathcal{P} (\mathfrak{J})$, $(b)$ $\sup \{ \| μ(p)\| \, : \, p \in \mathcal{P} (\mathfrak{J})\} < \infty$. In this paper we establish …

A bilinear form for spin$^c$ manifolds
Huijun Yang
https://arxiv.org/abs/2509.01979 https://arxiv.org/pdf/2509.01979

A bilinear form for spin$^c$ manifolds
For any $(8n {+} 2)$-dimensional closed oriented spin$^{c}$ manifold, denote by $[M]$ the fundamental class of $M$, $\langle \cdot ~, ~\cdot \rangle$ the Kronecker product, and $ρ_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ the $\bmod ~ 2$ reduction homomorphism. In this paper, we will prove that the following identity \[ \langle ρ_2(t) \cdot Sq^2 ρ_2 (t), [M] \rangle = \langle ρ_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] holds for any torsion class $t \in H^{4n}(M…

Zero Insertive Nil Clean Rings
Sanjiv Subba, Tikaram Subedi
https://arxiv.org/abs/2508.01333 https://arxiv.org/pdf/2508.01333

Zero Insertive Nil Clean Rings
This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element $x$ of a ring $R$ zero insertive if $x=arb$ for some $a,b,r\in R$ such that $ab=0$ and $ZI(R)$ denotes the set of all zero insertive elements of $R$. We establish that a ring $R$ is semicommutative if and only if $ZI(R) \subseteq E(R)$, and weakly semicommutative if and only if $ZI(R) \subseteq N(R)$, where $E(R)$ and $N(R)$ denote respectively t…

Sequences of surfaces in $4$-manifolds
Marina Ville
https://arxiv.org/abs/2509.00566 https://arxiv.org/pdf/2509.00566…

Sequences of surfaces in $4$-manifolds
Let $(Σ_n)$ be a sequence of surfaces immersed in a $4$-manifold $M$ which converges to a branched surface $Σ_0$ .\\ We denote by $k^T_p$ (resp. $k^N_p$) the amount of curvature of the tangent bundles $TΣ_n$ (resp. normal bundles $NΣ_n$) which concentrates around a branch point $p$ of $Σ_0$ when $n$ goes to infinity. Alternatively $k^T\pm k^N$ measures how much the twistor degrees drop when we go from $Σ_n$ to $Σ_0$. For complex algebraic curves, $k^T+k^N=0$..\\ In some instances - 1) if…

Binomial edge ideals of Cameron-Walker graphs
Takayuki Hibi, Sara Saeedi Madani
https://arxiv.org/abs/2509.01150 https://arxiv.org/pdf/2509.01150

Binomial edge ideals of Cameron-Walker graphs
Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is Cohen--Macaulay; (ii) $J_G$ is unmixed; (iii) $\dim (S/J_G) = n+1$; (iv) (a) $n = 3$ and $G$ is a path of length $2$ or (b) $n = 5$ and $G$ is a path of length $4$ or (c) $n=5$ and $G$ is obtained by attaching a path of length $2$ to a triangle. Moreover, the depth …

Reduced fidelities for free fermions out of equilibrium: From dynamical quantum phase transitions to Mpemba effect
Gilles Parez, Vincenzo Alba
https://arxiv.org/abs/2509.01608 h…

Reduced fidelities for free fermions out of equilibrium: From dynamical quantum phase transitions to Mpemba effect
We investigate the out-of-equilibrium dynamics after a quantum quench of the reduced fidelities between the states of a subregion $A$ at different times. Precisely, we consider the fidelity between the time-dependent state of $A$ and its initial value, as well as with the state at infinite time. We denote these fidelities as the reduced Loschmidt echo (RLE) and the final-state fidelity (FSF), respectively. If region $A$ is the full systems, the RLE coincide with the standard Loschmidt echo. We …

On the second partial Global Euler-Poincare characteristics for Galois cohomology
Yufan Luo
https://arxiv.org/abs/2509.03218 https://arxiv.org/pdf/2509.032…

On the second partial Global Euler-Poincare characteristics for Galois cohomology
Let $K$ be a number field, let $S$ be a finite set of primes of $K$ containing all archimedean primes, and let $G_{K,S}$ denote the Galois group of the maximal extension of $K$ unramified outside $S$. In this paper, we study the second partial Euler-Poincare characteristic $χ_{2}(G_{K,S},M)$ for a finite $G_{K,S}$-module $M$, without imposing the condition that the order of $M$ is an $S$-unit. By adjoining a further finite set of primes of $K$, which can be chosen to be disjoint from any presc…

On interwined polynomials
Fedor Pakovich
https://arxiv.org/abs/2510.01877 https://arxiv.org/pdf/2510.01877

On interwined polynomials
Let $A_1$ and $A_2$ be polynomials of degree at least two over $\mathbb C$. We say that $A_1$ and $A_2$ are intertwined if the endomorphism $(A_1, A_2)$ of $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$ given by $(z_1, z_2) \mapsto (A_1(z_1), A_2(z_2))$ admits an irreducible periodic curve that is neither a vertical nor a horizontal line. We denote by $\mathrm{Inter}(A)$ the set of all polynomials $B$ such that some iterate of $B$ is intertwined with some iterate of $A$. In this paper, we p…

yahoo_song: Yahoo song ratings (2011)
A bipartite network of users and songs they rated, as used in the 2011 KDD Cup and extracted from Yahoo! Music. Edge weights denote a rating scaled from 0 to 100. More information about this data set is available at http://konect.cc/networks/yahoo-song.
Th…

On minimal product-one sequences of maximal length over the non-abelian group of order $pq$
Danilo Vilela Avelar, Fabio Enrique Brochero Mart\'inez, S\'avio Ribas
https://arxiv.org/abs/2510.00070

On minimal product-one sequences of maximal length over the non-abelian group of order $pq$
Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with $p \mid q-1$ and let $C_q \rtimes C_p$ denote the non-abelian group of or…

Density Characterization with The Upper Bound of Density of Fibonacci Word
Duaa Abdullah, Jasem Hamoud
https://arxiv.org/abs/2509.00886 https://arxiv.org/p…

Density Characterization with The Upper Bound of Density of Fibonacci Word
This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote the units digit. Through analysis of these ratios and their convergence to powers of $φ$, we illustrate the intrinsic exponential growth rates characteristic of Fibonacci words. By considering the natural density concept for sets of positive integers, it is de…

Branching ratios and CP asymmetries of $B^0 \to \eta_c f_0$ in the improved perturbative QCD formalism
Min-Qi Li, Xin Liu, Zhi-Tian Zou, Ying Li, Zhen-Jun Xiao
https://arxiv.org/abs/2507.23578

Branching ratios and CP asymmetries of $B^0 \to η_c f_0$ in the improved perturbative QCD formalism
Motivated by the idea of fragmented scalar glueball, we investigate the decays $B^0 \to η_c f_0$ within the improved perturbative QCD (iPQCD) framework by including the known next-to-leading order corrections. Here, $B^0$ and $f_0$ denote the neutral $B_{d,s}^0$ mesons and the light scalar mesons $f_0(500, 980, 1370, 1500)$ under the $q\bar q$ assignment. The {\it CP}-averaged branching ratios (BRs) and the {\it CP} asymmetries of $B^0 \to η_c f_0$ are evaluated with the $f_0(500)[f_0(1370)]-…

Liftability of periodic mapping classes under alternating covers
Apeksha Sanghi, Kashyap Rajeevsarathy, Rajesh Dey
https://arxiv.org/abs/2509.02114 https://

Liftability of periodic mapping classes under alternating covers
Let $S_g$ be the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. Let $A_n$ denote the alternating group on $n$ letters. We derive necessary and sufficient conditions under which a periodic mapping class has a conjugate that lifts under the branched cover $S_g \to S_g/A_n$ induced by an action of $A_n$ on $S_g$. This provides a classification of the subgroups of $\mathrm{Mod}(S_g)$ that are isomorphic to $A_n \rtimes \mathbb{Z}_m$, …

Investigating Multi-layer Representations for Dense Passage Retrieval
Zhongbin Xie, Thomas Lukasiewicz
https://arxiv.org/abs/2509.23861 https://arxiv.org/p…

Investigating Multi-layer Representations for Dense Passage Retrieval
Dense retrieval models usually adopt vectors from the last hidden layer of the document encoder to represent a document, which is in contrast to the fact that representations in different layers of a pre-trained language model usually contain different kinds of linguistic knowledge, and behave differently during fine-tuning. Therefore, we propose to investigate utilizing representations from multiple encoder layers to make up the representation of a document, which we denote Multi-layer Represe…

Geometric conditions for bounded point evaluations in several complex variables
Stephen Deterding
https://arxiv.org/abs/2507.23688 https://arxiv.org/pdf/25…

Geometric conditions for bounded point evaluations in several complex variables
Let $U$ be a bounded domain in $\mathbb C^d$ and let $L^p_a(U)$, $1 \leq p < \infty$, denote the space of functions that are analytic on $\overline{U}$ and bounded in the $L^p$ norm on $U$. A point $x \in \overline{U}$ is said to be a bounded point evaluation for $L^p_a(U)$ if the linear functional $f \to f(x)$ is bounded in $L^p_a(U)$. In this paper, we provide a purely geometric condition given in terms of the Sobolev $q$-capacity for a point to be a bounded point evaluation for $L^p_a(U)$. T…

Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces
Feng Dai, Eero Saksman, Dachun Yang, Wen Yuan, Yangyang Zhang
https://arxiv.org/abs/2508.21269

Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces
Let $Λ_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{Λ_s}: = \inf_{g\in V} \|f-g\|_{Λ_s}$ from a function $f\in Λ_s$ to a non-dense subspace $V\subset Λ_s$ via the fractional semigroup $\{T_{α, t}: =e^{-t (-Δ)^{α/2}}: t\in (0, \infty)\}$ for any $α\in(0,\infty)$. Given an integer $ r >s/α$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $Λ_s$ if a…

Pseudo-Isometric Surgery
Matt Clay, Josh Thompson
https://arxiv.org/abs/2507.23666 https://arxiv.org/pdf/2507.23666…

Pseudo-Isometric Surgery
We introduce a type of surgery on metric spaces. This surgery, in some sense, seeks to replace a subspace $S$ of a metric space $X$ with another metric space $T$ via a function $f : S \to T$. When $T$ is a discrete space, this amounts to collapsing the subspace according to the function. This surgery results in a new metric space we denote $\widehat{X}_f$ and there is a natural function $F : X \to \widehat{X}_f$ induced from $f$. Our primary interest is investigating if properties of the origin…

On irrationals with Lagrange value exactly 3
Zhe Cao, Harold Erazo, Carlos Gustavo Moreira
https://arxiv.org/abs/2509.01867 https://arxiv.org/pdf/2509.0186…

On irrationals with Lagrange value exactly 3
For $c>0$, let $X_c$ denote the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ such that $\left| x-\frac{p}{q} \right|<\frac{1}{cq^2}$ has only finitely many rational solutions $\frac{p}{q}$. It is a classical fact, known since the 1950s, that $X_c$ is uncountable for $c>3$ and countable for $c<3$. However, the cardinality of $X_3$ does not appear to be present in the literature. We prove that $X_3$ is uncountable. More generally, we show that for any $n\in\mathbb{N}\cup\{\infty\}$, th…

Central limit theorems for the eigenvalues of graph Laplacians on data clouds
Chenghui Li, Nicol\'as Garc\'ia Trillos, Housen Li, Leo Suchan
https://arxiv.org/abs/2507.18803

Central limit theorems for the eigenvalues of graph Laplacians on data clouds
Given i.i.d.\ samples $X_n =\{ x_1, \dots, x_n \}$ from a distribution supported on a low dimensional manifold ${M}$ embedded in Eucliden space, we consider the graph Laplacian operator $Δ_n$ associated to an $\varepsilon$-proximity graph over $X_n$ and study the asymptotic fluctuations of its eigenvalues around their means. In particular, letting $\hatλ_l^\varepsilon$ denote the $l$-th eigenvalue of $Δ_n$, and under suitable assumptions on the data generating model and on the rate of decay …

Hook immanantal inequalities for totally nonnegative matrices
Mark Skandera
https://arxiv.org/abs/2510.00327 https://arxiv.org/pdf/2510.00327

Hook immanantal inequalities for totally nonnegative matrices
Given a weakly decreasing positive integer sequence $λ= (λ_1,\dotsc,λ_\ell)$ summing to $n$, let $χ^λ$ denote the irreducible character of the symmetric group $S_n$ indexed by $λ$. This representation has dimension $χ^λ(e)$, where $e$ is the identity element of $S_n$. Let $\mathrm{Imm}_{χ^λ}$ denote the corresponding irreducible character immanant, the function on $n \times n$ matrices $A = (a_{i,j})$ defined by $\mathrm{Imm}_{χ^λ}(A) := \sum_{w \in S_n} χ^λ(w) a_{1,w_1} \cdots a_…

Bayesian Separation Logic
Shing Hin Ho, Nicolas Wu, Azalea Raad
https://arxiv.org/abs/2507.15530 https://arxiv.org/pdf/2507.15530

Bayesian Separation Logic
Bayesian probabilistic programming languages (BPPLs) let users denote statistical models as code while the interpreter infers the posterior distribution. The semantics of BPPLs are usually mathematically complex and unable to reason about desirable properties such as expected values and independence of random variables. To reason about these properties in a non-Bayesian setting, probabilistic separation logics such as PSL and Lilac interpret separating conjunction as probabilistic independence …

Primes of the form $ax by$ in certain intervals with small solutions
Yuchen Ding, Takao Komatsu, Honghu Liu
https://arxiv.org/abs/2510.01781 https://arxiv.…

Primes of the form $ax+by$ in certain intervals with small solutions
Let $1

First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$
Elham Matinpour
https://arxiv.org/abs/2509.25617 https://

First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$
Consider $\mathbb{R}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $Σ$ be a complete embedded self-shrinker in $\mathbb{R}^3$ with the induced metric and weighted measure, and let $λ_1$ denote the first eigenvalue of the drift Laplacian in the weighted $L^2$ space. Inspired by Choe and Soret's estimate of the first eigenvalue of the Laplacian on symmetric minimal surfaces in $\mathbb{S}^3$, we prove that $λ_1$= 1/2 for self-shrinkers invariant under the dihedral group $…

On a Second-Order Version of Russellian Theory of Definite Descriptions
Yaroslav Petrukhin
https://arxiv.org/abs/2508.13928 https://arxiv.org/pdf/2508.1392…

On a Second-Order Version of Russellian Theory of Definite Descriptions
Definite descriptions are first-order expressions that denote unique objects. In this paper, we propose a second-order counterpart, designed to refer to unique relations between objects. We investigate this notion within the framework of Russell's theory of definite descriptions. While full second-order logic is incomplete, its fragment defined by Henkin's general models admits completeness. We develop our theory within this fragment and formalize it using a cut-free sequent calculus.

Generalized Holographic and Ricci Dark Energy: Cosmological Diagnostics and Scalar Field Realizations
Antonio Pasqua
https://arxiv.org/abs/2509.19386 https://

Generalized Holographic and Ricci Dark Energy: Cosmological Diagnostics and Scalar Field Realizations
In this work, we present two generalized formulations of the Holographic and Ricci dark energy (DE) models, given by $ ρ_{GH} = 3c^2M^{2}_{pl} \left[ 1-ε\left(1-\frac{R}{H^2}\right) \right]H^2$ and $ρ_{GR} = 3c^2M^{2}_{pl}\left[ 1-η\left(1-\frac{H^2}{R}\right) \right]R$ where $H$ and $R$ denote the Hubble parameter and the Ricci scalar, while $ε$ and $η$ are model parameters related by $ε= 1 - η$. We derived explicit analytical expressions for key cosmological quantities, including the …

Distinct Spatiotemporal Dynamics of Thermoelectric Transport Across Superconducting Transition
Rajae Malek, Qing-Dong Jiang, Haiwen Liu
https://arxiv.org/abs/2508.20792 https://…

Distinct Spatiotemporal Dynamics of Thermoelectric Transport Across Superconducting Transition
We investigate the relaxation dynamics of heat transport in superconductors, shaped by the interplay of diffusion, nonlinearity, and magnetic fields. Focusing on regimes near the critical temperature Tc, we analyze two classes of relaxation diffusion equations that give rise to qualitatively distinct dynamics, which we denote as Type I (linear) and Type II (nonlinear). Type I relaxation, characteristic of the normal state above Tc, results in a steady and spatially uniform heat current governed…

Commutators of n-cycles in the symmetric group
Philipp Bader
https://arxiv.org/abs/2509.22749 https://arxiv.org/pdf/2509.22749

Commutators of n-cycles in the symmetric group
We show that for $n \ge 6$ every even permutation on $n$ symbols is the commutator of two $n$-cycles. More precisely, let $S_n$ be the symmetric group and $A_n$ the alternating group. Let $C(n) \subset S_n$ denote the conjugacy class of $n$-cycles and $[\cdot, \cdot]$ be the commutator of two permutations. We prove: The map $C(n) \times C(n) \to A_n, \ (τ, π) \mapsto [τ, π]$ is surjective for all $n \ge 6$.

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

Quasi Divisor Topology of Modules over Domains
Mesut Bu\u{g}day, Dilara Erdemir, \"Unsal Tekir, Suat Ko\c{c}
https://arxiv.org/abs/2509.23743 https://…

Quasi Divisor Topology of Modules over Domains
Let $E$ be a module over a domain $A$, and $W(E)^{\#}=W(E)-ann(E)$ where $W(E)=\{a\in A:aE\neq E\}$. We define an equivalence relation $\sim$ on $W(E)^{\#}$ as follows: $a\sim b$ if and only if $aE=bE$ for any $a,b\in W(E)^{\#}$ and denote $EC(W(E)^{\#})$ to be the set of all equivalence classes $[a]$ of $W(E)^{\#}$. We first show that the family $\{U_a\}_{a\in W(E)^\#}$ generates a topology which we called the quasi divisor topology of $A$-module $E$ denoted by $qD_A(E)$ where $U_{a}=\{[b]\in …

Clique number of xor-powers of Kneser graphs
Zolt\'an F\"uredi, Andr\'as Imolay, \'Ad\'am Schweitzer
https://arxiv.org/abs/2510.01509 https://

Clique number of xor-powers of Kneser graphs
Let $f_\ell(n, k)$ denote the clique number of the xor-product of $\ell$ isomorphic Kneser graphs KG(n,k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed $f_\ell(n,1)\leq \ell n +1$. Imolay, Kocsis, and Schweitzer proved that $f_2(n,k)\leq n/k +c(k)$. Here, the order of magnitude of $c(k)$ is determined to be $Θ\left( k \binom{2k}{k} \right)$. By explicit constructions and by an algebraic proof, it is shown that $\ell n- 2\ell-1 \leq f_\ell(n…

Fano schemes of sub-maximal elementary symmetric functions
Alexandru Chirvasitu
https://arxiv.org/abs/2507.19163 https://arxiv.org/pdf/2507.19163

Fano schemes of sub-maximal elementary symmetric functions
Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective $(d-1)$-spaces contained in the zero locus of $E_{\dim V-1}$. Isolated points exist precisely for $\dim V=2d$, in which case they are in bijection with the $1\cdot 3\cdots (2d-1)$ pairings on a $2d$-element set. This, in particular, confirming a conjecture of Ambartsou…

Best-of-$\infty$ -- Asymptotic Performance of Test-Time Compute
Junpei Komiyama, Daisuke Oba, Masafumi Oyamada
https://arxiv.org/abs/2509.21091 https://arx…

Best-of-$\infty$ -- Asymptotic Performance of Test-Time Compute
We study best-of-$N$ for large language models (LLMs) where the selection is based on majority voting. In particular, we analyze the limit $N \to \infty$, which we denote as Best-of-$\infty$. While this approach achieves impressive performance in the limit, it requires an infinite test-time budget. To address this, we propose an adaptive generation scheme that selects $N$ based on answer agreement, thereby efficiently allocating inference-time computation. Beyond adaptivity, we extend the frame…

A family of non-uniform distributions on the set of parking functions generated by random permutations
Ross G. Pinsky
https://arxiv.org/abs/2509.24452 https://

A family of non-uniform distributions on the set of parking functions generated by random permutations
We introduce a rather natural family of non-uniform distributions on $PF_n$, $n\in\mathbb{N}$, the set of parking functions of length $n$. One of the motivations for this comes from a similar situation in the context of integer partitions. For a permutation $σ\in S_n$ and for $j\in[n]$, let $I_{n,

Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators
Long Huang, Yangzhi Zhang, Ciqiang Zhuo
https://arxiv.org/abs/2509.23839 htt…

Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators
Let $\mathcal H_{\infty}^δ$ denote the Hausdorff content of dimension $δ\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$-norm inequality with $p\in(1,\infty)$ and the weighted weak $L^1$-norm inequality on Hardy-Littlewood maximal operators associated with Hausdorff contents hold true. To achieve this, we introduce a class of capacitary Muckenhoupt weights depending on …

On cuts of small chromatic number in sparse graphs
Guillaume Aubian, Marthe Bonamy, Romain Bourneuf, Oscar Fontaine, Lucas Picasarri-Arrieta
https://arxiv.org/abs/2510.01791 htt…

On cuts of small chromatic number in sparse graphs
For a given integer $k$, let $\ell_k$ denote the supremum $\ell$ such that every sufficiently large graph $G$ with average degree less than $2\ell$ admits a separator $X \subseteq V(G)$ for which $χ(G[X]) < k$. Motivated by the values of $\ell_1$, $\ell_2$ and $\ell_3$, a natural conjecture suggests that $\ell_k = k$ for all $k$. We prove that this conjecture fails dramatically: asymptotically, the trivial lower bound $\ell_k \geq \tfrac{k}{2}$ is tight. More precisely, we prove that for every…

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
Alexander J. Barrios, Chung Pang Mok
https://arxiv.org/abs/2510.00926 htt…

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N=N_+N_-$, where $N_+$ and $N_-$ are coprime and $N_-$ is squarefree. Let $D$ be a positive fundamental discriminant satisfying the modified Heegner hypothesis with respect to $(N_+,N_-)$: primes dividing $N_+$ (resp. $N_-$) split (resp. are inert) in $\mathbb{Q}(\sqrt{D})$; we denote by $E^D/\mathbb{Q}$ the quadratic twist of $E/\mathbb{Q}$ by $D$. In the first half of the paper we consider the situation where $N_-$ is a squarefree produc…

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

Graphs with girth $2\ell$ and without longer even holes are $3$-colorable
Rong Chen
https://arxiv.org/abs/2509.01137 https://arxiv.org/pdf/2509.01137

Graphs with girth $2\ell$ and without longer even holes are $3$-colorable
For a number $\ell\geq 2$, let $\mathcal{H}_{\ell}$ denote the family of graphs which have girth $2\ell$ and have no even hole with length greater than $2\ell$. Wu, Xu, and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{H}_{\ell}$ is 3-colorable. In this paper, we prove that every graph in $\mathcal{H}_{\ell}$ is 3-colorable for any integer $\ell\geq5$.

A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices
Yangjie Yin, Gang Han
https://arxiv.org/abs/2509.25867

A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices
There exists a biderivation structure on the polynomial algebra $\mathscr{A}[n] = K[x_1,\dots,x_n],$ where $K$ is a field with $\operatorname{char}(K)\ne 2$, defined by $f \circ h = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\frac{\partial h}{\partial x_i}.$ Let $\mathscr{A}_k[n]$ denote the subspace of homogeneous polynomials of degree $k$. Then $(\mathscr{A}_2[n],\circ)$ is a Jordan algebra, isomorphic to the special Jordan algebra $H_n(K)$ of $n\times n$ symmetric matrices. Each $\mathscr…

Second order free cumulants: product, commutator, and anti-commutator
Daniel Munoz George, Daniel Perales
https://arxiv.org/abs/2507.21031 https://arxiv.or…

Second order free cumulants: product, commutator, and anti-commutator
Given two second order free random variables $a$ and $b$, we study the second order free cumulants of their product $ab$, their commutator $ab-ba$, and their anti-commutator $ab+ba$. Let $(κ_n^a)_{n\geq 1}$ and $(κ_{n,m}^a)_{n,m\geq 1}$ denote the sequence of free cumulants of first and second order, respectively, of a random variable $a$ in a second order non-commutative probability space $(\mathcal{A},φ,φ^2)$. Given $a$ and $b$ two second order freely independent random variables, we prov…

Invertible completions of FLI and FRI upper triangular operator matrices
Nikola Sarajlija
https://arxiv.org/abs/2508.20956 https://arxiv.org/pdf/2508.20956…

Invertible completions of FLI and FRI upper triangular operator matrices
If $A\in\mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B}(\mathcal{K})$ are given operators, denote by $M_C$ an operator matrix of the form $$M_C=\begin{pmatrix} A & C\\ 0 & B \end{pmatrix}\in\mathcal{B}(\mathcal{H}\oplus\mathcal{K})$$ acting on a direct sum of infinite dimensional separable Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, and $C\in\mathcal{B}(\mathcal{K},\mathcal{H})$ is unknown. In this article we solve the completion problem of $M_C$ to Fredholm left and Fredholm right inverti…

Polynomial properties of unipotent radicals of parabolic subgroups in classical groups
Qingchun Hao, Yang Yang
https://arxiv.org/abs/2508.20509 https://arx…

Polynomial properties of unipotent radicals of parabolic subgroups in classical groups
Let $R_u^{X,d}$ denote the unipotent radical of a (proper) maximal standard parabolic subgroup of the classical group $\mathrm{Sp}_{2n}(q)$, $\mathrm{SO}_{2n}(q)$, or $\mathrm U_{2n}({q^2})$. This paper establishes explicit formulas for the number of irreducible characters of $R_u^{X,d}$ with degree $q^e$.

Rephasing invariant formulae for general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles
Masaki J. S. Yang
https://arxiv.org/abs/2508.17866

Rephasing invariant formulae for general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles
In this letter, we present rephasing invariant formulae $δ^{(αi)} = \arg [ { V_{α1} V_{α2} V_{α3} V_{1i} V_{2i} V_{3i} / V_{αi }^{3} \det V } ] $ for CP phases $δ^{(αi)}$ associated with nine Euler-angle-like parameterizations of a flavor mixing matrix. Here, $α$ and $i$ denote the row and column carrying the trivial phases in a given parameterization. Furthermore, we show that the phases $δ^{(αi)}$ and the nine angles $Φ_{αi}$ of unitarity triangles satisfy compact sum rules $ δ^…

Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs
Yuqing Ji, Yue Wang, Yujun Yang, Xia Zhang
https://arxiv.org/abs/2509.00929 https://

Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs
A paraglider, house, 4-wheel, is the graph that consists of a cycle $C_4$ plus an additional vertex adjacent to three vertices, two adjacent vertices, all the vertices of the $C_4$, respectively. For a graph $G$, let $χ(G)$, $ω(G)$ denote the chromatic number, the clique number of $G$, respectively. Gerards and Seymour from 1995 conjectured that every graph $G$ has an odd $K_{χ(G)}$ minor. In this paper, based on the description of graph structure, it is shown that every graph $G$ with indep…

Time-frequency representations on Lorentz spaces over locally compact Abelian groups
Jun Liu, Yaqian Lu, Xianjie Yan, Chi Zhang
https://arxiv.org/abs/2509.23296 https://

Time-frequency representations on Lorentz spaces over locally compact Abelian groups
Let $G$ be a locally compact Abelian group with a fixed Haar measure and, denote by $\widehat{G}$ its dual group. In this article, the authors obtain various boundedness of the short-time Fourier transform on Lorentz spaces: $$L^{p_1,u}(G)\times L^{p_2,v}(G)\to L^{q,w}(G\times\widehat{G})$$ with the indexes satisfying appropriate relations. These results are then used to prove the corresponding boundedness of $τ$-Wigner transforms and $τ$-Weyl operators. As an application, the Lieb's uncertai…

$p$-adic hypoerbolicity for Shimura varieties and period images
Benjamin Bakker, Abhishek Oswal, Ananth N. Shankar, Zijian Yao
https://arxiv.org/abs/2509.25461 https://

$p$-adic hypoerbolicity for Shimura varieties and period images
We prove that Shimura varieties and geometric period images satisfy a $p$-adic extension property for large enough primes $p$. More precisely, let $\mathsf{D}^{\times}\subset \mathsf{D}$ denote the inclusion of the closed punctured unit disc in the closed unit disc. Let $X$ be either a Shimura variety or a geometric period image with torsion-free level structure. Let $F$ be a discretely valued $p$-adic field containing the number field of definition of $X$, where $p$ is a large enough prime. Th…

Smooth functions that split a Klein bottle into two M\"obius bands
Bohdan Mazhar, Sergiy Maksymenko
https://arxiv.org/abs/2508.19636 https://arxiv.org…

Smooth functions that split a Klein bottle into two Möbius bands
Given a compact surface $M$, consider the right action $\mathcal{C}^{\infty}(M)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M)$, $(f, h) \mapsto f\circ h$, of the group $\mathcal{D}(M)$ of $\mathcal{C}^{\infty}$ diffeomorphisms of $M$ on the space $\mathcal{C}^{\infty}(M)$ of $\mathcal{C}^{\infty}$ functions on $M$. For $f\in\mathcal{C}^{\infty}(M)$ denote by $\mathcal{O}(f)$ its orbit, and by $\mathcal{O}_f(f)$ the path component of $\mathcal{O}(f)$ containing $f$. The paper continues a serie…

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

Translates of completely normal elements and the Morgan-Mullen conjecture
Theodoulos Garefalakis, Giorgos Kapetanakis
https://arxiv.org/abs/2509.23245 https://

Translates of completely normal elements and the Morgan-Mullen conjecture
Denote by $\mathbb F_q$ the finite field of order $q$ and by $\mathbb F_{q^n}$ its extension of degree $n$. Some $a\in\mathbb F_{q^n}$ is called primitive if it generates the multiplicative group $\mathbb F_{q^n}^*$ and it is called $q^n/q$-normal if its $\mathbb F_q$-conjugates form an $\mathbb F_q$-basis of $\mathbb F_{q^n}$ if the latter is viewed as an $\mathbb F_q$-vector space. Furthermore, some $a\in\mathbb F_{q^n}$ is called $q^n/q$-completely normal if it is $q^n/q^d$-normal for all $d…

On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence
Boris Tsvelikhovskiy
https://arxiv.org/abs/2509.15825 https://arxiv.org/pd…

On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence
Let $G \subset SL_3(\mathbb{C})$ be a finite abelian subgroup, and let $Y = G\operatorname{-}\mathrm{Hilb}(\mathbb{C}^3)$ be the corresponding $G$-Hilbert scheme. Denote by $Ψ: D^b_G(\mathrm{Coh}(\mathbb{C}^3)) \to D^b(\mathrm{Coh}(Y))$ the Bridgeland--King--Reid derived equivalence. For a nontrivial character $χ$ of $G$, let $χ^!$ be the corresponding skyscraper sheaf supported at the origin. It is known that $Ψ(χ^!)$ is always a pure sheaf supported either in degree $0$ or in degree $-1$…

Study the decays of $\chi _{cJ}(J=0,1,2)$ to light meson pairs with SU(3) flavor symmetry/breaking analysis
Bo Lan, Qin-Ze Song, Jin-Huan Sheng, Yi Qiao, Ru-Min Wang
https://arxiv.org/abs/2508.16372

Study the decays of $χ_{cJ}(J=0,1,2)$ to light meson pairs with SU(3) flavor symmetry/breaking analysis
Based on available experimental results on $χ_{cJ}(J=0,1,2)$ decays, we investigate the $χ_{cJ}\to PP$, $VV$, $PV$, and $PT$ decays by using SU(3) flavor symmetry/breaking approach, where $P$, $V$, and $T$ denote light pseudoscalar, vector, and tensor mesons, respectively. With the decay amplitude relations determined by SU(3) flavor symmetry/breaking, we present the branching ratios for all $χ_{cJ}\to PP$ and $χ_{cJ}\to VV$ modes, including ones without experimental data. While theoretical…

The nucleus of the Grassmann graph $J_q(N,D)$
Jae-Ho Lee, Jongyook Park, Ian Seong
https://arxiv.org/abs/2509.15395 https://arxiv.org/pdf/2509.15395…

The nucleus of the Grassmann graph $J_q(N,D)$
Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with vertex set $X$ that consists of the $D$-dimensional subspaces of $\mathcal{V}$. Two vertices are adjacent whenever their intersection has dimension $D-1$. Fix a vertex $x$ in $X$. The Terwilliger algebra $T=T(x)$ of $J_q(N,D)$ with respect to $x$ is the subalgeb…

Improved bounds on the postage stamp problem for large numbers of stamps
Eric James Faust, Michael Tait
https://arxiv.org/abs/2507.23627 https://arxiv.org/…

Improved bounds on the postage stamp problem for large numbers of stamps
Let $F_h(n)$ denote the minimum cardinality of an additive {\em $h$-fold basis} of $\{1,2,\cdots,n\}$: a set $S$ such that any integer in $\{1,2,\cdots, n\}$ can be written as a sum of at most $h$ elements from $S$. While the trivial bounds $h!n \; \lesssim \; F_h(n)^h \; \lesssim \; h^h n$ are well-known, comparatively little has been established for $h>2$. In this paper, we make significant improvements to both of the best-known bounds on $F_h(n)$ for sufficiently large $h$. For the lower bou…

Multiplicities of fields of values of conjugacy classes in finite groups
Juan Mart\'inez Madrid, Marco Vergani
https://arxiv.org/abs/2508.16361 https://

Multiplicities of fields of values of conjugacy classes in finite groups
Given a group $G$ we write $h(G)$ to denote the maximum number of times that a field extension of $\mathbb{Q}$ appears as the field of values of a conjugacy class of a group. In this work, we prove that $|G|$ is bounded in terms of $h(G)$. Moreover, we classify the groups with $h(G)\leq 3$.

student_cooperation: Student cooperation (2012)
Network of cooperation among students in the "Computer and Network Security" course at Ben-Gurion University, in 2012. Nodes are students, and edges denote cooperation between students while doing their homework. The graph contains three types of links: Time, Computer, Partners.
This network has 185 nodes and 360 edges.
Tags: Social, Offline, Multigraph, Unweighted

A p-adic criterion for Lehmer's conjecture
Anup B. Dixit, Sushant Kala
https://arxiv.org/abs/2507.20141 https://arxiv.org/pdf/2507.20141

A p-adic criterion for Lehmer's conjecture
For a non-zero algebraic number $α$ of degree $d$, let $h(α)$ denote its logarithmic Weil height. It is known that when $h(α)$ is small, and $d$ is large, the conjugates of $α$ are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a $p$-adic analogue of this result by obtaining lower bounds for $h(α)$ in terms of the number of its conjugates that lie in a finite extension of $\mathbb{Q}_p$, for some prime $p$…

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

There are siblings of $\chi$ which are permutations for $n$ even
Bj\"orn Kriepke (University of Rostock), Gohar Kyureghyan (University of Rostock)
https://arxiv.org/abs/2509.24839

There are siblings of $χ$ which are permutations for $n$ even
Let $1$ be the all-one vector and $\odot$ denote the component-wise multiplication of two vectors in $\mathbb F_2^n$. We study the vector space $Γ_n$ over $\mathbb F_2$ generated by the functions $γ_{2k}:\mathbb F_2^n \to \mathbb F_2^n, k\geq 0$, where $$ γ_{2k} = S^{2k}\odot(1+S^{2k-1})\odot(1+S^{2k-3})\odot\ldots\odot(1+S) $$ and $S:\mathbb F_2^n\to\mathbb F_2^n$ is the cyclic left shift function. The functions in $Γ_n$ are shift-invariant and the well known $χ$ function used in several …

student_cooperation: Student cooperation (2012)
Network of cooperation among students in the "Computer and Network Security" course at Ben-Gurion University, in 2012. Nodes are students, and edges denote cooperation between students while doing their homework. The graph contains three types of links: Time, Computer, Partners.
This network has 185 nodes and 360 edges.
Tags: Social, Offline, Multigraph, Unweighted

On the Precise Asymptotics of $ex(n,n,n,K_{2,t})$ for even $t$
Zilin Luo
https://arxiv.org/abs/2509.21756 https://arxiv.org/pdf/2509.21756

On the Precise Asymptotics of $ex(n,n,n,K_{2,t})$ for even $t$
Let $K_{2,t}$ denote the complete bipartite graph. For an integer $n\ge 1$, let $ex(n,n,n,K_{2,t})$ be the maximum number of edges in an $n\times n\times n$ tripartite graph (that is, a 3-partite graph with three parts each of size $n$) containing no copy of $K_{2,t}$. In this paper we prove that, for even $t\ge 2$, $$ ex(n,n,n,K_{2,t}) \ge \frac{3\sqrt{t-1}}{\sqrt{2}}\, n^{3/2} + o(n^{3/2}). $$ Combining our construction with earlier work of Tait and Timmons, we obtain $$ \lim\limits_{n\to\inf…

Spectral Tur\'an-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability
Jian Zheng, Honghai Li, Li Su
https://arxiv.org/abs/2509.24354 https://

Spectral Turán-type problems for the $α$-spectral radius of hypergraphs with degree stability
An $r$-pattern $P$ is defined as an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $ϕ$: $V(H)\rightarrow [l]$ such that the $r$-multiset $\{ϕ(v_{1}),\ldots,ϕ(v_{r})\}$ is in $E$ for every edge $\{v_{1},\ldots,v_{r}\}\in E(H)$. Let $Col(P)$ denote the family of all $P$-colorable $r$-graphs. This paper establishes spectral extremal results for $α$-spec…

student_cooperation: Student cooperation (2012)
Network of cooperation among students in the "Computer and Network Security" course at Ben-Gurion University, in 2012. Nodes are students, and edges denote cooperation between students while doing their homework. The graph contains three types of links: Time, Computer, Partners.
This network has 185 nodes and 360 edges.
Tags: Social, Offline, Multigraph, Unweighted

Reduced complexities for sequences over finite alphabets
John M. Campbell, James Currie, Narad Rampersad
https://arxiv.org/abs/2509.16034 https://arxiv.org…

Reduced complexities for sequences over finite alphabets
Letting $w$ denote a finite, nonempty word, let $\text{red}(w)$ denote the word obtained from $w$ by replacing every subword $s$ of $w$ of the form $cc \cdots c$ for a given character $c$ (such that there is no character immediately to the left or right of $s$ equal to $c$) with $c$. Complexity functions for infinite words play important roles within combinatorics on words, and this leads us to introduce and investigate variants of the factor and abelian complexity functions using the given red…

Transcendency of variants of Mills' constant
Kota Saito
https://arxiv.org/abs/2508.16068 https://arxiv.org/pdf/2508.16068

Transcendency of variants of Mills' constant
Let $\lfloor x\rfloor$ denote the integer part of $x$. For every sequence $(C_k)_{k\ge 1}$ of positive integers, we define $ξ(C_k)$ as the smallest real number $ξ$ such that $\lfloor ξ^{C_k} \rfloor$ is a prime number for every positive integer $k$. The number $ξ(3^k)$ is called Mills' constant. Recently, the author showed that $ξ(3^k)$ is irrational, but the transcendency of this number is still open. In this paper, we give sufficient conditions on $(C_k)_{k\ge 1}$ to determine the transc…

Maximum Spread of Vertex Degrees in a Simple Graph
Sergey Dmitrievich Onishchenko
https://arxiv.org/abs/2509.21429 https://arxiv.org/pdf/2509.21429

Maximum Spread of Vertex Degrees in a Simple Graph
We consider the following problem: let $n>k$ be natural numbers, and let $G$ be a graph on $n$ vertices (undirected, without loops or multiple edges). Denote by $h_k(G)$ the number of unordered pairs of vertices in the graph $G$ whose degrees differ by less than $k$. We aim to determine the smallest possible value $f(n,k)$ of the quantity $h_k(G)$. Interest in this question is motivated by the fact that the bipartite analogue of the problem enabled S. Cichomski and F. Petrov to prove the Burdzy…

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

Evaluation of a determinant involving Legendre symbols
Chen-Kai Ren, Zhi-Wei Sun
https://arxiv.org/abs/2507.18589 https://arxiv.org/pdf/2507.18589

Evaluation of a determinant involving Legendre symbols
Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3 \pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely.

lesmis: Les Misérables coappearances
The network of scene coappearances of characters in Victor Hugo's novel "Les Miserables". Edge weights denote the number of such occurrences.
This network has 77 nodes and 254 edges.
Tags: Social, Fictional, Weighted
https://networks.skewed.de/net/lesmis

Planar Tur\'an number of disjoint union of $C_3$ and $C_5$
Luyi Li, Ping Li, Guiying Yan, Qiang Zhou
https://arxiv.org/abs/2507.16351 https://

Planar Turán number of disjoint union of $C_3$ and $C_5$
The planar Turán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k\geq 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\ell}$ denote the cycle of length $\ell$ and $C_{\ell}\cup C_t$ denote the union of disjoint cycles $C_{\ell}$ and $C_t$. The planar Turán number $ex_{\mathcal{P}}(n,H)$ is known if $H=C_{\ell}\cup C_k$, where $\ell,k\in \{3,4\}$. In this paper, we de…

Twisted periods of modular forms
Tianyu Ni, Hui Xue
https://arxiv.org/abs/2507.17041 https://arxiv.org/pdf/2507.17041

Twisted periods of modular forms
Let $S_k$ denote the space of cusp forms of weight $k$ and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $χ$ mod $D$, we introduce twisted periods $r_{t,χ}$ on $S_k$. We show that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$ and $D$, then any $n$ periods with the same twist but different indices are linearly independent. We also prove that if $k$ is sufficiently large relative to $D$ then any $n$ periods with the same index but different tw…

Spin Leonard pairs and the zero diagonal space
Kazumasa Nomura, Paul Terwilliger
https://arxiv.org/abs/2509.21520 https://arxiv.org/pdf/2509.21520

Spin Leonard pairs and the zero diagonal space
We consider a Leonard pair $A, A^*$ of linear maps on a vector space $V$ that has finite positive dimension. This Leonard pair $A,A^*$ is said to have spin whenever there exist invertible linear maps $W : V \to V$ and $W^* : V \to V$ such that $W A = A W$ and $W^* A^* = A^* W^*$ and $W A^* W^{-1} = (W^*)^{-1} A W^*$. Let $\{θ^*_i\}_{i=0}^d$ denote a standard ordering of the eigenvalues of $A^*$. There is a related sequence of scalars $\{a_i\}_{i=0}^d$ called intersection numbers. The Leonard p…

student_cooperation: Student cooperation (2012)
Network of cooperation among students in the "Computer and Network Security" course at Ben-Gurion University, in 2012. Nodes are students, and edges denote cooperation between students while doing their homework. The graph contains three types of links: Time, Computer, Partners.
This network has 185 nodes and 360 edges.
Tags: Social, Offline, Multigraph, Unweighted

Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
G. Kavya Keerthana, S. Ananya, Ranganatha D
https://arxiv.org/abs/2509.17705 https://arxi…

Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
Let $\overline{p}_{k}(n)$ denote the number of overpartition $k$-tuples of $n$. In 2023, Saikia \cite{saikia} conjectured the following congruences: \begin{align*} \overline{p}_{q}(8n+2)& \equiv 0 \pmod{4},\quad \overline{p}_{q}(8n+3)\equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+4) \equiv 0 \pmod{2},\\ \overline{p}_{q}(8n+5)& \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+6) \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+7)\equiv 0 \pmod{32}, \end{align*} where $n\geq0$ and $q$ is prime. Recently, Sell…

Raising and lowering maps for tridiagonal pairs
Paul Terwilliger
https://arxiv.org/abs/2507.19400 https://arxiv.org/pdf/2507.19400

Raising and lowering maps for tridiagonal pairs
Let $V$ denote a nonzero finite-dimensional vector space. A tridiagonal pair on $V$ is an ordered pair $A, A^*$ of maps in ${\rm End}(V)$ such that (i) each of $A, A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i + V_{i+1}$ $(0 \leq i \leq d)$, where $V_{-1} =0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i \rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \sub…

student_cooperation: Student cooperation (2012)
Network of cooperation among students in the "Computer and Network Security" course at Ben-Gurion University, in 2012. Nodes are students, and edges denote cooperation between students while doing their homework. The graph contains three types of links: Time, Computer, Partners.
This network has 185 nodes and 360 edges.
Tags: Social, Offline, Multigraph, Unweighted

Examples of diameter-2 graphs with no triangle or $K_{2,t}$
Sean Eberhard, Vladislav Taranchuk, Craig Timmons
https://arxiv.org/abs/2508.19646 https://arxi…

Examples of diameter-2 graphs with no triangle or $K_{2,t}$
For each $t \ge 1$ let $W_t$ denote the class of graphs other than stars that have diameter $2$ and contain neither a triangle nor a $K_{2,t}$. The famous Hoffman--Singleton Theorem implies that $W_2$ is finite. Recently Wood suggested the study of $W_t$ for $t > 2$ and conjectured that $W_t$ is finite for all $t \ge 2$. In this note we show that (1) $W_3$ is infinite, (2) $W_5$ contains infinitely many regular graphs, and (3) $W_7$ contains infinitely many Cayley graphs. Our $W_3$ and $W_5$ ex…

yahoo_song: Yahoo song ratings (2011)
A bipartite network of users and songs they rated, as used in the 2011 KDD Cup and extracted from Yahoo! Music. Edge weights denote a rating scaled from 0 to 100. More information about this data set is available at http://konect.cc/networks/yahoo-song.
Th…

An identity relating Catalan numbers to tangent numbers with arithmetic applications
Tongyuan Zhao, Zhicong Lin, Yongchun Zang
https://arxiv.org/abs/2507.20965 https://

An identity relating Catalan numbers to tangent numbers with arithmetic applications
We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$ \sum_{k=0}^{n-1}{2n\choose 2k+1}2^{2n-2k}(-1)^{k}E_{2k+1}=2^{2n+1}, $$ where $E_{2k+1}$ denote the tangent numbers. Interestingly, the latter identity can be applied to prove that $(n + 1)E_{2n+1}$ is divisible by $2^{2n}$ and the quotient is an odd number, a fact whose…

Necklaces, permutations, and periodic critical orbits for quadratic polynomials
Matthew Baker, Andrea Chen, Sophie Li, Matthew Qian
https://arxiv.org/abs/2508.12924 https://

Necklaces, permutations, and periodic critical orbits for quadratic polynomials
Let $G_n$ denote the $n^{\rm th}$ Gleason polynomial, whose roots correspond to parameters $c$ such that the critical point $0$ is periodic of exact period $n$ under iteration of $z^2 + c$, and let $\bar{G}_n$ denote the reduction of $G_n$ modulo $2$. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of $G_n$ is equal to the number of irreducible factors of $\bar{G}_n$ for all $n$. We provide a bijective proof for this result by first providing explicit …

Two results on set families: sturdiness and intersection
Yongjiang Wu, Zhiyi Liu, Lihua Feng, Yongtao Li
https://arxiv.org/abs/2508.19723 https://arxiv.org…

Two results on set families: sturdiness and intersection
This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness $β(\mathcal{F})$ is defined as the minimum $|\mathcal{F} (i,\bar{j})|$ over all $i\neq j$. A family $\mathcal{F}$ is called an IU-family if it satisfies the intersection constraint: $F\cap F'\neq \emptyset $ for all $F,F'\in \mathcal{F}$, as well as t…

An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles
Cheng Yeaw Ku, Kok Bin Wong
https://arxiv.org/abs/2508.17683 https://

An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles
Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if $π_1$ and $π_2$ do not have any common cycles for all distinct $π_1,π_2\in H$. The matching number of a family $\mathcal A\subseteq S_{n,k}$ is denoted by $ν_{p}(\mathcal A)$ and is defined to be the size of the largest matching in $\mathcal A$. In this…

Characterizing $A_\sigma$-minimizer graphs: given order and independence number
Shuchao Li, Jiaqi Zhang
https://arxiv.org/abs/2508.09770 https://arxiv.org/…

Characterizing $A_σ$-minimizer graphs: given order and independence number
For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_σ(G) \) matrix, proposed by Nikiforov, is defined as \( A_σ(G)=σD(G)+(1 - σ)A(G) \), where \( σ\in[0,1] \). This matrix captures the gradual transition from \( A(G) \) to \( Q(G) \). Let \( \mathcal{G}_{n,α} \) denote the family of all connected graphs with \( n \) vertices and independence number \( α\…

An eigenvalue interlacing approach to Garland's method
Alan Lew
https://arxiv.org/abs/2508.17279 https://arxiv.org/pdf/2508.17279

An eigenvalue interlacing approach to Garland's method
Let $X$ be a pure $d$-dimensional simplicial complex. For $0\le k\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let $\tilde{H}_k(X;\mathbb{R})$ be its $k$-dimensional reduced homology group with real coefficients. For $σ\in X$, let $\text{lk}(X,σ)$ be the link of $σ$ in $X$. For a matrix $M$, we denote by $\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that,…

Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups
Guoqing Wang
https://arxiv.org/abs/2509.18724 https://

Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups
For any finite abelian group $G$ and commutative unitary ring $R$, by $R[G]$ we denote the group algebra over $R$. Let $T=(g_1,\ldots,g_{\ell})$ be a sequence over the group $G$. We say $T$ is algebraically zero-sum free over R if $\prod\limits_{i=1}^r(X^{g_i}-a_i)\neq 0\in R[G]$ for all $a_1,\ldots,a_{\ell}\in R\setminus \{0\}$. Let $d(G,R)={\rm sup}\{|T|: T \mbox{ is an algebraically zero-sum free sequence over } R \mbox{ of terms from }G\}.$ This invariant of the group algebra $R[G…

Tops of graphs of projective codes
Edyta Bartnicka
https://arxiv.org/abs/2509.17958 https://arxiv.org/pdf/2509.17958

Tops of graphs of projective codes
Let $Γ_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. Denote by $Π[n,k]_q$ the subgraph of $Γ_k(V)$ formed by projective codes. We give a complete description of cliques $\langle U]^Π_{k}$ of $Π[n,k]_q$ consisting of all $k$-dimensional projective codes contained in a fixed $(k+1)$-dimensional subspace of $V$. We show when and in how…

Integral bases, perfect matchings, and the Petersen graph
Ahmad Abdi, Olha Silina
https://arxiv.org/abs/2508.15602 https://arxiv.org/pdf/2508.15602

Integral bases, perfect matchings, and the Petersen graph
Let $G=(V,E)$ be a matching-covered graph, denote by $P$ its perfect matching polytope, and by $L$ the integer lattice generated by the integral points in $P$. In this paper, we give polyhedral proofs for two difficult results established by Lovász (1987), and by Carvalho, Lucchesi, and Murty (2002) in a series of three papers. More specifically, we reprove that $L$ has a lattice basis consisting solely of incidence vectors of some perfect matchings of $G$, $2x\in L$ for all $x\in \mathrm{lin}…

A result on spanning trees with bounded total excess
Sizhong Zhou
https://arxiv.org/abs/2507.15139 https://arxiv.org/pdf/2507.15139…

A result on spanning trees with bounded total excess
Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $ρ(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{0,d_T(v)-k\}$. The total $k$-excess $\mbox{te}(T,k)$ is defined by $\mbox{te}(T,k)=\sum\limits_{v\in V(T)}{\max\{0,d_T(v)-k\}}$. A tree $T$ is said to be a $k$-tree if $d_T(v)\leq k$ for any $v\in V(T)$, that is to say, the maximum degree of a $k$-tree is at most $k$. In fact, $T$ is a spanning $k$-tree if and only if $\…

Tree-independence number VI. Thetas and pyramids
Maria Chudnovsky, Julien Codsi
https://arxiv.org/abs/2509.15458 https://arxiv.org/pdf/2509.15458

Tree-independence number VI. Thetas and pyramids
Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Let $W_{t\times t}$ be the $t$-by-$t$ hexagonal grid and let $\mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t \times t}$. We denote by $ω(G)$ the size of the largest clique in $G$. We prove that for every integer $t$ there exist integers $c_1(t)$, $c_2(t)$ and $d(t)$ such that …