Applications of an identity of Batır
Based on an interesting identity of Batır we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.
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Mathematics
arXiv preprints from January 1, 2026 through September 22, 2026 — 07:01:49 EST
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Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noises
In this paper, we study systems of $N$ interacting particles described by the classical and relativistic Langevin dynamics with singular forces and multiplicative noises. For the classical model, we prove the ergodicity, obtaining an exponential rate of convergence to the invariant Boltzmann-Gibbs distribution, and the small-mass...
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New gap principle for semiabelian varieties using globally valued fields
Hrushovski observed that the new gap principle of Gao-Ge-Kühne is essentially equivalent to the Bogomolov conjecture over arbitrary globally valued fields of characteristic $0$. Building on this observation, we prove a new gap principle for semiabelian varieties by reducing the Bogomolov conjecture for semiabelian varieties to the...
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Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms
We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric...
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On Navier-Stokes equations arising from the rotation of an obstacle in a fluid
We consider the modified Navier-Stokes equations in R3 describing the motion of a fluid in the presence of a rotating rigid body. Weighted Sobolev spaces are used to describe the behavior of solutions at large distances. Under suitable assumptions, w e prove the existence and regularity of solutions satisfying appropriate conditions...
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Pluriclosed 3-folds with vanishing Bismut Ricci form: General theory in the quasi-regular case
We study compact complex $3$-dimensional non-Kähler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special Kähler geometry in complex dimension $2$, recently obtained by Barbaro, Streets and the first and third authors. We show that in the quasi-regular case, the reduced geometry satisfies...
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A classification of coadjoint orbits carrying Gibbs ensembles
A coadjoint orbit $O_λ\subseteq {\mathfrak g}^*$ of a Lie group $G$ is said to carry a Gibbs ensemble if the set of all $x \in {\mathfrak g}$, for which the function $α\mapsto e^{-α(x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $Ω_λ$. We describe a classification of all coadjoint...
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Stability results for Berge-matching in hypergraphs
Given a graph $F$, a hypergraph is called a Berge-$F$ if it can be obtained by expanding each edge of $F$ into a hyperedge containing it. Let $M_{k}$ denote the matching of size $k$. Kang, Ni, and Shan [12] determined the Turán number of Berge-$M_k$. Our main result shows that if an $r$-uniform hypergraph $H$ on $n$ vertices has...
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An Extension of the Collatz Conjecture modulo $2^p+2^q$
In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new conjecture modulo $10$ which may be stated as follows. Starting from any positive integer, if it is...
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On the existence of meromorphic solutions of the complex Schrödinger equation with a q-shift
In this paper, we study the following complex Schrödinger equation with a $q$-difference term: \begin{align}\tag{†}\label{dagger} f'(z) = a(z)f(qz) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \end{align} where $a(z) \not\equiv 0$ is a small meromorphic function with respect to $f(z)$, and all the coefficient...
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Positive representations over real closed fields
We develop the theory of $Θ$-positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.
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Higher amalgamation in $\mathrm{ACFA}^{+}$
We show two results on higher amalgamation in the theory $\mathrm{ACFA}^{+}$, the model companion of the theory of difference fields with an additive character (added as a continuous logic predicate) on the fixed field in characteristic 0. On one hand, we show that the non-trivial condition for 3-amalgamation established in a...
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Horn inequalities on a quiver with an involution
Derksen and Weyman described the cone of semi-invariants associated with a quiver. We give an inductive description of this cone, followed by an example of refinement of the inequalities characterising anti-invariant weights in the case of a quiver equipped with an involution.
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A C*-cover lattice dichotomy
In this paper, we show that the lattice of C*-covers of a non-selfadjoint operator algebra is either one point or uncountable. We prove that there are non-selfadjoint operator algebras with a one-point lattice in two ways: as an explicit subalgebra of the C*-algebra of a universal contraction, and via a direct limit construction...
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Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions
This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence and uniqueness of maximal weak solutions in suitable weighted $Z$-spaces in the absence of...
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Isomorphism invariance of the girth of Artin groups
For all Artin groups, we characterise the girth (i.e. the length of a shortest cycle) of the defining graph algebraically, showing that it is an isomorphism invariant. Using this result, we prove that the Artin groups based on a cycle graph are isomorphically rigid. Alongside the girth, we introduce a new graph invariant, the...
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Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion
We investigate the rupture of invariant tubes in a class of nonautonomous dynamical systems arising from time-dependent Ermakov-type equations. Starting from an exactly tube-integrable reference system, we analyze a time-dependent invariant obtained from a positivity-preserving second-order perturbative construction, which provides a...
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A high order accurate and provably stable fully discrete continuous Galerkin framework on summation-by-parts form for advection-diffusion equations
We present a high-order accurate fully discrete numerical scheme for solving Initial Boundary Value Problems (IBVPs) within the Continuous Galerkin (CG)-based Finite Element framework. Both the spatial and time approximation in Summation-By-Parts (SBP) form are considered here. The initial and boundary conditions are imposed weakly...
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Dick and Liz Askey's visit to U.S.S.R. in 1987, and how the discrete Askey scheme also originated in Russia
This paper describes how the discrete Askey scheme independently arose in Russia and how Askey learned about this. In particular, Askey met main characters in this story, namely Gel'fand and Suslov as well as Nikiforov and Uvarov, during his trip to U.S.S.R, in September 1987. The paper describes this trip in some detail, in...
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Primeness property for regular gradings
Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite abelian group. For a $G$-graded $K$-algebra $A$, we define the primeness property for graded central polynomials: for any graded polynomials $f$ and $g$ in disjoint sets of variables, if $fg$ is graded central, then both $f$ and $g$ are graded central. Let...
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ZIVR: An Incremental Variance Reduction Technique For Zeroth-Order Composite Problems
This paper investigates zeroth-order (ZO) finite-sum composite optimization. Recently, variance reduction techniques have been applied to ZO methods to mitigate the non-vanishing variance of 2-point estimators in constrained/composite optimization, yielding improved convergence rates. However, existing ZO variance reduction methods...
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On the effects of protection zone and directed population flux in prey-predator dynamics
We study a spatial predator-prey model in which prey can enter a protection zone (refuge) inaccessible to predators, while predators exhibit directed movement toward prey-rich regions. The directed movement is modeled by a far-sighted population flux motivated by classical movement rules, in contrast to the more commonly analyzed...
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On the application of the factorized Fueter-Sce map to the slice hyperholomorphic Cauchy kernel
The Fueter-Sce theorem is one of the most important results in hypercomplex analysis, providing a two-step procedure for constructing axially monogenic functions starting from holomorphic functions of one variable. In the first step, the so-called slice operator is applied to holomorphic functions of one variable, producing the class...
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The Initial Value Problem for the Generalised Einstein Equations
We discuss the initial value problem for the Einstein equations in Hitchin's generalised geometry for the case of closed divergence (which correspond to the equations of motion in the bosonic part of the NS-NS sector in type II ten-dimensional supergravity) and establish the existence of a maximal globally hyperbolic development...
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Stochastic convergence of a class of greedy-type algorithms for Configuration Optimization Problems
Greedy Sampling Methods (GSMs) are widely used to construct approximate solutions of Configuration Optimization Problems (COPs), where a loss functional is minimized over finite configurations of points in a compact domain. While effective in practice, deterministic convergence analyses of greedy-type algorithms are often restrictive...