arXiv preprints from January 1, 2026 through July 20, 2026 — 14:08:20 EST
Let $q$ be an odd prime and let $d$ be an odd integer. We show that the arbitrary-step transition relation of the generalized Collatz map $T_{q,d}$ is not first-order definable in Base-2 Büchi Arithmetic ($BA_2$). We do this by demonstrating that if the transition relation were definable, the exponential set $P_q = \{q^y : y \in...
We show that the relativistic energy-momentum relation can emerge as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints, which are...
Let $K$ be a number field with ring of integers $\mathcal{O}$ and $α\in\mathcal{O}$. For any prime ideal $\mathfrak{p}$ of $\mathcal{O}$, we obtain its higher $α$-Wieferich property, which implies a nonexistence theorem for higher Wieferich unramified prime ideals. If $β\in\mathcal{O}$ is relatively prime to $α$ and all prime ideal...
A digraph $G$ is weightable if its edges can be weighted with real numbers such that the total weight in each directed cycle equals 1. There are several equivalent conditions: that $G$ admits a 0/1-weighting with the same property, or that $G$ contains no subdivided "double-cycle" as a subdigraph, or that for every triple of vertices,...
Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial...
Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the...
We introduce a novel spatial discretization technique for the reliable and efficient simulation of magnetization dynamics governed by the Landau-Lifshitz (LL) equation. The overall discretization error is systematically decomposed into temporal and spatial components. The spatial error analysis is conducted by formulating the LL...
We investigate the Keller--Segel--(Navier--)Stokes system posed in a smooth bounded domain \(Ω\subset \mathbb{R}^N\) with \(N = 2,3\): \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = Δn - \nabla \cdot \big( n S(n)\nabla c \big), \\[2mm] u \cdot \nabla c = Δc - c + n, \\[2mm] u_t + κ(u \cdot \nabla) u = Δu - \nabla P + n...
In this paper, we consider the following logarithmic Schrödinger equation \[ -Δu + V(x)u = u \log u^{2},\quad x\in\mathbb{R}^{N}. \] Assuming that \(V\in C(\mathbb{R}^{N},\mathbb R)\), \(V\) is bounded away from zero, and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the...
The article focuses on the differences in mathematics performance between girls and boys visible from the first four months of compulsory schooling in the French education system. The influence of gender stereotypes in the evaluation practices of teachers and the threat of the gender stereotype on student performance are questioned....
Highly oscillatory differential equations, commonly encountered in multi-scale problems, are often too complex to solve analytically. However, several numerical methods have been developed to approximate their solutions. Although these methods have shown their efficiency, the first part of the strategy often involves heavy...
A bi-Cayley graph over a cyclic group $\mathbb{Z}_n$ is called a bicirculant graph. Let $Γ=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=R^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T=T^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a...
We study bi-Lipschitz right-equivalence of holomorphic function germs $f:(\mathbb{C}^2,0)\to(\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $γ$ we consider the Newton polygon of $f_x(X+γ(Y),Y)$ and define its augmentation by adjoining the point $(0,\operatorname{ord} f(γ(y),y)-1)$. We prove that the resulting...
The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at $0\in\mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as...
Motivated by applications to the fractional quantum Hall effect and, in particular, to the Bernevig-Haldane conjectures, we investigates the behavior of Macdonald polynomials under specializations of the form q a t b = 1. Our main focus is to explain, in a simple and purely combinatorial way, why certain nonsymmetric Macdonald...
We propose a hierarchical splitting approach to differential equations that provides a design principle for constructing splitting methods for $N$-split systems by iteratively applying splitting methods for two-split systems. We analyze the convergence order, derive explicit formulas for the leading-order error terms, and investigate...
We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an explicit approach based on Poincaré residue calculus, allowing a uniform treatment of smooth,...
This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as...
We show that cutoff in separation occurs for Brownian motion in some families of compact harmonic manifolds. We compute the cutoff time and windows in four families of compact harmonic manifold namely S n , CP n , HP n and RP n (the first three families are the only families of compact simply connected harmonic manifolds, see [19])....
In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their...
The three-page index $α_3(L)$ is an invariant that measures the complexity of representing a link $L$ in a three-page book. It is known that $α_3(L)$ admits a linear upper bound in terms of the crossing number, with equality realized by the Hopf link. In this paper, we investigate the equality case of this bound from a diagrammatic...
Let $V$ be a Möbius vertex algebra and $G$ an abelian group of automorphisms of $V$. We construct $P(z)$-tensor product bifunctors for the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules (without $g$-actions) for $g\in G$ and the category of $C_{n}$-cofinite grading-restricted generalized...
Inverse problems are inherently ill-posed, suffering from non-uniqueness and instability. Classical regularization methods provide mathematically well-founded solutions, ensuring stability and convergence, but often at the cost of reduced flexibility or visual quality. Learned reconstruction methods, such as convolutional neural...
Berndtsson's famous theorem asserts that, for a compact Kähler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective...
In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^ω$, which serves as a leading order approximation to $\mathbf{S}_{\partial...