arXiv preprints from January 1, 2026 through July 20, 2026 — 11:56:20 EST
In this paper we derive new identities satisfied by Chebyshev polynomials of the first kind and big q-Jacobi polynomials. An immediate benefit of the derived identities is the achievement of closed-form expressions for the Laurent polynomials that identify minimum-support interpolating subdivision schemes reproducing finite sets of...
In this article, we consider the nonlinear Steklov eigenvalue problem in outward cuspidal domains. Using the compactness of the weighted trace embedding we obtain the variational characterization of the first non-trivial eigenvalue and prove the existence of a corresponding weak solution.
We prove quantum ergodicity and quantum mixing for sequences of finite Schreier graphs converging to an infinite Cayley graph whose adjacency operator has absolutely continuous spectrum. Under Benjamini-Schramm convergence (or strong convergence in distribution), we show that correlations between eigenvectors at distinct energies...
We study positive operator decompositions associated with rooted trees of orthogonal projections. In this sense, the refinement tree induces an ``MRA in $B\left(H\right)_{+}$''. To each node we assign a positive content operator, and these contents split along the tree and yield a positive decomposition at each fixed depth. The...
In this paper, we describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are "inner", i.e. given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson...
We investigate the asymptotic behavior of Halpern-type iterations applied to quasi-nonexpansive operators arising in best approximation problems over the intersection of finitely many closed convex sets in $\mathbb{R}^n$. Assuming a local decrease condition for the underlying operator and standard requirements on the stepsizes $(α_k)...
We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd integer. We first formulate an existence conjecture asserting that, for every prime number $p_n$,...
In these notes we describe the geometry of tropical balls in $\mathbb{R}^n$ equipped with the tropical metric. After defining the tropical length of rectifiable curves (and not just piecewise linear curves), we characterize compact tropically geodesic sets in $\mathbb{R}^n$. Next, we describe the tropical unit ball as a zonotope, via...
In this work, we deal with the stochastic counterpart of the nonlocal Cahn-Hilliard equation with regular potential in a smooth bounded one-, two- or three-dimensional domain. The problem is endowed with homogeneous Neumann boundary conditions and random initial data. Furthermore, the system is driven by cylindrical noise of...
We study ratio-induced mismatch costs of the form $c(s,o)=J(ι_S(s)/ι_O(o))$, built from positive scale maps $ι_S:S\to(0,\infty)$ and $ι_O:O\to(0,\infty)$ and a penalty $J:(0,\infty)\to[0,\infty)$. Assuming inversion symmetry, strict convexity, normalization $J(1)=0$, and a multiplicative d'Alembert identity, we show that...
We study Weyl elements in isotropic reductive groups over commutative rings. Our main result in an explicit formula for squares of such elements. We also classify these elements in rank one groups and prove basic properties of their loci.
We establish a multidimensional fractal transference principle for digit-restricted sets associated with subsets of $\mathbb{N}^d$, extending the one-dimensional framework of Nakajima--Takahasi, Adv. Math. (2025). We develop general Hausdorff-dimension tools via the singular value potential $φ^s(\mathbf a)$ and the multivariate...
This paper presents a static event-triggered control strategy for multivariable Newton-based extremum seeking. The proposed method integrates event-triggered actuation into the Newton-based optimization framework to reduce control updates while maintaining rapid convergence to the extremum. Unlike traditional gradient-based extremum...
The alternating direction method of multipliers (ADMM) has gained increasing popularity in embedded model predictive control (MPC) due to its code simplicity and pain-free parameter selection. However, existing ADMM solvers either target general quadratic programming (QP) problems or exploit sparse MPC formulations via Riccati...
We present a streamlined and simplified proof of the Kakeya set conjecture in $\mathbb{R}^3$.
In this work we introduce and analyse a new low-order method for the variable-density incompressible Navier-Stokes equations. The main novelty of the proposed method lies in the support of general meshes, possibly including polygonal or polyhedral elements as well as non-matching interfaces. We carry out a complete analysis, showing...
We present some thoughts on the relation between symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk. Among other things, use of this relation leads to a finite dimensional realization result for rational matrix functions in the Schur-class on the symmetrized bidisk and also to a...
Geometric models have emerged as an important tool in the representation theory of algebras. Surface models associated to gentle algebras have been particularly fruitful in advancing our understanding of their module and derived categories. We give an overview of some of the theoretical advances that geometric surface models for the...
This paper deals with the extension of partial actions of topological groups on topological spaces. Within this framework, we introduce a class of topological embeddings defined via the inverse semigroup of homeomorphisms between open subsets of a topological space. We describe several embeddings of this type, referred to as $Γ$-...
A total prime labeling of a graph of order $n$ is an extension of a prime labeling in which we distinctly label the vertices and edges. The goal of the labeling is for adjacent vertex labels to be relatively prime, and for each vertex of degree at least two, the greatest common divisor of the labels on its incident edges is equal to...
The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{μ^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right). $$ This formula expresses a non-trivial divisor sum involving the Möbius function $μ$ and Euler's...
This paper presents closed-form evaluations of two new Apéry-like series of weight $4$ that involve harmonic numbers of the form $H_{2k}$. Several key results are derived and subsequently used to establish connections to the main series.
Super-time-stepping (STS) methods provide an attractive approach for enabling explicit time integration of parabolic operators, particularly in large-scale, higher-dimensional kinetic simulations where fully implicit schemes are impractical. In this work, we present an explicit STS framework tailored for diffusion operators in...
In this paper, we extend the results of \cite{BCGS} on refined conjectures by Kurihara and Kolyvagin, allowing primes of any reduction type in the case of Kurihara's conjectures, and inert primes in the underlying imaginary quadratic field in the case of Kolyvagin's. The key innovation is a new approach to the computation of the...
Non-discrete semigroup $T_1$-topologies on the extended bicyclic semigroup $\mathscr{C}_\mathbb{Z}$ are constructed. Also, we present topological conditions, when a semigroup (shift-continuous) $T_1$-topology on $\mathscr{C}_\mathbb{Z}$ is discrete.