arXiv preprints from January 1, 2026 through September 6, 2026 — 01:51:17 EST
We study iterated weighted residual (WR) splittings generated by a positive operator $R_{0}\in B\left(H\right)_{+}$ and a finite family of contractions $C_{1},\dots,C_{m}$ in $B\left(H\right)$. The associated residual update $R\mapsto R^{1/2}(I-C^{*}_{j}C_{j})R^{1/2}$ produces an $m$-ary energy tree of residuals $\left\{ R_{w}\right\}...
In this paper, we give a classification of regular maps with Euler characteristic $-pq$ for distinct primes $q>p\geq 5$. This together with previous classification of regular maps with Euler characteristic $-2p,-3p$ and $-p^2$ completes the classification of regular maps with Euler characteristic $-pq$ for two primes $p$ and $q$. An...
We study an online linear programming (OLP) model in which inventory is not provided upfront but instead arrives gradually through an exogenous stochastic replenishment process. This replenishment-based formulation captures operational settings, such as e-commerce fulfillment, perishable supply chains, and renewable-powered systems,...
The Couette-Taylor instability occurs in a viscous fluid confined between two coaxial rotating cylinders. When the Taylor number surpasses a critical value, the stable Couette flow destabilizes, giving way to steady Taylor vortices. As the Taylor number increases further, these vortices themselves become unstable, transitioning into...
This paper considers submodular function minimization (SFM) restricted to a family of subsets. We show that SFM over complements of families with certain hierarchical structures can be solved in polynomial-time. This yields a polynomial-time algorithm for SFM over complements of various families, such as intersecting families,...
We introduce RANDSMAPs (Random-feature/multi-scale neural decoders with Mass Preservation), numerical analysis-informed, explainable neural decoders designed to explicitly respect conservation laws when solving the challenging ill-posed pre-image problem in manifold learning. We start by proving the equivalence of vanilla random...
The integrated theory of the probe and singular sources methods (IPS) is developed for an inverse obstacle problem governed by the stationary Schrödinger equation in a bounded domain. The unknown obstacles are penetrable, and their surface is modeled by a part of the support of the potential in the governing equation. The main results...
The Marcinkiewicz--Zygmund theorem is a fundamental result in probability theory that establishes rates of convergence in the strong law of large numbers (SLLN). Although numerous extensions have been developed for dependent sequences, many classes of processes, particularly those exhibiting strong dependence, remain unexplored. In...
In this paper, we apply an implicit Euler scheme to discretize the complex Ginzburg-Landau equation and prove the existence of a numerical attractor for the discrete Ginzburg-Landau system. We establish the upper semicontinuity of the numerical attractor with respect to the global attractor as the time step tends to zero. Furthermore,...
For a discrete dynamical system on $\R$ generated by a quadratic function, we show, using elementary computations, that the existence, number, and stability of 3-cycles are determined by a single parameter depending on the coefficients of the function.
We investigate the effect of planar univalent harmonic mappings on the Lebesgue measure of measurable sets in the complex plane. Motivated by Problem 3.25 of Koh and Kovalev (HQM2010), we establish sharp quantitative area distortion inequalities for disks and for arbitrary measurable sets under sense-preserving harmonic self-maps of...
This text evolves from the lecture notes for my course on Catalan's conjecture in winter term 2025/26. The ultimate goal is to give full details of Mihăilescu's proof. Current chapters: 1. Euler's theorem: $x^2-y^3=1$; 2. V. Lebesgue's theorem: $x^m-y^2=1$; 3. Chao Ko's theorem: $x^2-y^q=1$ with $q\ge5$; 4. Two relations of Cassels:...
In this manuscript, we introduce the quadratic--phase Fourier--Bessel transform and develop its foundational properties, including continuity, the Riemann--Lebesgue lemma, reversibility, and Parseval's identity. We define the associated translation operator and convolution product, establishing their main properties within this...
In this paper, we present a method capable of ensuring practical prescribed-time control with guaranteed performance for a class of nonlinear systems in the presence of time-varying parametric and dynamic uncertainties, and uncertain control coefficients. Our design consists of two key steps. First, we construct a performance-rate...
We study the structure of tensor products of $\mathfrak{gl}(\infty) = \varinjlim \mathfrak{gl}(n)$-modules $\mathbf L(\mathbf λ) \otimes \mathbf F$ where $\mathbf L(\mathbf λ)$ is a simple integrable highest weight module and $\mathbf F$ is a simple integrable weight multiplicity-free module. Both $\mathbf L(\mathbf λ)$ and $\mathbf...
We provide a complete description of the order isomorphisms between the self-adjoint parts of $C^*$-algebras. Furthermore, we characterize such isomorphisms between general operator intervals in $AW^*$-algebras. For the description, we use Jordan $^*$-isomorphisms and closed operators in the regular rings of $AW^{*}$-algebras. This...
We study a linear observation model with an unknown permutation called \textit{permuted/shuffled linear regression}, where responses and covariates are mismatched and the permutation forms a discrete, factorial-size parameter. The permutation is a key component of the data-generating process, yet its statistical investigation remains...
We consider a transmission problem for the Helmholtz equation across the boundary of an extension domain. A such boundary can be Lipschitz, fractal, or of varying Hausdorff dimension for instance. We generalise the notions of layer potential and Neumann-Poincar{é} operators, and of Calder{ó}n projectors in that context. Those boundary...
Inverse problems in computational physics often require matching high-dimensional spatio-temporal fields, leading to prohibitive computational costs and ill-conditioned optimizations. We introduce modal-centric field inversion (MCFI), a paradigm that reformulates inverse problems in the reduced space of proper orthogonal decomposition...
Let K be a finite Galois extension of Q. The normal basis theorem provides an element of K whose conjugates form a Q-basis of K. Here we obtain such an element with controlled size. This improves a recent result by Fukshansky and Jeong. By the way, we estimate Minkowski's minima of ideals of integers of number fields.
This communication contributes to research on proof validation as a lens for uncovering didactical phenomena related to proof and proving. We revisit the puzzling case of lower secondary students in France who validate circular proofs. That is proofs in which the conclusion is used within a step of the proof itself. While these...
This paper introduces a new class of geometric structures in almost contact metric geometry, which we call locally conformal almost generalized $f$-cosymplectic manifolds. These are almost contact metric structures $(φ, ξ, η, g)$ equipped with a closed Lee form $ω$ and a smooth function $f$ satisfying $$ dη= ω\wedge η, \;\; dΦ=...
We study the extended module category $m\operatorname{-mod}Λ$ of an algebra $Λ$, recently introduced by Gupta and Zhou. The existence of a postprojective component of $m\operatorname{-mod}Λ$ is shown for certain algebras $Λ$, and a knitting algorithm through cohomological dimension vectors is provided. In particular, the extended...
We consider the standard Darboux space equipped with the radial symmetric contact form. We study co-orientation preserving contactomorphisms between relatively compact domains up to the boundary. We determine the contactomorphism classes among all strict vertically convex domains over a round ball in the Liouville hyperplane that are...
This paper develops a unified framework for the analysis and design of dimension-varying control systems by constructing an intrinsic quotient state space, $Ω$. A significant challenge in non-fixed-dimensional systems is the lack of a common metric space that enables comparison of states across dimensions without relying on arbitrary...