Open Journal of Mathematical Sciences (OMS)

This paper investigates the coupled thermoelastic interactions within an \(n\)-dimensional rectangular parallelepiped domain under time-dependent boundary conditions, formulating a hyperbolic system based on the Cattaneo-Vernotte principle to account for finite-speed thermal wave propagation. The mixed boundary value problem, incorporating non-homogeneous Dirichlet conditions and Cauchy initial data for displacement and temperature fields, is solved analytically via the Generalized Fourier Principle, yielding a unified solution expressed as an \(n\)-dimensional eigenfunction expansion. To validate the analytical findings and address complex configurations, a Fibonacci Collocation Spectral Method (FCSM) evaluated at Chebyshev–Gauss–Lobatto nodes is developed. Rigorous error analysis in \(L^2\) and \(L^\infty\) norms confirms spectral convergence under appropriate regularity assumptions. Numerical experiments in one, two, and three dimensions demonstrate exponential error decay from \(\mathcal{O}(10^{-3})\) to \(\mathcal{O}(10^{-14})\) with moderate polynomial degrees, establishing a robust theoretical and computational framework for analyzing wave-like thermoelastic behavior in high-precision engineering and advanced materials applications.

We study the Diophantine problem of determining for which positive integers \(M\) the sum of \(M\) consecutive squares beginning at \(a^{2}\) can itself be a square, namely \[\sum\limits_{i=0}^{M-1}(a+i)^{2}=s^{2}.\] Using the necessary conditions established by Beeckmans, we derive sharper congruence restrictions on the parameter \(M\). In particular, we prove that no solution exists when \(M\equiv5,6,7,8\) or \(10\left(\text{mod}\,12\right)\). For the remaining congruence classes \(M\equiv0,1,2,4,9\) or \(11\left(\text{mod}\,12\right)\), we obtain refined necessary conditions, namely \(M\equiv0\) or \(24\left(\text{mod}\,72\right)\); \(M\equiv1,2\) or \(16\left(\text{mod}\,24\right)\); \(M\equiv9\) or \(33\left(\text{mod}\,72\right)\); or \(M\equiv11\left(\text{mod}\,12\right)\), together with the corresponding congruence restrictions on \(a\) and \(s\). These classes should be interpreted only as necessary compatibility conditions; they do not, on their own, establish the existence of solutions. The remaining residue class \(M\equiv3\left(\text{mod}\,12\right)\) is examined separately by means of a recursive residue-class sieve that yields computational evidence against solvability, although no complete symbolic exclusion is claimed. Finally, when \(M\) is itself a square and a solution exists, we show that necessarily \(M\equiv1\left(\text{mod}\,24\right)\) and \(\left(M-1\right)/24\) is a generalized pentagonal number.

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let f : (M, g) → (M̂, ĝ) be a harmonic map from a compact manifold. Instead of assuming global nonpositivity of the sectional curvature of the target manifold, we impose a curvature bound localized along the image f(M), expressed in terms of the maximal sectional curvature encountered along this image. We prove that if the minimal Ricci curvature of the domain dominates this image–dependent curvature bound through a quantitative curvature pinching inequality involving the maximal energy density of f, then the map must be constant. In the critical case of equality, we obtain a homothetic classification: the differential of f is parallel and the image f(M) is totally geodesic. Thus, the theorem replaces global curvature sign assumptions by an image–dependent curvature domination principle and provides a localized analogue of classical Yano–Ishihara–type rigidity results.

This work presents a novel investigation of the recently derived relativistic Burgers-FLRW model, a scalar hyperbolic balance law with nontrivial source terms, using the Moving Mesh Method (MMM). Building on an MMM framework originally developed for hyperbolic conservation laws, we examine a range of monitor and smoothing functions to identify effective combinations for accurately resolving key solution features while reducing computational error. Numerical experiments compare the MMM with Adaptive Mesh Refinement (AMR) and uniform mesh discretizations. An L¹-error analysis is used to study the effect of different monitor functions, explore the role of various β parameters, and directly compare the performance of the MMM and AMR strategies. The results show that both adaptive approaches provide higher accuracy and better efficiency than uniform meshes, while also offering a clear comparison between MMM and AMR and practical insight into mesh adaptation for scalar balance laws.

In the work, by establishing integral representations for a class of specific Maclaurin power series, the authors restate recently-published results related to the normalized remainder of the Maclaurin power series of the exponential function, alternatively prove some of these results, and pose some new problems in terms of the majorizing relations.

This article develops a formal proof of Castilgiano Theorem in an elasticity theory context. The results are based on standard tools of applied functional analysis and calculus of variations. It is worth mentioning such results here presented may be easily extended to a non-linear elasticity context. Finally, in the last section we present a numerical example in order to illustrate the results applicability.

In this paper, we introduce and investigate the concept of statistically bornological convergence for sequences of subsets in metric spaces. This notion combines the localization principle of bornological convergence with the asymptotic flexibility of statistical convergence. A sequence of sets is said to be statistically bornologically convergent if the bornological inclusion conditions hold for a set of indices with natural density one. We provide examples distinguishing this concept from classical bornological and Hausdorff convergence. Under appropriate boundedness assumptions, we establish a functional characterization using excess functionals. We prove stability under bi-Lipschitz embeddings using a direct inclusion-based approach with properly defined pushforward ideals, and establish a subsequence theorem via the diagonal density lemma. The relationship with Wijsman statistical convergence is clarified.

The theory of q-analogs develops combinatorial formulas for finite vector spaces over a finite field with q elements–in analogy with formulas for finite sets (the limiting case q = 1). A direct-sum decomposition of a finite vector space is the vector-space analogue of a set partition. This paper uses elementary counting methods to derive direct formulas for the number of direct-sum decompositions (DSDs) that play the role of the Stirling and Bell numbers for set partitions. In particular, we give a signature-based counting formula for DSDs and recover the standard set-partition formulas in the limit q → 1. We also develop new companion formulas that enumerate DSDs with m blocks in an n-dimensional vector space over GF(q) such that a specified nonzero vector lies in one of the blocks, together with the corresponding totals over all numbers of blocks. Initial computations are included for the case q = 2, with hand-checkable low-dimensional examples, internal consistency checks, and applications to the pedagogical model of quantum mechanics over ℤ₂ (QM/Sets). Four related sequences for q = 2 are recorded in the On-Line Encyclopedia of Integer Sequences

This paper presents several new integral formulas inspired by classical results and problems in the existing literature. We present a simpler alternative proof of a well-known technical result that avoids the use of the digamma function. Additionally, we derive a set of integral formulas involving finite series in the numerator, generalizing existing formulas.

This paper considers mathematical modelling and stability analysis of Varicella-Zoster Virus (VZV) disease model in a homogeneous population that is structured as a class of susceptible-exposed-quarantined-infected-hospitalized-recovered with immunity. In this paper, the infectious classes are the exposed, quarantined, infected and hospitalized. The infected class is further subdivided into three subclasses: incubation, prodromal and active classes of VZV. The infectious rate of VZV at the incubation, prodromal, active and hospitalization stages are discussed. The aim of this paper is to determine the significance of having the subclasses of the infected class, and the role these subclasses of the infected class and contact rate play in the spread of chickenpox in the population. The basic reproduction number of our VZV model is obtained. Also, we discuss the global stability of the disease-free equilibrium and the local stability of the endemic equilibrium in the feasible region of the VZV model. Some numerical simulations are carried out to valid the models in this paper, and it is found that the subclasses of the infected class and contact rate play distinct and significant role in the spread of chickenpox in a population.