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.

Axial flow of incompressible Burgers fluids in an infinite circular cylinder that slides along its symmetry axis with an arbitrary time-dependent velocity is analytically and numerically investigated in the presence of a constant magnetic field. Analytic expressions are established for the dimensionless velocity field and the non-trivial shear stress. For validation, distinct expressions are determined for the fluid velocity and their equivalence in a concrete case is graphically proved. The influence of relaxation time and of Burgers and magnetic parameters on the fluid velocity is graphically highlighted and discussed. It was found that, outside a small vicinity of the symmetry axis of cylinder, the fluid flows slower in the presence of the magnetic field. The oscillatory translational movement of the cylinder induces a motion with oscillating velocity of the fluid. If the cylinder velocity tends to a constant value for large values of the time t, the fluid motion becomes steady in time and the corresponding steady solutions are determined.

In this paper we establish a new nonlinear variable exponent Picone-type identities for p(x)-biharmonic operator on a general stratified Lie group. As applications, eigenvalue properties, domain monotonicity, Barta-type estimate are proved for p(x)-sub-biharmonic operator. Furthermore, a Díaz-Saa-type inequality is proved and applied to study results on uniqueness of positive solutions of quasilinear elliptic equations involving variable exponent p(x)-sub-biharmonic operator.

The concept of a classical structure provides a broad mathematical framework, whereas a Hyperstructure arises via the powerset construction, and an n-Superhyperstructure is obtained by iterating this construction n times. Intuitively, the n-th powerset corresponds to n successive applications of the powerset operator. Below, we recall the fundamental definitions and illustrate them with elementary examples. In the chemical sciences, various hyperstructural frameworks—such as Chemical Hyperstructures—have also been studied. In this paper, we introduce the notion of a Chemical Superhyperstructure and examine its foundational properties. We further extend the idea of a Weak Chemical Hyperstructure by defining the Weak Chemical Superhyperstructure via the Weak Superhyperstructure (SHv-Structure) framework, and provide an overview of its behavior. These constructions offer a concise and flexible means to represent hierarchical relationships in chemical systems.

The growing thermal demands of modern energy conversion, chemical processing, and micro-scale electronic devices necessitate advanced heat and mass transfer strategies, particularly for smart fluids operating in reactive environments. In this work, the flow behavior of an electrically conducting Oldroyd-B nanofluid containing motile microorganisms is analyzed within the Cattaneo-Christov double-diffusion (CCDD) paradigm, which accounts for finite thermal and solutal relaxation times beyond the classical Fourier-Fick theory. The model incorporates mixed convection, and Arrhenius-type chemical reactions to capture complex transport interactions. By employing the Chebyshev Collocation Method (CCM), the coupled nonlinear ordinary differential equations governing the system are solved with high spectral accuracy. The parametric analysis reveals that buoyancy-induced forces significantly strengthen convective transport. Distinct and contrasting influences of relaxation and retardation parameters are observed in the velocity field, highlighting the viscoelastic nature of the fluid. Moreover, thermophoresis and Dufour mechanisms promote thermal and concentration diffusion, while increasing Prandtl and Schmidt numbers, thermal relaxation time, and chemical reaction rate diminish the associated boundary layers. The combined effects of non-Fourier diffusion, and chemical activity lead to transport characteristics unattainable under classical assumptions. These findings offer valuable physical insight for the design and optimization of nanofluid-based thermal systems in advanced industrial and technological applications.