Open Journal of Mathematical Sciences (OMS)

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.

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.

For the first time, a nonlinear Schrödinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally generalizes a number of simpler nonlinear partial differential equations encountered in various fields of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. Two- and one-dimensional reductions are described, which reduce the studied nonlinear Schrödinger equation to simpler equations of lower dimension or ordinary differential equations (or systems of ordinary differential equations). In addition to the general Schrödinger equation with two arbitrary functions, related nonlinear partial differential equations are also examined, in which the dispersion function is specified arbitrarily while the potential function is expressed in terms of it. For all considered classes of nonlinear PDEs, using the methods of generalized and functional separation of variables, as well as the semi-inverse approach and the principle of structural analogy of solutions, many new exact solutions have been found, which are expressed in terms of elementary or special functions, or in the form of quadratures. Both Cartesian and polar coordinate systems are employed to analyze the equations under consideration. Special attention is paid to finding solutions with radial symmetry. It is shown that the nonlinear Schrödinger equation, in which the functions defining the potential and dispersion are linearly related (one of these functions can be chosen arbitrarily), can be reduced to a two-dimensional nonlinear PDE that admits exact linearization. The exact solutions obtained in this work can be used as test problems intended for verifying the adequacy and assessing the accuracy of numerical and approximate analytical methods for solving complex nonlinear PDEs of mathematical physics.

We study \(T\)-periodic solutions of cooperative non-autonomous systems of the form \[u'(t)=f(t,u(t))+F(t), \qquad t\in(0,T),\] in the ordered Banach space \(C_{\mathrm{per}}([0,T];\mathbb{R}^{m})\). Using the explicit periodic resolvent kernel \(K_\lambda\) associated with \(u'+\lambda u=g\), we recast the problem as a fixed-point equation \(u=\mathcal{T}u\) and work in a fully specified Carathéodory framework. More precisely, under assumptions (A1)–(A4) on measurability, regularity, cooperativity and local growth, and a structural condition (H\(_\lambda\)) on the diagonal derivatives of \(f\), we define a monotone, completely continuous operator \(\mathcal{T}\) that leaves invariant the order interval generated by a weak \(T\)-periodic sub- and supersolution. A monotone iteration scheme then yields the existence of weak \(T\)-periodic solutions trapped between the barriers, and we prove the existence of extremal (minimal and maximal) periodic solutions in this interval (Theorem 2). Under an additional Lipschitz condition (A5), we obtain a contraction property for \(\mathcal{T}\), which implies uniqueness and order-stability of the periodic orbit (Proposition 1). As an application, we revisit a water–solute cell-volume model with \(T\)-periodic influx and efflux, derive explicit parameter and bounding conditions ensuring the existence of a strictly positive periodic regime (Theorem 3), and illustrate the qualitative behaviour by a numerical simulation.