In this paper, we discuss the rate of convergence of the rational Fourier series and conjugate rational Fourier series of functions of generalized bounded variation. In particular, well- known Wiener’s and Siddiqi’s theorems for functions of p-bounded variation are proved in more general complete rational orthogonal system. Also some results are obtained for a class of functions wider than the class of functions of bounded variation and of {n^\alpha }-bounded variation.

A partition of a positive integer n is said to be simultaneously s-regular and t-distinct partition if none of the parts is divisible by s and parts appear fewer than t times. In this paper, we present some new congruences for simultaneously s-regular and t-distinct partition function denoted  by M ^d (n) with (s, t) (2, 5), (3, 4), (4, 9), (5\alpha, 5\beta ), (7\alpha, 7\beta ), (p, p), where\alpha and \beta are any positive integers and p is any prime.

Some variations of \pi -regular and nil clean rings were recently introduced in the works of the first author: “A generalization of \pi -regular rings, Turkish J. Math. 43 (2), 702–711 (2019)”, “A symmetrization in \pi -regular rings, Trans. A. Razmadze Math. Inst. 174 (3), 271–275 (2020)”, “A symmetric generalization of \pi -regular rings, Ric. Mat. 73 (1), 179–190 (2024)”. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that (m, n)-regularly nil clean rings are left-right symmetric and also show that the inclusions (D- regularly nil clean) (regularly nil clean) ((m, n)-regularly nil clean) hold, as well as we answer Questions 1, 2 and 3 posed in the third of the above-listed works. Moreover, we also consider other similar questions concerning the symmetric properties of certain classes of rings. For example, it is proven that centrally Utumi rings are always strongly \pi -regular.

The direct and inverse problems for a model mixed parabolic-hyperbolic equation with a characteristic line of type change have been studied. In the direct problem, a nonlocal problem for this equation with interior boundary conditions in the hyperbolic part of the domain has been investigated. The unknown of the inverse problem is the variable coefficient of the lower-order term in the parabolic equation. To determine it, the inverse problem is studied under the assumption that an integral overdetermination condition is known for the solution defined in the parabolic part of the domain. A theorem of unique solvability for the posed problems in the sense of a classical solution has been proven.

In the present paper we consider non-Volterra quadratic stochastic operators defined on the two-dimensional simplex depending on a parameter \alpha . We show that such an operator has a unique fixed point and all the trajectories converge to this unique fixed point.

The paper considers the problem of propagation of natural stress waves in a strip that is in contact with an unbounded isotropic viscoelastic medium made of another material. It is assumed that there are no external influences during the propagation of natural waves. In some cases, the physical properties of viscoelastic materials are described by linear hereditary Boltzmann–Voltaire relations with integral differences of heredity kernels. Some of the layers can be elastic. In this case, the heredity kernels describing the rheological properties of the layers are identically zero. A system in which the rheological properties of the layers are identical (the nuclei of heredity of elements are equal to each other) will be called dissipatively homogeneous. In the particular case, when there are no external influences, the propagation of damped waves of the system is considered; — in the presence of external influences — forced. The main problem is the study of the dissipative (damping) properties of the system as a whole, as well as its stress-strain state.

This paper establishes the existence of countably many positive solutions for a second order iterative system of two-point boundary value problems by using Holder’s inequality and Guo–Krasnoselskii’s fixed point theorem for operator on a cone.

In this paper we consider the family of operators μH:= ΔΔ— Vμ,     μ  > 0, that is,  a bilaplacian with  a finite-dimensional perturbation on  a one-dimensional lattice  Z , where Δ  is  a  discrete  Laplacian,  and  Vμ   is  an  operator  of  rank  two.  It  is  proved  that  for  any  μ  > 0 the discrete  spectrum  μH  is  two-element  e₁(μ ) < 0 and  e₂(μ ) < 0.  We find  convergent  expansions of the eigenvalues e_i(μ ), i = 1, 2 in a small neighborhood of zero for small μ  > 0.

A description has been obtained of all divisible, primary, as well as separable and algebraically compact torsion-free groups, the Lie endomorphism ring of which is solvable.

An extremely simplified transformation model of dynamic deformation of a rod-strip consisting of two sections along its length is constructed. It is based on the classical geometrically nonlinear Kirchhoff-Love model on an unfixed section, and the fixed section of finite length is considered to be connected to a rigid and fixed support element through elastic layers. On the fixed section, the deflections of the rod and interlayers are considered zero, and for displacements in the axial direction within the thicknesses of the rod and interlayers, approximations are adopted according to the shear model of S.P. Timoshenko, subject to the conditions of continuity at the points of their connection with each other and immobility at the points of connection of the interlayers with the support element. The conditions for the kinematic coupling of the unfixed and fixed sections of the rod are formulated, and based on these, using the d’Alembert-Lagrange variational principle, the corresponding equations of motion and boundary conditions, as well as the force conditions for coupling of the sections, are derived for the sections introduced into consideration.