The problem of developing the mathematical foundations of modular secret sharing in the special linear group over the ring of polynomials in one variable over the finite Galois field with $p$ elements is being solved. Secret sharing schemes should meet a large number of requirements: perfectness and ideality of a scheme, possibility of verification, changing a threshold without participation of a dealer, implementation of a non-threshold access structure and some others. Every secret sharing scheme developed to date does not fully satisfy all these requirements. The development of a scheme on a new mathematical basis is intended to expand the list of these configurations, thereby creating more possibilities for a user to choose the optimal option. A fundamental domain with respect to the action of the main congruence subgroup by right shifts in the special linear group of dimension 2 over the ring of polynomials is constructed. On this basis, methods for modular threshold secret sharing and its reconstruction are proposed.

The article is the fourth in a series of papers, where for a set $\pi$ consisting of odd primes, finite $\pi$-solvable  irreducible complex linear groups of  degree $2|H|+1$ are investigated, for which Hall $\pi$-subgroups are $TI$-subgroups and are not normal in groups. The purpose of the series is to prove solvability and to determine the conditions for factorization of such groups.

This work is dedicated to developing methods of the real Hardy-Sobolev space on the line for finding the best rational approximations in the $L_p$ space. The methods considered are based on representing a function of this space as a sum of simple functions and the application of a Cauchy-type integral. Sufficient conditions for a function's membership in the considered space have been obtained and inequalities for assessing the corresponding $\sigma$-norm have been proven. Using the obtained results, exact order estimates of the best rational approximations of certain functions have been found. In particular, from the obtained results, the well-known estimate of the best rational approximations of a function of bounded variation follows.

In this paper, based on the discrete analogue of comparison theorems and Jensen's inequality, blow-up conditions and upper bound of blow-up time of the solution of implicit finite-different problem which approximates Neumann problems for various nonlinear parabolic equations are obtained. Blow-up conditions and upper bound of blow-up time of approximated differential problems are given, which are obtained and based on comparison theorems and Jensen's inequality

In this paper, we consider the cone of completely positive matrices. Currently, some families of non-exposed polyhedral faces of this cone were constructed. Inspired by these results, in this paper, we continue the study of the existence and properties of non-exposed faces of the cone of completely positive matrices. We prove a criterion for a face of this cone to be non-exposed. We also provide sufficient conditions that can be easily checked numerically. We show that for any $p\geqslant 6$, there exist non-exposed non-polyhedral faces of the cone of $p\times p$ completely positive matrices. Illustrative examples are given

A necessary and sufficient condition is obtained for the coefficient matrix of a linear recurrence equation in the space of convex polygons, any two different solutions of which do not intersect, i. e. the values of the solutions for each argument are different

We consider a linear integro-differential equation on a closed curve located on the complex plane. The coefficients of the equation have a special structure. The equation is first reduced to the mixed Riemann-Carleman boundary value problem for analytic functions. Next, two differential equations are solved in areas of the complex plane with additional conditions. The conditions for the solvability of the original equation are indicated explicitly. When they executed, the solution is given in closed form. An example is given

The issues of constructing numerical algorithms based on the Chebyshev spectral method for approximate solution of elliptic equations with mixed derivatives in a rectangular domain with homogeneous Dirichlet boundary conditions are considered. To implement the spectral method, the biconjugate gradients stabilized method with preconditioners in the form of finite difference or spectral analogs of the Laplace operator is used. A comparison of the efficiency of processing the preconditioner using the iterative method of alternating directions and the Bartels-Stewart algorithm is carried out. The presented results show that the considered algorithms demonstrate computational characteristics comparable in computation time on grids of the same dimension with the characteristics of difference methods, but they are many times superior to the latter in accuracy in the case of sufficiently smooth solutions

We establish in a direct way, without involving the sigh function of permutations and matrice reducing to echelon form, the equivalence of the expansion of determinant along any row and any column. On base of this the rest of the theory of determinants is significantly simplified: determinant multiplicativity, the generalized Laplace expansion and Cauchy-Binet formula and so on