Articles

In the article, the asymptotics of the coefficients of the generating functions are found with a certain accuracy, which can be used to calculate the powers of layers of certain types of partially ordered sets, as well as to calculate the values of the sums of boundary functionals when estimating the number of antichains in such sets. In addition, applications of the obtained results are considered using examples.

The author considers an inverse coefficient problem for a model of sorption dynamics. The inverse problem is reduced to a nonlinear operator equation for an unknown coefficient. The differentiability of the nonlinear operator is proved. The Newton–Kantorovich method and the modified Newton–Kantorovich method are constructed for the numerical solution of the inverse problem. The results of numerical calculations are presented.

When describing the group behavior of high-frequency traders, a boundary value problem arises based on the concept of mean field games. The system consists of two coupled partial differential equations: the Hamilton–Jacobi–Bellman equation, which describes the evolution of the average payoff function in backward time, and the Kolmogorov–Fokker–Planck equation, which describes the evolution of the probability density distribution of traders in forward time. The system is ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to reduce the system to a set of Riccati equations; however, the question of the well-posedness of the reduced problem remains open. This work investigates this question, specifically the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.

Statistical inference often assumes that the distribution being sampled is normal. As observed, following the normal distribution assumption blindly may affect the accuracy of inference and estimation procedures. For this reason, many tests for normality have been proposed in the literature. This paper deals with the problem of testing normality in the case when data consists of a number of small independent samples such that in each small sample observations are independent and identically distributed while from sample to sample they have different parameters but the same type of distribution (call this multi-sample data). In this case it is necessary to use test statistics which do not depent on the parameters. A natural way to exclude the nuisance location parameter is to replace the observations within each small group by diferences. We obtain some estimates of stability of such a decomposition and study and compare the power of eight selected normality tests in the case of multi-sample data. The following tests are considered: the Pearson chi-square test, the Kolmogorov–Smirnov, the Cramer–von Mises, the Anderson-Darling, the Shapiro–Wilk, the Shapiro–Francia, the Jarque–Bera, and the adjusted Jarque–Bera tests. Power comparisons of these eight tests were obtained via the Monte Carlo simulation of sample data generated from several alternative distributions.

This article presents a new algorithm for calculating singularity (boundary) type of ribbon surface of generalized pseudo-Anosov homeomorphism using the surface’s combinatorial description provided with the so-called configuration. As an additional output the fundamental group relators of the ribbon surface are calculated for its co-presentation associated with a given ribbon partition. In comparison to a known algorithm, the one which is presented in this article does not involve any auxiliary sets nor recurrent functions.

The creation of cryptographic systems based on lattice theory is a promising direction in the field of post-quantum cryptography. The aim of this work is to obtain new properties of lattices through related objects — dense packings of equal spheres. The article proposes a method for constructing lattice packings of equal spheres corresponding to the packing density of the “Lambda” series in dimensions 1–24, using a series of coefficients to the height of a fundamental parallelepiped of dimension (n−1): 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2, √−1, 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2, √−1, 1/2, 1/3, 1/2, 0, 1/2, 1/3, 1/2. The construction of lattice packings of equal spheres using this method was carried out up to dimension 11 inclusive.

This paper solves the construction problem of embedding the complete rooted binary and ternary trees with k, k =1, 2, . . . , levels in rectangular lattices (RL) with minimum length and near minimum height. It is assumed that different vertices of the tree go to different (main) vertices of the RL, with the leaves of the tree going to the vertices of the RL located on its horizontal sides. It is also assumed that the edges of the tree go to simple (transit) chains of the RL, which connect the images of their end vertices and do not pass through other main vertices, with no more than 1 (correspondingly 2) transit chains passing through the same edge (the same vertex) of the RL.

Time optimal control problem with state constraint investigated in this article. The behavior of an object is described by a system of second-order linear differential equations. The coefficient matrix for state variables has various positive eigenvalues. The state constraint is linear. A admissible control is a piecewise continuous function that takes values from a given compact set. Sets of controllability to the origin are constructed for time intervals of various lengths. A study of the dependence of the solution of the problem on the parameter determining the state constraint was carried out.

In the work on real data, the modeling and assessment of the infection rate of a population of ixodid ticks with tick-borne encephalitis virus and Borrelia burgdorferi sensu lato is carried out using the maximum likelihood and moment methods, and their comparative analysis is given. A review of methods for solving direct and inverse problems of binary object propagation based on individual and group observations is made.