Finite groups with given systems of generalised σ-permutable subgroups

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables

The article is devoted to the latest results in metric theory of Diophantine approximation. One of the first major result in area of number theory was a theorem by academician Jonas Kubilius. This paper is dedicated to centenary of his birth. Over the last 70 years, the area of Diophantine approximation yielded a number of significant results by great mathematicians, including Fields prize winners Alan Baker and Grigori Margulis. In 1964 academician of the Academy of Sciences of BSSR Vladimir Sprindžuk, who was a pupil of academician J. Kubilius, solved the well-known Mahler’s conjecture on the measure of the set of S-numbers under Mahler’s classification, thus becoming the founder of the Belarusian academic school of number theory in 1962.

Steganographic method based on hidden messages embedding into Bezier curves of SVG images

The description of the steganographic method for embedding the digital watermark into image vector files of the SVG format is given. Vector images in SVG format can include elements based on Bezier curves. The proposed steganographic method is based on the splitting of cubic Bezier curves. Embedding hidden information involves splitting cubic Bezier curves according to the digital watermark given as numerical sequence. Algorithms of direct and reverse steganographic transformation are considered for proving the authenticity and integrity of a digital vector image. The StegoSVG library has been developed to implement forward and reverse steganographic transformations. The developed desktop application that implements the method is briefly described.

Synthesis of quantum circuits based on incompletely specified functions and if-decision diagrams

The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.

Designing a secure fail-safe cloud repository of paperworks of students and employees of educational institutions

The article discusses the main approaches to the design and development of an automated information system for a secure cloud repository of paperworks of students and employees of educational and research organisations (abstracts, essays, term papers and theses, master’s theses, deposited articles), providing secure storage and secure mobile access to stored data. The research and development of architectural solutions to ensure reliable and secure data storage using cloud technologies are described. The main problems of designing a secure repository and ways to solve them are considered. The cloud repository of written works is built on the basis of the Ceph distributed file system, which uses the NextCloud system and the virtual computing resources of the virtual network infrastructure of the Belarusian State University as a platform for building a cloud interface.

Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

Deterministic and stochastic models of infection spread and testing in an isolated contingent

The mathematical SIR model generalisation for description of the infectious process dynamics development by adding a testing model is considered. The proposed procedure requires the expansion of states’ space dimension due to variables that cannot be measured directly, but allow you to more adequately describe the processes that occur in real situations. Further generalisation of the SIR model is considered by taking into account randomness in state estimates, forecasting, which is achieved by applying the stochastic differential equations methods associated with the application of the Fokker – Planck – Kolmogorov equations for posterior probabilities. As COVID-19 practice has shown, the widespread use of modern means of identification, diagnosis and monitoring does not guarantee the receipt of adequate information about the individual’s condition in the population. When modelling real epidemic processes in the initial stages, it is advisable to use heuristic modelling methods, and then refine the model using mathematical modelling methods using stochastic, uncertain-fuzzy methods that allow you to take into account the fact that flow, decision-making and control occurs in systems with incomplete information. To develop more realistic models, spatial kinetics must be taken into account, which, in turn, requires the use of systems models with distributed parameters (for example, models of continua mechanics). Obviously, realistic models of epidemics and their control should include models of economic, sociodynamics. The problems of forecasting epidemics and their development will be no less difficult than the problems of climate change forecasting, weather forecast and earthquake prediction.

Random walks on Cayley graphs of complex reflection groups

Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

On rational Abel – Poisson means on a segment and approximations of Markov functions

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a > 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

Stabilised explicit Adams-type methods

In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.