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This paper is dedicated to the scientific and pedagogical activities of Nikolay Dmitrievich Kopachevsky (1940–2020) — a well-known mathematician, head of the Department of Mathematical Analysis of the V. I. Vernadsky Crimean Federal University, organizer and head of the Crimean Autumn Mathematical School-Symposium (KROMSH). [ABSTRACT FROM AUTHOR]

In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

Currently, one of the most urgent problems of digital image processing is the search for new mathematical approaches to the analysis and processing of multichannel images. The relevance of these studies is due to the need of optimization of available methods of digital image processing and increase the quality of the results obtained. In this paper, a new approach to processing of digital RGB-images based on the Blaschke theory of three-webs is proposed. Invariants of three-channel images with respect to the widest "topological" transformation group are determined. [ABSTRACT FROM AUTHOR]

This is an attachment to a letter of the author to Alexander Zvonkin (dated November 1, 1995). It is reproduced here with the kind permission of the author. [ABSTRACT FROM AUTHOR]

This is the first English translation of the paper originally published in the first issue of "Problems in Mathematical Analysis" in 1966 and reproduced to the 85th birthday of V. Maz'ya. [ABSTRACT FROM AUTHOR]

The existing notion of the Kostrikin radical as a radical in the Kurosh-Amitsur sense on classes of Mal'tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical. [ABSTRACT FROM AUTHOR]

The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized. [ABSTRACT FROM AUTHOR]

We study continuity conditions for the operator of harmonic reflection of functions over boundaries of simple Carathéodory domains. This operator is considered as one acting from a space of functions of Lipschitz–Hölder type defined by a general modulus of a continuity, into another space of such kind. The results obtained are based on the continuity criterion for the Poisson operator (acting in the same spaces of functions) in the domains in question. This criterion is also obtained in the paper. These results generalize and refine those obtained in the recent work by the second author and P. Paramonov (Analysis and Mathematical Physics, 2019). [ABSTRACT FROM AUTHOR]

The population size has far-reaching effects on the fitness of a population that in turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behavior. In this paper, we develop a rigorous mathematical analysis of the study of positive and negative effects of the increased population density in the classical nonlinear age-structured population model. One of the main results expresses the global stability of the system in terms of the newborn function only. We establish the existence of a threshold population size implying the population extinction, which is well known in the population dynamics as the Allee effect. [ABSTRACT FROM AUTHOR]

A characterization of the prime radical of loops as the set of strongly Engel elements was given in our earlier paper. In this paper, some properties of the prime radical of loops are considered. Also a connection between the prime radical of the loop of units of an alternative ring and the prime radical of this ring is given. [ABSTRACT FROM AUTHOR]

In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations (E,ϖ) and (E,ϖ). To put them in the context, I first very briefly recall the two previous stages of the method, an A-proof for E and an A-proof for E, first developed some 25 years ago by Alexei Stepanov, Eugene Plotkin, and myself (a definitive exposition was given in my paper 'A third look at weight diagrams') and an A-proof for E and E developed by Mikhail Gavrilovich and myself in early 2000. The first new twist outlined in this paper is an observation that the A-proof actually effectuates reduction to small parabolics, of corank 3 in E and corank 5 in E. This allows one to revamp proofs and to sharpen existing bounds in many applications. The second new variation is a D-proof for E, based on stabilization of columns with one zero. [I devised also a similar D-proof for E, based on stabilization of columns with two adjacent zeros, but it is too abstruse to be included in a casual exposition.] Also, I list several further variations. Actual detailed calculations will appear in my paper 'A closer look at weight diagrams of types (E,ϖ) and (E,ϖ).' [ABSTRACT FROM AUTHOR]

This paper presents a simple way for diagonalization of matrices in an arbitrary field, with which one can calculate functions of matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study universal equivalence of general and special linear groups over fields. We give the following criterion for this relation to hold: two groups Gn(K) and Gm(L) (G = GL, SL, K and L are infinite fields) are universally equivalent if and only if n = m and the fields K and L are universally equivalent. [ABSTRACT FROM AUTHOR]

This paper shows that isomorphisms of groups of invertible elements of incidence algebras imply isomorphisms of the partial orders defining the algebras. [ABSTRACT FROM AUTHOR]

In this paper, we describe properties of the Dedekind η-function, constructions arising from it, and their applications in various topics of the number theory and algebra. We discuss connections with the group representation theory and the study of the structure of spaces of modular forms. Special attention is paid to the special class of modular forms, namely, η-functions with multiplicative coefficients. [ABSTRACT FROM AUTHOR]

A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝn of order −n/2 are considered, and it is shown that F(ω)|x|−n/2, |ω| = 1, is an eigenfunction in the sense of the theory of distributions of the Fourier transform if and only if F(ω) is an eigenfunction of a certain singular integral operator on the unit sphere of ℝn. Since Ym,nkωx−n/2, where Ym,nk denote the spherical functions of order m in ℝn, are eigenfunctions of the Fourier transform, it follows that Ym,nk are eigenfunctions of the above-mentioned singular integral operator. In the planar case, all eigenfunctions of the Fourier transform of the form F(ω)|x|−1 are described by means of the Fourier coefficients of F(ω). [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a linearly ordered skewfield the groups of quotients of the semigroup Gn(픻) coincides with the group GLn(픻) for n ≥ 3. [ABSTRACT FROM AUTHOR]

Abstract. The remarkable class of measurable functions was introduced by classics of function theory. It has found different applications in various branches of mathematics. However this class turned out too restrictive for solving some natural mathematical problems because it is essentially connected with the property of countability. Therefore, along with it another remarkable class, essentially connected with the property of finiteness, was introduced. It is the class of uniform functions. Measurable functions are described both in the classical Lebesgue-Borel language of preimages and in the quite new language of covers. Uniform functions are described in the language of covers exclusively. Both the families of measurable functions and the families of uniform functions are determined by the rigid structure of their supports (descriptive spaces). For this reason, mathematicians weakened more that once the rigidity of the structure of descriptive spaces at the expense of using the additional property of negligence. The present paper is devoted to a contemporary formalization of the indicated ideas. Some applications of the introduced classes of functions to solving a number of known mathematical problems is traced in the paper. [ABSTRACT FROM AUTHOR]

Abstract. This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m( n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each color. In this paper, we obtain upper bounds of m( n) for small k and n, the exact value of m(8), and a lower bound for m(7). [ABSTRACT FROM AUTHOR]

An example of solving a boundary-value problem for a homogeneous Monge-Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. Cases of rather complicated foliations are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stolyarov, Vasyunin, and Zatitskiy. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate some uniform ergodicity and strong stability estimates for homogeneous markov chains, which may be considered as a refinement of those established by the authors with respect to a given weight norm. As a general rule, the initial parameter values of the most complex systems are approximately known (they are defined on the basis of statistical methods), which results in errors for the calculus of research characteristics for each studied system. For this, the uniform ergodicity and stability inequalities obtained in this paper make it possible to use them in order to estimate numerically the error of definition for the considered characteristics for small perturbations of the system's parameters. As an example of application, we study the well-known Lindley process, and a comparison with some results obtained by Kartashov is established. [ABSTRACT FROM AUTHOR]

In this paper stochastic Volterra equations of nonscalar type in a Hilbert space are studied. The aim of the paper is to provide some results on stochastic convolution and mild solutions of those Volterra equations. The motivation of the paper comes from a model of aging viscoelastic materials. The pseudo-resolvent approach is used. [ABSTRACT FROM AUTHOR]

This paper continues the previous work of the author and contains the results concerning the relations for a wide class of special functions related to multidimensional Poisson and polynomial random walks. [ABSTRACT FROM AUTHOR]

In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions. [ABSTRACT FROM AUTHOR]

In this paper, the capacity and module of a condenser, and also some functional classes on a sub-Finsler space are defined. Their general properties are studied, and the equality of the capacity and module of a condenser is proved. [ABSTRACT FROM AUTHOR]

The paper describes the congruent centralizer of the matrix $$ {\Delta}_n=\left(\begin{array}{llll}\hfill & \hfill & \hfill & 1\hfill \\ {}\hfill & \hfill & \dots \hfill & i\hfill \\ {}\hfill & 1\hfill & \dots \hfill & \hfill \\ {}1\hfill & i\hfill & \hfill & \hfill \end{array}\right) $$ , representing one of three blocks in the Horn-Sergeichuk canonical form, i.e., the set of matrices X such that X Δ X = Δ. [ABSTRACT FROM AUTHOR]

The paper introduces new subclasses, called P $$ \mathrm{\mathscr{H}} $$ N( π) and P $$ \mathrm{\mathscr{H}} $$ QN( π), of (nonsingular) $$ \mathrm{\mathscr{H}} $$ -matrices of order n dependent on a partition π of the index set {1, . . ., n}, which generalize the classes P $$ \mathrm{\mathscr{H}} $$ ( π), introduced previously, and contain, in particular, such subclasses as those of strictly diagonally dominant (SDD), Nekrasov, S-SDD, S-Nekrasov, QN, and P $$ \mathrm{\mathscr{H}} $$ ( π) matrices. Properties of the matrices introduced are studied, and upper bounds on their inverses in l norm are obtained. Block generalizations of the classes P $$ \mathrm{\mathscr{H}} $$ N( π) and P $$ \mathrm{\mathscr{H}} $$ QN( π) in the sense of Robert are considered. Also a general approach to defining subclasses $$ {\mathcal{K}}^{\pi } $$ of the class $$ \mathrm{\mathscr{H}} $$ containing a given subclass $$ \mathcal{K} $$ ⊂ $$ \mathrm{\mathscr{H}} $$ and dependent on a partition π is presented. [ABSTRACT FROM AUTHOR]

In a series of articles, V. S. Buldyrev considered interference head waves, called now Buldyrev waves. The main goal in the present paper is to obtain formulas describing Buldyrev waves using the locality principle. Similar formulas have been deduced (from other but also heuristic considerations) by Buldyrev earlier. A different point of view on the Buldyrev wave is described by V. M. Babich. Some formulas for waves of this class contain an illusory contradiction with the concept of locality. It is demonstrated that the contradiction is fictitious and the formulas deduced by Buldyrev and the results obtained later perfectly agree with the locality principle. [ABSTRACT FROM AUTHOR]

Control problems in dynamic systems allowing pulse forces due to the presence of unilateral active constraints are considered. This paper gives an analysis of controls used during impact interactions. A new kind of pulse controls arising at the time of impact is introduced. These problems are typical for the so-called vibroimpact systems with numerous applications, whose number is constantly increasing in recent years. The goal of the paper is to describe the phase of a controllable impact and to determine the constraint reactions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups. [ABSTRACT FROM AUTHOR]

In V. D. Belousov's papers, some properties of parastrophies were studied and some relations between parastrophies of a given quasigroup were obtained. Also some invariants of a parastrophy were found. This article continues our paper with V. V. Gushan, in which minimal sets of parastrophy systems for quasigroups of order 6 were obtained and some questions about orthogonality of parastrophies of a given quasigroup were studied as well. [ABSTRACT FROM AUTHOR]

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [, , ]. [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers of the author (same of them joint), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmam's classification). In this paper, the Yaneda algebra is described in terms of quivers with relations for algebras of semidihedral type of the family SD(2 $$ \mathcal{B} $$). Bibliography: 28 titles. [ABSTRACT FROM AUTHOR]

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL-factorizations of length 5 N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4 N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3 N. On the other hand, for SL( n, R), over a Bézout ring R, we further improve the bound to 2 N = n- n. Bibliography: 25 titles. [ABSTRACT FROM AUTHOR]

This paper is devoted to the discussion of some schemes of construction of radicals similar to special radicals, which generalize constructions of the basic special radicals of the algebras close to associative ones. [ABSTRACT FROM AUTHOR]

The paper concerns properties of fully indecomposable doubly stochastic matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of stabilization of an n-link inverted pendulum on a movable base (cart). A cart is allowed to move along the horizontal axis. A force applied to the cart is considered as a control. The problem is to minimize the mean square deviation of the pendulum from the vertical line. For the linearized model, we show that, for small deviations from the upper unstable equilibrium position, the optimal regime contains trajectories with more and more frequent switchings. Namely, the optimal trajectories with infinite number of switchings are shown to attain, in finite time, the singular surface and then continue these motion with singular control over the singular surface, approaching the origin in an infinite time. It is shown that the costructed solutions are globally optimal. [ABSTRACT FROM AUTHOR]

In the present paper, analogs of the Eilenberg-Watts theorem are proved for categories of finitely generated modules over finite-dimensional algebras for right exact and left exact functors. Furthermore, for left exact functors the corresponding bimodules are described explicitly. The main aim of this paper is to present how, with these versions of the Eilenberg-Watts theorem, we obtain some new descriptions of the Nakayama functor and the inverse Nakayama functor in the case of self-injective algebras. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorization G = UU UU ... U of a Chevalley group G = G(Φ, R) in terms of the unipotent radical U = U(Φ, R) of the standard Borel subgroup B = B(Φ, R) and the unipotent radical U = U(Φ, R) of the opposite Borel subgroup B = B(Φ, R). So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits the unitriangular factorization G = UU UU U of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper, we notice that from the work of Bass and Tavgen one immediately gets a much more general results, asserting that over any ring of stable rank 1 one has the unitriangular factorization G = UU UU of length 4. Moreover, we give a detailed survey of traingular factorizations, prove some related results, discuss prospects of generalization to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalized Riemann's Hypothesis, Chevalley groups over the ring $$ \mathbb{Z}\left[ {\frac{1}{p}} \right] $$ admit the unitriangular factorization G = UU UU UU of length 6. Otherwise, the best length estimate for Hasse domains with infinte multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors. Bibliography: 67 titles. [ABSTRACT FROM AUTHOR]

In the present paper, norm series for Honda formal groups are studied. The Steinberg relation for the classical Hilbert symbol is generalized to Honda formal groups. Necessary and sufficient conditions for a series to satisfy the generalized Steinberg relation are obtained. In the paper, such series are called norm series. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new estimate of the unknown pseudo periodic function which is observed in stationary noise. We compare the accuracy of the proposed estimate with the minimax risk. [ABSTRACT FROM AUTHOR]

In the paper, the structure of the $$ \mathcal{O} $$ [G]-module F( $$ \mathfrak{m} $$ ) is described, where M/L, L/K, and K/ℚ are finite Galois extensions (p is a fixed prime number), G = Gal(M/L), $$ \mathfrak{m} $$ is a maximal ideal of the ring of integers $$ \mathcal{O} $$ , and F is a Lubin-Tate formal group law over the ring $$ \mathcal{O} $$ for a fixed uniformizer π. [ABSTRACT FROM AUTHOR]

In the present paper, a generalization of the Hensel-Shafarevich basis for Lubin-Tate formal modules over a local field is presented. These formal modules are constructed on the maximal ideal of some extension of this field. The cases where the extension has a perfect residue field or an imperfect residue field are studied. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

In this paper we give a full description of regular elements of the semigroup that are defined by semilattices of the class, when Z ∩ Z ≠ ∅. Formulas are derived by means of which the number of regular elements of the semigroup is calculated when it is a finite set. [ABSTRACT FROM AUTHOR]

One of the most popular experimental techniques for investigation of brain activity is the so-called method of evoked potentials: the subject repeatedly makes some movements (by his/her finger), whereas brain activity and some auxiliary signals are recorded for further analysis. The key problem is the detection of points in the myogram that correspond to the beginning of the movements. The more precisely the points are detected, the more successfully the magnetoencephalogram is processed aiming at the identification of sensors that are closest to the activity areas. This paper proposes a statistical approach to this problem based on mixtures models that uses a specially modified method of moving separation of mixtures of probability distributions (MSMmethod) to detect the start points of the finger's movements. We demonstrate the correctness of the new procedure and its advantages as compared with the method based on the notion of the myogram window variance. [ABSTRACT FROM AUTHOR]

In this paper, necessary conditions for a minimum (the Pontryagin maximum principle) for an optimal control problem are proved on the basis of the concept of a mix, which enables one to reduce the study of the original problem to some approximation thereof, which is linear in the control. The study of the latter problem proves more simple. [ABSTRACT FROM AUTHOR]

In this paper, we study the question of the existence of a continuous right inverse mapping for a covering mapping. To describe covering mappings that have a continuous right inverse, a concept of strong covering is introduced. It is shown that the property of strong covering is stable under small Lipschitz perturbations. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 4, pp. 93-99, 2014. [ABSTRACT FROM AUTHOR]

In this paper, we examine the problem on the existence of best approximation elements in functional spaces with nonsymmetric norms and sign-sensitive weights. [ABSTRACT FROM AUTHOR]

This paper is devoted to further study of the theory of geodesic mappings and their generalizations, including conformal, holomorphically projective, F-planar, and almost geodesic mappings of affinely connected spaces. [ABSTRACT FROM AUTHOR]

It this paper, we examine the sums of double sine series with coefficients that are multiply monotonic with respect to subsequences. We obtain sufficient conditions under which these sums belong to the classes L , 0 < p < ∞. [ABSTRACT FROM AUTHOR]