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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]

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 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]

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 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 present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

In a previous paper, the order of the stable Grothendieck group of a symmetric special biserial algebra was computed. The present paper is the final part of that paper. The precise structure of K0(stmod (Λ)) is determined. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

This paper is one of the series of survey papers dedicated to the development of methods for solving problems of algebra for two-parameter polynomial matrices of general form. The paper considers the AB-algorithm and the ∇V-2 factorization algorithm, together with their applications. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

The paper is concerned with a new way of deriving computable estimates for the difference between the exact solutions of elliptic variational inequalities and arbitrary functions in the corresponding energy space that satisfy the main (Dirichlét) boundary conditions. Unlike the method derived earlier, the estimates are obtained by certain transformations of variational inequalities without using duality arguments. For linear elliptic and parabolic problems, this method was suggested by the author in previous papers. The present paper deals with two different types of variational inequalities (also called variational inequalities of the first and second kind). The techniques discussed can be applied to other nonlinear problems related to variational inequalities. Bibliography: 20 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 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]

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]

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]

In this paper, a simple shear flow in a half-space, which has interesting properties from the point of view of boundary regularity, is described. It is a solution with a bounded velocity field to both the homogeneous Stokes system and the Navier-Stokes equation, and satisfies the homogeneous initial and boundary conditions. The gradient of the solution may become unbounded near the boundary. The example significantly simplifies an earlier construction by K. Kang, and shows that the boundary estimates obtained in a recent paper by the first author are sharp. Bibliography: 4 titles. [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]

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]

In this paper, we describe the range of the Lp-norm of a function under fixed Lp-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two Lp-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse Hölder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5 [ABSTRACT FROM AUTHOR]

This paper is a slightly expanded text of our talk at the PCA-2014. There we announced two recent results concerning explicit polynomial equations defining exceptional Chevalley groups in microweight or adjoint representations. One of these results is an explicit characteristic-free description of equations on the entries of a matrix from the simply connected Chevalley group G( E, R) in the 56-dimensional representation V. Before, a similar description was known for the group G( E, R) in the 27-dimensional representation, whereas for the group of type E it was only known under the simplifying assumption that 2 ϵ R. In particular, we compute the normalizer of G( E, R) in GL(56, R) and establish that, as also the normalizer of the elementary subgroup E(E, R), it coincides with the extended Chevalley group $$ \overline{G}\left({\mathrm{E}}_7,R\right) $$. The construction is based on the works of J. Lurie and the first author on the E-invariant quartic forms on V. Another major new result is a complete description of quadratic equations defining the highest weight orbit in the adjoint representations of Chevalley groups of types E, E, and E. Part of these equations not involving zero weights, the so-called square equations (or π/2-equations) were described by the second author. Recently, the first author succeeded in completing these results, explicitly listing also the equations involving zero weight coordinates linearly (the 2 π/3-equations) and quadratically (the π-equations). Also, we briefly discuss recent results by S.Garibaldi and R.M.Guralnick on octic invariants for E. [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 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]

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]

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]

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]

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 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]

The classical results by Kräuter and Seifter concerning the divisibility of permanents for (±1)-matrices by large powers of 2 are useful in testing whether the permanent function is nonvanishing. This paper suggests a new approach to this problem, allowing one to obtain a short combinatorial proof of the results by Kräuter and Seifter. [ABSTRACT FROM AUTHOR]

The paper suggests upper bounds for the l norm of the inverses to block matrices belonging to certain subclasses of the class of block $$ \mathrm{\mathscr{H}} $$-matrices, which improve and supplement known results. [ABSTRACT FROM AUTHOR]

The classical criterion of asymptotic stability of the zero solution of equations x′ = f( t, x) is that there exists a function V ( t, x), a(∥ x∥) ≤ V ( t, x) ≤ b(∥ x∥) for some a, b ∈ K such that $$ \dot{V} $$( t, x) ≤ −c(∥ x∥) for some c ∈ K. In this paper, we prove that if $$ \mathop {V}\limits^{(m + {1})} $$( t, x) is bounded on some set [ tk − T, tk + T] × BH( tk → + ∞ as k → ∞), then the condition that $$ \dot{V} $$( t, x) ≤ −c(∥ x∥) can be weakened and replaced by that $$ \dot{V} $$( t, x) ≤ 0 and − (− $$ \dot{V} $$( tk, x)| + − $$ \ddot{V} $$( tk, x)| + ⋯ + − $$ \mathop {V}\limits^{(m)} $$( tk, x)|) ≤ −c′(∥ x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [-] [ABSTRACT FROM AUTHOR]

This paper is concerned with the optimal control of the sterilization of prepackaged food. The investigated system is constructed as an optimal control problem with free final horizon and phase constraints. Pontryagin's maximum principle, the necessary optimality condition for the system, is studied by the Dubovitskii and Milyutin functional analytical approach. The derived necessary condition is presented for the problem with both the control constraints and the state constraints. [ABSTRACT FROM AUTHOR]

This paper is a follow-up to a previous paper where the logical characterization of n-ary synchronous relations due to Eilenbeig, Elgot, and Shepherdson was investigated in the case where the alphabet has infinitely many letters. Here we show that modifying one of the predicates leads to a completely different picture for infinite alphabets, though it does not change the expressive power for finite alphabets. Indeed, roughly speaking, being able to express the fact that two words end with the same symbol leads to an undecidable theory, already for the Σ2 fragment. Finally, we show that the existential fragment is decidable. Bibliography: 19 titles. [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]

The aim of this paper is to provide an overview of results about classification of quantum groups which were obtained by the authors. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

In this paper, we obtain generating functions of three-variable Chebyshev polynomials (of the first as well as of the second type) associated with the root system of the A Lie algebra. Bibliography: 21 titles. [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]

In this paper, the investigation of groups with additional finiteness conditions is continued. Criteria of residual finiteness and of almost local solvability of a periodic F* -group are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we develop a theory of equipped graded graphs (or Bratteli diagrams) and an alternative theory of projective limits of finite-dimensional simplices. An equipment is an additional structure on the graph, namely, a system of 'cotransition' probabilities on the set of its paths. The main problem is to describe all probability measures on the path space of a graph with given cotransition probabilities; it goes back to the problem, posed by E. B. Dynkin in the 1960s, of describing exit and entrance boundaries for Markov chains. The most important example is the problem of describing all central measures, to which one can reduce the problems of describing states on AF-algebras or characters on locally finite groups. We suggest an unification of the whole theory, an interpretation of the notions of Martin, Choquet, and Dynkin boundaries in terms of equipped graded graphs and in terms of the theory of projective limits of simplices. In the last section, we study the new notion of 'standardness' of projective limits of simplices and of equipped Bratteli diagrams, as well as the notion of 'lacunarization.' [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]

Let TH be the space of ( T + H)-matrices of order n. The paper considers the following question: Which unitary matrices U satisfy the condition ∀A ∈ TH → U* AU ∈ TH? A criterion for verifying whether a given matrix U has this property is proposed. Bibliography: 4 titles. [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 construct noncommutative algebras over a square-closed field such that the unitary linear groups over these algebras decompose to nontrivial free products. In particular, we give an example in which the elementary unitary subgroup belongs to one of these factors. [ABSTRACT FROM AUTHOR]

In a recent author's paper, the cyclic behavior of maxima in a hierarchical summation scheme was discovered. In the present note, we show how the same phenomenon appears in the scheme of conventional summation: The distribution of the maximum of 2 independent copies of a sum of n i.i.d. random variables approaches, as n grows, some helix in the space of distributions. Bibliography: 4 titles. [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]

The paper proposes a Markovian approach to the performance evaluation of the ESSE (early slow start exit) modification of the TCP congestion control mechanism. ESSE takes advantage of estimating the optimal pipesize at the sender side to properly select initial slow start threshold. Previous simulative experiments have shown that ESSE allows one to speed-up TCP connections and significantly reduces the packet drop rate at the bottleneck. This work takes a step further in understanding the ESSE behavior by developing a model of TCP source to evaluate the influence of different settings of slow start threshold on a TCP performance. As confirmed by comparison with simulations, the model provides, significantly faster than simulations, accurate estimates of typical performance indicators such as the average completion time and average drop rate of short-lived TCP connections. [ABSTRACT FROM AUTHOR]

The derivation of inequalities with exact constants for stochastic models of complex systems is the specific character of the strong stability method. In this paper we obtain quantitative stability estimates for the M/G/1 //N system with multiple vacations and exhaustive service. [ABSTRACT FROM AUTHOR]