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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]

The paper extends an earlier result of G.V. Kalachev et al. (Sb. Math. 210(8):1129–1147, 2019) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on R n with kernel from some L q , 1 < q < ∞ . On the other hand, E. Lieb (Ann. of Math. 118:(2):349–374, 1983) proved the existence of a maximizer for the Hardy-Littlewood-Sobolev inequality and remarked that in general a convolution maximizer for a kernel from weak L q may not exist. In this paper we axiomatize some properties used in the proof of the Kalachev-Sadov 2019 theorem and obtain a more general result. As a consequence, we prove that the convolution maximizer always exists for kernels from a slightly more narrow class than weak L q , which contains all Lorentz spaces L q , s with q ≤ s < ∞ . [ABSTRACT FROM AUTHOR]

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

This paper discusses different approaches to constructing approximate mathematical models. [ABSTRACT FROM AUTHOR]

The inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable is found. A recurrence relation for the coefficients in terms of the Nørlund-Bernoulli polynomials is provided, and the half-plane of convergence is determined. The results obtained naturally supplement a number of previous investigations of the gamma ratios, which began in the 1930-ies. The expansion obtained in this paper plays a crucial role in the study of the behavior of the delta-neutral Fox’s H-function in the neighborhood of its finite singular point. A particular case of the inverse factorial series expansion is used in deriving a possibly new identity for the Nørlund-Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

The paper considers a Riemann surface (in a broad sense of the term in the Hurwitz-Courant terminology) and an open set with a compact closure on this surface. It is proved that, following Aikawa-Ohtsuka, with a condenser on a given open set one can associate a family of vector measures, whose modules are computed directly with the aid of the weighted capacity (with Muchenhoupt weight) of the condenser. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases. [ABSTRACT FROM AUTHOR]

Let matrices A,C ∈ Mn have eigenvalues α1,..., αn and γ1,..., γn, respectively. The set of complex numbers DC(A) = {det(A−UCU*) : U ∈ Mn, U*U = In} is called the C-determinantal range of A. The paper studies various conditions under which the relation DC(R S) = DC(S R) holds for some matrices R and S. [ABSTRACT FROM AUTHOR]

The concept of the minimal polyanalytic polynomial was introduced by M. Huhtanen in connection with the generalized Lanczos process as applied to a normal matrix. The paper discusses the possibility of finding an equivalent of the characteristic polynomial in the set of polyanalytic polynomials. [ABSTRACT FROM AUTHOR]

The paper continues the series of papers devoted to surveying and developing methods for solving problems for two-parameter polynomial and rational matrices. Different types of factorizations of two-parameter rational matrices (including irreducible and minimal ones), methods for computing them, and their applications to solving spectral problems are considered. Bibliography: 6 titles. [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]

The paper is a continuation of the paper where the so-called coordinate and spectral equations were derived for finding the edge Green's functions on a branched surface having branch points of order two. The coefficients of those equations contain unknown constants. To find these constants, it is necessary to state restrictions for the solutions of the equations. After that, finding the unknown constants becomes possible, for example, by a numerical procedure of determining zeros of discrepancies. The paper is devoted to the statement of the problem of finding the unknown constants. As an example, the problem of scattering by two perpendicular half-lines is considered. As the result of using a rather subtle property of the spectral equation (symmetry associated with the reciprocity theorem), one can give a set of restrictions, in which the number of unknowns is equal to the number of restrictions. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

This paper is a sequel to the author's papers [1–8] devoted to lower multiplicative bounds on L 1 norms and their applications. Now we give estimates for L 1 norms of exponential sums and prove the result announced in [8]. [ABSTRACT FROM AUTHOR]

This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π( R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense. This paper also contains a new proof of the Razmyslov-Kemer-Braun theorem on radical nilpotence of affine PI-algebras. This proof allows one to obtain some constructive estimates. The main goal of the paper is to develop a “virtual operator calculus.” Virtual operators (pasting, deleting, and transfer) depend not only on an element of the algebra but also on its representation. [ABSTRACT FROM AUTHOR]

The paper is devoted to two new directions in developing the classical geometric subjects related to studying the problem of unique determination of closed convex surfaces by their intrinsic metrics. The first of these directions is the study of unique determination of domains (i.e., open connected sets) in Euclidean spaces by relative metrics of the boundaries of these domains. It appeared about 25–30 years ago and was developed owing to the efforts of Russian scientists. The first part of the paper (Secs. 3–7) contains an overview of the results referring to this direction. The foundations of the second direction are presented in the second part of the paper, i.e., in Sec. 8, for the first time. This direction is closely related with the first one and consists of studying the problem of unique determination of conformal type. The main result of the section is the theorem on the unique determination of bounded convex domains by relative conformal moduli of their boundary conductors. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of time-periodic weak solutions for the wave equation with homogeneous boundary conditions. This paper deals with the cases where a nonlinear term has a superlinear and sublinear growth. [ABSTRACT FROM AUTHOR]

This paper compares sequences of independent, mean zero random variables in a rearrangement-invariant space X on [0, 1] with sequences of disjoint copies of individual terms in the corresponding rearrangement-invariant space Z on [0, ∞). The principal results of the paper show that these sequences are equivalent in X and Z, respectively, if and only if X possesses the (so-called) Kruglov property. We also apply our technique to complement well-known results concerning the isomorphism between rearrangement-invariant spaces on [0, 1] and [0, ∞). Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The paper adjoins the work of B. I. Plotkin and S. M. Vovsi, Varieties of Representations of Groups (Zinatne, Riga (1983)), and turns out to be, in a sense, its continuation. In the former, the varieties of representations have been considered. As a matter of fact, the varieties under consideration are action-type varieties. This paper studies other classes of representations, axiomatizable in a special action-type logic. [ABSTRACT FROM AUTHOR]

A new method, which enables us to compute rather efficiently the Galois group of a polynomial over ℚ or over ℤ, is presented. Reductions of this polynomial with respect to different prime modules are studied, and the information obtained is used for the calculation of the Galois group of the initial polynomial. This method uses an original modification of the Chebotarev density theorem, and it is in essence a probabilistic method. The irreducibility of the polynomial under consideration is not assumed. The appendix to this paper contains tables, which enable us to find the Galois group of polynomials of degree less than or equal to 10 as a subgroup of the symmetric group. Here the final part of the paper is published. The first part is contained in a previous issue (see Vol. 134, No. 6 (2006)). Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of R k, where R is a real closed field. We prove that the dimension k′ of a semi-algebraic set described by s polynomials of degree d in k variables can be computed in time . This result slightly improves the result by Vorobjov, who described an algorithm with complexity bound (sd)O(k′(k−k′)) for the same problem. The complexity bound of the algorithm described in this paper has a better dependence on the number s of polynomials in the input. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

The splitting method is one of the most powerful and well-studied approaches to solving various NP-hard problems. The main idea of this method is to split the input instance of a problem into several simpler instances (further simplified by certain simplification rules) such that when the solution for each of them is found, one can construct the solution for the initial instance in polynomial time. There exists a huge number of papers describing algorithms of this type, and usually a considerable part of such a paper is devoted to case analysis. In this paper, we present a program that, given a set of simplification rules, automatically generates a proof of an upper bound on the running time of a splitting algorithm using these rules. As an example, we report the results of experiments with such a program for the SAT, MAXSAT, and (n, 3)- MAXSAT (the MAXSAT problem for the case where every variable in the formula appears at most three times) problems. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

In a paper by Cook and Reckhow (1979), it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules. (The rule A/B is derivable if the formula A → B is derivable. The rule A/B is admissible if for all substitutions σ, if σ(A) is derivable, then σ(B) is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The present paper continues a series of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebras are described (in terms of quivers with relations) for the algebras of semidihedral type that form the family SD(3 $$\mathcal{K}$$ ). Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

We continue a series of papers in which the Yoneda algebra is computed for algebras of dihedral and semidihedral types. In this paper, the Yoneda algebra is computed for one more family of algebras, namely, for the family D(3 $$\mathcal{L}$$ ) (in the classification of K. Erdmann). In order to find the minimal resolutions of simple modules, we used for the first time a C++ program implemented by the second author. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models. [ABSTRACT FROM AUTHOR]

This paper is an English translation (made by V. Gelfreich) of V. F. Lazutkin’s work that was published in 1984 by VINITI and thus was not easily available for readers. In the paper, a formula for an exponentially small angle of separatrix splitting of the Chirikov standard map was obtained for the first time. Bibliography: 16 titles and 17 titles added by the translator. [ABSTRACT FROM AUTHOR]

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin$(n,R)$ over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that $n=2l$ and $|K|\backslash ge\; 9$. For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO$(n,R)$ were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles. [ABSTRACT FROM AUTHOR]

In this survey, we consider the papers devoted to elementary properties of groups with the main focus on linear groups. We also consider problems on groups that are not linear but not in detail. We do not pretend to aim for a complete treatment of the problem, but we hope that the main results of the subject will be considered. In the part on results obtained before 1984, we are guided by the second chapter of the survey of V. N. Remeslennikov and V. A. Roman'kov. The first section of the survey contains basic necessary information about model theory: definitions of model, first-order language, elementary and universal theories, axiomatizability and unaxiomatizability, saturation, elementary equivalence, and ultraproducts. We also present some main theorems concerned with the notions of stability, axiomatizability, saturation, ultrafilters, and ultraproducts. The second section is devoted to the classification of groups up to their elementary properties. We present the theorem W. Szmielewa theorem on the elementary equivalence of Abelian groups, some theorems of A. G. Myasnikov and V. N. Remeslennikov on nilpotent Q-groups of finite rank, and some results of F. Oger concerned with elementary equivalence of different (finitely generated, mainly finite-by- Abelian or finite-by-nilpotent) groups. Section 3 is the main one in the survey. It contains results of elementary classification of classical linear groups: the theorem of A. I. Maltsev on the elementary equivalence of groups GL, PGL, SL, and PSL over fields, results of C. I. Beidar and A. V. Mikhalev generalizing the previous theorem to the case of groups GLn over associative rings and skew-fields, the papers of E. I. Bunina on the elementary equivalence of unitary linear groups over fields, skew-fields, and rings and Chevalley groups over fields, and the results of V. Tolstykh on the elementary equivalence of infinite-dimensional linear groups over skew-fields. There are some other results on the elementary equivalence of groups and related structures, which are included in Sec. 4. These are results on partition lattices, lattices of subalgebras of free algebras, modules and their tensor products, filtered and wreath products of groups, and some other results. The fifth section is devoted to categoricity and stability. We describe some results on co-categorical and co-stable groups, ωi-categorical groups (including theorems of B. I. Zilber on ω1-categoricity of simple algebraic groups over algebraically closed fields). In the last, sixth section, we consider the main results on the decidability and undecidability of various group theories, including basic methods for proving the decidability and undecidability, and also results on universal group theories. [ABSTRACT FROM AUTHOR]

This paper contains a presentation of the author's main results obtained in constructing an algebraic theory of branched coverings over manifolds. The main inspiration that led the author to deal with this topic is the well-known result on represent ability of each compact orientable manifold as a branched covering over a sphere. This result is old, but for a long time, there was no remarkable progress made. First of all, this is related with the absence of necessary mathematical techniques that allow one to find the interrelation between characteristics of the covering space and the base. The purpose of the author is to fill in the existing gap. In this paper, the obtained solution is presented in the simplest case. [ABSTRACT FROM AUTHOR]

One of the central places in modern differential geometry is occupied by connection theory (parallel translation theory). As is well known, a nondegenerate Riemannian metric generates a parallel translation of the tangent space of a manifold. In other words, it generates an affine connection called the Levi-Civita connection, which is uniquely determined by the torsionless requirement and by the vanishing of the covariant derivative of the metric. In the case of a degenerate metric, these requirements, being imposed on the connection, do not determine it uniquely. This, to all appearances, explains the fact that the theory of Riemannian spaces with degenerate metric has not received a proper development effort. But manifolds with degenerate metric appear in other fields of differential geometry and its applications: in the theory of almost-product manifolds, web theory, the theory of lightlike submanifolds, and others. From our point of view, for the development of the theory, it is more advantageous to attract not the means of affine-connect ion theory, but the means of pseudoconnect ion theory, which, in a certain sense, allows us to realize the idea of partial parallel translation. The notion of a pseudoconnection (general connection) extending the notion of an affine connection was introduced and studied by Otsuki and by many other authors. In the present paper, we review the papers devoted to pseudoconnection theory and its applications to the manifolds with degenerate metric. [ABSTRACT FROM AUTHOR]

The minimax signal detection problem for an admissible sets of signals forming a ball with a removed domain around its center has been considered in detail in the recent author's papers. In the present paper, we study additional possibilities arising under the assumption of positivity of the signal. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider linear semi-infinite programming problems and the possibility of having no active constraint at a boundary point of the corresponding feasible set. This phenomenon causes serious troubles when one intends to construct the cone of feasible directions at this point. The main feature of our approach consists of exploiting the (sub)differential properties of the sup-function that allows replacing infinitely many constraints with a unique convex constraint. In the case where the set of active constraints is empty, this sup-function can present a quite abnormal behavior. To face this situation, two families of relaxed active constraint sets are introduced in the paper, and the relationship between them is studied in detail. Under the so-called strong Slater condition, we obtain two different formulas for the polar of the cone of feasible directions. They are derived through an extension of Valadier's formula for the subdifferential of a sup-function. Finally, a stability result is given for the most relevant mapping in our context. [ABSTRACT FROM AUTHOR]

The paper is a continuation of the authors' previous paper on the topology of higher local fields. A useful class of systems of topological generators for the additive and multiplicative groups is described. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

A discrete string with fixed endpoints carrying a finite number of beads is determined by the masses of beads and the distances between them. The string possesses a set of simple eigenfrequencies corresponding to harmonic eigenmodes. In this paper, the following problem is studied: to find a discrete string carrying seven beads such that its eigenfrequencies coincide with the freqiencies of the notes of the first octave of the musical scale. The problem is solved in two steps. First, the spectral inverse problem is considered, i.e., we recover the string from its spectrum and a set of constants related to the normalized eigenmodes. A procedure of solving this problem is described. One of the main results of the paper is a necessary and sufficient condition for the solvability of the spectral inverse problem. The second step is numerical realization of the procedure. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

This paper is devoted to the further development of the theory of almost geodesic mappings of spaces with affine connection. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some new properties and applications of the linking number of two space curves and of families of planes tangent to a given curve. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the subclass of the invertible T-algebras by a second-order formula. [ABSTRACT FROM AUTHOR]

The classification problem for finite and infinite algebraic lattices has also been extensively addressed. A wild classification problem contains the classification problem of pairs of matrices up to simultaneous similarity. In this paper, we prove that the classification problem for graphs is wild by reducing the classification problem for finite 2-nilpotent p-groups to the classification problem for graphs. [ABSTRACT FROM AUTHOR]

In this paper, we give a full description of idempotent elements of semigroups B ( D) defined by semilattices of the class Σ( X, 7). For the case where X is a finite set, we derive formulas by means of which we can calculate the numbers of idempotent elements of the respective semigroup. [ABSTRACT FROM AUTHOR]

For a standard Wiener process on a certain probability space we consider the family of all probability measures on the same space such that each one is a result of Girsanov's change of the original measure. The one of those measures that maximizes the local time at zero (properly normalized) of the corresponding process is constructed in this paper. That process turns out to be a non-Markovian one. To compare the optimal process with others, we evaluate the values of the payoff functions for some Markov processes of the type. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

In this paper, we study the local structure of geometrical quantities on the Lobachevski plane. This structure is used for the description of metric differential invariants on the Lobachevski plane. [ABSTRACT FROM AUTHOR]

In this paper, we study almost C(λ)-manifolds. We obtain necessary and sufficient conditions for an almost contact metric manifold to be an almost C(λ)-manifold. We prove that contact analogs of A. Gray's second and third curvature identities on almost C(λ)-manifolds hold, while a contact analog of A. Gray's first identity holds if and only if the manifold is cosymplectic. It is proved that a conformally flat, almost C(λ)-manifold is a manifold of constant curvature λ. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed proof of two theorems of geodesic uniqueness in the whole of compact, in some sense generally recurrent, Riemannian spaces with a positively defined metric. Our studies are based on the H. Hopf theorem. [ABSTRACT FROM AUTHOR]

This paper shows that the stability or nonstability of switched systems does not depend on the eigenvalues of the matrices. The result is obtained by giving examples of no stable (resp. no unstable) switched linear systems consisting of stable (resp. unstable) matrices. Moreover, for the first time all kinds of eigenvalues are considered, as well as two general results for establishing the stability or nonstability of switched linear systems are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the functional differential equation with impulsive perturbationsCriteria on uniform asymptotic stability of sets are established for the above system using Lyapunov functions and the Razumikhin technique. Some examples are also discussed to illustrate the theorems. [ABSTRACT FROM AUTHOR]

In this paper, we considered a power of a non-Hermitian random matrix and prove that the empirical distribution of its singular values converges to the Fuss-Catalan distribution almost surely. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem with time-delay. The state and the control variables contain various constant time-delays. This allows us to represent the necessary conditions in an explicit form. Solution of this problem with infinite terminal time is also given. [ABSTRACT FROM AUTHOR]

This paper continues investigations into the database of queries of first-order language theory. It is known that for many decidable theories, the collapse result holds: each locally generic query is equivalent to some restricted query. But, until now, the problem of effective construction of this query remains almost unexplored. We use earlier results of the author on the construction of a method of effective obtaining this query. The method is rather general and applicable, for example, to the Presburger arithmetic and the real number theory. [ABSTRACT FROM AUTHOR]

This paper concerns BMO-regularity and AK-stability for couples ( X, Y) of quasi-Banach lattices of measurable functions on the measure space $ \left( {\mathbb{T},m} \right) \times \left( {\Omega, \mu } \right) $, where $ \left( {\mathbb{T},m} \right) $ is the unit circle with Lebesgue measure. In an earlier work, S. Kislyakov introduced a weaker version of BMO-regularity and conjectured that this property is the same as the 'strong' one in the case of couples of lattices having the Fatou property. Here we prove that these properties are indeed equivalent, thus verifying that BMO-regularity for couples is a self-dual property stable under division by a lattice. We also study another refinement of the AK-stability property and develop some techniques which allow us to slightly enlarge the class of weighted l-valued lattices for which AK-stability implies BMO-regularity. Finally, we discuss some points which might be relevant to the yet unanswered question about the relationship between AK-stability and BMO-regularity in general. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

In this paper, a methodological example of task-solving with the help of radical media is considered and technological problems of solving operational lifetime tasks of a complex system are discussed. [ABSTRACT FROM AUTHOR]