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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.

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.

A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.

UDC 519.48 We establish a relationship between autostable and point-definable congruences on countable universal algebras. Bibliography :9 titles. The source of this paper is two-fold. First, since the main objects studied in a universal algebra are universal algebras up to an isomorphism, it is natural to require, working with derived objects (subalgebras, endomorphisms, automorphisms, congruences, and so on), that the derived objects (for isomorphic algebras) should be in a one-to-one correspondence independently of a particular isomorphism between the algebras. This leads to consideration of those derived objects on universal algebras that are stable under automorphisms of these algebras (in particular, autostable congruences). Second, it is natural to consider derived objects that are definable on the algebra in some language of logic L (L-definable congruences). In this paper, we study the main simplest properties of autostable and L-definable (for different languages L of logic) congruences and their interaction. A congruence θ on a universal algebra A = � A; σ� is autostable if for any a, b ∈ A and

An example of solving a boundary-value problem for a homogeneous Monge–Ampere 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.

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles.

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. Recently, the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009)., Comment: 13 pages

The purpose of this paper is to determine the improve system reliability by introducing the redundancy of components. This paper considers one warm stand-by unit with one main unit. We study the reliability of such systems using probability models in terms of Sedyakin’s model. A parametric family of Birnbaum-Saunders distributions is considered as the distribution of failure times both of main and stand-by units. Parametric point and interval estimation is obtained from censored data. Bibliography: 17 titles.

The present paper continues a cycle 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 algebra is described (in terms of quivers with relations) for algebras of semidihedral type, namely, of the family SD(3\(\mathcal{B}\))1. Bibliography: 16 titles.

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, 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 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bezout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.

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.

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.

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.

Let μ be a Borel charge (i.e., a real Borel measure) on ℝ, and let $ {P_{(y)}}(t) = \frac{y}{\pi \left( {{y^2} + {t^2}} \right)},y > 0 $ , t ∈ ℝ, denote the Poisson kernel. Bourgain proved that for a nonnegative μ and for many points t ∈ ℝ, the variation of the function $ y \mapsto \left( \mu * {P_{ {(y)}}} \right)(x) $ on (0, 1] is finite. This is true, in particular, for so-called B-points x introduced in a previous author’s paper, In the present paper, we give new descriptions of B-points which are adjusted to some applications of this notion. Bibliography: 5 titles.

The paper deals with the inverse problem of reconstructing a Riemannian manifold from its boundary data. This problem has been solved by the boundary control method, and at the moment there are several variants of solving it. In the paper, one more version of the procedure, which recovers the manifold from scalar spectral or dynamical data, is proposed. This version is the simplest one in regard to the devices in use: geometrical optics, polar representation of operators, etc. are not employed and only a controllability property of a relevant dynamical system is applied. Without substantial changes, this version is applicable to a more complicated (vector) problem of electrodynamics for the Maxwell system. The simplicity of the procedure proposed provides additional chances for its numerical realizability. At the end of the paper, a plan of numerical experiment is discussed. To draw attention to such new options is one of the main aims of the paper. Bibliography: 9 titles.

The paper deals with some additional details concerning the parametrization of the highest Weyl orbit of equations on the highest weight orbit in the adjoint representations of Chevalley groups of types E 7 and E 8 , as given in the author’s paper “Numerology of square equations.” Bibliography: 25 titles.

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L2 with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.

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 K 0(stmod (Λ)) is determined. Bibliography: 2 titles.

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.

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 Holder 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

The paper is devoted to Koosis' approach to the Carleson theorem. We show that this approach also works for other related questions. The main emphasis in this paper is on the method. Bibliography: 10 titles.

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory. The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization to multidimensional problems. Bibliography: 31 titles.

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 (Dirichlet) 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.

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 X 2 on [0, ∞). The principal results of the paper show that these sequences are equivalent in X and Z X 2 , 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.

In the present paper, we characterize ⋀n(GL(n, R)) over any commutative ring R as the connected component of the stabilizer of the Plucker ideal. This folk theorem is classically known for algebraically closed fields and should also be well known in general. However, we are not aware of any obvious reference, so we produce a detailed proof, which follows a general scheme developed by W.C.Waterhouse. The present paper is a technical preliminary to a subsequent paper, where we construct the decomposition of transvections in polyvector representations of GL n. Bibliography: 50 titles.

The present paper is one in a series of papers devoted to the classification of some classes of tame algebras up to stable category equivalence. In this paper, we study symmetric algebras (their stable categories have a structure of triangulated categories) and the simplest class of tame algebras-the class of special biserial algebras (SB-algebras). In the paper, we give a relevant version of the “diagrammatic method” and study the structure of the triangulated category “in a neighborhood” of the periodic part (with respect to Ω) of the stable category. Thus we prove the invariance of the collection of lengths of G-cycles under equivalence of stable categories (see Theorem 2.12). Then we use the invariance stated above, together with some properties of the Cartan matrix of a symmetric SB-algebra, to prove that the number of A-cycles (but not their lengths!) is also an invariant of stable equivalence. Bibliography: 8 titles.

In one of his early papers, D. Grigoriev analyzed the decidability and computational complexity of different physical theories. This analysis was motivated by the hope that it would help physicists. In this paper, we survey several similar ideas that may be of help to physicists. We hope that further research may lead to useful physical applications. Bibliography: 41 titles.

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.

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles.

The paper presents a survey of new results in general equilibrium theory with linear vector lattice commodity space (Kantorovich space). The importance of order structures and the Riesz-Kantorovich formula is clarified. The main novelty of the paper is new characterizations of elements of the fuzzy core in an exchange economy. Then we apply these characterizations to prove a new theorem on the existence of quasi-equilibrium for a linear vector lattice economy. This theorem, based on the E-properness of preferences by Podczeck-Florenzano-Marakulin, develops the Florenzano-Marakulin approach and generalizes previous Tourky's results. Bibliography: 29 titles.

The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles.

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.

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.

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.

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 \(\left| K \right| \geqslant 9\). 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.

In the preceding paper, the authors published an existence theorem for basic submodules of right modules over right Noetherian, serial rings. The aim of the present paper is to prove a uniqueness theorem for basic submodules over such rings. Bibliography: 13 titles.

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations., Several changes in the last section. In the original version of this paper we claimed that any regular triangulation of a convex d-polytope has at least d ears. For a proof we used the same arguments as in Schatteman's paper [22]. Since this paper has certain gaps (see our paper [1]), the d -ears problem of a regular triangulation is still open

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.

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.

Any link in ℝ3 is isotopic to a link lying on the union T of three half-planes with common boundary line. A nontrivial theory of knots and links on T was developed by the same author in an earlier paper. In the present paper, the results obtained are interpreted in the context of M. Gusarov's theory of invariants of finite degree (cubic space theory). Bibliography: 6 titles.

This paper is a continuation of a paper published by the author in 1983 where the question How bad can a Banach space with approximation property be? is discussed. Some results obtained there are generalized. Bibliography: 17 titles.

The paper is an addition to the paper of Yu. I. Babenko and V. A. Zalgaller published in the same volume. It gives a condition under which the set of all vertices of several coaxial prisms inscribed in a sphere in \(\mathbb{R}_3\) has power invariants I1,...,In. A finite set in \(\mathbb{R}_3\) with 11 invariants is constructed. It is also proved that unions of prisms yield finite sets in \(\mathbb{R}_3\) with any preassigned number n of invariants with alternating signs. Bibliography: 5 titles.

The objective of the paper is to find simple but rather general conditions giving the opportunity to extend the Kolmogorov theorem on the law of the iterated logarithm to the case of unbounded summands with finite variances. The results obtained in the paper include both the Kolmogorov and the Hartman-Wintner theorems. Bibliography: 11 titles.

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles.

In the previous paper by the authors, it is shown that a sense-preserving homeomorphism, absolutely continuous on lines (ACL), under certain integral conditions on the complex dilation is conformal at a point. This result includes the Teichmuller-Wittich-Belinski theorem. In the present paper, it is shown that under weakened conditions the local behavior of such a homeomorphism is similar to that of a quasiconformal mapping. Bibliography: 2 titles.

The paper continues the author's previous paper and deals with the case where the existence of the exponential moment of the distribution under consideration is not assumed. Bibliography: 11 titles.

The paper continues the author's studies of the question on the existence of quadratic differentials Q(z)dz2 having given structure of trajectories and poles of high orders. It is shown that such differentials can be considered as the limits of sequences of quadratic differentials that have poles of second order with trajectories asymptotically similar to logarithmic spirals and realize extremal configurations in suitable families of nonoverlapping domains. It is established that there exist differentials Q(z)dz2 of indicated form having given initial terms of the Laurent expansions in the vicinities of the poles of Q(z)dz2 of order not smaller than three. Some discrepancies in an earlier paper are corrected. Bibliography: 9 titles.

This paper contiues the series of papers devoted to designing algorithms for computing the second and subsequent terms of the ray series for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media. The method proposed in this paper essentially differs from the previous ones. It is based on expanding the amplitudes, the eikonal, and the given problem parameters into power series with respect to the coordinates transversal to the direction of propagation of the waves considered. The methods for computing the correction term of the ray expansion for the vector of longitudinal displacements are compared. Bibliography: 10 titles.

This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered. Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers) are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography: 15 titles.

The transfer matrix of the 6-vertex model of two-dimensional statistical physics commutes with many (more complicated) transfer matrices, but these latter, generally, do not commute between each other. The studying of their action in the eigenspaces of the 6-vertex model transfer matrix becomes possible due to a ``multiplicative property'' of the {\em vacuum curves} of $\cal L$-operators from which transfer matrices are built. This approach allowed, in particular, to discover for the first time the fact that the dimensions of abovementioned eigenspaces must be multiples of (big enough) degrees of the number 2., An English version of the author's article which recently appeared in Zapiski Nauvhnyh Seminarov POMI (S-Petersburg) and is based on two older papers (of 1987) mentioned in the bibliography. The author's Russian paper of 1986 from the bibliography will soon appear in English, too