Showing papers in "Russian Mathematics in 2018"
TL;DR: In this paper, the authors systematically develop methods for approximation and interpolation by simple partial fractions (SPFs) and their modifications, and they formulate principal results and outline methods to prove them whenever possible.
Abstract: In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPFs) as suggested by Dolzhenko. Many solved and unsolved extremal problems, related to SPFs, are traced back to works of Boole, Macintyre, Fuchs, Marstrand, Gorin, Gonchar, and Dolzhenko. Now many authors systematically develop methods for approximation and interpolation by SPFs and their modifications. Simultaneously, related problems, being of independent interest, arise for SPFs: obtaining inequalities of different metrics, estimation of derivatives, separation of singularities, etc. In introduction to this survey, we systematize some of these problems. In themain part, we formulate principal results and outline methods to prove them whenever possible.
25 citations
TL;DR: In this article, a boundary value problem for degenerating pseudoparabolic equations with variable coefficients and with Gerasimov-Caputo fractional derivative is considered and a priori estimates in differential and difference settings are obtained.
Abstract: In this paper we consider a boundary-value problems for degenerating pseudoparabolic equation with variable coefficients and with Gerasimov–Caputo fractional derivative. To solve the problem we obtain a priori estimates in differential and difference settings. These a priori estimates imply uniqueness and stability of the solution with respect to the initial data and the right-hand side on the layer, as well as the convergence of the solution of each of the difference problem to the solution of the corresponding differential problem.
23 citations
TL;DR: In this paper, the authors generalize the Lomov regularization method to partial integro-differential equations and develop an algorithm for constructing a regularized asymptotic solution and carry out its full substantiation.
Abstract: We generalize the Lomov’s regularization method to partial integro-differential equations. It turns out that the procedure for regularization and the construction of a regularized asymptotic solution essentially depend on the type of the integral operator. The most difficult is the case, when the upper limit of the integral is not a variable of differentiation. In this paper, we consider its scalar option. For the integral operator with the upper limit coinciding with the variable of differentiation, we investigate the vector case. In both cases, we develop an algorithm for constructing a regularized asymptotic solution and carry out its full substantiation. Based on the analysis of the principal term of the asymptotic solution, we study the limit in solution of the original problem (with the small parameter tending to zero) and solve the so-called initialization problem about allocation of a class of input data, in which the passage to the limit takes place on the whole considered period of time, including the area of boundary layer.
13 citations
TL;DR: In this article, a covariant functor from the category of an arbitrary partially ordered set into a category of C*-algebras and their *-homomorphisms is considered.
Abstract: We consider a covariant functor from the category of an arbitrary partially ordered set into the category of C*-algebras and their *-homomorphisms. In this case one has inductive systems of algebras over maximal directed subsets. The article deals with properties of inductive limits for those systems. In particular, for a functor whose values are Toeplitz algebras, we show that each such an inductive limit is isomorphic to a reduced semigroup C*-algebra defined by a semigroup of rationals. We endow an index set for a family of maximal directed subsets with a topology and study its properties. We establish a connection between this topology and properties of inductive limits.
12 citations
TL;DR: In this article, it was shown that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids).
Abstract: We study the lattice of varieties of monoids, i.e., algebras with two operations, namely, an associative binary operation and a 0-ary operation that fixes the neutral element. It was unknown so far, whether this lattice satisfies some non-trivial identity. The objective of this paper is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids, and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.
11 citations
TL;DR: In this article, the inverse scattering method was applied to the integration of the Korteweg-de Vries equation with a self-consistent source in the class of complex-valued rapidly decreasing functions.
Abstract: We apply the inverse scattering method to the integration of the Korteweg–de Vries equation with a self-consistent source in the class of complex-valued rapidly decreasing functions.
11 citations
TL;DR: A simple rule is proposed for the step-size choice in the conditional gradient method, which does not require any line-search procedure and is established under the same assumptions as those for the previously known methods.
Abstract: We propose a simple rule for the step-size choice in the conditional gradient method, which does not require any line-search procedure. It takes into account the current behavior of the method. Its convergence is established under the same assumptions as those for the previously known methods.
10 citations
TL;DR: In this article, a non-contradictory aggregate preference relation is based on the weighted majority graph that characterizes the degree of superiority of one alternative over another, and satisfies requirements to group decisions, namely, the monotony, the preservation of the Pareto relation, the minimality of the distance to expert preferences.
Abstract: We study the problem of collective choice. The profile of individual preferences of experts is defined by relations of strict order. A non-contradictory aggregate preference relation is based on the weighted majority graph that characterizes the degree of superiority of one alternative over another. The aggregate relation also defines a strict order and satisfies requirements to group decisions, namely, the monotony, the preservation of the Pareto relation, the minimality of the distance to expert preferences.
7 citations
TL;DR: In this article, it was shown that in general case total axial strain consists from residual (irreversible) strain of lateral degradation, reversible strain of material's micro-rearrangement, and reversible creep strain.
Abstract: For cross-ply fiber reinforced plastics under short- and long-term loading conditions we prove the defining relations. It was shown that in general case total axial strain consists from residual (irreversible) strain of lateral degradation, reversible strain of material’s microrearrangement, and reversible creep strain. We consider issue of experimental determination and problems of identification of mechanical parameters taken into consideration.
7 citations
TL;DR: In this article, the authors studied two bisingular Dirichlet problems with the additional boundary layer and constructed asymptotic solutions to the three-zone, bisingularity Dirichlets by using the generalized method of boundary functions.
Abstract: We study two bisingular Dirichlet problem with the additional boundary layer: 1) for the second order linear elliptic equation in a ring, 2) for linear ordinary differential equations of second order in a segment. We construct asymptotic solutions to the three-zone, bisingular Dirichlet problems by using the generalized method of boundary functions and obtain estimates for the residual functions.
7 citations
TL;DR: The main purpose of as discussed by the authors is to analyze the classes of disjoint hypercyclic and topologically mixing abstract degenerate (multi-term) fractional differential equations in Banach and Frechet function spaces.
Abstract: The main purpose of this paper is to analyze the classes of disjoint hypercyclic and disjoint topologically mixing abstract degenerate (multi-term) fractional differential equations in Banach and Frechet function spaces. We focus special attention on the analysis of abstract degenerate differential equations of first and second order, when we also consider disjoint chaos as a linear topological dynamical property. We provide several illustrative examples and applications of our abstract results.
TL;DR: In this article, the authors obtained Lp-versions of theorems proved by J. L. Fernandez and J. M. Rodriguez in the paper "The Exponent of Convergence of Riemann Surfaces, Bass RiemANN Surfaces" by using the isoperimetric profile of hyperbolic domains.
Abstract: We obtain Lp-versions of theorems proved by J. L. Fernandez and J. M. Rodriguez in the paper “The Exponent of Convergence of Riemann Surfaces, Bass Riemann Surfaces”, Ann. Acad. Sci. Fenn. Ser.A. I.Mathematica 15, 165–182 (1990). An important role in the proof of our results is due to the approach of V. M. Miklyukov and M. K. Vuorinen. In particular, we use the isoperimetric profile of hyperbolic domains.
TL;DR: In this paper, Ivanov and Tsirul'skii this paper solved two direct problems for logarithmic potential and for simple layer potential in the case of polygonal contour.
Abstract: We describe possibilities of realizing V K Ivanov’s method for solving inverse problems of logarithmic potential and that of A V Tsirul’skii’s method for solving direct problems of logarithmic potential We solve two direct problems for logarithmic potential and for simple layer potential in the case of polygonal contour
TL;DR: In this paper, the existence and uniqueness conditions for solution to one Robin type problem for inhomogeneous biharmonic equation in the unit ball were investigated and a polynomial solution to the problem when the boundary functions of the problems are polynomials.
Abstract: We investigate existence and uniqueness conditions for solution to one Robin type problem for inhomogeneous biharmonic equation in the unit ball. We construct polynomial solution to the problem when the boundary functions of the problems are polynomials.
TL;DR: In this article, the authors studied symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X and proved that it is the case for all spaces, which satisfy an analogue of the classical Dunford-Pettis theorem on relatively weakly compact subsets in L 1.
Abstract: A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L1. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.
TL;DR: In this article, the authors considered a set of two squares built on primitive periods 1 and i and sufficiently close to each other and investigated a four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms.
Abstract: We consider a totality of two squares built on primitive periods 1 and i and “sufficiently close to each other“. In a vicinity of this set we investigate four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms. We seek a solution in a class of functions, which are analytic outside this set and vanish at infinity. The equation is applicable to the moments problem for entire functions of exponential type.
TL;DR: In this paper, integral representations for solutions of some types of the Beltrami equations were obtained and analogs of some classical complex analysis for these solutions were shown to be equivalent.
Abstract: We obtain integral representations for solutions of some types of the Beltrami equations. These representations allow us to prove analogs of some classical complex analysis for these solutions.
TL;DR: In this paper, the authors describe algebras of distributions of binary isolating formulas for theories of abelian groups and some of their ordered enrichments, and give Cayley tables for algaes that correspond to theories of basic abelians.
Abstract: We describe algebras of distributions of binary isolating formulas for theories of abelian groups and some of their ordered enrichments. The base of this description is the general theory of algebras of isolating formulas. We also take into account the specificity of the basedness of theories of abelian groups on Szmielew invariants. We give Cayley tables for algebras that correspond to theories of basic abelian groups and their ordered enrichments and propose a technique for transforming algebras for theories of basic abelian groups into algebras for arbitrary theories of abelian groups.
TL;DR: In this paper, a differential operator of the sixth order with an alternating weight function is considered, and the boundary conditions are separated, where the potential of the operator has a first-order discontinuity at some point of the segment where the operator is being considered.
Abstract: We study a differential operator of the sixth order with an alternating weight function. The potential of the operator has a first-order discontinuity at some point of the segment, where the operator is being considered. The boundary conditions are separated. We study the asymptotics of solutions to the corresponding differential equations and the asymptotics of eigenvalues of the considered differential operator.
TL;DR: In this paper, a principal G-bundle with G-invariant Riemannian metric on its total space was investigated and the Levi-Civita connection and curvatures in two-dimensional directions were derived.
Abstract: We investigate a principal G-bundle with G-invariant Riemannian metric on its total space. We derive formulas describing the Levi-Civita connection and curvatures in two-dimensional directions. We obtain estimates of the influence of properties of sectional curvatures to topological invariants of the bundle.
TL;DR: In this paper, the authors studied the invariant ideals of graded C*-algebras with respect to the representation of a compact group G in the automorphism group of this algebra and proved that the invariance of the ideal is equivalent to the fact that this ideal is graded C *-algebra.
Abstract: We present general results about graded C*-algebras and continue the previously initiated research of the C*-algebra generated by the left regular representation of an abelian semigroup. We study the invariant ideals of this C*-algebra invariant with respect to the representation of a compact group G in the automorphism group of this algebra. We prove that the invariance of the ideal is equivalent to the fact that this ideal is graded C*-algebra, that there is a maximum of all invariant ideals, and it is the commutator ideal. Separately we study a class of graded primitive ideals generated by a single projector.
TL;DR: For loaded abstract Legendre equation, the authors finds sufficient conditions of solvability of the Cauchy problem and the boundary control problem, and also considers nonlocal problem that contains fractional integral of a function with respect to another function.
Abstract: For loaded abstract Legendre equation we find sufficient conditions of solvability of the Cauchy problem and the boundary control problem. We also consider nonlocal problem that contains fractional integral of a function with respect to another function.
TL;DR: In this article, the authors investigated a four-element functional equation in a class of solutions holomorphic outside a quadrangle and vanishing at infinity and proposed equivalent regularization method of such quadrangles.
Abstract: We investigate a four-element functional equation in a class of solutions holomorphic outside a quadrangle and vanishing at infinity. We propose equivalent regularization method of such quadrangles. We construct examples with only single condition of solvability for such an equation. We show applications to moments problem for entire functions of exponential type.
TL;DR: In this paper, the sum of the values of an entire function at the zeros of the other entire function was calculated by means of the formula of logarithmic residue.
Abstract: We calculate the sum of the values of an entire function at the zeros of the other entire function by means of the formula of logarithmic residue. As a result, we can answer the question whether these functions have common zeros or not. Thus, we developed an approach to the determination of the resultant of two entire functions.
TL;DR: In this article, the authors obtained unimprovable effective oscillation conditions for all solutions of linear first-order differential and difference equations with several delays, and showed that known results of the kind are consequences of the new results.
Abstract: We obtain new unimprovable effective oscillation conditions for all solutions of linear first-order differential and difference equations with several delays. We show that known results of the kind are consequences of the new results. We reveal the reasons for the impossibility to obtain oscillation conditions for equations with several delays, as sharp as the conditions for the equation with one delay, in the case when only known approaches are used.
TL;DR: The technique of quadratic and cubic summation of power series in the perturbation method was first used for finding exact solutions to nonlinear evolution equations as mentioned in this paper, where the series were constructed with the use of exponential partial solutions to linearized equations.
Abstract: The technique of quadratic and cubic summation of power series in the perturbation method was first used for finding exact solutions to nonlinear evolution equations. The series were construction with the use of exponential partial solutions to linearized equations. The solution of both classic and modified nonintegrable Korteweg-de Vries equations, the modified Burgers equation, and the Fisher one allows one to demonstrate specific features of the mentioned method. We obtain exact solitary wave solutions to the mentioned equations in the form of a wave impulse and a wave front and show that summation parameters depend on the pole orders of the desired solutions.
TL;DR: For the Cauchy problem associated with an evolutionary operator equation of the first kind with an additional controlled term which nonlinearly depends on the phase variable, in a Banach space, this paper established conditions for the total (on the set of admissible controls) preservation of unique global solvability under variation of the control parameter.
Abstract: For the Cauchy problem associated with an evolutionary operator equation of the first kind with an additional controlled term which nonlinearly depends on the phase variable, in a Banach space, we establish conditions for the total (on the set of admissible controls) preservation of unique global solvability under variation of the control parameter. We also establish the uniform bound for solutions. As examples, we consider initial-boundary value problems that are associated with a pseudoparabolic equation and a system of Oskolkov equations.
TL;DR: For any natural number n, the degree spectrum of such relations of some computable linear orders contains exactly all n-computable enumerable degrees as mentioned in this paper, and the interconnections of these relations among themselves.
Abstract: We give the collection of relations on computable linear orders. For any natural number n, the degree spectrum of such relations of some computable linear orders contains exactly all n-computable enumerable degrees. We also study interconnections of these relations among themselves.
TL;DR: In this paper, the Riemann tensor is shown to be an invariant geometric object with respect to the geodesic mappings of affinely connected spaces, and conditions which ensure that it is invariant to the studied mappings.
Abstract: This paper is devoted to geodesic and almost geodesic mappings of affinely connected spaces. We find conditions which ensure that the Riemann tensor is an invariant geometric object with respect to the studied mappings. In this work we present an example of a non-trivial geodesic mapping between the flat spaces.
TL;DR: In terms of coefficients of an equation under consideration, this paper derived 16 variants of collections of conditions of resolvability of this equation in quadratures, each collection consists of three identities that interconnect five coefficients appearing in the left-hand side of the equation.
Abstract: In terms of coefficients of an equation under consideration, we derive 16 variants of collections of conditions of resolvability of this equation in quadratures. Every collection consists of three identities that interconnect five coefficients appearing in left-hand side of an equation.