Showing papers in "Mathematics of The Ussr-sbornik in 1971"
TL;DR: In this paper, a study of infinite Hankel matrices and approximation problems connected with them is presented, with a focus on the problem of finding the optimal solution to a given problem.
Abstract: This article is a study of infinite Hankel matrices and approximation problems connected with them. Bibliography: 22 items.
685 citations
TL;DR: In this article, a generalization of the Rouche theorem on the logarithmic residue for meromorphic operator-functions has been obtained, based on a theorem concerning a special factorization of a meromorphic operation at a point.
Abstract: We obtain the operator generalization of the theorem on the logarithmic residue for meromorphic operator-functions. The proof of the generalization is based on a theorem concerning a special factorization of a meromorphic operator-function at a point. This theorem also allows us to generalize, to the case of meromorphic operator-functions, the formula of M. V. Keldys for the principal part of the resolvent as well as several other theorems.A definition is given for the multiplicity of a pole for a meromorphic operator-function. The basic properties of the multiplicity of a pole are proved, and also a generalization of the Rouche theorem.Bibliography: 16 items.
366 citations
TL;DR: In this paper, it was shown that any birational mapping between smooth hypersurfaces of degree four is an isomorphism, and since B. Segre constructed examples of smooth unirational quartics, this leads to a negative resolution of the Luroth problem.
Abstract: In this paper we prove that any birational mapping between smooth hypersurfaces of degree four is an isomorphism. Since B. Segre constructed examples of smooth unirational quartics, this leads to a negative resolution of the three-dimensional Luroth problem. Bibliography: 13 items.
360 citations
TL;DR: In this article, the Wick and anti-Wick operator symbols are studied in connection with expansion into normal and antinormal series in terms of generation and annihilation operators, and a series of characteristic spectral properties are identified for.
Abstract: In this paper Wick and anti-Wick operator symbols are studied in connection with expansion into normal and antinormal series in terms of generation and annihilation operators. By the aid of the Wick and anti-Wick symbol of the operator a series of characteristic spectral properties are identified for . In particular, results are presented concerning necessary and sufficient conditions (separately) for to belong to the classes of bounded operators, completely continuous operators and nuclear operators, and also concerning bounds on the spectrum of , and the asymptotic behavior of the number of eigenvalues below ; and for positive selfadjoint operators a bound is obtained for the trace of the Green function: Bibliography: 14 titles.
107 citations
TL;DR: In this article, the authors investigated elliptic pseudodifferential operators p(x, D) which are degenerate on a submanifold Γ of any codimension.
Abstract: There are investigated elliptic pseudodifferential operators p(x, D) which are degenerate on a submanifold Γ of any codimension. Under certain further assumptions, for the operator which is obtained by adjoining to p(x, D) boundary and coboundary conditions on the submanifold Γ, there are constructed left and right regularizers, and theorems on hypoellipticity and local solvability are proved. In case p(x, D) is defined on a smooth compact manifold it is shown to be noetherian on special weighted spaces of Sobolev type. Bibliography: 24 references.
107 citations
TL;DR: In this paper, the connection between the theory of one-dimensional formal groups and unitary cobordism has been discussed, and two new algebraic concepts are introduced: formal power systems and two-valued formal groups.
Abstract: This paper provides a systematic presentation of the connection between the theory of one-dimensional formal groups and the theory of unitary cobordism. Two new algebraic concepts are introduced: formal power systems and two-valued formal groups. A presentation of the general theory of formal power systems is given, and it is shown that cobordism theory gives a nontrivial example of a system which is not a formal group. A two-valued formal group is constructed whose ring of coefficients is closely related to the bordism ring of a symplectic manifold. Finally, applications of formal groups and power systems are made to the theory of fixed points of periodic transformations of quasicomplex manifolds. Bibliography: 17 citations
77 citations
TL;DR: In this article, the authors constructed the set, open and everywhere dense in, of -stable mappings, which is totally disconnected and is topologically conjugate to the topological Markov chain with a finite number of states.
Abstract: In this article is constructed the set , open and everywhere dense in , of -stable mappings. is totally disconnected and is topologically conjugate to the topological Markov chain with a finite number of states; for we have and topologically conjugate to . For there exists a hyperbolic structure on .Diagrams: 1Bibliography: 9 items
71 citations
TL;DR: In this paper, the authors studied weak solutions of elliptic equations of the form in a bounded domain, where it is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts.
Abstract: In this paper the authors study weak solutions of elliptic equations of the form in a bounded domain . It is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts. It is assumed moreover that an estimate on the -norm of the solution holds on some subdomain . Summability of such solutions with a weight function that vanishes at the boundary is established, and with the use of the results of Ja. A. Roĭtberg integral representations are given in terms of the Green's function for the Dirichlet problem.Bibliography: 8 titles.
55 citations
TL;DR: In this paper, it was shown that the trajectory of a lattice of complete measure under the action of a one-parameter group of unipotent linear transformations does not tend to infinity in the lattice space.
Abstract: We prove that the trajectory of a lattice of complete measure under the action of a one-parameter group of unipotent linear transformations does not tend to infinity in the lattice space. Bibliography: 3 items.
45 citations
TL;DR: In this article, a criterion for hypoellipticity is proved which is formulated in terms of certain estimates in the norms, and which is a generalization of a criterion of Treves.
Abstract: In this paper a criterion for hypoellipticity is proved which is formulated in terms of certain estimates in the norms, and which is a generalization of a criterion of Treves. With the use of this criterion it is possible to prove the hypoellipticity of certain operators that do not satisfy Hormander's criterion. It is proved, for example, that the operator is hypoelliptic, where is an infinitely differentiable function that is not equal to zero for and has a zero of infinite order at .Bibliography: 10 titles.
44 citations
TL;DR: In this paper, the connections between the regular differentials defined on a manifold and its zero-dimensional cycles equivalent under several relations of equivalence directly generalizing rational equivalence were investigated.
Abstract: In the paper we investigate the connections between the regular differentials defined on a manifold and its zero-dimensional cycles equivalent under several relations of equivalence directly generalizing rational equivalence. Bibliography: 3 items.
TL;DR: In this article, the authors proved that any two polar factorizations of an effective Lie group are conjugate under an inner automorphism, which is called polar factorization.
Abstract: A Lie group is said to be effective if it is connected and contains no compact normal divisors. A factorization of a connected Lie group into the product of two connected subgroups, the first of which is maximally compact and the second completely solvable is called a polar factorization. In this article the following theorem is proved. Theorem. Any two polar factorizations of an effective Lie group are conjugate under an inner automorphism. Bibliography: 5 items.
TL;DR: In this paper, it was shown that an intersection of quadrics passing through a canonical curve is a reduced variety and that the possible cases when the intersection does not coincide with the curve itself are also examined.
Abstract: The principal result of the present work consists in the proof that an intersection of quadrics passing through a canonical curve is a reduced variety. The possible cases when the intersection of quadrics does not coincide with the curve itself are also examined in this article. Figures: 1. Bibliography: 8 references.
TL;DR: In this article, it was proved that similar properties are also possessed by generalized solutions of elliptic equations in a bounded region of n-dimensional space, and the main results were announced earlier (see MR 41 #3968).
Abstract: It is known that if an analytic function on the disk has exponential growth approaching the boundary, then it has generalized boundary values with which by means of Poisson kernels one can regenerate the function. In this work it is proved that similar properties are also possessed by the generalized solutions of elliptic equations in a bounded region of n-dimensional space. The main results were announced earlier (see MR 41 #3968). Bibliography: 18 references.
TL;DR: Petrovski et al. as discussed by the authors considered problems of the form A(x,∂/∂x,p)u(x) = f(x), where p is a complex parameter.
Abstract: The paper is concerned with problems of the form A(x,∂/∂x,p)u(x) = f(x) in G, B(x,∂/∂x,p)u(x) = g(x) on Γ. Here G is a region in Rnx with smooth boundary Γ; A and B are matrices of linear partial differential operators with smooth coefficients, depending polynomially on the complex parameter p. The operator A is obtained by replacing ∂/∂x by p in the operator A(x,∂/∂x, ∂/∂t), which is strongly hyperbolic in the sense of I. G. Petrovskiĭ. Under some supplementary assumptions, the existence and uniqueness of a strong solution in the spaces Hqs is demonstrated, and an a priori estimate in norms involving the parameter p is obtained for large values of Re p. Bibliography: 30 references.
TL;DR: In this article, the authors considered diffeomorphisms of a closed surface which satisfy Smale's axiom A and an acyclicity condition and showed that if one of its basis sets is one-dimensional, then there is also a zero-dimensional source or sink.
Abstract: Diffeomorphisms of a closed surface are considered which satisfy Smale's axiom A and an acyclicity condition It is shown that if one of its basis sets is one-dimensional, then there is also a zero-dimensional source or sink As a preliminary, some auxiliary propositions of general character are established concerning sources and sinks of diffeomorphisms satisfying the axiom and the condition above Bibliography: 10 entries
Abstract: In this paper, difference analogs of Sobolev–Slobodeckiĭ spaces are studied: , where is either the whole space or a halfspace. One obtains difference analogs of imbedding theorems, on traces and on extensions of network functions from a halfspace to the whole space with preservation of class. Bibliography: 19 items.
TL;DR: For suitable sets of axioms, uniqueness theorems are proved in the following categories: a) countable locally finite polyhedra and proper mappings; b) metrizable compacta; c) locally compact second-countable spaces; d) Hausdorff spaces whose compact subspaces are metrizability; e) paracompact weakly locally contractible spaces.
Abstract: For suitable sets of axioms, uniqueness theorems are proved in the following categories: a) countable locally finite polyhedra and proper mappings; b) metrizable compacta; c) locally compact second-countable spaces and proper mappings; d) Hausdorff spaces whose compact subspaces are metrizable; e) paracompact weakly locally contractible spaces.Bibliography: 11 items.
TL;DR: In this article, the action of the Galois group on rational cohomology classes of type is investigated, where is an abelian variety defined over a field of characteristic zero.
Abstract: In the paper the action of the Galois group on is investigated, where is an abelian variety defined over a field of characteristic zero. We prove that the Galois group acts on the rational cohomology classes of type as far as they are algebraic.Bibliography: 10 items.
TL;DR: In this article, the connection between completely linear functionals and measures on bases in a -space is studied, and a realization of spaces of regular functionals is established, where the connections between regular functions and completely linear functions are analyzed.
Abstract: In the first two sections the representation of completely linear functionals in -spaces and the connection between completely linear functionals and measures on bases in a -space is studied. In §3 a realization of spaces of regular functionals is established.Bibliography: 15 titles.
TL;DR: In this article, the question of whether a homomorphism can be lifted to a lattice for the case that the lattice is in a Lie group of type T is investigated.
Abstract: Let and be simply connected Lie groups, and let be a lattice in . In the present article we investigate the question whether the homomorphism can be lifted to a homomorphism for the case that or is a Lie group of type . Incidentally we prove some of the properties of lattices in such groups.Bibliography: 13 items.
TL;DR: In this paper, an analog to the Cauchy-Weil formula for differential forms on analytic polyhedra is presented, and a proof for the Martinelli-Bochner formula is given.
Abstract: In the paper one constructs and proves an analog to the Cauchy-Weil formula for differential forms on analytic polyhedra. On the way one obtains a proof for the Martinelli-Bochner formula for differential forms on domains in .Bibliography: 3 items.
TL;DR: In this paper, a theory for perturbations of Hermitian operators with respect to completely continuous operators is presented, where singular and characteristic numbers are derived for operators from the von Neumann algebra and their minimax properties are derived.
Abstract: By means of the concept of Segal measure, defined on projectors (and thus on subspaces associated with a von Neumann algebra) we introduce the concept of relative compactness of sets and, on this basis, the concept of operators completely continuous with respect to the von Neumann algebra and the Segal measure The article is concerned with the formal structure of the theory of this class of operators: the general theorem of Calkin is obtained on the uniqueness of the ideal with respect to completely continuous operators; a theory is constructed for perturbations of Hermitian operators with respect to completely continuous ones; singular and characteristic numbers are introduced for operators from the von Neumann algebra and their minimax properties are derived; some characterizations are introduced in terms of completely continuous operatorsBibliography: 10 items
TL;DR: In this paper, the normalizer of an element in the braid broup is shown to be finitely generated, and a method for finding generators for is indicated, where the generator is given.
Abstract: Let be the normalizer of an element in the braid broup . It is shown that is finitely generated, and a method for finding generators for is indicated. Bibliography: 4 items.
TL;DR: In this article, the authors investigate abelian groups that are the group-theoretical analog of symplectic linear spaces and show that they can be viewed as a form of group theorems.
Abstract: In the paper one investigates symplectic abelian groups that are the group-theoretical analog of symplectic linear spaces. Bibliography: 3 items.
TL;DR: In this article, the authors consider a nonlinear equation in a Banach space which is invariant relative to a continuous group and give conditions which allow them to reduce Ljapunov-Schmidt branch equations in both the number of equations and the unknowns, which makes it possible to simplify significantly the search for multi-parameter families of solutions of the given problem.
Abstract: We consider a nonlinear equation in a Banach space which is invariant relative to a continuous group. We give conditions which allow us to reduce Ljapunov-Schmidt branch equations in both the number of equations and the number of unknowns, which makes it possible to simplify significantly the search for multi-parameter families of solutions of the given problem.Bibliography: 15 items.
TL;DR: In this article, the authors construct the theory of extensions of Hermitian operators which are initially defined on a manifold in Hilbert space, and the extension is accompanied by a result in the Hilbert space of ideal elements.
Abstract: In this paper we construct the theory of extensions of Hermitian operators which are initially defined on a manifold in Hilbert space. The operators may have infinite defect numbers, and the manifold may fail to be dense. The extension is accompanied by a result in the Hilbert space of ideal elements (generalized functions which are defined on the Hilbert space of elements which belong to the basic Hilbert space: ). We conduct a detailed analysis of extended generalized resolvents and corresponding spectral functions. We explain the connection between functions of the form , where is an extended generalized resolvent, and the theory of -functions.Bibliography: 14 titles.
TL;DR: In this article, the authors prove the Weyl inequalities for the set of operators of range dimension not greater than a given factor in the Banach space of a continuous operator acting on the space and the complete system of its eigenvalues.
Abstract: Let be a completely continuous operator acting on the Banach space , let be the complete system of its eigenvalues (with regard for multiplicity) and let be the distance from to the set of all operators of range dimension not greater than . If (1)then , where is a functional which is linear on the set of operators satisfying condition (1) (and continuous in a certain topology) and which coincides with its trace for finite-dimensional . The proof of this theorem is based on certain analogs of the famous Weyl inequalities.Bibliography: 14 titles.
TL;DR: In this paper, the authors consider Markov processes with continuous time, where the switching of the controls takes place at random (independent of the future) moments of time, and derive Bellman's cost equation and the existence of optimal strategies, prove the measurability of cost and give an excessive characterization of cost.
Abstract: We consider Markov processes with continuous time, where the switching of the controls takes place at random (independent of the future) moments of time We derive Bellman's cost equation and the existence of optimal strategies, prove the measurability of cost and give an excessive characterization of cost Bibliography: 9 items
TL;DR: In this paper, the authors studied the asymptotic properties of the Green function for the Neumann problem in the exterior of a planar convex domain for the Helmholtz equation.
Abstract: First part of paper. The paper is concerned with the study of asymptotic properties of the Green function for the Neumann problem in the exterior of a planar convex domain for the Helmholtz equation. Figures: 1. Bibliography: 10 items.