Showing papers in "Mathematics of The Ussr-sbornik in 1970"
TL;DR: In this paper, a theory of generalized solutions in the large Cauchy's problem for the equations in the class of bounded measurable functions is constructed, and the existence, uniqueness and stability theorems for this solution are proved.
Abstract: In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the "vanishing viscosity method"; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity.Bibliography: 22 items.
1,799 citations
TL;DR: In this article, it was shown that in a neighborhood of a trajectory which is doubly asymptotic to a rough state of equilibrium of saddle-focus type, under addition assumptions on the dynamical system, there exists a subsystem whose trajectories are in one-to-one correspondence with a set which is the quotient space of a topological Bernoulli process with an infinite set of symbols.
Abstract: It is shown that in a neighborhood of a trajectory which is doubly asymptotic to a rough state of equilibrium of saddle-focus type, under addition assumptions on the dynamical system, there exists a subsystem whose trajectories are in one-to-one correspondence with a set which is the quotient space of a topological Bernoulli process with an infinite set of symbols. Bibliography: 10 items.
421 citations
TL;DR: In this article, a complete description of the polyhedra of finite volume with dihedral angles not exceeding 90° in three-dimensional Lobacevski space is given, where the dihedral angle is fixed.
Abstract: The paper contains a complete description of the polyhedra of finite volume with dihedral angles not exceeding 90° in three-dimensional Lobacevskiĭ space. Bibliography: One item.
254 citations
TL;DR: In this article, a complete description of convex bounded polyhedra with dihedral angles not exceeding 90, in a 3D Lobacevski space is given, which permits a description of the discrete groups generated by reflections acting in the 3D lobster space with a compact fundamental region.
Abstract: This paper is concerned with the investigation of properties of convex polyhedra in Lobacevskiĭ spaces; it gives a complete description of convex bounded polyhedra with dihedral angles not exceeding 90, in a three-dimensional Lobacevskiĭ space. This in turn permits a description of the discrete groups generated by reflections acting in the three-dimensional Lobacevskiĭ space with a compact fundamental region.6 figures; bibliography: 6 items.
224 citations
TL;DR: In this article, a necessary and sufficient condition for operators of this class to be hypoelliptic was found, namely that the equation,, has no nontrivial solutions in.
Abstract: Let the variables in be broken up into two groups , where and . We consider differential operators with polynomial symbols of the form where . We assume that the symbol is quasihomogeneous: and that is elliptic for . We have found a necessary and sufficient condition for operators of this class to be hypoelliptic: namely, that the equation , , have no nontrivial solutions in . Thus for example, the operator is hypoelliptic for any integers and , and the operator is hypoelliptic if and only if is not an eigenvalue of the operator in . These results are partially extended to operators with variable coefficients and to pseudodifferential operators.Bibliography: 22 references.
178 citations
TL;DR: The integral representation obtained for smooth functions defined in strictly pseudoconvex domains in -dimensional complex space can be considered as the "natural" extension of the well-known Cauchy-Green formula as mentioned in this paper.
Abstract: The integral representation obtained for smooth functions defined in strictly pseudoconvex domains in -dimensional complex space can be considered as the "natural" extension of the well-known Cauchy-Green formula. Using this representation we succeed in obtaining a formula and uniform bound for solutions of the -problem in strictly pseudoconvex domains.Bibliography: 11 titles.
99 citations
TL;DR: In this paper, the authors study analogs of Lie algebras and formal Lie groups, which differ from usual Lie groups in that they admit anticommuting canonical parameters.
Abstract: In this paper we study analogs of Lie algebras and formal Lie groups. These analogs of groups differ from usual Lie groups, roughly speaking, in that they admit anticommuting canonical parameters. The analogs of Lie algebras differ from usual Lie algebras by properties of the commutator. In the definition of these objects an essential role is played by the gradient. In case it is trivial they become Lie groups and algebras in the usual sense. To these generalized objects we carry over classical theorems on the connection between Lie groups and algebras and the basic representation theory. Bibliography: 11 references.
93 citations
TL;DR: In this paper, orthogonal and unitary K1-functors over an associative ring with involution are examined, which are similar to general K1 -functors associated with a general linear group over a ring.
Abstract: Orthogonal and unitary groups over an associative ring with involution are examined. Using these groups, orthogonal and unitary K1-functors are introduced which are similar to general K1-functors associated with a general linear group over a ring. Results are obtained about stabilization of orthogonal and unitary groups. Bibliography: 9 entries
78 citations
TL;DR: The existence or nonexistence of eigenfunctions for quasilinear elliptic equations of arbitrary even order with homogeneous Dirichlet boundary conditions was examined in this paper.
Abstract: The existence or nonexistence of eigenfunctions is examined for quasilinear elliptic equations of arbitrary even order with homogeneous Dirichlet boundary conditions. Bibliography: 12 items
73 citations
TL;DR: In this paper, the authors studied pseudodifferential operators which are elliptic outside an (n - 1)-dimensional submanifold ω of a closed n-dimensional manifold.
Abstract: This article studies pseudodifferential operators which are elliptic outside an (n - 1)-dimensional submanifold ω of a closed n-dimensional manifold Γ. It is assumed that at those points of the cotangent bundle at which the ellipticity condition is violated the gradient of the determinant of the symbol is nonzero and transversal to ω. On ω a number of boundary conditions are prescribed, and a number of potential operators with unknown densities are adjoined to the original equation; the normal solvability of this boundary value problem is then established. Bibliography: 21 items.
65 citations
TL;DR: In this paper, the authors derived formulas for the Chern-Dold character chU and chU in the theory of universal bordisms and cobordisms U*, respectively.
Abstract: The fundamental results of the paper are: derivation of formulas for the Chern-Dold character chU and chU in the theory of universal bordisms U* and cobordisms U*, respectively; derivation of the formula of a series over the ring Ω*U which gives addition in the formal group of "geometric" cobordisms, and derivation of the formula for the series kΨUk(u), where uU2() and the ΨkU are Adams operators in U*-theory. Bibliography: 13 items.
TL;DR: In this paper, the authors established the existence of a non-random limit for a wide class of stationary ergodic potentials under the assumption that the potential is Markovian, and the argument is based on the well-known theorems of Sturm.
Abstract: Let be the number of eigenvalues not exceeding for the selfadjoint boundary problem with random potential , and let Our problem is to clarify the conditions under which this function will exist and to indicate methods for calculating it.In the present article we establish the existence of a nonrandom limit for a wide class of stationary ergodic potentials. This limit is calculated under the assumption that the potential is Markovian, and the argument is based on the well-known theorems of Sturm.At the end of the article we consider an example in which is a Markov process with two states. In this case the calculations can all be carried out completely in a practical way, with the result that we obtain a formula expressing by means of integrals of elementary functions.Bibliography: 9 items.
TL;DR: In this article, complex representations of the group GL(n, Fq ) are studied, where Fq is a field of order q and n is the number of vertices in the group.
Abstract: Complex representations of the group GL(n, Fq ) are studied, where Fq is a field of order q. The concept of analytic representations of GL(n, Fq ) is introduced and is used to construct all the irreducible representations of GL(n, Fq ). Results are obtained on the irreducible polynomials of 3rd degree over Fq . Bibliography: 21 items.
TL;DR: In this article, it was proved that for (?F?? F) = 1, 2 and 3 a birational class of these surfaces is uniquely determined by the standard pencil of rational curves.
Abstract: In the paper standard G-surfaces with a pencil of rational curves and with (?F ? ?F) > 0 are examined up to birational equivalence. It is proved that for (?F ? ?F) = 1 , 2 and 3 a birational class of these surfaces is uniquely determined by the birational class of their standard pencil of rational curves. For (?F ? ?F) > 4 each of these surfaces is birationally equivalent to either the plane P2 or some G-surface which is a biregular form of the surface P1?P1. Bibliography: 6 items.
TL;DR: In this article, the notion of dimension of embedding of compacta in En is introduced and the main theorem states that an embedding is "wild" if and only if the complementary space is not 1-ULC.
Abstract: Recently the fundamental importance of the 1-ULC property of the complementary space in describing a given embedding in En has become clear. "Wild" embeddings in En are characterized by the absence of the 1-ULC property. In this paper "tame" and "wild" embeddings in En of arbitrary compacta in codimension at least 3 are defined. For this purpose the notion of the "dimension of embedding" of compacta in En is introduced. The main theorem asserts that an embedding of a compactum in En, n≥6, is "wild" if and only if the complementary space is not 1-ULC. Bibliography: 23 items.
TL;DR: In this article, the relative homological dimension of a normed module over a Banach algebra is defined and the results obtained are applicable to finding cohomology groups of certain Banach algebras.
Abstract: In this work there is introduced a notion of relative homological dimension of a normed module over a Banach algebra. This dimension is computed for several cases. The results obtained are applicable to finding cohomology groups of certain Banach algebras and to solving the question of the strong decomposability of certain classes of extended Banach algebras. Bibliography: 13 references.
TL;DR: In this article, the authors established imbedding theorems pertaining to arbitrary classes of functions of a single variable (L), L(L), Hpω(δ), and Lvlnβ(1 + L).
Abstract: In the first part of the work are established imbedding theorems pertaining to arbitrary classes of functions of a single variable (L), L(L), Hpω(δ) and Lvlnβ(1 + L). The second part contains estimates for best approximations (moduli of continuity) in different metrics. It is shown that in certain cases these estimates cannot be strengthened. Bibliography: 14 titles.
TL;DR: In this paper, explicit formulas for zonal spherical functions on the group GLn(D) are derived, where D is a division algebra of finite rank over a discrete normed field with finite residue class field.
Abstract: In ?? 1 and 2 explicit formulas for zonal spherical functions on the group GLn(D) are derived. Here D is a division algebra of finite rank over a discrete normed field with finite residue class field. In ? 3 these formulas are applied to summation of multiple Hecke series and zeta-functions in n variables on the group GLn(D). In ? 4 the results of ? 3 are applied to summation of Hecke series and zeta-functions on the symplectic group of genus n over local fields. Furthermore, the following conjectures are proved: the conjecture of Satake about the form of the denominator of zeta-functions, and the conjecture of Shimura about the degrees of the numerator and the denominator. Bibliography: 7 items.
TL;DR: In this paper, the authors construct examples disproving Samuel's conjecture that the ring A is factorial for a complete factorial local ring A and prove a theorem under some restrictions.
Abstract: We construct examples disproving Samuel's conjecture stating that the ring A[[T]] is factorial for a complete factorial local ring A. We also prove a theorem asserting (under some restrictions) that the ring A[[T]] is factorial for a geometrically factorial ring A. The bibliography contains 16 items.
TL;DR: In this paper, the method of averaging of N. Bogoljubov is applied to abstract parabolic equations of the form (1) where is a linear, in general unbounded, operator generating an analytic semigroup, and is an operator subordinate to, in general a nonlinear map, possessing the mean Other conditions on the mapping are formulated in terms of the theory of semigroups.
Abstract: In this paper the method of averaging of N.N. Bogoljubov is applied to abstract parabolic equations of the form (1)where is a linear, in general unbounded, operator generating an analytic semigroup, and is an operator subordinate to , in general a nonlinear map, possessing the mean Other conditions on the mapping are formulated in terms of the theory of semigroups.The main results are contained in two theorems.Theorem 1 relates the initial value problem for equation (1) with the equation (2)Theorem 2, in the case of periodic dependence of the mapping on time, establishes a connection between the stability of the stationary solution to equation (2) and the stability of the corresponding periodic solution of (1).Bibliography: 5 items.
TL;DR: In this paper, the structure of the invariant subspaces, irreducibility, the operators which commute with the group (the intertwining operators), invariant Hermitian forms, and unitarity are studied.
Abstract: We study representations of the group , 1$ SRC=http://ej.iop.org/images/0025-5734/10/3/A04/tex_sm_2158_img2.gif/>, 1$ SRC=http://ej.iop.org/images/0025-5734/10/3/A04/tex_sm_2158_img3.gif/>, in the spaces , ( is a complex number; or 1), of -functions on the cone , of homogeneous degree and parity . We consider the structure of the invariant subspaces, irreducibility, the operators which commute with the group (the intertwining operators), invariant Hermitian forms, and unitarity.Bibliography: 13 items. 1 figure.
TL;DR: In this paper, it was proved that the -modulus of continuity dominates any deviation, constructed with the help of a measure with compact support, orthogonal to polynomials of degree.
Abstract: The following theorem is proved:Theorem. Let , where is a convex domain in . Then where the on the left is taken over all degree polynomials, and the norm on the right is taken over the set in which the th difference is defined. The constant depends only on , , and the ratio of the diameter of to its width.H. Whitney proved this theorem in the case and . As a corollary, it is proved that the -modulus of continuity dominates any "deviation", constructed with the help of a measure with compact support, orthogonal to polynomials of degree .Bibliography: 10 items.
TL;DR: In this paper, the formal stability of periodic solutions for a Hamiltonian system in two degrees of freedom is investigated for the case of a resonance of order q ≥ 3, and the results are applied to the restricted problem of three bodies, which allows us to explain qualitatively the nature of all gaps with q ≤ 3 in the distribution of asteroids.
Abstract: The formal stability of periodic solutions is investigated for a Hamiltonian system in two degrees of freedom. The nature of the zones of instability is exhibited in the case of a resonance of order q≥3. In contrast to classical theory, an isoenergetic reduction is not carried out. This permits unstable solutions close to periodic solutions to be studied in full. The results are applied to the restricted problem of three bodies, which allows us to explain qualitatively the nature of all gaps with q≥3 in the distribution of asteroids. 19 figures. Bibliography: 37 titles.
TL;DR: In this paper, the authors considered the parabolic equation with a characteristic form which is strongly convex, and derived the asymptotic behavior of its Green's function for is derived.
Abstract: The form of order , which is a function of the variables , where and is called strongly convex if the quadratic form (in a space of dimension equal to the number of the multi-indices with ) is positive definite. All even-order differentials of a strongly convex form are positive definite forms.The paper considers the parabolic equation , with a characteristic form which is strongly convex, and the asymptotic behavior of its Green's function for is derived. It is an unexpected property that this asymptotic behavior is dependent not on all saddle points of the corresponding integral with , but only on some of these. (This effect has not been observed for the previously known cases, with or .)The asymptotic behavior of the Green's function (for ) is derived also for the corresponding elliptic equation . It is suggested that analogous results hold for all convex forms , i.e. all forms having a positive definite second differential.Bibliography: 4 items.
TL;DR: In this paper, it was shown that a ring A has a Discrete Group of Classes (DGC) if the group of divisor classes is preserved in going to the ring of formal power series.
Abstract: We shall say that a ring A has a DGC (discrete group of classes) if the group of divisor classes is preserved in going to the ring of formal power series, i.e. C(A) → C(A[[T]]) is an isomorphism. We prove the localness and faithfully flat descent of the DGC property. We establish a connection between the DGC property of a ring and its depth. We also give a characterization of two-dimensional rings with DGC and characteristic zero rings with DGC. Finally, it is shown that the discreteness of the group of divisor classes is preserved under regular extensions of rings such as A[T1,⋯, Tn], A[[T1 ,⋯, Tn ]], completions, etc. Bibliography: 13 items.
TL;DR: In this paper, the existence and uniqueness of the solution in Sobolev spaces for the first boundary value problem for elliptic (parabolic) equations of the form Here and in the elliptic case, in the parabolic case, was proved.
Abstract: The existence and uniqueness of the solution in Sobolev spaces () is proved for the first boundary value problem for elliptic (parabolic) equations of the form Here and in the elliptic case, in the parabolic case. The subscript takes any values close to two.Bibliography: 10 items.
TL;DR: In this article, it was shown that any function, analytic in a convex region, is representable in Dirichlet series by an operator with characteristic function from the class of analytic functions.
Abstract: In the author's paper (On the representation of analytic functions by Dirichlet series, Math. USSR Sb. 9 (1969), 111-150) a theorem was proved stating that every function , analytic in a finite convex region and continuous in , can be represented in by a Dirichlet series. Here we have obtained a definitive result: any function , analytic in , is representable in by a Dirichlet series. The proof is based on the following assertion: let be a function analytic in a finite convex region . There exist a function , analytic in and continuous in , and an operator with characteristic function from the class , such that .Bibliography: 4 items.
TL;DR: In this article, a local variant of Lavrent'ev's theorem concerning a global homeomorphism proved earlier by us is established, where the coefficient of quasiconformality in the region of the deleted sphere is defined.
Abstract: With a view toward the preparation of the apparatus for the investigation of quasiconformal mappings of manifolds, in this work we establish the following local variant of M.A.?Lavrent'ev's theorem concerning a global homeomorphism proved earlier by us:Theorem. Let be a locally homeomorphic mapping of the deleted sphere into . Let be the coefficient of quasiconformality of in the region . Then the following assertions are valid.1) When and , the mapping is homeomorphic in some deleted neighborhood of the point , and can be continued up to homeomorphism to the whole neighborhood of this point.2) In the sense of the admissible order of the growth of , the assertion 1) is correct.Bibliography: 3 items.
TL;DR: In this article, inequalities are established for the moduli of families of curves corresponding with each other under a certain, not necessarily homeomorphic, quasiconformal mapping, and these inequalities are applied to the study of the relation of dilatation with the minimal multiplicity of a ramification of such mappings.
Abstract: The modulus method is one of the most effective methods in the theory of quasiconformal homeomorphisms. Over the course of a long time there has been no success, however, in applying this method to the analysis of nonhomeomorphic quasiconformal mappings of spatial domains. In the present paper inequalities are established for the moduli of families of curves corresponding with each other under a certain, not necessarily homeomorphic, quasiconformal mapping. These inequalities are applied to the study of the relation of dilatation with the minimal multiplicity of a ramification of such mappings. Bibliography: 7 items.
TL;DR: In this article, the existence of generalized wave operators for the Friedrichs model with discontinuous kernel and for differential operators with potentials of Coulomb type was shown. But the conditions for the existence were not given.
Abstract: In this paper we generalize the concepts of wave operators and scattering operators. We find sufficient conditions for the existence of generalized wave operators for the Friedrichs model with discontinuous kernel, and for differential operators with potential of Coulomb type. Bibliography: 16 items.