Showing papers in "Journal of Mathematical Sciences in 1981"

Stochastic evolution equations

Abstract: The theory of strong solutions of Ito equations in Banach spaces is expounded. The results of this theory are applied to the investigation of strongly parabolic Ito partial differential equations.

Computational methods of linear algebra

Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Chronological algebras and nonstationary vector fields

Abstract: A calculus is developed, reflecting the most general group-theoretic properties of flows, defined by nonstationary differential equations on manifolds.

Finite-zone, almost-periodic solutions in WKB approximations

Abstract: It is shown that the recently discovered finite-zone, almost-periodic solutions may, on the one hand, serve as the foundation for the development of the multiphase WKB method in nonlinear equations (the method of Whitham) and, on the other hand, serve to define Lagrangian manifolds with complex germs which can be (second) quantized in the quasiclassical approximation.

Mathematical theory of optimal control

Abstract: Papers on optimal, control theory, reviewed in Referativnyi Zhurnal “Matematika” during 1971–1975, are surveyed.

Modern methods of the statistical theory of nonequilibrium processes

Abstract: Possible approaches to the construction of nonequilibrium equations from first principles are analyzed: the prescription of the initial state at time t=0 and the investigation of the asymptotics as t→∞ as well as methods of averaging over the initial times or boundary conditions in the distant past. In simple examples, a comparison is made of the methods of the projection operator and the nonequilibrium statistical operator (in two versions) of Kubo-Yokota-Nakajima, Mori, Robertson, Kawasaki-Gunton, Kadanoff-Martin, etc.

Geometry of nonlinear differential equations

Abstract: The paper contains a survey of certain contemporary concepts and results connected with the geometric foundations of the theory of nonlinear partial differential equations. At the base of the account is situated the geometry and analysis on jet spaces, finite and infinite.

Closed categories and the theory of proofs

Abstract: The main aim of this article is to suggest a translation of the simplest concepts of category theory into the language of (structural) theory of proofs, to use this translation to simplify the proofs of some known results [1], and to obtain two new ones: the coherence theorem for canonical morphisms in (nonmonoidal, nonsymmetric) closed categories [2], and the solution of the problem of equality of canonical morphisms. Extensions of these results to monoidal closed, symmetric closed, and monoidal symmetric closed categories are sketched. The decision procedure is obtained by means of a correct and faithful translation of canonical morphisms into an expansion of the λ-language, which has the tools for a special account of “superfluous” premises of implications (the thinning rule). The expansions of the λ-language which have so far appeared in the literature have not possessed this facility.

Linear operators in spaces with indefinite metric and their applications

Abstract: The paper contains a survey of papers reviewed in Ref. Zh. Matematika from 1953–1978 on the theory of linear operators in (mainly Hibert) spaces with indefinite metric and their applications to various domains of mathematics and mechanics. As a preliminary, the needed results on the geometry of spaces with indefinite metric are described.

Hardy spaces of vector-valued functions

Abstract: The purpose of this paper is to consider the relations among the various definitions of the spaces Hp of analytic functions with values in a Banach space and to investigate the problem of the structure of the conjugates of these spaces. In particular, one constructs an example of a reflexive separable Banach space χ, for which the equality Hp(X)*=Hp′(X*) (1

Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations

Abstract: Various aspects of the calculus of functions of ordered self-adjoint operators are considered. Passage to the commutative limit in the case of general nonlinear commutation relations is studied. An asymptotic solution of the Cauchy problem and asymptotically self-similar solutions are constructed for unitary-nonlinear operator equations. Asymptotic solutions are found for the Hartree equation with Coulomb interaction.

Regular convergence of operators and approximate solution of equations

Abstract: The survey is devoted to different concepts of the convergence of linear and nonlinear operators (discrete, regular, compact, stable convergence, etc.). Attention is paid to the applications of these concepts to numerical analysis methods.

Schur multiplier of a group of elementary matrices of finite order

Abstract: If a ring Λ is finitely generated as a module over its center, then for the Schur multiplier coincides with . For a representation of is obtained which is analogous to van der Kallen's representation for a commutative ring.

Markov and Lagrange spectra (survey of the literature)

Abstract: A survey is given of investigations on the Markov problem of the arithmetic minima of indeterminate, binary, quadratic forms and on the Lagrange-Hurwitz problem of Diophantine approximations of irrational numbers by rational numbers.

Absolutely continuous spectrum of a nondissipative operator and the functional model. I

Abstract: In the paper one investigates nondissipative operators L in a Hilbert space; for them one constructs a model representation similar to the Nagy-Foias model for dissipative operators. In this representation one succeeds to calculate the action of the resolvent of a nondissipative operator on selected subspaces. This allows us to relinquish the consideration of the -self-adjoint dilation of the operator, whose spectral representation involves considerable difficulties. Isolated results are new also for the dissipative case which is not excluded. In part I one considers the “triangular” factorization of the characteristic funcion of the operator L and one carries out the proof of the fundamental theorem which gives a formula for the calculation of (L-No)−1(JmNoO TmNo

Stabilization theorem for the Milnor K2-functor

Abstract: Let Λ be an associative ring. For every natural number n there is a canonical homomorphismψ n: K2,n(Λ)→K2(Λ), where K2 is the Milnor functor and K2,n(Λ) the associated unstable K-group. Dennis and Vasershtein have proved that if n is larger than the stable rank of Λ,ψ n is an epimorphism. It is proved in the article that if n − 1 is greater than the stable rank of Λ, the homomorphismψ n is an isomorphism.

Theta-series of even unimodular 24-dimensional lattices

Abstract: It is proved that the twelve theta-series of the even unimodular 24-dimensional lattices are linearly independent and one finds all the relations among the k-th theta-series of these lattices for k≤4.

One theorem of Cohn

Abstract: Let F be the field of algebraic functions of one variable over the field of constants k, v be a point of field F/k, and Av be the ring of functions not having poles outside point v. It is proved that Av is a GE2-ring if and only if it coincides with the ring k[X] of polynomials of one variable over field k.

Subgroups of the unitary group that contain the group of diagonal matrices

Abstract: In the groupU(n,ℂ) of all complex unitary matrices we obtain a description of the lattice of those subgroups H that contain the group of diagonal unitarymatrices. This lattice is finite, and the connected intermediate subgroups H are in bijective correspondence with the equivalence relations on a set ofn elements.

Evaluation of the normal distribution function

Abstract: Known methods are described for evaluating the multivariate normal distribution function in general and in a number of special cases. Methods are presented for evaluation of the one-dimensional normal distribution function and its inverse.

Primes are nonnegative values of a polynomial in 10 variables

Abstract: A polynomial with integer coefficients of 10 variables is constructed, whose set of all nonnegative values (for positive integer values of the variables) is precisely the set of all prime numbers.

Time asymptotics for some degenerate models of evolution of systems with an infinite number of particles

Abstract: The paper is devoted to the problem of convergence to the equilibrium state in the motion of infinite systems of classical particles. Two models of the motion are considered: free motion of point particles in Euclidean spaceR v , ν ≥1, and motion of solid rods on the line R1. The paper contains new results obtained by the authors and also a survey of previous results in this direction.

Geometry of Lobachevskii in connection with certain questions of arithmetic

Abstract: A new geometric method is developed for investigating the Markov problem of the arithmetic minima of indeterminate binary quadratic forms and problems related to them.

Complexity of algorithms and computations

Abstract: A survey is given of some results on the complexity of algorithms and computations published up to 1973.

A description of the subgroups of the complete linear group that contain the group of diagonal matrices

Abstract: Let K be a field containing at least seven elements. In the group G=GL(n, K) we describe the subgroups containing the group D of all diagonal matrices. This description is given in terms of the concept of a D-net subgroup, defined as a subgroup of G composed of matrices (a ij) with zero elementsa ij in some prescribed cells outside the main diagonal (the set of cells is subordinated to some condition of agreement). The main theorem is: Every subgroup of G containing D is contained between a uniquely determined D-net subgroup and its normalizer in G. The structure of all subgroups of G containing D is finite and does not depend on the field K (when card K ≥ 7).

Algebras with general commutation relations and their applications. I. Pseudodifferential equations with increasing coefficients

Abstract: Operators of the regular representation for commutation relations of general form are constructed. On the basis of the technique developed, conditions for the existence of a mixed (with respect to smoothness and rate of growth at infinity) asymptotic expression for a solution of a pseudodifferential equation with increasing coefficients are obtained. The work contains a survey of the general theory of quasiinversion of functions of ordered operators.

Spaces with “small” annihilators

Abstract: In this note one investigates the properties of subspaces G of C(S), such that G1 is “not a very large part” of the space C(S)*. The fundamental result is: if G1 is reflexive, then every operator from G* into l2 is absolutely summable.

Harmonic analysis on reductive Lie groups

Abstract: This is a survey on harmonic analysis on real reductive Lie groups for the period 1950–1978. Contained is a description of the fundamental series of representations of a reductive group G, the theory of characters, a description of the Plancherel measure, a description of dual spaces with respect to the Fourier transform on G, the classification of irreducible representations of the group G, questions of harmonic analysis on symmetric spaces of noncompact type.

Two methods for avoiding hereditary completeness

Abstract: A family of vectors Open image in new window of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {xn′;n≥1}((xn,xk′)=δnk) and if any elementx, xe H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′n)xn:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families Open image in new window of the same type but already complete in H.. Clearly, in this case Open image in new window cannot be hereditarily complete. The second method consists in considering linear deformationn:n⩾1 of the orthogonal basesn: n⩾1; here A is an unbounded operator of a special type.

Complex representations of the group GL(2, Z/pnZ)

Abstract: We obtain a description of all irreducible complex representations of the group indicated in the title (p ≠ 2 is a prime) Namely, for each n ≥ 2 we distinguish three series of representations of degrees (p + 1)pn−1, (p2 − 1)pn−2, (p − 1)pn−1 The other representations of\(GL(2,Z_{p^n } )\) are obtained from representations of\(GL(2,Z_{p^{n - 1} } )\) by tensor multiplication by one-dimensional representations