Showing papers in "Journal of Mathematical Sciences in 1985"
TL;DR: A survey of the theory of Kats-Moody algebras is given in this paper, which contains a description of the connection between the infinite-dimensional Lie algebra of Kats and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the inverse scattering problem.
Abstract: The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the method of the inverse scattering problem. A survey of the theory of Kats-Moody algebras is also given.
1,288 citations
TL;DR: In this paper, conjectures regarding the value of L-functions of motives are formulated and some computations are presented corroborating them, and some conjectures are formulated regarding the L-values of motives.
Abstract: In the work conjectures are formulated regarding the value of L-functions of motives and some computations are presented corroborating them.
540 citations
TL;DR: In this article, the authors present a compilation of the most promising and fruitful results on the graph isomorphism problem, combining the works of Babai and Luks with the theory of permutation groups, and present an algorithm for testing the isomorphisms of graphs of genus g in time O(vO(g), where v is the number of vertices.
Abstract: The article is a creative compilation of certain papers devoted to the graph isomorphism problem, which have appeared in recent years. An approach to the isomorphism problem is proposed in the first chapter, combining, mainly, the works of Babai and Luks. This approach, being to the survey's authors the most promising and fruitful of results, has two characteristic features: the use of information on the special structure of the automorphism group and the profound application of the theory of permutation groups. In particular, proofs are given of the recognizability of the isomorphism of graphs with bounded valences in polynomial time and of all graphs in moderately exponential time. In the second chapter a free exposition is given of the Filotti-Mayer-Miller results on the isomorphism of graphs of bounded genus. New and more complete proofs of the main assertions are presented, as well as an algorithm for the testing of the isomorphism of graphs of genus g in time O(vO(g)), where v is the number of vertices. In the third chapter certain extended means of the construction of algorithms testing an isomorphism are discussed together with probabilistically estimated algorithms and the Las Vegas algorithms. In the fourth chapter the connections of the graph isomorphism problem with other problems are examined.
132 citations
TL;DR: In this article, a proof of Milnor's conjecture that for any field F and natural number n > 1, there is the isomorphismR.............. n,F.............. :K..............2(F)/nK>>\s 2(F)
Abstract: Recent results on the structure of the group K2 of a field and its connections with the Brauer group are presented. The K-groups of Severi-Brauer varieties and simple algebras are computed. A proof is given of Milnor's conjecture that for any field F and natural number n > 1 there is the isomorphismR
n,F
:K
2(F)/nK
2(F)
→
∼
n
Br(F). Algebrogeometric applications of the main results are presented.
100 citations
TL;DR: Results of recent investigations at the juncture of coding theory, the theory of computability, and algebraic geometry over finite fields are presented.
Abstract: Results of recent investigations at the juncture of coding theory, the theory of computability, and algebraic geometry over finite fields are presented. The basic problems of the asymptotic theory of codes and Goppa's construction of codes on the basis of algebraic curves are presented, and a detailed algorithmic analysis is given of the codes arising on the modular curves of elliptic modules of V. G. Drinfel'd.
99 citations
TL;DR: In this article, the inverse scattering method applies to both the classical and the quantum Goryachev-Chaplygin top, and a new method, based on the R-matrix formalism, is proposed for deriving the equations determining the spectrum of the quantum integrals of motion.
Abstract: It is shown that the inverse scattering method applies to both the classical and the quantum Goryachev-Chaplygin top. A new method, based on theR-matrix formalism, is proposed for deriving the equations determining the spectrum of the quantum integrals of motion. This method is of a rather general nature and may serve as an alternative to the so-called algebraic Bethe Ansatz.
94 citations
TL;DR: In this article, the main ideas of global "finite zone integration" are presented, and a detailed analysis is given of applications of the technique developed to some problems based on the theory of elliptic functions.
Abstract: The main ideas of global “finite-zone integration” are presented, and a detailed analysis is given of applications of the technique developed to some problems based on the theory of elliptic functions. In the work the Peierls model is integrated as an important application of the algebrogeometric spectral theory of difference operators.
79 citations
TL;DR: A survey of the spectral properties of matrix finite-zone operators can be found in this paper, where conditions of the type of J-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained.
Abstract: A survey is given of the spectral properties of matrix finite-zone operators. Conditions of the type of J-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained. The simplest examples of such J-self-adjoint, finite-zone operators turn out to be connected with the theory of ovals of plane, real, algebraic curves.
75 citations
TL;DR: In this paper, a new variant of the theory of periodic approximations of dynamical systems and C*-algebras is presented, namely the construction for each automorphism of the Lebesgue space of a Markov tower (or adic model) of periodic automorphisms.
Abstract: One presents a new variant of the theory of periodic approximations of dynamical systems and C*-algebras, namely the construction for each automorphism of the Lebesgue space of a Markov tower (or adic model) of periodic automorphisms. One gives several examples.
71 citations
TL;DR: In this paper, the completeness of the multiplet system constructed from the Bethe vectors is proved for the Heisenberg model of arbitrary spin and for the generalized Kondo model.
Abstract: The completeness of the multiplet system constructed from the Bethe vectors is proved for the Heisenberg model of arbitrary spin and for the generalized Kondo model.
67 citations
TL;DR: In this paper, a class of problems connected with the description of the motion of an attracted quantum particle in possibly time-dependent, periodic, external fields is studied on the basis of a development of the method of the inverse problem.
Abstract: A class of problems connected with the description of the motion of an attracted quantum particle in possibly time-dependent, periodic, external fields is studied on the basis of a development of the method of the inverse problem.
TL;DR: A survey of single-weighted and doubleweighted estimates of strong and weak types for the Hardy-Littlewood maximal function, Riesz potentials, singular integral operators, and harmonic functions can be found in this article.
Abstract: We give a survey of research on the problem of single-weighted and double-weighted estimates of strong and weak types for the Hardy-Littlewood maximal function, Riesz potentials, singular integral operators, and harmonic functions. Necessary and sufficient conditions on the weight are given under which weighted estimates are valid (Muckenhoupt's Ap-condition, Sawyer's condition, etc.). Special attention is given to papers which appeared after 1980 and the latest results, published as reports and preprints.
TL;DR: In this article, it was proved that the problemdet(uxx)=f(x,u,ux)⩾Ν>0, is solvable in spaces c^{\kappa + 2 + \propto } (\bar \Omega ),\kappa \geqslant 2, 0< \proplo< 1\).
Abstract: It is proved that the problemdet(uxx)=f(x,u,ux)⩾Ν>0, is solvable in spaces\(C^{\kappa + 2 + \propto } (\bar \Omega ),\kappa \geqslant 2, 0< \propto< 1\), provided a natural connection between the curvature of the closed surface∂Ω and the growth of the functionf(x,u,p) in¦p¦ is valid.
TL;DR: In this paper, the authors present a survey of the study of multidimensional web and the geometric and algebraic structures connected with them, which is a continuation of the first part of the survey by V. D. Belousov and V. V. Ryzhkov, published in 1972.
Abstract: We consider papers devoted to the study of multidimensional webs and also the geometric and algebraic structures connected with them. The survey is a continuation of the first part of the survey by V. D. Belousov and V. V. Ryzhkov “The Geometry of Webs,” published in 1972.
TL;DR: In this paper, the finite dimensionality of a bounded set M of a Hilbert space H, negatively invariant relative to a transformation V, has been shown for two-dimensional Navier-Stokes equations.
Abstract: One proves the finite-dimensionality of a bounded set M of a Hilbert space H, negatively invariant relative to a transformation V, possessing the following properties: For any points υ and
$$\tilde \upsilon$$
of the set M one has
$$\left\| {V(\upsilon ) - V(\tilde \upsilon )} \right\| \leqslant \ell \left\| {\upsilon - \tilde \upsilon } \right\|$$
, while
$$\left\| {Q_n V(\upsilon ) - Q_n V(\tilde \upsilon )} \right\| \leqslant \delta \left\| {\upsilon - \tilde \upsilon } \right\|, \delta< 1$$
,δ<1 where Qn is the orthoprojection onto a subspace of codimension n. With the aid of this result and of the results found in O. A. Ladyzhenskaya's paper “On the dynamical system generated by the Navier-stokes equations” (J. Sov. Math.,3, No. 4 (1975)) one establishes the finite-dimensionality of the complete attractor for two-dimensional Navier-Stokes equations. The same holds for many other dissipative problems.
TL;DR: A systematic survey of the theory of linear evolution equations in Banach spaces, reviewed in the period 1968-1982 in Ref Zh Matematika, is presented in this paper, with a focus on linear evolution.
Abstract: A systematic survey of the theory of linear evolution equations in Banach spaces, reviewed in the period 1968–1982 in Ref Zh Matematika, is presented
TL;DR: The convolution is defined as the sum of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian as discussed by the authors.
Abstract: The convolution is defined as the sum
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where
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for n≠0
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and W0,W1 are arbitrary smooth functions Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function
TL;DR: In this article, the authors proved the global unique solvability of the initial-boundary-value problem for the quasilinear system in the case of the elastic-viscous Kelvin-Voigt fluids.
Abstract: One proves the global unique solvability in class\(W_\infty ^1 (0,T;C^{2,d} (\bar \Omega ) \cap H(\Omega ))\) of the initial-boundary-value problem for the quasilinear system
$$\frac{{\partial \vec \upsilon }}{{\partial t}} + \upsilon _k \frac{{\partial \vec \upsilon }}{{\partial x_k }} - \mu _1 \frac{{\partial \Delta \vec \upsilon }}{{\partial t}} - \int\limits_0^t {K(t - \tau )\Delta \vec \upsilon (\tau )d\tau + grad p = \vec f,di\upsilon \bar \upsilon = 0,\upsilon , > 0.}$$
This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation
$$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$
TL;DR: In this paper, the mathematical structures at the basis of supersymmetry were described based on the theory of supertwistors, where the symmetry groups mix bosons and fermions.
Abstract: The work is devoted to a description of the mathematical structures at the basis of supersymmetry — field theory in which the symmetry groups mix bosons and fermions. The approach developed is based on the theory of supertwistors.
TL;DR: A short survey is given of some directions in probability theory that have developed most intensively in recent years.
Abstract: A short survey is given of some directions in probability theory that have developed most intensively in recent years. Separable statistics and criteria for the verification of statistical hypotheses based on them, various schemes for distributing particles among cells, and problems connected with estimating the unknown size of a finite collection are considered.
TL;DR: In this article, a description of Hamiltonian structure and integrals of motion for nonlinear equations with two space variables is given, in a general group-theoretic setting.
Abstract: A description of Hamiltonian structure and of the integrals of motion is given, in a general group-theoretic setting, for nonlinear equations with two space variables. The connections with a nonlocal Riemann problem, the multitime formalism, and the Hamiltonian structure of stationary problems are discussed.
TL;DR: In this article, a survey of algebraic-geometric methods for computing lower bounds on multiplicative complexity of polynomials is presented. But it is not a complete survey.
Abstract: The present article is a survey of selected methods for obtaining lower bounds in algebraic complexity. We present the contents. Introduction. i. Basic concepts. Chapter I. Algebraic-geometric approach to obtaining lower bounds of computational complexity of polynomials. 2. Evaluating a polynomial with "general" coefficients. 3. Computational complexity of individual polynomials. 4. The degree method and its generalizations (the case of an infinite ground field). 5. The degree method (the case of a finite ground field). 6. Additive complexity and real roots. Chapter II. Lower bounds on multiplicative complexity for problems of linear algebra. 7. Multiplicative complexity and rank. 8. Rank of a pair of bilinear forms. 9. Multiplicative complexity of a bilinear form over a commutative ring. i0. Bounds on the rank of algebras, ii. Linearized multiplicative complexity. Chapter III. Complexity for straight-line programs of nonstandard types. 12. Irrational computational complexity of algebraic functions. 13. Monotone programs. 14. Time-space tradeoffs. 15. Graph-theoretic methods in algebraic complexity. 16. Additive complexity in triangular and directed computations and Bruhat decomposition.
TL;DR: Automorphisms of unitary linear groups over the ring Λ are found to be standard on an elementary unitary subgroup in the case when the hyperbolic rank of the form q is strictly greater than one and n⩾5 for a commutative ring.
Abstract: Automorphisms of unitary linear groups u (n,Λ,q) over the ring Λ are found to be standard on an elementary unitary subgroup in the case when the hyperbolic rank of the form q is strictly greater than one and n⩾5 (for a commutative ring, n⩾4).
TL;DR: In this paper, a nonlocal Riemann problem is used to construct multidimensional nonlinear integrable systems and their solutions are obtained by means of a non-local RCP.
Abstract: A new method for constructing multidimensional nonlinear integrable systems and their solutions by means of a nonlocal Riemann problem is presented. This is the natural generalization of the method of the local Riemann problem to the case of several space variables and includes the well-known Zakharov-Shabat method of dressing by Volterra operators.
TL;DR: In this paper, a systematic study of the distributions of functionals of stochastic processes by the fibering method and a survey of results obtained in this direction in recent years are presented.
Abstract: The paper is devoted to a systematic study of the distributions of functionals of stochastic processes by the fibering method and to a survey of results obtained in this direction in recent years. Principal attention is given to distinguishing conditions ensuring: a) absolute continuity; b) the existence of a bounded density; c) applicability of the local limit theorem for the distributions of functionals. Smooth, convex functionals and functionals of integral type are considered in detail.
TL;DR: In this paper, the authors construct strongly elliptic second-order differential equations in the divergence form with measurable bounded complex coefficients in Rn, n⩾3, whose generalized solutions are not bounded in any neighborhood of the origin.
Abstract: One constructs examples of strongly elliptic second-order differential equations in the divergence form with measurable bounded complex coefficients in Rn, n⩾3, whose generalized solutions are not bounded in any neighborhood of the origin.
TL;DR: In this article, a spectral theory for a special class of operators in a Banach space over a non-Archimedean field is constructed, and a functional calculus and perturbation theory are constructed.
Abstract: A spectral theory is constructed for a special class of operators in a Banach space over a non-Archimedean field. A spectral theorem is proved, and a functional calculus and perturbation theory are constructed.
TL;DR: In this article, a relation between Hamiltonian structures on polynomial bundles of various degrees is established using the relation between the Legendre-Ostrogradskii transformation and the Kirillov form on the corresponding orbit.
Abstract: A relation between Hamiltonian structures on polynomial bundles of various degrees is established Using this relation it is shown that the symplectic form on the space of stationary solutions, previously defined in terms of the Legendre-Ostrogradskii transformation, is identical to the Kirillov form on the corresponding orbit
TL;DR: The Grothendieck group K 0(δ∞) of the group of finite permutations of a countable set is described in this paper, where all semidefinite characters of δ ∞ are described.
Abstract: The Grothendieck group K0(δ∞) of the group of finite permutations of a countable set is described. All semifinite characters of δ∞ are described and with their help the cone of representations K
+
0
(δ∞) is characterized.
TL;DR: In this article, the structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined.
Abstract: The structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined. Applications to Einstein's theory of gravitation are given.