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• Open/close Integrable maps

  A P Veselov 1991 Russ. Math. Surv. 46 1 Tag this article

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  CONTENTS
  Introduction
  Chapter I. Integrable Lagrange systems with discrete time
  § 1. Hamiltonian theory of Lagrange systems with discrete time
  § 2. Discrete analogues of classical integrable systems and factorization of matrix polynomials
  § 3. Spectral theory of difference operators and discrete systems
  Chapter II. Integrability in dynamics of general algebraic maps
  § 1. Commuting polynomial maps and simple Lie groups
  § 2. Integrable polynomial automorphisms of the plane
  § 3. Algebraic relations and the Yang-Baxter equation. The growth of the number of images of iterations of multivalued maps
  References

• Open/close Padé-Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the -property of stationary compact sets

  Andrei A Gonchar et al 2011 Russ. Math. Surv. 66 1015 Tag this article

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  Padé-Chebyshev approximants are considered for multivalued analytic functions that are real-valued on the unit interval . The focus is mainly on non-linear Padé-Chebyshev approximants. For such rational approximations an analogue is found of Stahl's theorem on convergence in capacity of the Padé approximants in the maximal domain of holomorphy of the given function. The rate of convergence is characterized in terms of the stationary compact set for the mixed equilibrium problem of Green-logarithmic potentials.

  Bibliography: 79 titles.

• Open/close Unstable invariant sets of semigroups of non-linear operators and their perturbations

  A V Babin and M I Vishik 1986 Russ. Math. Surv. 41 1 Tag this article

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  CONTENTS
  Introduction
  § 1. Attractors of semigroups in a normed space
  § 2. Examples of non-stationary differential equations having attractors
  § 3. An invariant manifold
  § 4. Examples of equations having a regular attractor
  § 5. Semigroups of operators depending on a parameter
  § 6. The existence of a Lyapunov function for a perturbed semigroup
  § 7. The behaviour of local unstable manifolds near a critical point
  References

• Open/close Trajectory attractors of equations of mathematical physics

  Marko I Vishik and Vladimir V Chepyzhov 2011 Russ. Math. Surv. 66 637 Tag this article

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  In this survey the method of trajectory dynamical systems and trajectory attractors is described, and is applied in the study of the limiting asymptotic behaviour of solutions of non-linear evolution equations. This method is especially useful in the study of dissipative equations of mathematical physics for which the corresponding Cauchy initial-value problem has a global (weak) solution with respect to the time but the uniqueness of this solution either has not been established or does not hold. An important example of such an equation is the 3D Navier-Stokes system in a bounded domain. In such a situation one cannot use directly the classical scheme of construction of a dynamical system in the phase space of initial conditions of the Cauchy problem of a given equation and find a global attractor of this dynamical system. Nevertheless, for such equations it is possible to construct a trajectory dynamical system and investigate a trajectory attractor of the corresponding translation semigroup. This universal method is applied for various types of equations arising in mathematical physics: for general dissipative reaction-diffusion systems, for the 3D Navier-Stokes system, for dissipative wave equations, for non-linear elliptic equations in cylindrical domains, and for other equations and systems. Special attention is given to using the method of trajectory attractors in approximation and perturbation problems arising in complicated models of mathematical physics.

  Bibliography: 96 titles.

• Open/close New integral representations of Whittaker functions for classical Lie groups

  A A Gerasimov et al 2012 Russ. Math. Surv. 67 1 Tag this article

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  The present paper proposes new integral representations of -Whittaker functions corresponding to an arbitrary semisimple Lie algebra with the integrand expressed in terms of matrix elements of the fundamental representations of . For the classical Lie algebras , , and a modification of this construction is proposed, providing a direct generalization of the integral representation of -Whittaker functions first introduced by Givental. The Givental representation has a recursive structure with respect to the rank of the Lie algebra , and the proposed generalization to all classical Lie algebras retains this property. It was observed elsewhere that an integral recursion operator for the -Whittaker function in the Givental representation coincides with a degeneration of the Baxter -operator for -Toda chains. In this paper -operators for the affine Lie algebras , and a twisted form of are constructed. It is then demonstrated that the relation between integral recursion operators for the generalized Givental representations and degenerate -operators remains valid for all classical Lie algebras.

  Bibliography: 33 titles.

• Open/close Topology of quasi-periodic functions on the plane

  I A Dynnikov and S P Novikov 2005 Russ. Math. Surv. 60 1 Tag this article

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  In this paper the topological theory of quasi-periodic functions on the plane is presented. The development of this theory was started (in another terminology) by the Moscow topology group in the early 1980s, motivated by needs of solid state physics which led to the necessity of investigating a special (non-generic) case of Hamiltonian foliations on Fermi surfaces with a multivalued Hamiltonian function [1]. These foliations turned out to have unexpected topological properties, discovered in the 1980s ([2], [3]) and 1990s ([4]-[6]), which led finally to non-trivial physical conclusions ([7], [8]) by considering the so-called geometric strong magnetic field limit [9]. A reformulation of the problem in terms of quasi-periodic functions and an extension to higher dimensions in 1999 [10] produced a new and fruitful approach. One can say that for monocrystalline normal metals in a magnetic field the semiclassical trajectories of electrons in the quasi-momentum space are exactly the level curves of a quasi-periodic function with three quasi-periods which is the restriction of the dispersion relation to the plane orthogonal to the magnetic field. The general study of topological properties of level curves for quasi-periodic functions on the plane with arbitrarily many quasi-periods began in 1999 when some new ideas were formulated in the case of four quasi-periods [10]. The last section of this paper contains a complete proof of these results based on the technique developed in [11] and [12]. Some new physical applications of the general problem were found recently [13].

• Open/close About the impact factor

  S M Gusein-Zade 2004 Russ. Math. Surv. 59 1005 Tag this article

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• Open/close Lévy-based spatial-temporal modelling, with applications to turbulence

  O E Barndorff-Nielsen and J Schmiegel 2004 Russ. Math. Surv. 59 65 Tag this article

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  This paper involves certain types of spatial-temporal models constructed from Lévy bases. The dynamics is described by a field of stochastic processes , on a set of sites  , defined as integrals

  

  where denotes a Lévy basis. The integrands are deterministic functions of the form , where has a special form and is a subset of . The first topic is OU (Ornstein-Uhlenbeck) fields , which represent certain extensions of the concept of OU processes (processes of Ornstein-Uhlenbeck type); the focus here is mainly on the potential of for dynamic modelling. Applications to dynamical spatial processes of Cox type are briefly indicated. The second part of the paper discusses modelling of spatial-temporal correlations of SI (stochastic intermittency) fields of the form

  

  This form is useful when explicitly computing expectations of the form

  

  which are used to characterize correlations. The SI fields can be viewed as a dynamical, continuous, and homogeneous generalization of turbulent cascades. In this connection an SI field is constructed with spatial-temporal scaling behaviour that agrees with the energy dissipation observed in turbulent flows. Some parallels of this construction are also briefly sketched.

• Open/close First steps in symplectic topology

  V I Arnol'd 1986 Russ. Math. Surv. 41 1 Tag this article

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  CONTENTS
  Introduction
  § 1. Is there such a thing as symplectic topology?
  § 2. Generalizations of the geometric theorem of Poincaré
  § 3. Hyperbolic Morse theory
  § 4. Intersections of Lagrangian manifolds
  § 5. Legendrian submanifolds of contact manifolds
  § 6. Lagrangian and Legendrian knots
  § 7. Two theorems of Givental' on Lagrangian embeddings
  § 8. Odd-dimensional analogues
  § 9. Optical Lagrangian manifolds
  References

• Open/close CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY

  Ya B Pesin 1977 Russ. Math. Surv. 32 55 Tag this article

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  CONTENTS
  Part I
  § 1. Introduction
  § 2. Prerequisites from ergodic theory
  § 3. Basic properties of the characteristic exponents of dynamical systems
  § 4. Properties of local stable manifolds
  Part II
  § 5. The entropy of smooth dynamical systems
  § 6. "Measurable foliations". Description of the π-partition
  § 7. Ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure. The K-property
  § 8. The Bernoullian property
  § 9. Flows
  § 10. Geodesic flows on closed Riemannian manifolds without focal points
  References