Showing papers in "Russian Mathematical Surveys in 2004"

Asymptotic behaviour and expansions of solutions of an ordinary differential equation

Abstract: An ordinary differential equation of quite general form is considered. It is shown how to find the following near a finite or infinite value of the independent variable by using algorithms of power geometry: (i) all power-law asymptotic expressions for solutions of the equation; (ii) all power-logarithmic expansions of solutions with power-law asymptotics; (iii) all non-power-law (exponential or logarithmic) asymptotic expressions for solutions of the equation; (iv) certain exponentially small additional terms for a power-logarithmic expansion of a solution that correspond to exponentially close solutions. Along with the theory and algorithms, examples are presented of calculations of the above objects for one and the same equation. The main attention is paid to explanations of algorithms for these calculations.

Logarithmic equivalence of Welschinger and Gromov-Witten invariants

Abstract: The Welschinger numbers, a kind of a real analogue of the Gromov-Witten numbers that count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any generic collection of real points. Logarithmic equivalence of sequences is understood to mean the asymptotic equivalence of their logarithms. Such an equivalence is proved for the Welschinger and Gromov-Witten numbers of any toric Del Pezzo surface with its tautological real structure, in particular, of the projective plane, under the hypothesis that all, or almost all, the chosen points are real. A study is also made of the positivity of Welschinger numbers and their monotonicity with respect to the number of imaginary points.

Random Metric Spaces and Universality

Abstract: The notion of random metric space is defined, and it is proved that such a space is isometric to the Urysohn universal metric space with probability one. The main technique is the study of universal and random distance matrices; properties of metric (in particular, universal) spaces are related to properties of distance matrices. Examples of other categories in which randomness and universality coincide (graphs, and so on) are given.

Superdiffusions and positive solutions of non-linear partial differential equations

Abstract: By using super-Brownian motion, all positive solutions of the non-linear differential equation with in a bounded smooth domain are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case was treated by B. Mselati in 2002.

Lévy-based spatial-temporal modelling, with applications to turbulence

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

Cellular cochain algebras and torus actions

Abstract: . Cellular cochains do notadmit a functorial associative multiplication because a proper cellular diag-onal approximation does not exist in general. The construction of moment-angle complexes is a functor from the category of simplicial complexes to thecategory of spaces with torus action. We show that in this special case theproposed cellular approximation of the diagonal is associative and functorialwith respect to those maps of moment-angle complexes which are inducedby simplicial maps.The face ring of a complex K on the vertex set [m] = {1,...,m} isthe graded quotient ring Z[K] = Z[v

Kolmogorov and Gödel's approach to intuitionistic logic: current developments

Abstract: Intuitionistic mathematics was created by Brouwer on the basis of constructive reasoning, where the existence of a proof was the criterion for truth. Kolmogorov and Godel proposed interpreting intuitionistic logic on the basis of classical notions of a problem's solution and of provability. In 1933 Godel made the first substantial step toward the building of such an interpretation. Despite much progress in the understanding of intuitionism, this task was not complete before the author's 1995 paper. This survey will cover the results of the past decade obtained within this framework.

Lectures on mirror symmetry, derived categories, and D-branes

Abstract: This paper, mainly intended for a mathematical audience, is an introduction to homological mirror symmetry, derived categories, and topological D-branes. Mirror symmetry from the point of view of physics is explained, along with the relationship between symmetry and derived categories, and the reason why the Fukaya category must be extended by using co-isotropic A-branes. There is also a discussion of how to extend the definition of the Floer homology to these objects and a description of mirror symmetry for flat tori. The paper consists of four lectures given at the Institute of Pure and Applied Mathematics (Los Angeles) in March 2003, as a part of the programme “Symplectic Geometry and Physics”.

On Hilbert's thirteenth problem and related questions

Abstract: Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.

Some problems of the qualitative Sturm-Liouville theory on a spatial network

Abstract: An analogue of the Sturm oscillation theory of the distribution of the zeros of eigenfunctions is constructed for the problem (*)on a spatial network (in other terms, is a metric graph, a CW complex, a stratified locally one-dimensional manifold, a branching space, a quantum graph, and so on), where is the family of boundary vertices of . At interior points of the edges of the quasi-derivative has the classical form , and at interior nodes it is assumed that where the summation is taken over the edges incident to the node and, for an edge , stands for the `endpoint' derivative of the restriction of the function to . Despite the branching argument, which is a kind of intermediate type between the one-dimensional and multidimensional cases, the outward form of the results turns out to be quite classical. The classical nature of the operator is clarified, and exact analogues of the maximum principle and of the Sturm theorem on alternation of zeros are established, together with the sign-regular oscillation properties of the spectrum of the problem (*) (including the simplicity and positivity of the points of the spectrum and also the number of zeros and their alternation for the eigenfunctions).

Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras

Abstract: In this paper a global operator approach to the Wess-Zumino-Witten-Novikov theory for compact Riemann surfaces of arbitrary genus with marked points is developed. The term `global' here means that Krichever-Novikov algebras of gauge and conformal symmetries (that is, algebras of global symmetries) are used instead of loop algebras and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous paper of the authors (Russ. Math. Surveys 54:1 (1999)). The present paper gives a construction of the conformal blocks and of a projectively flat connection on the bundle formed by them.

Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs

Abstract: This is a survey of recent investigations of quasi-periodic localization on lattices (of both methods based on perturbation theory and non-perturbative methods) and of applications of KAM theories in connection with infinite-dimensional Hamiltonian systems. The focus is on applications of these investigations to the Schrodinger equation and the wave equation with periodic boundary conditions, and to non-linear random Schrodinger equations with short-range potentials.

Rings of continuous functions, symmetric products, and Frobenius algebras

Abstract: A constructive proof is given for the classical theorem of Gel'fand and Kolmogorov (1939) characterising the image of the evaluation map from a compact Hausdorff space into the linear space dual to the ring of continuous functions on . Our approach to the proof enabled us to obtain a more general result characterising the image of the evaluation map from the symmetric products into . A similar result holds if and leads to explicit equations for symmetric products of affine algebraic varieties as algebraic subvarieties in the linear space dual to the polynomial ring. This leads to a better understanding of the algebra of multisymmetric polynomials. The proof of all these results is based on a formula used by Frobenius in 1896 in defining higher characters of finite groups. This formula had no further applications for a long time; however, it has appeared in several independent contexts during the last fifteen years. It was used by A. Wiles and R.L. Taylor in studying representations and by H.-J. Hoehnke and K.W. Johnson and later by J. McKay in studying finite groups. It plays an important role in our work concerning multivalued groups. Several properties of this remarkable formula are described. It is also used to prove a theorem on the structure constants of Frobenius algebras, which have recently attracted attention due to constructions taken from topological field theory and singularity theory. This theorem develops a result of Hoehnke published in 1958. As a corollary, a direct self-contained proof is obtained for the fact that the 1-, 2-, and 3-characters of the regular representation determine a finite group up to isomorphism. This result was first published by Hoehnke and Johnson in 1992.

Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants

Abstract: This survey is devoted to investigations concerning topological, algebraic, and combinatorial characteristics as well as metric invariants for arbitrary groups of homeomorphisms of the line and the circle. Relationships between these characteristics are established, the most important metric invariants are studied (in the form of invariant, projectively invariant, and ω-projectively invariant measures), and the main 'obstructions' to the existence of metric invariants of this kind are described.

Hypergeometric functions over an arbitrary field

Abstract: This paper is devoted to the definition and investigation of hypergeometric functions over an arbitrary continuous locally compact field or a finite field, based on the author's earlier definition of generalized hypergeometric function over the field .

Passive scalar equation in a turbulent incompressible Gaussian velocity field

Abstract: The time evolution of a passive scalar in a turbulent homogeneous incompressible Gaussian flow is considered. The turbulent nature of the flow results in non-smooth coefficients of the corresponding evolution equation. A strong solution (in the probabilistic sense) of the equation is constructed by using the Wiener Chaos expansion, and properties of the solution are studied. In particular, a certain -regularity of the solution and a representation formula of Feynman-Kac type (or a Lagrangian formula) are among the results obtained. The results can be applied to both viscous and conservative flows.

Some physical models of the reaction-diffusion equation, and coupled map lattices

Abstract: A number of models are surveyed which appear in physics, biology, chemistry, and other areas and which are described by a reaction-diffusion equation. The corresponding coupled map lattice (CML) system is obtained by discretizing this equation. These CMLs are classified by the type of the dynamics of the local map. Several different types of behavior are observed: Morse-Smale type systems, systems with attractors, and systems with Smale horseshoes.

Mathematical models of turbulent motion of an incompressible viscous fluid

Abstract: The paper was apparently written at the end of the 1950s. The text is in its final form, but the list of references was not worked out in detail. Ya.G.?Sinai added some comments to the text and included in the list of references some items related to the content of the paper.

Partial observation control in an anticipating environment

Abstract: A study is made of a controlled stochastic system whose state at time is described by a stochastic differential equation driven by Levy processes with filtration . The system is assumed to be anticipating, in the sense that the coefficients are assumed to be adapted to a filtration with for all . The corresponding anticipating stochastic differential equation is interpreted in the sense of forward integrals, which naturally generalize semimartingale integrals. The admissible controls are assumed to be adapted to a filtration such that for all . The general problem is to maximize a given performance functional of this system over all admissible controls. This is a partial observation stochastic control problem in an anticipating environment. Examples of applications include stochastic volatity models in finance, insider influenced financial markets, and stochastic control of systems with delayed noise effects. Some particular cases in finance, involving optimal portfolios with logarithmic utility, are solved explicitly.

Functional identities in rings and their applications

Abstract: In the present survey the main results of the theory of functional identities are presented and an analysis of the current state of this theory is given. Applications obtained in this direction include the solution of all the Herstein problems concerning maps of Lie type, the description of Lie-admissible multiplications, and characterizations of commutativity-preserving maps and of maps preserving normal elements.

Everywhere divergent Fourier series with respect to the Walsh system and with respect to multiplicative systems

Abstract: In this paper a new construction of everywhere divergent Fourier-Walsh series is presented. This construction enables one to halve the gap in the Lebesgue-Orlicz classes between the Schipp-Moon lower bound established by using Kolmogorov's construction and the Sjolin upper bound obtained by using Carleson's method. Fourier series which are everywhere divergent after a rearrangement are constructed with respect to the Walsh system (and to more general systems of characters) with the best lower bound for the Weyl factor. Some results related to an upper bound of the majorant for partial sums of series with respect to rearranged multiplicative systems are established. The results thus obtained show certain merits of harmonic analysis on the dyadic group in clarifying and overcoming fundamental difficulties in the solution of the main problems of Fourier analysis.

Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem

Abstract: A new approach to the construction of the analytic theory of difference equations with rational and elliptic coefficients is proposed, based on the construction of canonical meromorphic solutions which are analytic along "thick" paths. The concept of these solutions leads to the definition of local monodromies of difference equations. It is shown that, in the continuous limit, these local monodromies converge to monodromy matrices of differential equations. In the elliptic case a new type of isomonodromy transformations changing the periods of elliptic curves is constructed.

On solvability and unsolvability of equations in explicit form

Abstract: In this survey the classical results of Abel, Liouville, Galois, Picard, Vessiot, Kolchin, and others on the solvability and unsolvability of equations in explicit form are discussed. The one-dimensional topological version of Galois theory is presented in detail (this version describes topological obstructions to the representability of functions by quadratures).

Hausdorff distance and image processing

Abstract: Mathematical methods for image processing make use of function spaces which are usually Banach spaces with integral norms. The corresponding mathematical models of the images are functions in these spaces. There are discussions here involving the value of for which the distance between two functions is most natural when they represent images, or the metric in which our eyes measure the distance between the images. In this paper we argue that the Hausdorff distance is more natural to measure the distance (difference) between images than any norm.

Rigidity for circle diffeomorphisms with singularities

Abstract: This paper reviews recent results related to rigidity theory for circle diffeomorphisms with singularities. Both diffeomorphisms with a break point (sometimes called a `fracture-type singularity' or `weak discontinuity') and critical circle maps are discussed. In the case of breaks, results are presented on the global hyperbolicity of the renormalization operator; this property implies the existence of an attractor of the Smale horseshoe type. It is also shown that for maps with singularities rigidity is stronger than for diffeomorphisms, in the sense that rigidity is not violated for non-generic rotation numbers, which are abnormally well approximable by rationals. In the case of critical rotations of the circle it is proved that any two such rotations with the same order of the singular point and the same irrational rotation number are -smoothly conjugate.

Splitting of a multiple eigenvalue in a problem on concentrated masses

Abstract: . (9)Forthecaseofasimpleeigenvaluesee[5]. Wealsomentionthattheadditionalconditionoforthogonalityarosein[6]foraproblemwithafrequentalternationinthetypeofboundarycondition.Bibliography[1]G.A.Chechkin,Boundaryvalueproblemsfornon-classicalpartialdifferentialequations,Inst.Mat.Sibirsk.Otdel.Akad.NaukSSSR,Novosibirsk1989,pp.197–200;Englishtransl.,Mathematicalproblemsinelasticityandhomogenization,North-Holland,Amsterdam1992.[2]O.A.Oleinik,G.A.Iosif’yan,andA.S.Shamaev,Mathematicalproblemsinthetheoryofstronglyinhomogeneouselasticmedia,MoscowStateUniv.Press,Moscow1990.(Russian)[3]A.M.Il’in,Matchingofasymptoticexpansionsofsolutionsofboundaryvalueproblems,Nauka,Moscow1989;Englishtransl.,Amer.Math.Soc.,Providence,RI1992.[4]R.R.Gadyl’shin,AlgebraiAnaliz10 (1998),3–19;Englishtransl.,St.PetersburgMath.J.10 (1999),1–14.[5]G.AChechkin,“Onvibrationofapartiallyfastenedmembranewithmany‘light’concentratedmassesontheboundary”,C.R.Acad.Sci.ParisS´er.IIb332 (2004)(toappear).[6]D.I.Borisov,Izv.Ross.Akad.NaukSer.Mat.67:6(2003),23–70;Englishtransl.,Izv.Math.67 (2003),1101–1148.MoscowStateUniversityE-mail:chechkin@mech.math.msu.ru Received24/MAY/04[6] D. I. Borisov, Izv. Ross. Akad. Nauk Ser. Mat. 67:6 (2003), 23 70; English transl., Izv.Math. 67 (2003), 1101