Reviews in Mathematics and Mathematical Physics – Cambridge Scientific Publishers Ltd

Quantization Conditions on Riemannian Surfaces and Spectral Properties of Non-Selfadjoint Differential Operators

Janielw — Sat, 16 Oct 2021 19:20:20 +0000

It is well known that, for self-adjoint operators, asymptotic properties of spectra are deeply connected with real Lagrangian geometry and theory of Hamiltonian systems. For example, semi-classical eigenvalues can be computed from Maslov quantization conditions on Lagrangian manifolds; these manifolds have to be invariant with respect to Hamiltonian systems, defined by symbols of initial operators. For non-selfadjoint operators, the corresponding theory is not well developed. We describe known results in this direction (including quite recent ones); the main idea of the new theory is to replace the real geometry by the complex one and to describe spectral characteristics of operators via geometrical properties of complex manifolds. We study this correspondence for a number of operators, which are popular in mathematical physics, and discuss physical applications.

Differential-Topological Theory of Webs and its Applications

Janielw — Sat, 16 Oct 2021 19:10:22 +0000

The most studied are three-webs and their special classes and characteristics. The three-web formed by foliations of condimension p, q, r on a manifold of dimension P+Q is denoted by W (p, q, r). In case p= q= r, the closure of sufficiently small configurations of a certain type on the web W (r, r, r) corresponds to some identity in the coordinate quasigroup f. It was W. Blaschke and his colleagues K. Reidemeister, G. Thomsen, G. Bol, and others, who began the study of the differential-topological theory of webs in 1920s-1930s. The study of multidimensional three-webs continued with the work of G. Bol (1935-1936), S.S. Chern (1936), M. Akivis (since 1969), V. V. Goldberg (since 1973) and others. In this review the author also describes the configurations that arise at the boundary of a curved three-web and fractal structures that naturally arise on a curvilinear three-web

Spectral Expansion of the Transfer Matrices of Gibbs Fields (SECOND EDITION)

Janielw — Sat, 16 Oct 2021 19:00:09 +0000

Second Edition - R.A. Minlos, Institute for Information Transmission Problems, Moscow This survey presents investigations of the structures of the spectrum of transfer matrices (stochastic operators) of lattice Gibbs fields and considers cluster expansion of the transfer matrix, invariant cluster R-particle subspaces of the transfer matrix and cluster operators in pre-representation. 2019 Pbk ISBN: 978-1-904868-99-6 70pp

Parity and Patterns in Low Dimensional Topology

Janielw — Sat, 16 Oct 2021 18:36:21 +0000

The parity theory initiated in 2009 by the second named author (V.O. Manturov) argues that if there is a smart way to distinguish between even and odd nodes (crossings, letters) in a way consistent with moves then this allows one to construct functional mappings between objects of the theory, construct various powerful invariants, reduce problems about objects (say knots) to problems about their diagrams, refine many existing invariants. Over the last four years, parity theory has experienced a rapid growth: investigations were undertaken by dozens of scientists worldwide. Various problems in low-dimensional topology were solved by using parity

Intermittency, Diffusion and Generation in a Nonstationary Random Medium

Janielw — Thu, 14 Oct 2021 00:11:33 +0000

Integration over random paths is used, along with asymptomatic behaviour of the product of a large number of independent identically distributed random matrices. The most interesting effect is the appearance of concentrated structures (intermittency) of a smooth initial distribution of the transported quantity. The occurrence of intermittent distributions in the linear problem is due to the fact that the coefficients of the transport equation are stochastic. The intermittency shows itself in the rates of exponential growth of the successive moments (Lyapunov exponents) as the moment number increases. Moment equations are obtained for the scalar and vector, and are used to study temperature evolution and magnetic-field generation in a random fluid flow. These equations are differential in a medium with short time correlations and integral in the general case. The range of application of the diffusion description is analysed. The behaviour of the diffusion coefficients in the case of time reversal is examined. The properties of an individual realization of a scalar and vector are also explained, and a dynamo theorem is given on the exponential growth of the magnetic field in a random flow with renewal

Introduction to the Theory of Representations of Finitely Presented *-Algebras

Janielw — Wed, 13 Oct 2021 23:47:55 +0000

The theory is illustrated with numerous examples of *-algebras. The examples, in particular, include *-algebras with two self-adjoint generators that satisfy a quadratic or a more general relation, *-algebras with three and four generators, *-algebras that arise from one and many-dimensional discrete dynamical systems, Wick *-algebras, various *-wild algebras. This review is intended for graduate students and researchers who specialize in this area of mathematics

Perturbation Theory in Periodic Problems for Two-Dimensional Integrable Systems (SECOND EDITION)

Janielw — Wed, 13 Oct 2021 23:39:27 +0000

The contents of the review include:

Perturbation Theory of Finite Zone Solutions of Evolution Lax-type Equations
Spectral Theory of Non-Stationary Schrodinger Operators
Periodic Problem for Kadomtsev-Petviashvilli-Type Equations
Spectral Theory of Two-Dimensional Periodic Schrodinger Operators

This review was first published in 1992 and in the course of the last twenty years, these ideas, concepts and methods of the theory of integrable systems have been developed and applied to a wide range of contemporary problems of geometry and topology

Singularities of Functions, Wave Fronts, Caustics, and Multidimensional Integrals

Janielw — Wed, 13 Oct 2021 23:31:20 +0000

The authors discuss the basic ideas, concepts and methods of the theory of singularities and the survey is presented in three sections:

Section 1: Singularities of Functions, Caustics and Wave Fronts
Section 2: Integrals of the Stationary Phase Method
Section 3: The Geometry of Formulas

The survey provides a useful source of reference for students, postgraduates and researchers in these areas of mathematics

Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations (SECOND EDITION)

Janielw — Wed, 13 Oct 2021 23:22:17 +0000

This review is devoted to the differential-geometric theory of homogenous forms and other different homogenous structures (mainly, Poisson and symplectic structures) on loop spaces of smooth manifolds, their natural generalizations and applications in mathematical physics and field theory

Multidimensional Monge-Ampere Equation

Janielw — Wed, 13 Oct 2021 23:16:29 +0000

A. V. Pogorelov, Institute of Low Temperature Physics and Engineering, Kharkov, Ukraine This review presents a detailed exposition of the results concerning the existence and uniqueness of the solutions of the general Monge-Ampere multidimensional equations of elliptic type 2009 Pbk ISBN: 978-1-904868-81-8 110pp