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Reviews in Mathematics and Mathematical Physics

Reviews in Mathematics and Mathematical Physics

Edited by A.T.Fomenko


Reviews in Mathematics and Mathematical Physics publishes review articles in mathematics and mathematical physicsfrom the former Soviet Union. Topics include soliton theory, the theory of quantum topological models and their applications for algebraic and differential geometry and topology.


Ordering information: Reviews in Mathematics and Mathematical Physics is a journal. Each volume consists of an irregular number of issues, depending on extent. Issues are available individually (see following details) as well as by subscription. For subscription information click on Journals.

Geometry of Singularities of Integrable Systems on Lie Algebras 
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Pbk    56pp     2006

YU.  A.Brailov, Moscow State University, Russia

Integrable systems generated by consistent Poisson brackets on semi-simple Lie Algebras are related to the structure of ground algebra.  The most important properties of Liouvilles foliation are expressed in algebraic term.  The exact relation is also established between the singularities of momentum mapping and degenerations of the corresponding spectral curve.


Introduction To The Theory of Representations of Finitely Presented *- Algebra. I. Representations By Bounded Operators 
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Pb        262pp              1999

V.Ostrovskyi and Yu. Samoilenko

Institute of Mathematics, Kiev, Ukraine


This volume provides the fundamentals of representations of finitely presented *-algebras by bounded operators. The theory is illustrated with numerous examples of *-algebra. The examples, in particular, include *-algebra with two self-adjoint generators that satisfy a quadratic or a more general relation, *-algebra that arise from one- and many-dimensional discrete dynamical systems, Wick *-algebra, various *-wild algebras.


Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations (Second Edition) 
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Pb        204pp              2009


L.D.Landau Institute for Theoretical Physics, Moscow, Russia and

University of Paderborn, Germany


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.


Invariant Kahler Structures on the Cotangent Bundle of Symmetric Spaces and Reduction  
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100pp Pbk 2005

I.V.Mykytyuk, Institute of Applied Problems of Mathematics and Mechanics, Lviv, Ukraine


The main objects of investigation are G-invariant polarizations on domains D in the cotangent bundles T*(G/K).

The book comprises two parts. Part I: Invariant Kahler structures and  Part II: Invariant hyperkahler structures.


Basis Properties and Completeness of Certain System of Elementary Functions 
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2008 112pp  Pbk


E. I. Moiseev, A. P. Prudnikov and A. M. SedletskiiA. A. Dorodnitsyn Computational Center, Russian Academy of Sciences, Moscow

The possibility of approximating functions of a sufficiently general form by relatively simple functions and therefore more convenient for analysis and computation is the initial reason for approximation theory. The basic fact of this theory is the Weierstrasse theorem. In the first part of this book the authors consider the problem of description of complete power systems. In the second part, the authors study the problem on bases in Sobolev spaces, in their subspaces and also in weighted spaces and finally consider continuous analogs of basis expansions (the generalized sine and cosine Fourier transforms).


Isometric Immersions and Embeddings of Locally Euclidean Metrics 
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2008 112pp  Pbk

I. Kh. Sabitov, Moscow State University, Russia


The aim of this volume is to review the results on isometric immersions of locally Euclidean metrics into Euclidean spaces along with the description of the extrinsic geometry of these immersions. The review begins with the consideration of a problem specific only for constant curvature metrics, namely the problem of “natural” realization of locally Euclidean metrics by Euclidean-space domains of corresponding dimension with the standard Euclidean metric and then studies their isometric immersions into Euclidean spaces of greater dimension with emphasis on the problems of smoothness. The author presents many results with their original proofs, and also formulates a large number of unsolved problems of theory.(G/K).


Multidimensional Monge-Ampère Equation 
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2009 110pp Pb

Multidimensional Monge-Ampère Equation by A.V.Pogorelov presents a detailed exposition of the results concerning the existence and uniqueness of the solutions of the general Monge-Ampère multidimensional equations of elliptic type. This division of the theory of partial differential equations is closely connected with geometry. This review is intended for students, postgraduates and researchers in geometry and differential equations.


Inverse Function Theorems and Their Applications to the Theory of Polyhedra 
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V. A. Alexandrov, Sobolev Institute of Mathematics, Novosibirsk, Russia

2011 120pp Pbk

This volume reviews how the implicit and inverse function theorems operate in polyhedron theory, and how they are used to deduce classical and new theorems on polyhedra, for example, the existence, uniqueness and rigidity of a convex polyhedron with a given development; construction of flexible polyhedra; existence and uniqueness of a convex polyhedron with given areas and directions of faces; generalization of theorems for nonconvex polyhedra.


Topology of Integrable Systems  
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D. B. Zotev, Volgograd State Technical University, Russia

2011                180pp              Pbk


The topological theory of integrable Hamiltonian systems was created and developed by A. T. Fomenko and his group. This review briefly describes the theory on a level of strictness sufficient for self-dependent applications. Some new results are presented, illustrating the theses of the theory and also a method of A.V.Bolsinov.


Spectral Expansion of the Transfer Matrices of Gibbs Fields (second edition) 
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R.A. Minlos, Institute for Information Transmission Problems, Moscow

2013                70pp              Pbk

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 p-representation. This edition of a classic review provides a useful source of reference for students, postgraduates and researchers in these areas of mathematics.


Singularities of Functions, Wave Fronts, Caustics and Multidimensional Integrals  
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V.I. Arnold, A.N. Varchenko, A.B. Givental and A.G. Khovanskii

2012                120pp              Pbk


This classic survey paper is an introduction to some difficult contemporary fields of study in mathematics known under the rubric of Catastrophe Theory, which encompasses the theory of "typical" singularities of functions and mappings. 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.


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