Yuri E. Gliklikh – författare

The geometrical methods in modem mathematical physics and the developments in Geometry and Global Analysis motivated by physical problems are being intensively worked out in contemporary mathematics. In particular, during the last decades a new branch of Global Analysis, Stochastic Differential Geometry, was formed to meet the needs of Mathematical Physics. It deals with a lot of various second order differential equations on finite and infinite-dimensional manifolds arising in Physics, and its validity is based on the deep inter-relation between modem Differential Geometry and certain parts of the Theory of Stochastic Processes, discovered not so long ago. The foundation of our topic is presented in the contemporary mathematical literature by a lot of publications devoted to certain parts of the above-mentioned themes and connected with the scope of material of this book. There exist some monographs on Stochastic Differential Equations on Manifolds (e. g. [9,36,38,87]) based on the Stratonovich approach. In [7] there is a detailed description of It6 equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and survey papers on the Lagrange approach to Hydrodynamics [2,31,73,88], etc. , give good presentations of the use of infinite-dimensional ordinary differential geometry in ideal hydrodynamics. We should also refer here to [89,102], to the previous books by the author [53,64], and to many others.

Methods of global analysis and stochastic analysis are most often applied in mathematical physics as separate entities, thus forming important directions in the field. However, while combination of the two subject areas is rare, it is fundamental for the consideration of a broader class of problems.This book develops methods of Global Analysis and Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as divergent and requiring different methods of investigation.Global and Stochastic Analysis with Applications to Mathematical Physics covers branches of mathematics that are currently absent in monograph form. Through the demonstration of new topics of investigation and results, both in traditional and more recent problems, this book offers a fresh perspective on ordinary and stochastic differential equations and inclusions (in particular, given in terms of Nelson's mean derivatives) on linear spaces and manifolds. Topics covered include classical mechanics on non-linear configuration spaces, problems of statistical and quantum physics, and hydrodynamics.A self-contained book that provides a large amount of preliminary material and recent results which will serve to be a useful introduction to the subject and a valuable resource for further research. It will appeal to researchers, graduate and PhD students working in global analysis, stochastic analysis and mathematical physics.

Methods of global analysis and stochastic analysis are most often applied in mathematical physics as separate entities, thus forming important directions in the field. However, while combination of the two subject areas is rare, it is fundamental for the consideration of a broader class of problems.This book develops methods of Global Analysis and Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as divergent and requiring different methods of investigation.Global and Stochastic Analysis with Applications to Mathematical Physics covers branches of mathematics that are currently absent in monograph form. Through the demonstration of new topics of investigation and results, both in traditional and more recent problems, this book offers a fresh perspective on ordinary and stochastic differential equations and inclusions (in particular, given in terms of Nelson's mean derivatives) on linear spaces and manifolds. Topics covered include classical mechanics on non-linear configuration spaces, problems of statistical and quantum physics, and hydrodynamics.A self-contained book that provides a large amount of preliminary material and recent results which will serve to be a useful introduction to the subject and a valuable resource for further research. It will appeal to researchers, graduate and PhD students working in global analysis, stochastic analysis and mathematical physics.

This volume (a sequel to LNM 1108, 1214, 1334 and 1453)continues the presentation to English speaking readers ofthe Voronezh University press series on Global Analysis andIts Applications. The papers are selected fromtwo Russianissues entitled "Algebraic questions of Analysis andTopology" and "Nonlinear Operators in Global Analysis".CONTENTS: YuE. Gliklikh: Stochastic analysis, groups ofdiffeomorphisms and Lagrangian description of viscousincompressible fluid.- A.Ya. Helemskii: From topologicalhomology: algebras with different properties of homologicaltriviality.- V.V. Lychagin, L.V. Zil''bergleit: Duality instable Spencer cohomologies.- O.R. Musin: On some problemsof computational geometry and topology.- V.E. Nazaikinskii,B.Yu. Sternin, V.E.Shatalov: Introduction to Maslov''soperational method (non-commutative analysis anddifferential equations).- Yu.B. Rudyak: The problem ofrealization of homology classes from Poincare up to thepresent.- V.G. Zvyagin, N.M. Ratiner: Oriented degree ofFredholm maps of non-negativeindex and its applications toglobal bifurcation of solutions.- A.A. Bolibruch: Fuchsiansystems with reducible monodromy and the Riemann-Hilbertproblem.- I.V. Bronstein, A.Ya. Kopanskii: Finitely smoothnormal forms of vector fields in the vicinity of a restpoint.- B.D. Gel''man: Generalized degree of multi-valuedmappings.- G.N. Khimshiashvili: On Fredholmian aspects oflinear transmission problems.- A.S. Mishchenko: Stationarysolutions of nonlinear stochastic equations.- B.Yu. Sternin,V.E. Shatalov: Continuation of solutions to ellipticequations and localisation of singularities.- V.G. Zvyagin,V.T. Dmitrienko: Properness of nonlinear ellipticdifferential operators in H|lder spaces.

This volume (a sequel to LNM 1108, 1214, 1334 and 1453)continues the presentation to English speaking readers ofthe Voronezh University press series on Global Analysis andIts Applications. The papers are selected fromtwo Russianissues entitled "Algebraic questions of Analysis andTopology" and "Nonlinear Operators in Global Analysis".CONTENTS: YuE. Gliklikh: Stochastic analysis, groups ofdiffeomorphisms and Lagrangian description of viscousincompressible fluid.- A.Ya. Helemskii: From topologicalhomology: algebras with different properties of homologicaltriviality.- V.V. Lychagin, L.V. Zil'bergleit: Duality instable Spencer cohomologies.- O.R. Musin: On some problemsof computational geometry and topology.- V.E. Nazaikinskii,B.Yu. Sternin, V.E.Shatalov: Introduction to Maslov'soperational method (non-commutative analysis anddifferential equations).- Yu.B. Rudyak: The problem ofrealization of homology classes from Poincare up to thepresent.- V.G. Zvyagin, N.M. Ratiner: Oriented degree ofFredholm maps of non-negativeindex and its applications toglobal bifurcation of solutions.- A.A. Bolibruch: Fuchsiansystems with reducible monodromy and the Riemann-Hilbertproblem.- I.V. Bronstein, A.Ya. Kopanskii: Finitely smoothnormal forms of vector fields in the vicinity of a restpoint.- B.D. Gel'man: Generalized degree of multi-valuedmappings.- G.N. Khimshiashvili: On Fredholmian aspects oflinear transmission problems.- A.S. Mishchenko: Stationarysolutions of nonlinear stochastic equations.- B.Yu. Sternin,V.E. Shatalov: Continuation of solutions to ellipticequations and localisation of singularities.- V.G. Zvyagin,V.T. Dmitrienko: Properness of nonlinear ellipticdifferential operators in H|lder spaces.

The geometrical methods in modem mathematical physics and the developments in Geometry and Global Analysis motivated by physical problems are being intensively worked out in contemporary mathematics. In particular, during the last decades a new branch of Global Analysis, Stochastic Differential Geometry, was formed to meet the needs of Mathematical Physics. It deals with a lot of various second order differential equations on finite and infinite-dimensional manifolds arising in Physics, and its validity is based on the deep inter-relation between modem Differential Geometry and certain parts of the Theory of Stochastic Processes, discovered not so long ago. The foundation of our topic is presented in the contemporary mathematical literature by a lot of publications devoted to certain parts of the above-mentioned themes and connected with the scope of material of this book. There exist some monographs on Stochastic Differential Equations on Manifolds (e. g. [9,36,38,87]) based on the Stratonovich approach. In [7] there is a detailed description of It6 equations on manifolds in Belopolskaya-Dalecky form. Nelson's book [94] deals with Stochastic Mechanics and mean derivatives on Riemannian Manifolds. The books and survey papers on the Lagrange approach to Hydrodynamics [2,31,73,88], etc. , give good presentations of the use of infinite-dimensional ordinary differential geometry in ideal hydrodynamics. We should also refer here to [89,102], to the previous books by the author [53,64], and to many others.