The Index Theorem And The Heat Equation Method
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The Index Theorem And The Heat Equation Method
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Author : Yanlin Yu
language : en
Publisher: World Scientific
Release Date : 2001-07-02
The Index Theorem And The Heat Equation Method written by Yanlin Yu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-07-02 with Science categories.
This book provides a self-contained representation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first order geometric elliptic operators by using the heat equation method. The proofs are up to the standard of pure mathematics. In addition, a Chern root algorithm is introduced for proving the local index theorems, and it seems to be as efficient as other methods.
Invariance Theory
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Author : Peter B. Gilkey
language : en
Publisher: CRC Press
Release Date : 1994-12-22
Invariance Theory written by Peter B. Gilkey and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-12-22 with Mathematics categories.
This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.
Elliptic Operators Topology And Asymptotic Methods
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Author : John Roe
language : en
Publisher: Longman Scientific and Technical
Release Date : 1988
Elliptic Operators Topology And Asymptotic Methods written by John Roe and has been published by Longman Scientific and Technical this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.
Heat Kernels And Dirac Operators
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Author : Nicole Berline
language : en
Publisher: Springer Science & Business Media
Release Date : 2003-12-08
Heat Kernels And Dirac Operators written by Nicole Berline and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-12-08 with Mathematics categories.
In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive paperback.
The Index Theorem And The Heat Equation
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Author : Peter B. Gilkey
language : en
Publisher: Publish or Perish
Release Date : 1974
The Index Theorem And The Heat Equation written by Peter B. Gilkey and has been published by Publish or Perish this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with Mathematics categories.
Random Walk And The Heat Equation
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Author : Gregory F. Lawler
language : en
Publisher: American Mathematical Soc.
Release Date : 2010-11-22
Random Walk And The Heat Equation written by Gregory F. Lawler and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-22 with Mathematics categories.
The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
Heat Kernel And Analysis On Manifolds
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Author : Alexander Grigoryan
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Heat Kernel And Analysis On Manifolds written by Alexander Grigoryan and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Education categories.
The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace-Beltrami operator and the associated heat equation. The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectral-theoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels. Grigor'yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research.Titles in this series are co-published with International Press, Cambridge, MA, USA.
Finite Difference Methods For Ordinary And Partial Differential Equations
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Author : Randall J. LeVeque
language : en
Publisher: SIAM
Release Date : 2007-01-01
Finite Difference Methods For Ordinary And Partial Differential Equations written by Randall J. LeVeque and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-01 with Mathematics categories.
This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.
The Laplacian On A Riemannian Manifold
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Author : Steven Rosenberg
language : en
Publisher: Cambridge University Press
Release Date : 1997-01-09
The Laplacian On A Riemannian Manifold written by Steven Rosenberg and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-09 with Mathematics categories.
This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.