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The Index Theorem And The Heat Equation Method
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Book Synopsis The Index Theorem And The Heat Equation Method by : Yanlin Yu
Download or read book The Index Theorem And The Heat Equation Method written by Yanlin Yu and published by World Scientific. This book was released on 2001-07-02 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Invariance Theory by : Peter B. Gilkey
Download or read book Invariance Theory written by Peter B. Gilkey and published by CRC Press. This book was released on 2018-05-02 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Invariance Theory by : Peter B. Gilkey
Download or read book Invariance Theory written by Peter B. Gilkey and published by CRC Press. This book was released on 1994-12-22 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Index Theorem and the Heat Equation by : Peter B. Gilkey
Download or read book The Index Theorem and the Heat Equation written by Peter B. Gilkey and published by Publish or Perish. This book was released on 1974 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Laplacian on a Riemannian Manifold by : Steven Rosenberg
Download or read book The Laplacian on a Riemannian Manifold written by Steven Rosenberg and published by Cambridge University Press. This book was released on 1997-01-09 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.
Book Synopsis The Atiyah-Patodi-Singer Index Theorem by : Richard Melrose
Download or read book The Atiyah-Patodi-Singer Index Theorem written by Richard Melrose and published by CRC Press. This book was released on 1993-03-31 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies.
Book Synopsis Elliptic Operators, Topology, and Asymptotic Methods by : John Roe
Download or read book Elliptic Operators, Topology, and Asymptotic Methods written by John Roe and published by Longman Scientific and Technical. This book was released on 1988 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Heat Kernel and Analysis on Manifolds by : Alexander Grigoryan
Download or read book Heat Kernel and Analysis on Manifolds written by Alexander Grigoryan and published by American Mathematical Soc.. This book was released on 2009 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Partial Differential Equations II by : Michael E. Taylor
Download or read book Partial Differential Equations II written by Michael E. Taylor and published by Springer Science & Business Media. This book was released on 2010-11-02 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centred about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion.
Download or read book The Heat Equation written by D. V. Widder and published by Academic Press. This book was released on 1976-01-22 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Heat Equation
Book Synopsis Partial Differential Equations II by : Michael Taylor
Download or read book Partial Differential Equations II written by Michael Taylor and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 547 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centred about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion.
Book Synopsis Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem by : Peter B. Gilkey
Download or read book Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem written by Peter B. Gilkey and published by . This book was released on 1984 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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, & 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.
Book Synopsis Random Walk and the Heat Equation by : Gregory F. Lawler
Download or read book Random Walk and the Heat Equation written by Gregory F. Lawler and published by American Mathematical Soc.. This book was released on 2010-11-22 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Partial Differential Equations and Boundary-Value Problems with Applications by : Mark A. Pinsky
Download or read book Partial Differential Equations and Boundary-Value Problems with Applications written by Mark A. Pinsky and published by American Mathematical Soc.. This book was released on 2011 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.
Book Synopsis Partial Differential Equations by : Walter A. Strauss
Download or read book Partial Differential Equations written by Walter A. Strauss and published by John Wiley & Sons. This book was released on 2007-12-21 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
Book Synopsis Some Applications of Topological K-Theory by :
Download or read book Some Applications of Topological K-Theory written by and published by Elsevier. This book was released on 1980-01-01 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Some Applications of Topological K-Theory
Download or read book Manifolds written by Paul Bracken and published by BoD – Books on Demand. This book was released on 2017-01-18 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential geometry is a very active field of research and has many applications to areas such as physics and gravity, for example. The papers in this book cover a number of subjects which will be of interest to workers in these areas. It is hoped that the papers here will be able to provide a useful resource for researchers with regard to current fields of research in this important area.
