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Heat Kernel Method and its Applications

Author: Ivan Avramidi

Publisher: Birkhäuser

Published: 2015-11-26

Total Pages: 402

ISBN-13: 3319262661

The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.

Heat Kernel on Lie Groups and Maximally Symmetric Spaces

Author: Ivan G. Avramidi

Publisher: Springer Nature

Published: 2023-05-27

Total Pages: 197

ISBN-13: 3031274512

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This monograph studies the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. It introduces many original ideas, methods, and tools developed by the author and provides a list of all known exact results in explicit form – and derives them – for the heat kernel on spheres and hyperbolic spaces. Part I considers the geometry of simple Lie groups and maximally symmetric spaces in detail, and Part II discusses the calculation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces in various dimensions. This text will be a valuable resource for researchers and graduate students working in various areas of mathematics – such as global analysis, spectral geometry, stochastic processes, and financial mathematics – as well in areas of mathematical and theoretical physics – including quantum field theory, quantum gravity, string theory, and statistical physics.

Heat Kernels for Elliptic and Sub-elliptic Operators

Author: Ovidiu Calin

Publisher: Springer Science & Business Media

Published: 2010-10-10

Total Pages: 444

ISBN-13: 0817649956

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This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes. Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.

Introduction to Quantum Field Theory with Applications to Quantum Gravity

Author: Iosif L. Buchbinder

Publisher: Oxford University Press, USA

Published: 2021-03

Total Pages: 536

ISBN-13: 019883831X

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This textbook presents a detailed introduction to the general concepts of quantum field theory, with special emphasis on principal aspects of functional methods and renormalization in gauge theories, and includes an introduction to semiclassical and perturbative quantum gravity in flat and curved spacetimes.

Proceedings Of The Julian Schwinger Centennial Conference

Author: Englert Berthold-georg

Publisher: World Scientific

Published: 2019-10-30

Total Pages: 336

ISBN-13: 9811213151

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The Julian Schwinger Centennial Conference of 2018 assembled many of Schwinger's students, colleagues, and friends to celebrate this towering figure of twentieth century physics one hundred years after his birth. This proceedings volume collects talks delivered on this occasion. They cover a wide range of topics, all related to Schwinger's rich scientific legacy — supplemented by personal recollections about Julian Schwinger, the physicist, the teacher, and the gentleman.Also included are an essay of 1985, co-authored by Schwinger but not published previously, as well as the transcripts of speeches by distinguished colleagues at the 1978 gathering when Schwinger's sixtieth birthday was celebrated.

Laguerre Calculus and Its Applications on the Heisenberg Group

Author: Carlos A. Berenstein

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 333

ISBN-13: 0821827618

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For nearly two centuries, the relation between analytic functions of one complex variable, their boundary values, harmonic functions, and the theory of Fourier series has been one of the central topics of study in mathematics. The topic stands on its own, yet also provides very useful mathematical applications. This text provides a self-contained introduction to the corresponding questions in several complex variables: namely, analysis on the Heisenberg group and the study of the solutions of the boundary Cauchy-Riemann equations. In studying this material, readers are exposed to analysis in non-commutative compact and Lie groups, specifically the rotation group and the Heisenberg groups-both fundamental in the theory of group representations and physics. Introduced in a concrete setting are the main ideas of the Calderón-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of harmonic and complex analysis, in particular, the Morera and Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering. The book was borne of graduate courses and seminars held at the University of Maryland (College Park), the University of Toronto (ON), Georgetown University (Washington, DC), and the University of Georgia (Athens). Readers should have an advanced undergraduate understanding of Fourier analysis and complex analysis in one variable.

Journal of Mathematical Sciences, the University of Tokyo

Author:

Publisher:

Published: 2004

Total Pages: 534

ISBN-13:

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The Method of Layer Potentials for the Heat Equation in Time-Varying Domains

Author: John L. Lewis

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 170

ISBN-13: 0821803603

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This memoir consists of three papers in which we develop the method of layer potentials for the heat equation in time-varying domains. In Chapter I we show certain singular integral operators on [italic]L[superscript italic]p are bounded. in Chapter II, we develop a modification of the David buildup scheme to obtain [italic]L[superscript italic]p boundedness of the double layer heat potential on the boundary of our domains. In Chapter III, we use the results of the first two chapters to show the mutual absolute continuity of parabolic measure and a certain projective Lebesgue measure.

Graph-Based Methods in Computer Vision: Developments and Applications

Author: Bai, Xiao

Publisher: IGI Global

Published: 2012-07-31

Total Pages: 395

ISBN-13: 1466618922

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Computer vision, the science and technology of machines that see, has been a rapidly developing research area since the mid-1970s. It focuses on the understanding of digital input images in many forms, including video and 3-D range data. Graph-Based Methods in Computer Vision: Developments and Applications presents a sampling of the research issues related to applying graph-based methods in computer vision. These methods have been under-utilized in the past, but use must now be increased because of their ability to naturally and effectively represent image models and data. This publication explores current activity and future applications of this fascinating and ground-breaking topic.

The Ubiquitous Heat Kernel

Author: Jay Jorgenson

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 410

ISBN-13: 0821836986

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The aim of this volume is to bring together research ideas from various fields of mathematics which utilize the heat kernel or heat kernel techniques in their research. The intention of this collection of papers is to broaden productive communication across mathematical sub-disciplines and to provide a vehicle which would allow experts in one field to initiate research with individuals in another field, as well as to give non-experts a resource which can facilitate expanding theirresearch and connecting with others.