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Introduction To Heat Potential Theory
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Book Synopsis Introduction to Heat Potential Theory by : N. A. Watson
Download or read book Introduction to Heat Potential Theory written by N. A. Watson and published by American Mathematical Soc.. This book was released on 2012 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the first to be devoted entirely to the potential theory of the heat equation, and thus deals with time dependent potential theory. Its purpose is to give a logical, mathematically precise introduction to a subject where previously many proofs were not written in detail, due to their similarity with those of the potential theory of Laplace's equation. The approach to subtemperatures is a recent one, based on the Poisson integral representation of temperatures on a circular cylinder. Characterizations of subtemperatures in terms of heat balls and modified heat balls are proved, and thermal capacity is studied in detail. The generalized Dirichlet problem on arbitrary open sets is given a treatment that reflects its distinctive nature for an equation of parabolic type. Also included is some new material on caloric measure for arbitrary open sets. Each chapter concludes with bibliographical notes and open questions. The reader should have a good background in the calculus of functions of several variables, in the limiting processes and inequalities of analysis, in measure theory, and in general topology for Chapter 9.
Book Synopsis Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction by : Dang D. Ang
Download or read book Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction written by Dang D. Ang and published by Springer. This book was released on 2004-10-13 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: Moment Theory is not a new subject; however, in classical treatments, the ill-posedness of the problem is not taken into account - hence this monograph. Assuming a "true" solution to be uniquely determined by a sequence of moments (given as integrals) of which only finitely many are inaccurately given, the authors describe and analyze several regularization methods and derive stability estimates. Mathematically, the task often consists in the reconstruction of an analytic or harmonic function, as is natural from concrete applications discussed (e.g. inverse heat conduction problems, Cauchy's problem for the Laplace equation, gravimetry). The book can be used in a graduate or upper undergraduate course in Inverse Problems, or as supplementary reading for a course on Applied Partial Differential Equations.
Book Synopsis Potential Theory in Gravity and Magnetic Applications by : Richard J. Blakely
Download or read book Potential Theory in Gravity and Magnetic Applications written by Richard J. Blakely and published by Cambridge University Press. This book was released on 1996-09-13 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text bridges the gap between the classic texts on potential theory and modern books on applied geophysics. It opens with an introduction to potential theory, emphasising those aspects particularly important to earth scientists, such as Laplace's equation, Newtonian potential, magnetic and electrostatic fields, and conduction of heat. The theory is then applied to the interpretation of gravity and magnetic anomalies, drawing on examples from modern geophysical literature. Topics explored include regional and global fields, forward modeling, inverse methods, depth-to-source estimation, ideal bodies, analytical continuation, and spectral analysis. The book includes numerous exercises and a variety of computer subroutines written in FORTRAN. Graduate students and researchers in geophysics will find this book essential.
Book Synopsis Foundations of Potential Theory by : Oliver Dimon Kellogg
Download or read book Foundations of Potential Theory written by Oliver Dimon Kellogg and published by Barman Press. This book was released on 1929 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: FOUNDATIONS OF POTENTIAL THEORY by OLIVER DIMON KELLOGG. Originally published in 1929. Preface: The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to - the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the ok may present sound ideals to the student, and also serve the ma pmatician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem Gauss, or Greens theorem on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Pirichlet problem. Exercises are introduced in the conviction that no mastery of a mathematical subject is possible without working with it. They are designed primarily to illustrate or extend the theory, although the desirability of requiring an occasional concrete numerical result has not been lost sight of. O. D. Kellogg. August, 1929. Contents include: Chapter 1. The Force of Gravity. 1. The Subject Matter of Potential Theory 1 2. Newtons Law 2 3. Interpretation of Newtons Law for Continuously Distributed Bodies . 3 4. Forces Due to Special Bodies 4 5. Material Curves, or Wires 8 6 Material Surfaces or Lammas 10 7. Curved Lammas 12 8. Ordinary Bodies, or Volume Distributions 15 9 The Force at Points of the Attracting Masses 17 10. Legitimacy of the Amplified Statement of Newtons Law Attraction between Bodies 22 11. Presence of the Couple Centrobaric Bodies Specific Force 26 Chapter II. Fields of Force. 1. Fields of Force and Other Vector Fields 28 2. Lines of Force 28 3. Velocity Fields 31 4. Expansion, or Divergence of a Field 34 5. The Divergence Theorem 37 6. Flux of Force Solenoidal Fields 40 7. Gauss Integral 42 8. Sources and Sinks 44 9. General Flows of Fluids Equation of Continuity 45 Chapter III The Potential. 1. Work and Potential Energy 48 2 Equipotential Surfaces 54 3. Potentials of Special Distributions 55 4. The Potential of a Homogeneous Circumference 58 5. Two Dimensional Problems The Logarithmic Potential 62 6. Magnetic Particles 65 7. Magnetic Shells, or Double Distributions 66 8. Irrotational Flow 69 . Stokes Theorem 72 10. Flow of Heat 76 11. The Energy of Distributions 79 12...
Book Synopsis The Theory of the Heat Potential (installment I). Direct Values of the Heat Potentials and the Reverse Problem in the Theory of the Heat Potential by : Wang Tung
Download or read book The Theory of the Heat Potential (installment I). Direct Values of the Heat Potentials and the Reverse Problem in the Theory of the Heat Potential written by Wang Tung and published by . This book was released on 1966 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Introduction to the Mathematical Theory of the Conduction of Heat in Solids by : Horatio Scott Carslaw
Download or read book Introduction to the Mathematical Theory of the Conduction of Heat in Solids written by Horatio Scott Carslaw and published by . This book was released on 1906 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to the Mathematical Theory of the Conduction of Heat in Solids by Horatio Scott Carslaw, first published in 1945, is a rare manuscript, the original residing in one of the great libraries of the world. This book is a reproduction of that original, which has been scanned and cleaned by state-of-the-art publishing tools for better readability and enhanced appreciation. Restoration Editors' mission is to bring long out of print manuscripts back to life. Some smudges, annotations or unclear text may still exist, due to permanent damage to the original work. We believe the literary significance of the text justifies offering this reproduction, allowing a new generation to appreciate it.
Book Synopsis Theory of Heat by : James Clerk Maxwell
Download or read book Theory of Heat written by James Clerk Maxwell and published by . This book was released on 1872 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classic sets forth the fundamentals of thermodynamics and kinetic theory simply enough to be understood by beginners, yet with enough subtlety to appeal to more advanced readers, too.
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 by : Michael Shearer
Download or read book Partial Differential Equations written by Michael Shearer and published by Princeton University Press. This book was released on 2015-03-01 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible yet rigorous introduction to partial differential equations This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis. Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. The book also covers fundamental solutions, Green's functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs. Provides an accessible yet rigorous introduction to partial differential equations Draws connections to advanced topics in analysis Covers applications to continuum mechanics An electronic solutions manual is available only to professors An online illustration package is available to professors
Book Synopsis Classical Potential Theory and Its Probabilistic Counterpart by : J. L. Doob
Download or read book Classical Potential Theory and Its Probabilistic Counterpart written by J. L. Doob and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 865 pages. Available in PDF, EPUB and Kindle. Book excerpt: Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
Book Synopsis Partial Differential Equations by : Emmanuele DiBenedetto
Download or read book Partial Differential Equations written by Emmanuele DiBenedetto and published by Springer Nature. This book was released on 2023 with total page 768 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate textbook provides a self-contained introduction to the classical theory of partial differential equations (PDEs). Through its careful selection of topics and engaging tone, readers will also learn how PDEs connect to cutting-edge research and the modeling of physical phenomena. The scope of the Third Edition greatly expands on that of the previous editions by including five new chapters covering additional PDE topics relevant for current areas of active research. This includes coverage of linear parabolic equations with measurable coefficients, parabolic DeGiorgi classes, Navier-Stokes equations, and more. The “Problems and Complements” sections have also been updated to feature new exercises, examples, and hints toward solutions, making this a timely resource for students. Partial Differential Equations: Third Edition is ideal for graduate students interested in exploring the theory of PDEs and how they connect to contemporary research. It can also serve as a useful tool for more experienced readers who are looking for a comprehensive reference.
Book Synopsis University Physics by : Samuel J. Ling
Download or read book University Physics written by Samuel J. Ling and published by . This book was released on 2017-12-19 with total page 818 pages. Available in PDF, EPUB and Kindle. Book excerpt: University Physics is designed for the two- or three-semester calculus-based physics course. The text has been developed to meet the scope and sequence of most university physics courses and provides a foundation for a career in mathematics, science, or engineering. The book provides an important opportunity for students to learn the core concepts of physics and understand how those concepts apply to their lives and to the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Coverage and Scope Our University Physics textbook adheres to the scope and sequence of most two- and three-semester physics courses nationwide. We have worked to make physics interesting and accessible to students while maintaining the mathematical rigor inherent in the subject. With this objective in mind, the content of this textbook has been developed and arranged to provide a logical progression from fundamental to more advanced concepts, building upon what students have already learned and emphasizing connections between topics and between theory and applications. The goal of each section is to enable students not just to recognize concepts, but to work with them in ways that will be useful in later courses and future careers. The organization and pedagogical features were developed and vetted with feedback from science educators dedicated to the project. VOLUME II Unit 1: Thermodynamics Chapter 1: Temperature and Heat Chapter 2: The Kinetic Theory of Gases Chapter 3: The First Law of Thermodynamics Chapter 4: The Second Law of Thermodynamics Unit 2: Electricity and Magnetism Chapter 5: Electric Charges and Fields Chapter 6: Gauss's Law Chapter 7: Electric Potential Chapter 8: Capacitance Chapter 9: Current and Resistance Chapter 10: Direct-Current Circuits Chapter 11: Magnetic Forces and Fields Chapter 12: Sources of Magnetic Fields Chapter 13: Electromagnetic Induction Chapter 14: Inductance Chapter 15: Alternating-Current Circuits Chapter 16: Electromagnetic Waves
Book Synopsis Topological Modular Forms by : Christopher L. Douglas
Download or read book Topological Modular Forms written by Christopher L. Douglas and published by American Mathematical Soc.. This book was released on 2014-12-04 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory. This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss-Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.
Book Synopsis Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties by : Jorg Jahnel
Download or read book Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties written by Jorg Jahnel and published by American Mathematical Soc.. This book was released on 2014-12-02 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type--both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer-Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.
Book Synopsis Geometry of Isotropic Convex Bodies by : Silouanos Brazitikos
Download or read book Geometry of Isotropic Convex Bodies written by Silouanos Brazitikos and published by American Mathematical Soc.. This book was released on 2014-04-24 with total page 618 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.
Book Synopsis The Octagonal PETs by : Richard Evan Schwartz
Download or read book The Octagonal PETs written by Richard Evan Schwartz and published by American Mathematical Soc.. This book was released on 2014-07-03 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.
Book Synopsis Grid Homology for Knots and Links by : Peter S. Ozsváth
Download or read book Grid Homology for Knots and Links written by Peter S. Ozsváth and published by American Mathematical Soc.. This book was released on 2015-12-04 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt: Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.
