Graphs With Potential Theory
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Author | : Lucio Prado |
Publisher | : LAP Lambert Academic Publishing |
Total Pages | : 156 |
Release | : 2010-02 |
Genre | : |
ISBN | : 9783838334967 |
Graphs with Potential Theory This book is intended for very broad group of graduate students who wish to have a systematic introduction into the theory of nonlinear potential theory applied to graphs for future work. These objects are similar in many ways to Riemannian manifolds. The author focuses on topics such as p- Laplacian, p-harmonicity, p-Dirichlet spaces, p- capacity, extended divergence formula, p-Harnack inequality, p-hyperbolicity, and p-Poisson equations, while always using variational approach on discrete settings of graphs. The aim is to introduce the basic concepts and results coherently, and show how they are interconnected and interplayed. The treatment presupposes an introductory course on real analysis, and the knowledge of basic facts from potential theory. In the introduction, the author includes viable information on basics facts from graph theory, and the construction of spaces of functions on graphs.
Author | : John Wermer |
Publisher | : Springer Science & Business Media |
Total Pages | : 156 |
Release | : 2013-06-29 |
Genre | : Mathematics |
ISBN | : 366212727X |
Potential theory grew out of mathematical physics, in particular out of the theory of gravitation and the theory of electrostatics. Mathematical physicists such as Poisson and Green introduced some of the central ideas of the subject. A mathematician with a general knowledge of analysis may find it useful to begin his study of classical potential theory by looking at its physical origins. Sections 2, 5 and 6 of these Notes give in part heuristic arguments based on physical considerations. These heuristic arguments suggest mathematical theorems and provide the mathematician with the problem of finding the proper hypotheses and mathematical proofs. These Notes are based on a one-semester course given by the author at Brown University in 1971. On the part of the reader, they assume a knowledge of Real Function Theory to the extent of a first year graduate course. In addition some elementary facts regarding harmonic functions are aS$umed as known. For convenience we have listed these facts in the Appendix. Some notation is also explained there. Essentially all the proofs we give in the Notes are for Euclidean 3-space R3 and Newtonian potentials ~.
Author | : Paolo M. Soardi |
Publisher | : Springer |
Total Pages | : 199 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 3540487980 |
The aim of the book is to give a unified approach to new developments in discrete potential theory and infinite network theory. The author confines himself to the finite energy case, but this does not result in loss of complexity. On the contrary, the functional analytic machinery may be used in analogy with potential theory on Riemann manifolds. The book is intended for researchers with interdisciplinary interests in one of the following fields: Markov chains, combinatorial graph theory, network theory, Dirichlet spaces, potential theory, abstract harmonic analysis, theory of boundaries.
Author | : Lucio M-G. Prado |
Publisher | : |
Total Pages | : 286 |
Release | : 2004 |
Genre | : |
ISBN | : |
Author | : M. Picardello |
Publisher | : Cambridge University Press |
Total Pages | : 378 |
Release | : 1999-11-18 |
Genre | : Mathematics |
ISBN | : 9780521773126 |
Comprehensive and interdisciplinary text covering the interplay between random walks and structure theory.
Author | : Lester L. Helms |
Publisher | : Springer Science & Business Media |
Total Pages | : 494 |
Release | : 2014-04-10 |
Genre | : Mathematics |
ISBN | : 1447164229 |
Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region. The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion. In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics and engineering.
Author | : Anders Björn |
Publisher | : European Mathematical Society |
Total Pages | : 422 |
Release | : 2011 |
Genre | : Harmonic functions |
ISBN | : 9783037190999 |
The $p$-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book.
Author | : Oliver Dimon Kellogg |
Publisher | : Courier Corporation |
Total Pages | : 404 |
Release | : 1953-01-01 |
Genre | : Science |
ISBN | : 9780486601441 |
Introduction to fundamentals of potential functions covers the force of gravity, fields of force, potentials, harmonic functions, electric images and Green's function, sequences of harmonic functions, fundamental existence theorems, the logarithmic potential, and much more. Detailed proofs rigorously worked out. 1929 edition.
Author | : Victor Anandam |
Publisher | : Springer Science & Business Media |
Total Pages | : 152 |
Release | : 2011-06-27 |
Genre | : Mathematics |
ISBN | : 3642213995 |
Random walks, Markov chains and electrical networks serve as an introduction to the study of real-valued functions on finite or infinite graphs, with appropriate interpretations using probability theory and current-voltage laws. The relation between this type of function theory and the (Newton) potential theory on the Euclidean spaces is well-established. The latter theory has been variously generalized, one example being the axiomatic potential theory on locally compact spaces developed by Brelot, with later ramifications from Bauer, Constantinescu and Cornea. A network is a graph with edge-weights that need not be symmetric. This book presents an autonomous theory of harmonic functions and potentials defined on a finite or infinite network, on the lines of axiomatic potential theory. Random walks and electrical networks are important sources for the advancement of the theory.
Author | : Matthew Baker |
Publisher | : American Mathematical Soc. |
Total Pages | : 466 |
Release | : 2010 |
Genre | : Mathematics |
ISBN | : 082187540X |