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Author: Lucio Prado Publisher: LAP Lambert Academic Publishing ISBN: 9783838334967 Category : Languages : en Pages : 156
Book Description
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: Lucio Prado Publisher: LAP Lambert Academic Publishing ISBN: 9783838334967 Category : Languages : en Pages : 156
Book Description
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: Paolo M. Soardi Publisher: Springer ISBN: 3540487980 Category : Mathematics Languages : en Pages : 199
Book Description
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: Lester L. Helms Publisher: Springer Science & Business Media ISBN: 1447164229 Category : Mathematics Languages : en Pages : 494
Book Description
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: Josef Kral Publisher: Walter de Gruyter ISBN: 3110818574 Category : Mathematics Languages : en Pages : 513
Book Description
The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
Author: Matthew Baker Publisher: American Mathematical Soc. ISBN: 0821849247 Category : Mathematics Languages : en Pages : 466
Book Description
The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and ``elementary'' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices--on analysis, $\mathbb{R}$-trees, and Berkovich's general theory of analytic spaces--are included to make the book as self-contained as possible. The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of $p$-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szego theorem and Bilu's equidistribution theorem.
Author: Anders Björn Publisher: European Mathematical Society ISBN: 9783037190999 Category : Harmonic functions Languages : en Pages : 422
Book Description
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: Victor Anandam Publisher: Springer Science & Business Media ISBN: 3642213995 Category : Mathematics Languages : en Pages : 152
Book Description
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: Paolo Maurizio Soardi Publisher: Springer Verlag ISBN: Category : Mathematics Languages : en Pages : 187
Book Description
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 Prado
Author: Lucio Prado
Author: Paolo M. Soardi
Author: Lester L. Helms
Author: Lucio M-G. Prado
Author: M. Picardello
Author: Josef Kral
Author: Matthew Baker
Author: Anders Björn
Author: Victor Anandam
Author: Paolo Maurizio Soardi