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Potential Theory in the Complex Plane

Potential Theory in the Complex Plane Author Thomas Ransford
ISBN-10 0521466547
Release 1995-03-16
Pages 232
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Ransford provides an introduction to the subject, concentrating on the important case of two dimensions, and emphasizing its links with complex analysis. This is reflected in the large number of applications, which include Picard's theorem, the Phragmén-Lindelöf principle, the Radó-Stout theorem, Lindelöf's theory of asymptotic values, the Riemann mapping theorem (including continuity at the boundary), the Koebe one-quarter theorem, Hilbert's lemniscate theorem, and the sharp quantitative form of Runge's theorem. In addition, there is a chapter on connections with functional analysis and dynamical systems, which shows how the theory can be applied to other parts of mathematics and gives a flavor of some recent research in the area.



Potential Theory and Dynamics on the Berkovich Projective Line

Potential Theory and Dynamics on the Berkovich Projective Line Author Matthew Baker
ISBN-10 9780821849248
Release 2010-03-10
Pages 428
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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.



Linear Holomorphic Partial Differential Equations and Classical Potential Theory

Linear Holomorphic Partial Differential Equations and Classical Potential Theory Author Dmitry Khavinson
ISBN-10 9781470437800
Release 2018-07-09
Pages 214
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Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? Is there a reflection principle for harmonic functions in higher dimensions similar to the Schwarz reflection principle in the plane? How far outside of their natural domains can solutions of the Dirichlet problem be extended? Where do the continued solutions become singular and why? This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results illustrating a nice interplay between parts of modern analysis and themes in “physical” mathematics of the nineteenth century. To make the book accessible to a wide audience including students, the authors do not assume expertise in the theory of holomorphic PDE, and most of the book is accessible to anyone familiar with multivariable calculus and some basics in complex analysis and differential equations.



Modern Trends in Constructive Function Theory

Modern Trends in Constructive Function Theory Author E. B. Saff
ISBN-10 9781470425340
Release 2016-03-31
Pages 297
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This volume contains the proceedings of the conference Constructive Functions 2014, held from May 26-30, 2014, at Vanderbilt University, Nashville, TN, in honor of Ed Saff's 70th birthday. The papers in this volume contain results on polynomial approximation, rational approximation, Log-optimal configurations on the sphere, random continued fractions, ratio asymptotics for multiple orthogonal polynomials, the bivariate trigonometric moment problem, minimal Riesz energy, random polynomials, Pade and Hermite-Pade approximation, orthogonal expansions, hyperbolic differential equations, Bergman polynomials, the Meijer $G$-function, polynomial ensembles, and integer lattice points.



Introduction to Banach Spaces and Algebras

Introduction to Banach Spaces and Algebras Author Graham R. Allan
ISBN-10 9780199206537
Release 2011
Pages 371
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A graduate level text in functional analysis, with an emphasis on Banach algebras. Based on lectures given for Part III of the Cambridge Mathematical Tripos, the text will assume a familiarity with elementary real and complex analysis, and some acquaintance with metric spaces, analytic topology and normed spaces (but not theorems depending on Baire category, or any version of the Hahn-Banach theorem).



Complex Dynamics

Complex Dynamics Author Lennart Carleson
ISBN-10 9781461243649
Release 2013-11-11
Pages 192
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A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.



Representation Theorems in Hardy Spaces

Representation Theorems in Hardy Spaces Author Javad Mashreghi
ISBN-10 9780521517683
Release 2009-03-19
Pages 372
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This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane.



Dynamical Systems and Ergodic Theory

Dynamical Systems and Ergodic Theory Author Mark Pollicott
ISBN-10 0521575990
Release 1998-01-29
Pages 179
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This book is essentially a self-contained introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. Parts of the book are suitable for a final year undergraduate course or for a master's level course. A number of applications are given, principally to number theory and arithmetic progressions (through van der Waerden's theorem and Szemerdi's theorem).



Set Theory

Set Theory Author A. Hajnal
ISBN-10 1107362555
Release 2014-05-14
Pages 325
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This is a classic introduction to set theory, from the basics through to the modern tools of combinatorial set theory.



Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Author Rick Miranda
ISBN-10 9780821802687
Release 1995
Pages 390
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The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.



The Prime Number Theorem

The Prime Number Theorem Author G. J. O. Jameson
ISBN-10 0521891108
Release 2003-04-17
Pages 252
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The prime numbers appear to be distributed in a very irregular way amongst the integers, but the prime number theorem provides a simple formula that tells us (in an approximate but well-defined sense) how many primes we can expect to find that are less than any integer we might choose. This is indisputably one of the the great classical theorems of mathematics. Suitable for advanced undergraduates and beginning graduates, this textbook demonstrates how the tools of analysis can be used in number theory to attack a famous problem.



Journal Canadien de Math matiques

Journal Canadien de Math  matiques Author
ISBN-10 UCSD:31822036024362
Release 2007
Pages
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Journal Canadien de Math matiques has been writing in one form or another for most of life. You can find so many inspiration from Journal Canadien de Math matiques also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Journal Canadien de Math matiques book for free.



Complex Algebraic Curves

Complex Algebraic Curves Author Frances Clare Kirwan
ISBN-10 0521423538
Release 1992-02-20
Pages 264
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This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.



CMFT

CMFT Author
ISBN-10 UOM:39015072634119
Release 2008
Pages
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CMFT has been writing in one form or another for most of life. You can find so many inspiration from CMFT also informative, and entertaining. Click DOWNLOAD or Read Online button to get full CMFT book for free.



Proceedings

Proceedings Author
ISBN-10 UCSD:31822036334803
Release 2009
Pages
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Proceedings has been writing in one form or another for most of life. You can find so many inspiration from Proceedings also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Proceedings book for free.



Mathematics of Computation

Mathematics of Computation Author
ISBN-10 UCSD:31822035033281
Release 2006
Pages
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Mathematics of Computation has been writing in one form or another for most of life. You can find so many inspiration from Mathematics of Computation also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Mathematics of Computation book for free.



Automorphisms of Surfaces After Nielsen and Thurston

Automorphisms of Surfaces After Nielsen and Thurston Author Andrew J. Casson
ISBN-10 0521349850
Release 1988
Pages 104
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A comprehensive introduction to selected aspects of modern low-dimensional topology for readers with a knowledge of basic algebra.