Linear Holomorphic Partial Differential Equations And Classical Potential Theory

Author : Dmitry Khavinson,Erik Lundberg
Publisher : American Mathematical Soc.
Page : 214 pages
File Size : 42,7 Mb
Release : 2018-07-09
Category : Differential equations, Linear
ISBN : 9781470437800

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Linear Holomorphic Partial Differential Equations and Classical Potential Theory by Dmitry Khavinson,Erik Lundberg Pdf

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.

Author : Dmitry Khavinson
Publisher : Unknown
Page : 123 pages
File Size : 53,5 Mb
Release : 1996-06-01
Category : Potential theory (Mathematics)
ISBN : 8460093239

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Holomorphic Partial Differential Equations and Classical Potential Theory by Dmitry Khavinson Pdf

Author : Josef Kral,Jaroslav Lukes,Ivan Netuka,Jiri Vesely
Publisher : Springer
Page : 276 pages
File Size : 45,8 Mb
Release : 2007-02-08
Category : Mathematics
ISBN : 9783540459521

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Potential Theory, Surveys and Problems by Josef Kral,Jaroslav Lukes,Ivan Netuka,Jiri Vesely Pdf

The volume comprises eleven survey papers based on survey lectures delivered at the Conference in Prague in July 1987, which covered various facets of potential theory, including its applications in other areas. The survey papers deal with both classical and abstract potential theory and its relations to partial differential equations, stochastic processes and other branches such as numerical analysis and topology. A collection of problems from potential theory, compiled on the occasion of the conference, is included, with additional commentaries, in the second part of this volume.

Author : K. GowriSankaran,J. Bliedtner,D. Feyel,M. Goldstein,W.K. Hayman,I. Netuka
Publisher : Springer Science & Business Media
Page : 467 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401111386

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Classical and Modern Potential Theory and Applications by K. GowriSankaran,J. Bliedtner,D. Feyel,M. Goldstein,W.K. Hayman,I. Netuka Pdf

Proceedings of the NATO Advanced Research Workshop, Château de Bonas, France, July 25--31, 1993

Author : Lester Helms
Publisher : Springer Science & Business Media
Page : 442 pages
File Size : 55,5 Mb
Release : 2009-05-27
Category : Mathematics
ISBN : 9781848823198

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Potential Theory by Lester Helms Pdf

The ?rst six chapters of this book are revised versions of the same chapters in the author’s 1969 book, Introduction to Potential Theory. Atthetimeof the writing of that book, I had access to excellent articles,books, and lecture notes by M. Brelot. The clarity of these works made the task of collating them into a single body much easier. Unfortunately, there is not a similar collection relevant to more recent developments in potential theory. A n- comer to the subject will ?nd the journal literature to be a maze of excellent papers and papers that never should have been published as presented. In the Opinion Column of the August, 2008, issue of the Notices of the Am- ican Mathematical Society, M. Nathanson of Lehman College (CUNY) and (CUNY) Graduate Center said it best “. . . When I read a journal article, I often ?nd mistakes. Whether I can ?x them is irrelevant. The literature is unreliable. ” From time to time, someone must try to ?nd a path through the maze. In planning this book, it became apparent that a de?ciency in the 1969 book would have to be corrected to include a discussion of the Neumann problem, not only in preparation for a discussion of the oblique derivative boundary value problem but also to improve the basic part of the subject matter for the end users, engineers, physicists, etc.

Author : David Hoff
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 53,7 Mb
Release : 2020-11-18
Category : Education
ISBN : 9781470461614

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Linear and Quasilinear Parabolic Systems: Sobolev Space Theory by David Hoff Pdf

This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.

Author : Walter K. Hayman,Eleanor F. Lingham
Publisher : Springer Nature
Page : 284 pages
File Size : 52,7 Mb
Release : 2019-09-07
Category : Mathematics
ISBN : 9783030251659

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Research Problems in Function Theory by Walter K. Hayman,Eleanor F. Lingham Pdf

In 1967 Walter K. Hayman published ‘Research Problems in Function Theory’, a list of 141 problems in seven areas of function theory. In the decades following, this list was extended to include two additional areas of complex analysis, updates on progress in solving existing problems, and over 520 research problems from mathematicians worldwide. It became known as ‘Hayman's List’. This Fiftieth Anniversary Edition contains the complete ‘Hayman's List’ for the first time in book form, along with 31 new problems by leading international mathematicians. This list has directed complex analysis research for the last half-century, and the new edition will help guide future research in the subject. The book contains up-to-date information on each problem, gathered from the international mathematics community, and where possible suggests directions for further investigation. Aimed at both early career and established researchers, this book provides the key problems and results needed to progress in the most important research questions in complex analysis, and documents the developments of the past 50 years.

Author : Lars Hörmander
Publisher : Springer Science & Business Media
Page : 424 pages
File Size : 40,9 Mb
Release : 2007-06-25
Category : Mathematics
ISBN : 9780817645854

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Notions of Convexity by Lars Hörmander Pdf

The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.

Author : John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce
Publisher : American Mathematical Soc.
Page : 300 pages
File Size : 51,6 Mb
Release : 2019-04-12
Category : Geometry, Differential
ISBN : 9781470450144

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Virtual Fundamental Cycles in Symplectic Topology by John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce Pdf

The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov's original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds.

Author : Bernard Host,Bryna Kra
Publisher : American Mathematical Soc.
Page : 427 pages
File Size : 45,5 Mb
Release : 2018-12-12
Category : Ergodic theory
ISBN : 9781470447809

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Nilpotent Structures in Ergodic Theory by Bernard Host,Bryna Kra Pdf

Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behavior of multiple ergodic averages. This book is a comprehensive treatment of their role in ergodic theory, covering development of the abstract theory leading to the structural statements, applications of these results, and connections to other fields. Starting with a summary of the relevant dynamical background, the book methodically develops the theory of cubic structures that give rise to nilpotent groups and reviews results on nilsystems and their properties that are scattered throughout the literature. These basic ingredients lay the groundwork for the ergodic structure theorems, and the book includes numerous formulations of these deep results, along with detailed proofs. The structure theorems have many applications, both in ergodic theory and in related fields; the book develops the connections to topological dynamics, combinatorics, and number theory, including an overview of the role of nilsystems in each of these areas. The final section is devoted to applications of the structure theory, covering numerous convergence and recurrence results. The book is aimed at graduate students and researchers in ergodic theory, along with those who work in the related areas of arithmetic combinatorics, harmonic analysis, and number theory.

Author : Alexandru Aleman,Haakan Hedenmalm,Dmitry Khavinson,Mihai Putinar
Publisher : Springer
Page : 281 pages
File Size : 52,8 Mb
Release : 2019-05-30
Category : Mathematics
ISBN : 9783030146405

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Analysis of Operators on Function Spaces by Alexandru Aleman,Haakan Hedenmalm,Dmitry Khavinson,Mihai Putinar Pdf

This book contains both expository articles and original research in the areas of function theory and operator theory. The contributions include extended versions of some of the lectures by invited speakers at the conference in honor of the memory of Serguei Shimorin at the Mittag-Leffler Institute in the summer of 2018. The book is intended for all researchers in the fields of function theory, operator theory and complex analysis in one or several variables. The expository articles reflecting the current status of several well-established and very dynamical areas of research will be accessible and useful to advanced graduate students and young researchers in pure and applied mathematics, and also to engineers and physicists using complex analysis methods in their investigations.

Author : Karim Belabas,Henri Cohen
Publisher : American Mathematical Soc.
Page : 429 pages
File Size : 48,5 Mb
Release : 2021-06-23
Category : Education
ISBN : 9781470463519

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Numerical Algorithms for Number Theory: Using Pari/GP by Karim Belabas,Henri Cohen Pdf

This book presents multiprecision algorithms used in number theory and elsewhere, such as extrapolation, numerical integration, numerical summation (including multiple zeta values and the Riemann-Siegel formula), evaluation and speed of convergence of continued fractions, Euler products and Euler sums, inverse Mellin transforms, and complex L L-functions. For each task, many algorithms are presented, such as Gaussian and doubly-exponential integration, Euler-MacLaurin, Abel-Plana, Lagrange, and Monien summation. Each algorithm is given in detail, together with a complete implementation in the free Pari/GP system. These implementations serve both to make even more precise the inner workings of the algorithms, and to gently introduce advanced features of the Pari/GP language. This book will be appreciated by anyone interested in number theory, specifically in practical implementations, computer experiments and numerical algorithms that can be scaled to produce thousands of digits of accuracy.

Author : Lester L. Helms
Publisher : Springer Science & Business Media
Page : 494 pages
File Size : 46,5 Mb
Release : 2014-04-10
Category : Mathematics
ISBN : 9781447164227

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Potential Theory by Lester L. Helms Pdf

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 : Nicola Arcozzi,Richard Rochberg,Eric T. Sawyer,Brett D. Wick
Publisher : American Mathematical Soc.
Page : 536 pages
File Size : 49,5 Mb
Release : 2019-09-03
Category : Dirichlet principle
ISBN : 9781470450823

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The Dirichlet Space and Related Function Spaces by Nicola Arcozzi,Richard Rochberg,Eric T. Sawyer,Brett D. Wick Pdf

The study of the classical Dirichlet space is one of the central topics on the intersection of the theory of holomorphic functions and functional analysis. It was introduced about100 years ago and continues to be an area of active current research. The theory is related to such important themes as multipliers, reproducing kernels, and Besov spaces, among others. The authors present the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem. The first part of this book is an introduction to the function theory and operator theory of the classical Dirichlet space, a space of holomorphic functions on the unit disk defined by a smoothness criterion. The Dirichlet space is also a Hilbert space with a reproducing kernel, and is the model for the dyadic Dirichlet space, a sequence space defined on the dyadic tree. These various viewpoints are used to study a range of topics including the Pick property, multipliers, Carleson measures, boundary values, zero sets, interpolating sequences, the local Dirichlet integral, shift invariant subspaces, and Hankel forms. Recurring themes include analogies, sometimes weak and sometimes strong, with the classical Hardy space; and the analogy with the dyadic Dirichlet space. The final chapters of the book focus on Besov spaces of holomorphic functions on the complex unit ball, a class of Banach spaces generalizing the Dirichlet space. Additional techniques are developed to work with the nonisotropic complex geometry, including a useful invariant definition of local oscillation and a sophisticated variation on the dyadic Dirichlet space. Descriptions are obtained of multipliers, Carleson measures, interpolating sequences, and multiplier interpolating sequences; estimates are obtained to prove corona theorems.

Author : Pramod N. Achar
Publisher : American Mathematical Soc.
Page : 562 pages
File Size : 46,6 Mb
Release : 2021-09-27
Category : Education
ISBN : 9781470455972

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Perverse Sheaves and Applications to Representation Theory by Pramod N. Achar Pdf

Since its inception around 1980, the theory of perverse sheaves has been a vital tool of fundamental importance in geometric representation theory. This book, which aims to make this theory accessible to students and researchers, is divided into two parts. The first six chapters give a comprehensive account of constructible and perverse sheaves on complex algebraic varieties, including such topics as Artin's vanishing theorem, smooth descent, and the nearby cycles functor. This part of the book also has a chapter on the equivariant derived category, and brief surveys of side topics including étale and ℓ-adic sheaves, D-modules, and algebraic stacks. The last four chapters of the book show how to put this machinery to work in the context of selected topics in geometric representation theory: Kazhdan-Lusztig theory; Springer theory; the geometric Satake equivalence; and canonical bases for quantum groups. Recent developments such as the p-canonical basis are also discussed. The book has more than 250 exercises, many of which focus on explicit calculations with concrete examples. It also features a 4-page “Quick Reference” that summarizes the most commonly used facts for computations, similar to a table of integrals in a calculus textbook.