Analysis

Author: Elliott H. Lieb,Michael Loss

Publisher: American Mathematical Soc.

ISBN: 0821827839

Category: Mathematics

Page: 346

View: 9778

This is an excellent textbook on analysis and it has several unique features: Proofs of heat kernel estimates, the Nash inequality and the logarithmic Sobolev inequality are topics that are seldom treated on the level of a textbook. Best constants in several inequalities, such as Young's inequality and the logarithmic Sobolev inequality, are also included. A thorough treatment of rearrangement inequalities and competing symmetries appears in book form for the first time. There is an extensive treatment of potential theory and its applications to quantum mechanics, which, again, is unique at this level. Uniform convexity of $L^p$ space is treated very carefully. The presentation of this important subject is highly unusual for a textbook. All the proofs provide deep insights into the theorems. This book sets a new standard for a graduate textbook in analysis. --Shing-Tung Yau, Harvard University For some number of years, Rudin's ``Real and Complex'', and a few other analysis books, served as the canonical choice for the book to use, and to teach from, in a first year grad analysis course. Lieb-Loss offers a refreshing alternative: It begins with a down-to-earth intro to measure theory, $L^p$ and all that ... It aims at a wide range of essential applications, such as the Fourier transform, and series, inequalities, distributions, and Sobolev spaces--PDE, potential theory, calculus of variations, and math physics (Schrodinger's equation, the hydrogen atom, Thomas-Fermi theory ... to mention a few). The book should work equally well in a one-, or in a two-semester course. The first half of the book covers the basics, and the rest will be great for students to have, regardless of whether or not it gets to be included in a course. --Palle E. T. Jorgensen, University of Iowa

Semiclassical Analysis

Author: Maciej Zworski

Publisher: American Mathematical Soc.

ISBN: 0821883208

Category: Mathematics

Page: 431

View: 5513

This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. --Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.

Applied Asymptotic Analysis

Author: Peter David Miller

Publisher: American Mathematical Soc.

ISBN: 0821840789

Category: Mathematics

Page: 467

View: 7598

"The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects."--BOOK JACKET.

Higher Order Fourier Analysis

Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821889869

Category: Mathematics

Page: 187

View: 4603

Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemeredi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems. This book, which is the first monograph in this area, aims to cover all of these topics in a unified manner, as well as to survey some of the most recent developments, such as the application of the theory to count linear patterns in primes. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one's knowledge.

Fourier Analysis

Author: Javier Duoandikoetxea Zuazo

Publisher: American Mathematical Soc.

ISBN: 9780821883846

Category: Mathematics

Page: 222

View: 1447


A Companion to Analysis

A Second First and First Second Course in Analysis

Author: Thomas William Körner

Publisher: American Mathematical Soc.

ISBN: 0821834479

Category: Mathematics

Page: 590

View: 4085

This book not only provides a lot of solid information about real analysis, it also answers those questions which students want to ask but cannot figure how to formulate. To read this book is to spend time with one of the modern masters in the subject. --Steven G. Krantz, Washington University, St. Louis One of the major assets of the book is Korner's very personal writing style. By keeping his own engagement with the material continually in view, he invites the reader to a similarly high level of involvement. And the witty and erudite asides that are sprinkled throughout the book are a real pleasure. --Gerald Folland, University of Washingtion, Seattle Many students acquire knowledge of a large number of theorems and methods of calculus without being able to say how they hang together. This book provides such students with the coherent account that they need. A Companion to Analysis explains the problems which must be resolved in order to obtain a rigorous development of the calculus and shows the student how those problems are dealt with. Starting with the real line, it moves on to finite dimensional spaces and then to metric spaces. Readers who work through this text will be ready for such courses as measure theory, functional analysis, complex analysis and differential geometry. Moreover, they will be well on the road which leads from mathematics student to mathematician. Able and hard working students can use this book for independent study, or it can be used as the basis for an advanced undergraduate or elementary graduate course. An appendix contains a large number of accessible but non-routine problems to improve knowledge and technique.

A Course in Abstract Analysis

Author: John B. Conway

Publisher: American Mathematical Soc.

ISBN: 0821890832

Category: Mathematics

Page: 367

View: 7164

This book covers topics appropriate for a first-year graduate course preparing students for the doctorate degree. The first half of the book presents the core of measure theory, including an introduction to the Fourier transform. This material can easily be covered in a semester. The second half of the book treats basic functional analysis and can also be covered in a semester. After the basics, it discusses linear transformations, duality, the elements of Banach algebras, and C*-algebras. It concludes with a characterization of the unitary equivalence classes of normal operators on a Hilbert space. The book is self-contained and only relies on a background in functions of a single variable and the elements of metric spaces. Following the author's belief that the best way to learn is to start with the particular and proceed to the more general, it contains numerous examples and exercises.

Functional Analysis

An Introduction

Author: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

Publisher: American Mathematical Soc.

ISBN: 0821836463

Category: Mathematics

Page: 322

View: 4169

The goal of this textbook is to provide an introduction to the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators, and spectral theory of self-adjoint operators. It also presents the basic theorems and methods of abstract functional analysis and a few applications of these methods to Banach algebras and the theory of unbounded self-adjoint operators. The text corresponds to material for two semester courses (Part I and Part II, respectively), and it is as self-contained as possible. The only prerequisites for the first part are minimal amounts of linear algebra and calculus. However, for the second course (Part II), it is useful to have some knowledge of topology and measure theory. Each chapter is followed by numerous exercises, whose solutions are given at the end of the book.

Global Analysis

Differential Forms in Analysis, Geometry, and Physics

Author: Ilka Agricola,Thomas Friedrich

Publisher: American Mathematical Soc.

ISBN: 0821829513

Category: Mathematics

Page: 343

View: 4762

This book introduces the reader to the world of differential forms and their uses in geometry, analysis, and mathematical physics. It begins with a few basic topics, partly as review, then moves on to vector analysis on manifolds and the study of curves and surfaces in $3$-space. Lie groups and homogeneous spaces are discussed, providing the appropriate framework for introducing symmetry in both mathematical and physical contexts. The final third of the book applies the mathematical ideas to important areas of physics: Hamiltonian mechanics, statistical mechanics, and electrodynamics. There are many classroom-tested exercises and examples with excellent figures throughout. The book is ideal as a text for a first course in differential geometry, suitable for advanced undergraduates or graduate students in mathematics or physics.

Stochastic Analysis on Manifolds

Author: Elton P. Hsu

Publisher: American Mathematical Soc.

ISBN: 0821808028

Category: Mathematics

Page: 281

View: 9253

Concerned with probability theory, Elton Hsu's study focuses primarily on the relations between Brownian motion on a manifold and analytical aspects of differential geometry. A key theme is the probabilistic interpretation of the curvature of a manifold

A First Course in Sobolev Spaces

Author: Giovanni Leoni

Publisher: American Mathematical Soc.

ISBN: 0821847686

Category: Mathematics

Page: 607

View: 3408

Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis. The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables. The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces. The book contains over 200 exercises.

Mathematics of Probability

Author: Daniel W. Stroock

Publisher: American Mathematical Soc.

ISBN: 1470409070

Category: Mathematics

Page: 284

View: 2485

This book covers the basics of modern probability theory. It begins with probability theory on finite and countable sample spaces and then passes from there to a concise course on measure theory, which is followed by some initial applications to probability theory, including independence and conditional expectations. The second half of the book deals with Gaussian random variables, with Markov chains, with a few continuous parameter processes, including Brownian motion, and, finally, with martingales, both discrete and continuous parameter ones. The book is a self-contained introduction to probability theory and the measure theory required to study it.

Complex Made Simple

Author: David C. Ullrich

Publisher: American Mathematical Soc.

ISBN: 0821844792

Category: Mathematics

Page: 489

View: 6209

Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from? In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work. Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains.Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem. This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.

A Course in Operator Theory

Author: John B. Conway

Publisher: American Mathematical Soc.

ISBN: 0821820656

Category: Mathematics

Page: 372

View: 3559

A new volume in the marquee series of the AMS, featuring broad mathematical topics written by some of the best and brightest that the mathematics field has to offer. All titles have attractive hardcovers and market-oriented prices.

Introduction to Fourier Analysis and Wavelets

Author: Mark A. Pinsky

Publisher: American Mathematical Soc.

ISBN: 082184797X

Category: Mathematics

Page: 376

View: 5100

This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs-Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere. Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces $L^p(\mathbb{R}^n)$. Chapter 4 gives a gentle introduction to these results, using the Riesz-Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry-Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the $L_2$ theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis. The text contains numerous examples and more than 200 exercises, each located in close proximity to the related theoretical material.

Teaching Mathematics in Colleges and Universities

Case Studies for Today's Classroom

Author: Solomon Friedberg,Conference Board of the Mathematical Sciences

Publisher: American Mathematical Soc.

ISBN: 9780821828236

Category: Mathematics

Page: 67

View: 5046

Progress in mathematics frequently occurs first by studying particular examples and then by generalizing the patterns that have been observed into far-reaching theorems. Similarly, in teaching mathematics one often employs examples to motivate a general principle or to illustrate its use. This volume uses the same idea in the context of learning how to teach: By analyzing particular teaching situations, one can develop broadly applicable teaching skills useful for the professional mathematician. These teaching situations are the Case Studies of the title. Just as a good mathematician seeks both to understand the details of a particular problem and to put it in a broader context, the examples presented here are chosen to offer a serious set of detailed teaching issues and to afford analysis from a broad perspective. Each case raises a variety of pedagogical and communication issues that may be explored either individually or in a group facilitated by a faculty member. The methodology of Case Studies is widely used in areas such as business and law. The consideration of the mathematics cases presented here will help readers to develop teaching skills for their own classrooms. See the faculty edition at Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom: Faculty Edition

Global Calculus

Author: S. Ramanan

Publisher: American Mathematical Soc.

ISBN: 0821837028

Category: Mathematics

Page: 316

View: 6010

The power that analysis, topology and algebra bring to geometry has revolutionised the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on Global Analysis.

Functional Analysis

An Elementary Introduction

Author: Markus Haase

Publisher: American Mathematical Society

ISBN: 0821891715

Category: Mathematics

Page: 372

View: 7648

This book introduces functional analysis at an elementary level without assuming any background in real analysis, for example on metric spaces or Lebesgue integration. It focuses on concepts and methods relevant in applied contexts such as variational methods on Hilbert spaces, Neumann series, eigenvalue expansions for compact self-adjoint operators, weak differentiation and Sobolev spaces on intervals, and model applications to differential and integral equations. Beyond that, the final chapters on the uniform boundedness theorem, the open mapping theorem and the Hahn-Banach theorem provide a stepping-stone to more advanced texts. The exposition is clear and rigorous, featuring full and detailed proofs. Many examples illustrate the new notions and results. Each chapter concludes with a large collection of exercises, some of which are referred to in the margin of the text, tailor-made in order to guide the student digesting the new material. Optional sections and chapters supplement the mandatory parts and allow for modular teaching spanning from basic to honors track level.

A Course in Complex Analysis and Riemann Surfaces

Author: Wilhelm Schlag

Publisher: American Mathematical Society

ISBN: 0821898477

Category: Mathematics

Page: 384

View: 634

Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.

Problems in Real and Functional Analysis

Author: Alberto Torchinsky

Publisher: American Mathematical Soc.

ISBN: 1470420570

Category: Functional analysis

Page: 467

View: 5630

It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating the needed definitions and basic results in the area and closes with a short description of the problems. - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpuf It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating the needed definitions and basic results in the area and closes with a short description of the problems. The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most "natural" rather than the most elegant solution is presented. The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most “natural” rather than the most elegant solution is presented. - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpufhe Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpuft is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpufIt is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpuf