Analysis

Analysis
by Elliott H. Lieb, Michael Loss

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

How to Think about Analysis
by Lara Alcock

Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the student’s existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these.

The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.


The Way of Analysis
by Robert S. Strichartz

The Way of Analysis gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction of the Lebesgue integral. The text provides proofs of all main results, as well as motivations, examples, applications, exercises, and formal chapter summaries. Additionally, there are three chapters on application of analysis, ordinary differential equations, Fourier series, and curves and surfaces to show how the techniques of analysis are used in concrete settings.

Analysis I
by Herbert Amann, Joachim Escher

Logical thinking, the analysis of complex relationships, the recognition of und- lying simple structures which are common to a multitude of problems — these are the skills which are needed to do mathematics, and their development is the main goal of mathematics education. Of course, these skills cannot be learned ‘in a vacuum’. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential, without being distracted by appearances and irrelevancies. The present book strives for clarity and transparency. Right from the beg- ning, it requires from the reader a willingness to deal with abstract concepts, as well as a considerable measure of self-initiative. For these e?orts, the reader will be richly rewarded in his or her mathematical thinking abilities, and will possess the foundation needed for a deeper penetration into mathematics and its applications. Thisbookisthe?rstvolumeofathreevolumeintroductiontoanalysis.It- veloped from courses that the authors have taught over the last twenty six years at theUniversitiesofBochum,Kiel,Zurich,BaselandKassel.Sincewehopethatthis book will be used also for self-study and supplementary reading, we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses, not only elegance and inner beauty, but also provides e?cient methods for the solution of concrete problems.

Analysis I
by Terence Tao

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.


Handbook of Water Analysis
by Leo M.L. Nollet, Leen S. P. De Gelder

This work details water sampling and preservation methods by enumerating the different ways to measure physical, chemical, organoleptical, and radiological characteristics. It provides step-by-step descriptions of separation, residue determination, and cleanup techniques for a variety of fresh- and salt-waters. It also discusses information regarding the analysis and detection of bacteria and algae.

Data Collection and Analysis
by Roger Sapsford, Victor Jupp

In simple and non-technical terms, the Second Edition of Data Collection and Analysis illustrates a wide range of techniques and approaches used in social research projects. Always accessible and engaging, this comprehensive text covers both quantitative and qualitative approaches to data collection and analysis in social research, considering both the structure and logic of research projects and the ethics and politics of research.

A wide range of examples illustrate the text and a set of exercises runs throughout the book to aid the reader in understanding and planning research projects.

Building on the strengths of the First Edition, this new and expanded version includes:

– The addition of chapter introductions, summaries and key terms to guide the reader through the text

– Three brand new chapters focusing on: research and information on the Net; discourse research; ethnographic and discursive qualitative analysis

– Up-to-date examples of research in action

– New material on questionnaire design, composite measurement and techniques of quantitative and qualitative interviewing

An invaluable guide for students from across the social sciences, this wide-ranging volume is also a key resource for practitioners in a variety of applied areas including nursing, social work, the criminal justice system, teaching and education.


Analysis I
by Roger Godement

Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.


Dictionary of the Theatre
by Patrice Pavis

Patrice Pavis is one of France’s most brilliant academics and a leading expert internationally in the theory of theatre. Dictionary of the Theatre is an English translation of Pavis’s acclaimed Dictionnaire du théâtre, now in its second printing in France.

This encyclopedic dictionary includes theoretical, technical, and semiotic terms and concepts. Alphabetical entries range from ‘absurd’ to ‘word scenery’ and treat the reader to a vast panoply of theatre and theory. The extended discussions are supported by useful examples drawn from the international repertoire of plays and playwrights, both classic and contemporary. The Foreword is by Marvin Carlson.

This dictionary is remarkably well integrated, partly because of its excellent system of cross-referencing, but also because it represents the vision and scholarship of a single, recognized authority. There is no other source like it available and it will be warmly welcomed by the English-language theatre world.



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