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It offers many features rarely treated in introductory p-adic texts such as topological models of p-adic spaces inside Euclidian space, a special case of Hazewinkel's functional equation lemma, and a treatment of analytic elements.

Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory.

This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat's Last Theorem, and Iwasawa's theory of Z_p-extensions.

All three theorems can be viewed as special cases of a general theorem about functions qJ on abelian semigroups with involution (S, +, *) which are positive definite in the sense that the matrix (qJ(sJ + Sk" is positive definite for all finite choices of elements St, .

This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field.

Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from basic notions of combinatorial enumeration to a variety of topics, ranging from algebra to statistical physics.

Suitable for a one-semester graduate course in integration theory.

The origins of the mathematics in this book date back more than two thou sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu clid.

This textbook on homology and cohomology theory is geared towards the beginning graduate student.

This book provides a rigorous but elementary introduction to the theory of Markov Processes on a countable state space. Topics covered include: Doeblin's theory, general ergodic properties, and continuous time processes.

This is the third version of a book on differential manifolds. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations.

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses.

Banach algebras are Banach spaces equipped with a continuous multipli- tion. Much of operator theory is best approached from a Banach algebra point of view and many questions in complex analysis (such as approximation by polynomials or rational functions in speci?c - mains) are best understood within the framework of Banach algebras.

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites.

Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.

This is a book about physics, written for mathematicians. appreciate mathematical elarity, but are willing to accept physical motiva tions for the mathematics in place of mathematical ones 4.

The main additions consist of a section on convergence of probabilities on metric spaces and a chapter whose first section on domains of attrac tion completes the study of the Central limit problem, while the second one is devoted to random walks.

This text has as its object an introduction to elements of the theory of random processes. Previous knowledge of probability theory is not required since the discussion starts with the basic notions of probability theory.

In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH.

Oxtoby Bryn Mawr, April 1980 Preface to the First Edition This book has two main themes: the Baire category theorem as a method for proving existence, and the "duality" between measure and category.

The coverage of topics and exposition style are designed to leave no gaps in understanding and stimulate further study.This third edition includes new Sections 3.5, 4.4, 4.5 as well as a new chapter on "Weighted Inequalities," which has been moved from GTM 250, 2nd Edition.

This text is aimed at graduate students in mathematics and to interested researchers who wish to acquire an in depth understanding of Euclidean Harmonic analysis.

Chapters in the remainder of the text can be read independently and explore specific aspects of the theory of binomial ideals, including edge rings and edge polytopes, join-meet ideals of finite lattices, binomial edge ideals, ideals generated by 2-minors, and binomial ideals arising from statistics.

Developed over years of classroom use, this textbook provides a clear and accessible approach to real analysis. This modern interpretation is based on the author's lecture notes and has been meticulously tailored to motivate students and inspire readers to explore the material, and to continue exploring even after they have finished the book. The definitions, theorems, and proofs contained within are presented with mathematical rigor, but conveyed in an accessible manner and with language and motivation meant for students who have not taken a previous course on this subject.The text covers all of the topics essential for an introductory course, including Lebesgue measure, measurable functions, Lebesgue integrals, differentiation, absolute continuity, Banach and Hilbert spaces, and more. Throughout each chapter, challenging exercises are presented, and the end of each section includes additional problems. Such an inclusive approach creates an abundance of opportunities for readers to develop their understanding, and aids instructors as they plan their coursework. Additional resources are available online, including expanded chapters, enrichment exercises, a detailed course outline, and much more.Introduction to Real Analysis is intended for first-year graduate students taking a first course in real analysis, as well as for instructors seeking detailed lecture material with structure and accessibility in mind. Additionally, its content is appropriate for Ph.D. students in any scientific or engineering discipline who have taken a standard upper-level undergraduate real analysis course.

Since all the specific functions that one studied were determined by simple analytic expressions, Baire delineated those functions that can be constructed starting from con tinuous functions and iterating the operation 0/ pointwise limit on a se quence 0/ functions.

Chapters in the remainder of the text can be read independently and explore specific aspects of the theory of binomial ideals, including edge rings and edge polytopes, join-meet ideals of finite lattices, binomial edge ideals, ideals generated by 2-minors, and binomial ideals arising from statistics.

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted."