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Mod Two Homology and Cohomology

This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers.

Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions.

§1 Faced by the questions mentioned in the Preface I was prompted to write this book on the assumption that a typical reader will have certain characteristics. He will presumably be familiar with conventional accounts of certain portions of mathematics and with many so-called mathematical statements, some of which (the theorems) he will know (either because he has himself studied and digested a proof or because he accepts the authority of others) to be true, and others of which he will know (by the same token) to be false. He will nevertheless be conscious of and perturbed by a lack of clarity in his own mind concerning the concepts of proof and truth in mathematics, though he will almost certainly feel that in mathematics these concepts have special meanings broadly similar in outward features to, yet different from, those in everyday life; and also that they are based on criteria different from the experimental ones used in science. He will be aware of statements which are as yet not known to be either true or false (unsolved problems). Quite possibly he will be surprised and dismayed by the possibility that there are statements which are "definite" (in the sense of involving no free variables) and which nevertheless can never (strictly on the basis of an agreed collection of axioms and an agreed concept of proof) be either proved or disproved (refuted).

This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set-measures of symmetry-and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric-the phenomenon of stability.

Thistext is an introduction to the spectral theory of the Laplacian oncompact or finite area hyperbolic surfaces. For some of thesesurfaces, called ΓÇ£arithmetic hyperbolic surfacesΓÇ¥, theeigenfunctions are of arithmetic nature, and one may use analytictools as well as powerful methods in number theory to study them.Afteran introduction to the hyperbolic geometry of surfaces, with aspecial emphasis on those of arithmetic type, and then anintroduction to spectral analytic methods on the Laplace operator onthese surfaces, the author develops the analogy between geometry(closed geodesics) and arithmetic (prime numbers) in proving theSelberg trace formula. Along with important number theoreticapplications, the author exhibits applications of these tools to thespectral statistics of the Laplacian and the quantum uniqueergodicity property. The latter refers to the arithmetic quantumunique ergodicity theorem, recently proved by Elon Lindenstrauss.Thefruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results andthen to be led towards very active areas in modern mathematics.

This book presents a wide range of well-known and less common methods used for estimating the accuracy of probabilistic approximations, including the Esseen type inversion formulas, the Stein method as well as the methods of convolutions and triangle function.

This textbook covers the general theory of Lie groups. By first considering the case of linear groups (following von Neumann''s method) before proceeding to the general case, the reader is naturally introduced to Lie theory.Written by a master of the subject and influential member of the Bourbaki group, the French edition of this textbook has been used by several generations of students. This translation preserves the distinctive style and lively exposition of the original. Requiring only basics of topology and algebra, this book offers an engaging introduction to Lie groups for graduate students and a valuable resource for researchers.

Proofs of theorems such as the Uniform Boundedness Theorem, the Open Mapping Theorem, and the Closed Graph Theorem are worked through step-by-step, providing an accessible avenue to understanding these important results.

Written by an expert on the topic and experienced lecturer, this textbook provides an elegant, self-contained introduction to functional analysis, including several advanced topics and applications to harmonic analysis.Starting from basic topics before proceeding to more advanced material, the book covers measure and integration theory, classical Banach and Hilbert space theory, spectral theory for bounded operators, fixed point theory, Schauder bases, the Riesz-Thorin interpolation theorem for operators, as well as topics in duality and convexity theory.Aimed at advanced undergraduate and graduate students, this book is suitable for both introductory and more advanced courses in functional analysis. Including over 1500 exercises of varying difficulty and various motivational and historical remarks, the book can be used for self-study and alongside lecture courses.

This book gives a systematic introduction to the basic theory of financial mathematics, with an emphasis on applications of martingale methods in pricing and hedging of contingent claims, interest rate term structure models, and expected utility maximization problems.

This textbook on combinatorial commutative algebra focuses on properties of monomial ideals in polynomial rings and their connections with other areas of mathematics such as combinatorics, electrical engineering, topology, geometry, and homological algebra.

This introduction covers Markov Chains, Birth and Death processes, Brownian motion and Autoregressive models, using the Maple computer-algebra system to simplify both the underlying mathematics and the conceptual understanding of random processes.