Bücher der Reihe Universitext

Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease.

This textbook is devoted to the study of some simple but representative nonlinear boundary value problems by topological methods.

"I do not think at all that I am able to present here any procedure of investiga tion that was not perceived long ago by all men of talent;

This book is a textbook for graduate or advanced undergraduate students in mathematics and (or) mathematical physics. It is not primarily aimed, therefore, at specialists (or those who wish to become specialists) in integra­ tion theory, Fourier theory and harmonic analysis, although even for these there might be some points of interest in the book (such as for example the simple remarks in Section 15). At many universities the students do not yet get acquainted with Lebesgue integration in their first and second year (or sometimes only with the first principles of integration on the real line ). The Lebesgue integral, however, is indispensable for obtaining a familiarity with Fourier series and Fourier transforms on a higher level; more so than by us­ ing only the Riemann integral. Therefore, we have included a discussion of integration theory - brief but with complete proofs - for Lebesgue measure in Euclidean space as well as for abstract measures. We give some emphasis to subjectsof which an understanding is necessary for the Fourier theory in the later chapters. In view of the emphasis in modern mathematics curric­ ula on abstract subjects (algebraic geometry, algebraic topology, algebraic number theory) on the one hand and computer science on the other, it may be useful to have a textbook available (not too elementary and not too spe­ cialized) on the subjects - classical but still important to-day - which are mentioned in the title of this book.

In this book, the author adopts a state space approach to time series modeling to provide a new, computer-oriented method for building models for vector-valued time series.

Several applications to quantum information are also included.Introduction to Matrix Analysis and Applications is appropriate for an advanced graduate course on matrix analysis, particularly aimed at studying quantum information.

This book 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.

Rather, business cycle theory turned into stabilization theory which investigated theoretical possibilities of stabilizing a fluctuating economy.

This book provides the mathematical foundations of networks of linear control systems, developed from an algebraic systems theory perspective. Part I can be used as the basis for a first course in Algebraic System Theory, while Part II serves for a second, advanced, course on linear systems.

Analysis Volume IV introduces the reader to functional analysis (integration, Hilbert spaces, harmonic analysis in group theory) and to the methods of the theory of modular functions (theta and L series, elliptic functions, use of the Lie algebra of SL2).

Presents a survey of the field of dynamical systems and its significance for research in complex systems and other fields, based on an analysis of specific examples. This book explains the fundamental underlying mathematical concepts, with a focus on invariants of dynamical systems, including a systematic treatment of Morse-Conley theory.

Contains lecture notes on four topics at the forefront of research in computational mathematics. This book presents a self-contained guide to a research area, an extensive bibliography, and proofs of the key results. It is suitable for professional mathematicians who require an accurate account of research in areas parallel to their own.

Mathematical Methods in Biology and Neurobiology

The primary intent of the book is to introduce an array of beautiful problems in a variety of subjects quickly, pithily and completely rigorously to graduate students and advanced undergraduates. The book takes a number of specific problems and solves them, the needed tools developed along the way in the context of the particular problems.

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.

Based on a graduate course by the celebrated analyst Nigel Kalton, this well-balanced introduction to functional analysis makes clear not only how, but why, the field developed.

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.

On the application side, it demonstrates how Levy traffic models arise when modelling current queueing-type systems (as communication networks) and includes applications to finance.Queues and Levy Fluctuation Theory will appeal to postgraduate students and researchers in mathematics, computer science, and electrical engineering.

Translated from the popular French edition, the goal of the book is to provide a self-contained introduction to mean topological dimension, an invariant of dynamical systems introduced in 1999 by Misha Gromov.

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.

Researchers in algebra, algebraic combinatorics, automata theory, and probability theory, will find this text enriching with its thorough presentation of applications of the theory to these fields.

This text develops the necessary background in probability theory underlying diverse treatments of stochastic processes and their wide-ranging applications.

The reader is assumed to only have knowledge of basic real analysis, complex analysis, and algebra.The latter part of the text provides an outstanding treatment of Banach space theory and operator theory, covering topics not usually found together in other books on functional analysis.

This text is a rigorous treatment of the basic qualitative theory of ordinary differential equations, at the beginning graduate level. Designed as a flexible one-semester course but offering enough material for two semesters, A Short Course covers core topics such as initial value problems, linear differential equations, Lyapunov stability, dynamical systems and the Poincaré¿Bendixson theorem, and bifurcation theory, and second-order topics including oscillation theory, boundary value problems, and Sturm¿Liouville problems. The presentation is clear and easy-to-understand, with figures and copious examples illustrating the meaning of and motivation behind definitions, hypotheses, and general theorems. A thoughtfully conceived selection of exercises together with answers and hints reinforce the reader's understanding of the material. Prerequisites are limited to advanced calculus and the elementary theory of differential equations and linear algebra, making the text suitable for seniorundergraduates as well.

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.