Bücher der Reihe CMS Books in Mathematics

This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. The appendices include a selection of original papers that encompass the most important milestones in the evolution of theory connected to the Riemann Hypothesis.

Reflection groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.

One of the fundamental questions of Banach space theory is whether every Banach space has a basis. A space with a basis gives us the feeling of familiarity and concreteness, and perhaps a chance to attempt the classification of all Banach spaces and other problems.The main goals of this book are to: introduce the reader to some of the basic concepts, results and applications of biorthogonal systems in infinite dimensional geometry of Banach spaces, and in topology and nonlinear analysis in Banach spaces; to do so in a manner accessible to graduate students and researchers who have a foundation in Banach space theory; expose the reader to some current avenues of research in biorthogonal systems in Banach spaces; provide notes and exercises related to the topic, as well as suggesting open problems and possible directions of research.The intended audience will have a basic background in functional analysis. The authors have included numerous exercises, as well as open problems that point to possible directions of research.

The theory of convex optimization has been constantly developing over the past 30 years.  Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "e;anticonvex"e; and "e;convex-anticonvex"e; optimizaton problems.  This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity.  This manuscript will be of great interest for experts in this and related fields.

The pioneering work of French mathematician Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth, providing readers with an overview of the many properties of Fermat numbers and demonstrating their applications in areas such as number theory, probability theory, geometry, and signal processing. This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers.

This book introduces the basic principles of functional analysis and areas of Banach space theory that are close to nonlinear analysis and topology. The text can be used in graduate courses or for independent study. It includes a large number of exercises of different levels of difficulty, accompanied by hints.

This book evolved from notes originally developed for a graduate course, "e;Best Approximation in Normed Linear Spaces,"e; that I began giving at Penn State Uni- versity more than 25 years ago. It soon became evident. that many of the students who wanted to take the course (including engineers, computer scientists, and statis- ticians, as well as mathematicians) did not have the necessary prerequisites such as a working knowledge of Lp-spaces and some basic functional analysis. (Today such material is typically contained in the first-year graduate course in analysis. ) To accommodate these students, I usually ended up spending nearly half the course on these prerequisites, and the last half was devoted to the "e;best approximation"e; part. I did this a few times and determined that it was not satisfactory: Too much time was being spent on the presumed prerequisites. To be able to devote most of the course to "e;best approximation,"e; I decided to concentrate on the simplest of the normed linear spaces-the inner product spaces-since the theory in inner product spaces can be taught from first principles in much less time, and also since one can give a convincing argument that inner product spaces are the most important of all the normed linear spaces anyway. The success of this approach turned out to be even better than I had originally anticipated: One can develop a fairly complete theory of best approximation in inner product spaces from first principles, and such was my purpose in writing this book.

1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.

This book is intended as a text for a problem-solving course at the first- or second-year university level, as a text for enrichment classes for talented high-school students, or for mathematics competition training. It can also be used as a source of supplementary material for any course dealing with algebraic equations or inequalities, or to supplement a standard elementary number theory course. There are already many excellent books on the market that can be used for a problem-solving course. However, some are merely collections of prob- lems from a variety of fields and lack cohesion. Others present problems according to topic, but provide little or no theoretical background. Most problem books have a limited number of rather challenging problems. While these problems tend to be quite beautiful, they can appear forbidding and discouraging to a beginning student, even with well-motivated and carefully written solutions. As a consequence, students may decide that problem solving is only for the few high performers in their class, and abandon this important part of their mathematical, and indeed overall, education.

This research monograph provides an introduction to the theory of nonautonomous semiflows with applications to population dynamics.

This book examines results of convex analysis and optimization in Hilbert space, presenting a concise exposition of related theory that allows for algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions and more.

This book is devoted exclusively to Lebesgue spaces and their direct derived spaces. Unique in its sole dedication, this book explores Lebesgue spaces, distribution functions and nonincreasing rearrangement.

Using the theory of impulsive differential equations, this book focuses on mathematical models which reflect current research in biology, population dynamics, neural networks and economics.

This book is intended to serve as a textbook for a course in Representation Theory of Algebras at the beginning graduate level. The main tool for describing the representation theory of a finite-dimensional algebra is its Auslander-Reiten quiver, and the text introduces these quivers as early as possible.

Optimization is a rich and thriving mathematical discipline, and the underlying theory of current computational optimization techniques grows ever more sophisticated. This new edition adds material on semismooth optimization, as well as several new proofs.

The main purpose of this book is to show how ideas from combinatorial group theory have spread to two other areas of mathematics: the theory of Lie algebras and affine algebraic geometry. Some of these ideas, in turn, came to combinatorial group theory from low-dimensional topology in the beginning of the 20th Century.

The book is intended for those who love mathematics, including undergraduate students of mathematics, more experienced students and the vast unseen host of amateur mathematicians.

We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set.

This monograph leads the reader to the frontiers of the very latest research developments in what is regarded as the central zone of discrete geometry. It is constructed around four classic problems in the subject, including the Kneser-Poulsen Conjecture.

The authors provide a friendly introduction to the delights of algebraic number theory via Pell's Equation. The only prerequisites are knowledge of elementary number theory and abstract algebra. There are references for those following up on various topics.

Borwein is an authority in the area of mathematical optimization, and his book makes an important contribution to variational analysisProvides a good introduction to the topic

He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals.

An overview of the dramatic reorganization in reaction to N. Karmakar's seminal 1984 paper on algorithmic linear programming in the area of algorithmic differentiable optimization and equation-solving, or, more simply, algorithmic differentiable programming. Aimed at readers familiar with advanced calculus and numerical analysis.