Proceedings of Symposia in Applied Mathematics Volume: 74
2018; 224 pp; Hardcover
MSC: Primary 37; 65; 35; 34;
Print ISBN: 978-1-4704-2814-3
This volume is based on lectures delivered at the 2016 AMS Short Course gRigorous Numerics in Dynamicsh, held January 4?5, 2016, in Seattle, Washington.
Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics, which draws inspiration from ideas in scientific computing, numerical analysis and approximation theory.
The articles included in this volume present novel techniques for the rigorous study of the dynamics of maps via the Conley-index theory; periodic orbits of delay differential equations via continuation methods; invariant manifolds and connecting orbits; the dynamics of models with unknown nonlinearities; and bifurcations diagrams.
Graduate students and researchers interested in theoretical aspects and applications of numerical methods in dynamics.
CBMS Regional Conference Series in Mathematics Volume: 126
2018; 88 pp; Softcover
MSC: Primary 52;
Print ISBN: 978-1-4704-4359-7
A co-publication of the AMS and CBMS
Theory of valuations on convex sets is a classical part of convex geometry which goes back at least to the positive solution of the third Hilbert problem by M. Dehn in 1900. Since then the theory has undergone a multifaceted development. The author discusses some of Hadwiger's results on valuations on convex compact sets that are continuous in the Hausdorff metric. The book also discusses the Klain-Schneider theorem as well as the proof of McMullen's conjecture, which led subsequently to many further applications and advances in the theory. The last section gives an overview of more recent developments in the theory of translation-invariant continuous valuations, some of which turn out to be useful in integral geometry.
This book grew out of lectures that were given in August 2015 at Kent State University in the framework of the NSF CBMS conference gIntroduction to the Theory of Valuations on Convex Setsh. Only a basic background in general convexity is assumed.
Graduate students and researchers interested in the theory of valuations on convex sets.
AMS Chelsea Publishing Volume: 383
1987; 368 pp; Hardcover
MSC: Primary 08; 03; 06;
Print ISBN: 978-1-4704-4295-8
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
This book presents the foundations of a general theory of algebras. Often called guniversal algebrah, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding.
The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras.
There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text.
Graduate students and researchers interested in algebra and its applications to lattices, logic, and category theory.
Graduate Studies in Mathematics Volume: 190
2018; 240 pp; Hardcover
MSC: Primary 11;
Print ISBN: 978-1-4704-4289-7
The theory of finite fields encompasses algebra, combinatorics, and number theory and has furnished widespread applications in other areas of mathematics and computer science. This book is a collection of selected topics in the theory of finite fields and related areas. The topics include basic facts about finite fields, polynomials over finite fields, Gauss sums, algebraic number theory and cyclotomic fields, zeros of polynomials over finite fields, and classical groups over finite fields. The book is mostly self-contained, and the material covered is accessible to readers with the knowledge of graduate algebra; the only exception is a section on function fields. Each chapter is supplied with a set of exercises. The book can be adopted as a text for a second year graduate course or used as a reference by researchers.
Graduate students and researchers interested in number theory, in particular, the theory of finite fields.
Asterisque Volume: 396
2018; 360 pp; Softcover
MSC: Primary 22; Secondary 11
Print ISBN: 978-2-85629-871-8
This volume elaborates the idea that harmonic analysis on a spherical variety X X is intimately connected to the Langlands program.
In the local setting, the key conjecture is that the spectral decomposition of L2(X)L2(X)is controlled by a dual group attached to XX.
Guided by this, the authors develop a Plancherel formula for L2(X)L2(X), formulated in terms of simpler spherical varieties which model the geometry of
XXat infinity. This local study is then related to global conjectures?namely, conjectures about period integrals of automorphic forms over spherical subgroups.
Graduate students and research mathematicians.
Hindustan Book Agency
Volume: 74; 2018; 358 pp; Softcover
MSC: Primary 55; Secondary 54
Print ISBN: 978-93-86279-67-5
This is the second (revised and enlarged) edition of the book originally published in 2003. It introduces the first concepts of algebraic topology such as general simplicial complexes, simplicial homology theory, fundamental groups, covering spaces and singular homology theory in detail. The text has been designed for undergraduate and beginning graduate students of mathematics. It assumes a minimal background of linear algebra, group theory and topological spaces.
The author deals with the basic concepts and ideas in a very lucid manner, giving suitable motivations and illustrations. As an application of the tools developed in this book, some classical theorems such as Brouwer's fixed point theorem, the Lefschetz fixed point theorem, the Borsuk-Ulam theorem, Brouwer's separation theorem, and the theorem on invariance of domain are proved and illustrated. Most of the exercises are elementary, but some are more challenging and will help readers with their understanding of the subject.
Undergraduate and graduate students interested in algebraic topology.