Bao, D., University of Houston, TX, USA
Chern, S.S., University of California at Berkeley, CA, USA
Shen, Z., Purdue University at Indianapolis, IN, USA
An Introduction to Riemann-Finsler Geometry
2000. Approx. 400 pp. 10 figs.
0-387-98948-X
In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are
represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one.
Contents: From the contents: Finsler methods and the fundamentals of Minkowski norms.- The Chern
connection.- Curvature and Schur's Lemma.- Finsler surfaces and a generalized Gauss-Bonnet theorem.-
Variations of arc length, Jacobi fields, and the effect of curvature.- The Gauss lemma and the Hopf-Rinow
theorem.- The index form and the Bonnet-Myers theorem.- The cut and conjugate loci, and Synge's theorem.-
The Cartan-Hadamard theorem and Rauch's first theorem.- Berwald spaces and Szabo's theorem for Berwald
spaces.- Randers spaces and a theorem from the Japanese school.- Constant flag curvature spaces, and ther
Andar-Zadeh theorem.- Riemannian manifolds and two theorems of Hopf's.- Minkowski spaces, the theorems of Dickie and Brickell.
Series: Graduate Texts in Mathematics.VOL. 200
Fields: Geometry
Written for: Graduate mathematics students
Book category: Graduate Textbook
Publication language: English
Cederberg, J.N., St. Olaf College, Northfield, NM, USA
A Course in Modern Geometries, 2nd ed.
2000. Approx. 285 pp. 151 figs.
0-387-98972-2
Designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. The first chapter presents several finite geometries in an axiomatic framework, while Chapter 2 continues the synthetic approach in introducing both Euclid and ideas of non-Euclidean geometry. There follows a new introduction to symmetry and hands-on explorations of isometries that precedes an extensive analytic treatment of similarities and affinities. Chapter 4 presents plane projective geometry both synthetically and analytically, and the new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software explorations of the Poincar? model, isometries, projectivities, conics and fractals. Parallel versions are available for "Cabri Geometry" and "Geometer Sketchpad".
Contents: * Axiomatic systems and finite geometries * Non-Euclidean geometry * Geometric transformations of the Euclidean plane * Projective geometry * Chaos to symmetry: an introduction to fractal geometry
Series: Undergraduate Texts in Mathematics.
Publication date: July 2000
Fields: Geometry
Written for: Undergraduate math students
Book category: Undergraduate Textbook
Publication language: English
Edited by: James Cogdell, Oklahoma State University, Stillwater, OK,
Simon Gindikin, Rutgers University, Piscataway, NJ,
and Peter Sarnak, Princeton University, NJ
Selected Works of Ilya Piatetski-Shapiro
Description
This selection of papers of I. Piatetski-Shapiro represents almost 50 years of his mathematical activity. Included are many of his major papers in harmonic analysis, number theory, discrete groups, bounded homogeneous domains, algebraic geometry, automorphic forms, and automorphic $L$-functions.
The papers in the volume are intended as a representative and accurate reflection of both the breadth and depth of Piatetski-Shapiro's work in mathematics. Some of his early works, such as those on the prime number theorem and on sets of uniqueness for trigonometric series, appear for the first time in English. Also included are several commentaries by his close colleagues. This volume offers an elegant representation of the contributions made by this renowned mathematician.
Contents
Autobiographical papers
Automorphic functions and discrete groups
Bounded homogeneous domains
Applied
Automorphic $L$-functions
Theta lifts and applications to generalized Ramanujan conjectures
Details:
Series: Collected Works Volume: 15
Publication Year: 2000
ISBN: 0-8218-0930-X
Paging: approximately 856 pp.
Binding: Hardcover
Mitsuru Ikawa, Kyoto University, Japan
Hyperbolic Partial Differential Equations and Wave Phenomena
Description
The familiar wave equation is the most fundamental hyperbolic partial differential equation. Other hyperbolic equations, both linear and nonlinear, exhibit many wave-like phenomena. The primary theme of this book is the mathematical investigation of such wave phenomena.
The exposition begins with derivations of some wave equations, including waves in an elastic body, such as those observed in connection with earthquakes. Certain existence results are proved early on, allowing the later analysis to concentrate on properties of solutions. The existence of solutions is established using methods from functional analysis. Many of the properties are developed using methods of asymptotic solutions. The last chapter contains an analysis of the decay of the local energy of solutions. This analysis shows, in particular, that in a connected exterior domain, disturbances gradually drift into the distance and the effect of a disturbance in a bounded domain becomes small after sufficient time passes.
The book is geared toward a wide audience interested in PDEs. Prerequisite to the text are some real analysis and elementary functional analysis. It would be suitable for use as a text in PDEs or mathematical physics at the advanced undergraduate and graduate level.
Contents
Wave phenomena and hyperbolic equations
The existence of a solution for a hyperbolic equation and its properties
The construction of asymptotic solutions
Local energy of the wave equation
Perspectives on current research in mathematics
Bibliography
Solutions to the exercises
Index
Details:
Series: Translations of Mathematical Monographs, Volume: 189
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2000
ISBN: 0-8218-1021-9
Paging: approximately 194 pp.
Binding: Softcover
K. Peeva , Sofia, Bulgaria, / H.-J. Vogel , Potsdam, Germany,
R. Lozanov , Montreal, Quebec, Canada, / P.N. Peeva , Sofia, Bulgaria
Elsevier's Dictionary of Mathematics
In English, German, French and Russian
Description
Elsevier's Dictionary of Mathematics contains 11,652 entries with more than 4,750 cross-references. Selection of the terms was based either on their significance or on their frequency of use according to authoritative encyclopedias, dictionaries and textbooks. Included are both modern developments and contemporary changes in terminology as well as recently established terms.
The terminology covers all the major branches from elementary to advanced subjects: arithmetic, algebra, geometry, set theory, discrete mathematics, logic, Boolean algebra, linear algebra, matrix algebra, calculus, differential equations, vector algebra, field theory, probability theory and statistics, optimization, numerical
methods, mathematical programming, modern algebra, algebraic structures, computer algebra, category theory, applied mathematics, theory of automata and formal languages, theory of games, theory of graphs, as well as some commonly used entries in computer architecture, hardware, communications, system and application software, microprogramming, etc.
This work will provide readers, writers and translators with a guide of the most widely used terms and collections in the area, and will prove to be a useful tool for all professionals exploring the multilingual scientific terminology.
Hardbound
ISBN: 0-444-82953-9
1006 pages
11,652 terms
Walter G. Kelley University of Oklahoma, Norman
Allan C. Peterson University of Nebraska, Lincoln
Difference Equations, 2E
An Introduction with Applications
ISBN: 012403330X
Cover: CaseBound
This book uses elementary analysis and linear algebra to investigate solutions of difference equations. Some of the techniques discussed are summation methods, generating functions, z-transforms, theory of linear equations, matrix methods, stability, chaos, asymptotic methods, Green's functions, finite Fourier analysis, variational methods, fixed point theorems, and connections with differential equations. Applications of difference equations to combinatorics, geometry, epidemiology, special functions, economics, population biology, numerical analysis, circuit analysis, differential equations, and other fields have been included. Many examples of the theory are given, and there are a large number of exercises, with difficulty ranging from elementary calculation to investigation of new ideas or applications. The new edition also includes an appendix on the use of computer algebra systems.
NEW TO THIS EDITION
Phase plane analysis for systems of two linear equations
Use of equations of variation to approximate solutions
Fundamental matrices and Floquet theory for periodic systems
LaSalle invariance theorem
Additional applications: secant line method, Bison problem, juvenile-adult population model, probability theory
Appendix on the use of Mathematica for analyzing difference equaitons
Exponential generating functions
Many new examples and exercises
CONTENTS:
Preface. Introduction. The Difference Calculus. Linear Difference Equations. Stability Theory. Asymptotic Methods. The Self-Adjoint Second Order Linear Equation. The
Sturm-Liouville Problem. Discrete Calculus of Variations. Boundary Value Problems for Nonlinear Equations. Partial Difference Equations. Appendix. Answers to Selected
Problems. References. Index.
H. Ted Davis University of Minnesota, Minneapolis
Kendall T. Thomson Purdue University, West Lafayette, Indiana
Linear Algebra and Linear Operators in Engineering
With Applications in Mathematica
ISBN: 012206349X
Cover: CaseBound
A Volume in the Process Systems Engineering Series
Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical sciences. Also included are several numerical applications, complete with Mathematica solutions and code, giving the student a "hands-on" introduction
to numerical analysis. Linear Algebra and Linear Operators in Engineering is ideally suited as the main text of an introductory graduate course, and is a fine instrument for self-study or as a general reference for those applying mathematics.
KEY FEATURES * Contains numerous Mathematica examples complete with full code and solutions *
Provides complete numerical algorithms for solving linear and nonlinear problems * Spans elementary notions to the functional theory of linear integral and differential equations * Includes over 130 examples, illustrations, and exercises and over 220 problems ranging from basic concepts to challenging applications * Presents real-life applications from chemical, mechanical, and electrical engineering and the physical sciences