Kleiner, Israel

History of Abstract Algebra

2006, Approx. 200 p., 10 illus., Hardcover
ISBN: 0-387-34050-5
Due: December 2006

About this textbook

This exposition provides a comprehensive historical account of the intellectual lineage behind the basic concepts, results, and theories of abstract algebra. Prior to the 19th century, the concept of "algebra" strictly referred to the study of the solution of polynomial equations. By the 20th century it came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields.

Abstract algebra came into existence largely because mathematicians were unable to solve classical problems by classical means. A major theme of this book is to demonstrate how abstract algebra can illuminate the solution to some of these concrete problems. By focusing on the history of the sources of the subject and considering the context in which the originator of an idea was operating, the reader can better comprehend the "burning problem" that spurred the innovation, while gaining a deeper appreciation of the mathematics involved.

Table of contents

Maeda, Y.; Michor, P.; Ochiai, T.; Yoshioka, A. (Eds.)

From Geometry to Quantum Mechanics
In Honor of H. Omori

Series: Progress in Mathematics , Vol. 252
2007, Approx. 320 p., 10 illus., Hardcover
ISBN: 0-8176-4512-8
Due: December 2006

About this book

This volume is composed of invited expository articles by well-known mathematicians in differential geometry and mathematical physics that have been arranged in celebration of Hideki Omori’s recent retirement from Tokyo University of Science and in honor of his fundamental contributions to these areas.

The papers focus on recent trends and future directions in symplectic and Poisson geometry, global analysis, infinite-dimensional Lie group theory, quantizations and noncommutative geometry, as well as applications of partial differential equations and variational methods to geometry. These articles will appeal to graduate students in mathematics and quantum mechanics, as well as researchers, differential geometers, and mathematical physicists.

Contributors include: M. Cahen, D. Elworthy, M. Goto, J. Grabowski, S. Gutt, A. Inoue, S. Kaneyuki, M. Karasev, O. Kobayashi, Y. Maeda, G. Marmo, Y. Matsuyama, P.W. Michor, K. Mikami, N. Miyazaki, T. Mizutani, T. Morimoto, H. Moriyoshi, T. Nagano, Y. Nakanishi, M. Obata, H. Omori, T. Ratiu, T. Sasai, D. Sternheimer, A. Weinstein, K. Yamaguchi, and A. Yoshioka.

Table of contents

Preface.- About Hideki Omori.-Global Analysis on Infinite-Dimensional Lie Groups.- Riemannian Geometry.-Symplectic Geometry and Poisson Geometry.-Quantizations and Noncommutative Geometry.

Bove, Antonio; Colombini, Ferruccio; Santo, Daniele Del (Eds.)

Phase Space Analysis of Partial Differential Equations

Series: Progress in Nonlinear Differential Equations and Their Applications , Vol. 69
2006, Approx. 330 p., 10 illus., Hardcover
ISBN: 0-8176-4511-X
Due: December 2006

About this book

This collection of original articles and surveys treats the linear and nonlinear aspects of the theory of partial differential equations. Phase space analysis methods, including microlocal analysis, have yielded striking results in past years and have become one of the main tools of investigation. Equally important is their role in many applications to physics, for example, in quantum and spectral theories.

Table of contents

Preface.-Trace theorem on the Heisenberg group on homogeneous Hypersurfaces.-Strong unique continuation and finite jet determination for Cauchy--Riemann mappings.-On the Cauchy problem for some hyperbolic operator with double characteristics.-On the differentiability class of the admissible square roots of regular nonnegative function.-The Benjamin--Ono equation in energy space.-Instabilities in Zakharov equations for laser propagation in a plasma.-Symplectic strata and analytic hypoellipticity.-On the backward uniqueness property for a class of parabolic operators.-Inverse problems for hyperbolic equations.-On the optimality of some Observability inequalities for plate systems with potentials.-Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach.-Non-effectively hyperbolic operators and bicharacteristics.-On the Fefferman?Phong inequality for systems of PDEs.-Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations.-An elementary proof of Fedii’s theorem and extensions.-Outgoing parametrices and global Strichartz estimates for Schrodinger equations with variable coefficients.-On the analyticity of solutions of sum of squares of vector fields.-Index.

Calter, Paul

Squaring the Circle
Geometry in Art and Architecture

2007, 400 p., 1200 illus., Softcover
ISBN: 1-930190-82-4
A Key College Publishing book
Due: January 2007

About this textbook

This truly unique new title should appeal to both mathematicians and mathematics educators. It should also find a small market among professional and reference book buyers: mathematical professionals with interest in travel, art, architecture. The title is intended for math students who are interested in art, or art students with an interest (or requirement) in mathematics, or professionals with interest in mathematics and art. Geometry concepts are introduced by analyzing well known buildings and works of art. The book is packaged with an access code which allows the reader into a protected site, which will contain most of the fine art from the book in full color as well as teaching resources. The text appeals both to mathematicians and to artists and will generally be used in courses that bridge the two subjects.

Written for:

As part of the NSF-funded Mathematics Across the Curriculum Project at Dartmouth College, this text was initially intended for a cross-curricular mathematics course; appeals to mathematicians and to artists and will generally be used in courses that bridge the two subjects.

Table of contents

Music of the Spheres.- The Golden Ratio.- The Triangle.- Ad Quadratum and the Sacred Cut.- Polygons, Tilings, and Sacred Geometry.- The Circle.- Circular Designs in Architecture.- Squaring the Circle.- The Ellipse and the Spiral.- The Solids.- The Sphere and Celestial Themes in Art & Architecture.- Brunelleschi’s Peepshow and the Origins of Perspective.- Fractals.- Appendices.- General Index.

Cox, David A., Little, John, O'Shea, Donal

Ideals, Varieties, and Algorithms
An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e

Series: Undergraduate Texts in Mathematics
3rd ed., 2007, Approx. 575 p., 93 illus., Hardcover
ISBN: 0-387-35650-9
Due: January 2007

About this textbook

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?

The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.

In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes: A significantly updated section on Maple in Appendix C; Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR; A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3.

From the 2nd Edition:

"I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry." -The American Mathematical Monthly

Table of contents

Preface to the First Edition.- Preface to the Second Edition.- Preface to the Third Edition.- Geometry, Algebra, and Algorithms.- Groebner Bases.- Elimination Theory.- The Algebra-Geometry Dictionary.- Polynomial and Rational Functions on a Variety.- Robotics and Automatic Geometric Theorem Proving.- Invariant Theory of Finite Groups.- Projective Algebraic Geometry.- The Dimension of a Variety.- Appendix A. Some Concepts from Algebra.- Appendix B. Pseudocode.- Appendix C. Computer Algebra Systems.- Appendix D. Independent Projects.- References.- Index