The Shape of Algebra is the authors' attempt to share their mathematical experiences with readers who have more than a passing interest in mathematics, but have only a traditional exposure to elementary algebra. Secondary school and college teachers and students who want to expand their horizons in the field will find a fresh presentation of familiar concepts and some unexpected results.
This book serves as a text for an gappreciationh course in modern mathematics designed for non-mathematics majors or for first-year students who are considering the possibility of studying mathematics or related disciplines. It can also serve as a source of computer-supported activities that could supplement traditional courses in algebra, multivariable calculus, and complex variable.
This book gives the reader a sense of the visual nature of mathematics. Mathematical experiments with universal mapping software VisuMatica, designed by Vladimir Nodel'man, form the very core of the book. Readers are encouraged to reproduce, play with, and expand on these experiments. Numerous problems are interspersed throughout the text to guide the reader.
Our treatment of standard algebra is visual and computational. By introducing visual computational environments like VisuMatica, our book promotes this geometric approach to algebra and makes it accessible to readers a great deal earlier.
The book will enable our readers to approach its content on three levels: the first one which requires only some fluency with elementary algebraic manipulations; the second one which also presumes familiarity with the notions of derivatives and tangent lines to plane curves, and the third one which uses some basic concepts of multivariable calculus. All three levels are clearly marked in the text, and will allow for a smooth reading and virtual experiments, regardless of the level that our readers will find comfortable.
Introduction Maps upon Maps
Maps, Models, and the Coordinate Plane
The Universe of Quadratic Polynomials
The Complex Numbers
The Geometry of Complex Linear Polynomial Maps
Complex Polynomial Maps, Closed Plane Curves and their Invariants
Experimenting with the Geometry of Complex Polynomial Maps of Degrees Two and Three
The Old Art of Solving Cubic Equations and the New Science
Complex Polynomial Maps of Higher Degrees and the Fundamental Theorem of Algebra
Local and Global Geometry of Complex Polynomial Maps
The Fundamental Theorem and the Complex Viete Map
Appendices:
Estimating Polynomial Expressions
More about the Taylor Formula for Polynomials
The Geometry of Discriminants
Line Integrals and an Analytical Proof of the Loop Theorem
Deformations of Complex Polynomials and the Braids
Readership: Undergraduates, school teachers, college professors interested in algebra and its geometric approach.
500pp (approx.) Pub. date: Scheduled Winter 2010
ISBN: 978-981-4313-59-9
This textbook, set for a one or two semester course in commutative algebra, provides an introduction to commutative algebra at the postgraduate and research levels. The main prerequisites are familiarity with groups, rings and fields. Proofs are self-contained.
The book will be useful to beginners and experienced researchers alike. The material is so arranged that the beginner can learn through self-study or by attending a course. For the experienced researcher, the book may serve to present new perspectives on some well-known results, or as a reference.
Rings and Ideals
Modules and Algebras
Polynomials and Power Series Rings
Homological Tools I
Finiteness Conditions
Primary Decomposition
Filtrations and Completion
Numerical Functions
Dimension of Local Rings
Integral Extensions
Normal Domains
Integral Closure in a Finite Field Extension
Transcendental Extensions
Affine and Graded Rings
Valuation Rings and Valuations
Derivations and Differentials
Homological Tools II
Depth and Cohen?Macaulay Rings
Regular Rings
Divisor Class Groups
Readership: Students and researchers in mathematics at the graduate level.
350pp (approx.) Pub. date: Nov 2010
ISBN: 978-981-4313-61-2
ISBN: 978-981-4313-62-9(pbk)
This book provides the mathematical definition of white noise and gives its significance. White noise is in fact a typical class of idealized elemental (infinitesimal) random variables. Thus, we are naturally led to have functionals of such elemental random variables that is white noise. This book analyzes those functionals of white noise, particularly the generalized ones called Hida distributions, and highlights some interesting future directions. The main part of the book involves infinite dimensional differential and integral calculus based on the variable which is white noise.
The present book can be used as a supplementary book to Lectures of White Noise Functionals published in 2008, with detailed background provided.
White Noise and Its Probability Distribution
Linear Functionals of White Noise
Gaussian Systems and Their Representations
Nonlinear Functionals of White Noise
Hida Distributions (Generalized Functionals of White Noise)
Applications to Quantum Dynamics, Statistics and Biology
Readership: Researchers in probability and statistics, in particular stochastic analysis.
200pp (approx.) Pub. date: Scheduled Spring 2011
ISBN: 978-981-283-688-5
The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Perron integral; in particular, it includes the powerful Lebesgue integral. This book presents an introduction of the multiple Henstock-Kurzweil integral. Along with the classical results, this book contains some recent developments connected with measures, multiple integration by parts, and multiple Fourier series. The book can be understood with a prerequisite of advanced calculus.
The One-Dimensional Henstock-Kurzweil Integral
The Multiple Henstock-Kurzweil Integral
Lebesgue Integrable Functions
Further Properties of Henstock-Kurzweil Integrable Functions
The Henstock Variational Measure
Multipliers for the Henstock-Kurzweil Integral
Some Selected Topics in Trigonometric Series
Some Applications of the Henstock-Kurzweil Integral to Double Trigonometric Series
Readership: Graduate students and researchers in real analysis.
300pp (approx.) Pub. date: Scheduled Spring 2011
ISBN: 978-981-4324-58-8
This book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of the Gelfand?Graev type, or of the Fourier?Jacobi type to certain residual Eisenstein series. An account of this automorphic descent, with complete, detailed proofs, leads to a thorough understanding of important ideas and techniques. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars.
Certain Residual Eisenstein Series
Fourier Coefficients of Gelfand?Graev Type and Fourier?Jacobi Type
Jacquet Modules Corresponding to Gelfand?Graev Models
Jacquet Modules Corresponding to Fourier?Jacobi Models
The Tower Property
Non-Vanishing of the Descent
On the Global Genericity of the Descent and Rankin?Selberg Integrals
The Descent and Langlands Functoriality from Classical Groups to GL(n)
Applications of the Descent (Generalized Endoscopy, Base Change)
Readership: Graduate students and mathematicians in algebra and number theory.
350pp (approx.) Pub. date: Scheduled Fall 2011
ISBN: 978-981-4304-98-6