Series in Algebra - Vol. 11
672pp Pub. date: Feb 2011
ISBN: 978-981-4329-71-2
This monograph is concerned with exchange rings in various conditions related to stable range. Diagonal reduction of regular matrices and cleanness of square matrices are also discussed. Readers will come across various topics: cancellation of modules, comparability of modules, cleanness, monoid theory, matrix theory, K-theory, topology, amongst others. This is a first-ever book that contains many of these topics considered under stable range conditions. It will be of great interest to researchers and graduate students involved in ring and module theories.
Readership: Mathematicians and graduate students interested in ring and module theory.
Notational Conventions
Stable Range One
Unit 1-Stable Range
On m-Fold Stable Rings
Strongly Stable Rings
Weakly Stable Rings
Related Comparability
Generalized Stable Rings
QB-Rings
PB-Rings
Power-Cancellation
Stably Euclidean Rings
The n-Stable Range Condition
Stable Range for Ideals
Diagonal Reduction
Clean Properties, I
Clean Properties, II
Abelian Rings and Exchange
520pp (approx.) Pub. date: Mar 2011
ISBN: 978-981-4335-64-5
ISBN: 978-981-4340-28-1(pbk)
This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya?Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.
Considerable emphasis is placed on the algebraic system consisting of congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced ? via congruence classes ? to the idea of cosets and factor groups. This enables the transition to cosets and factor objects in a more abstract setting to be relatively painless. The chapters dealing with applications help to reinforce the concepts and methods developed in the context of more down-to-earth problems.
Most introductory texts in abstract algebra either avoid cosets, factor objects and homomorphisms completely or introduce them towards the end of the book. In this book, these topics are dealt with early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible, even if there are alternative methods of proof.
Readership: Undergraduates from approximately 2nd to 4th year. Familarity with linear algebra is required.
Logic and Proofs
Set Theory
Cartesian Products and Relations, Maps and Binary Operations
The Integers with a Thorough Treatment of Congruences
Groups (including the Sylow Theorems)
Permutation Groups
Rings, Integral, Domains and Fields
Latin Squares
Polya?Burnside Enumeration
Group Codes
Polynomial Codes
and other chapters
World Scientific Series on Nonlinear Science, Series A
350pp (approx.) Pub. date: Scheduled Summer 2011
ISBN: 978-981-4329-06-4
Differential equations with random perturbations are the mathematical models of real-world processes that cannot be described via deterministic laws, and their evolution depends on the random factors. The modern theory of differential equations with random perturbations is on the edge of two mathematical disciplines: random processes and ordinary differential equations. Consequently, the sources of these methods come both from the theory of random processes and from the classic theory of differential equations.
This work focuses on the approach to stochastic equations from the perspective of ordinary differential equations. For this purpose, both asymptotic and qualitative methods which appeared in the classical theory of differential equations and nonlinear mechanics are developed.
Readership: Graduate students and researchers in mathematics and physics.
Differential Equations with Random Right Hand Side and Random Impulse Action
Invariant Sets of Systems with Random Perturbations
Stability of Invariant Sets and the Reduction Principle for Ito Systems, Linear and Quasilinear Stochastic Ito Systems
Exponential Dichotomy in the Quadratic Mean
Asymptotic Equivalence of Linear
Extension of Ito Systems on Torus
Stability of Invariant Tori
Stochastic Invariant Tori of Nonlinear Analysis of the Equations with Random Perturbations Using Averaging
320pp (approx.) Pub. date: Mar 2011
ISBN: 978-981-4335-31-7
This book provides a self-contained and accessible introduction to linear and multilinear algebra. Besides the standard techniques for linear and multilinear algebra many advanced topics are included. Emphasis is placed on the Kronecker product and tensor product. The Kronecker product has widespread applications in signal processing, discrete wavelets, statistical physics, computer graphics, fractals, quantum mechanics and quantum computing. All these fields are covered in detail. A key feature of the book is the many detailed worked-out examples. Computer algebra applications are also given. Each chapter includes useful exercises. The book is well suited for pure and applied mathematicians as well as theoretical physicists and engineers.
New topics added to the second edition are: braid-like relations, Clebsch?Gordan expansion, nearest Kronecker product, Clifford and Pauli group, universal enveloping algebra, computer algebra and Kronecker product.
Readership: Students, engineers, researchers, and scientists in mathematical physics, applied mathematics, algebra & number theory, and numerical analysis.
Matrix Calculus
Kronecker Product
Applications
Tensor Product
Braid-like Relations
Clebsch?Gordan Expansion
Nearest Kronecker Product
Clifford and Pauli Group
Universal Enveloping Algebra
Computer Algebra Implementation
450pp (approx.) Pub. date: Scheduled Fall 2011
ISBN: 978-981-4335-62-1
This book presents, in a unitary frame and from a new perspective, the main concepts and results of one of the most fascinating branches of modern mathematics, namely differential equations, and offers the reader another point of view concerning a possible way to approach the problems of existence, uniqueness, approximation, and continuation of the solutions to a Cauchy problem. In addition, it contains simple introductions to some topics which are not usually included in classical textbooks: the exponential formula, conservation laws, generalized solutions, Caratheodory solutions, differential inclusions, variational inequalities, viability, invariance, gradient systems.
In this new edition we have corrected several small errors and added the following new topics: Volterra Integral Equations and Elements of Calculus of Variations. Some problems and exercises, referring to these two new topics are also included. The bibliography has been updated and expanded.
Readership: Graduate or undergraduate students dealing with analysis and differential equations, and mathematical modeling.
Generalities
The Cauchy Problem
Approximation Methods
Systems of Linear Differential Equations
Elements of Stability
Prime Integrals
Extensions and Generalizations
Auxiliary Results
Series on Knots and Everything
350pp (approx.) Pub. date: Scheduled Winter 2011
ISBN: 978-981-4340-83-0
The aim of the book is to provide a new and fruitful approach to the challenging problems of modern physics, astrophysics, and cosmology. The well-known observations of the flat rotation curves of spiral galaxies and of the gravitational lensing effect greatly exceeding the expectations based on the classical GRT can be explained without bringing in the notion of dark matter. The Tully-Fisher law and the unusual features of globular clusters' motion become clear. It also turns out that new features appear in the cosmological picture that involves the Universe expansion and the acceleration of the latter.
The theory and the first observational results of the specific galactic scale experiment based on the optical-metrical parametric resonance are also discussed in the book. Instead of the direct measurements of the extremely small gravitational waves, it appears sufficient just to register their action on the radiation of the space masers. It can be done for special cases when the source of the gravitational wave is strictly periodic and presents a close binary system. When the amount of data obtained in such observations is large enough, it would be possible to judge upon the geometrical properties of the space-time region enveloping our galaxy, the Milky Way.
The foundations of the new approach stem from the equivalence principle which is the basics of the classical GRT. In order to make the presentation self-contained, the roots of century-old ideas are discussed again. This makes the book interesting not only to the specialists in the field but also to graduates and ambitious undergraduate students.
Readership: Researchers in the fields of astrophysics, theoretical physics, geometry and topology.
Physics and Geometry ? Brief History of Relations
Causality
Relativity and Covariance
Paradoxical Observations and Notions
Anisotropic Metric and Space
Gravitation Potential and Classical GRT Tests on the Galaxy Scale
Cosmological Consequences
Theory and Observations of the Optical-Metrical Parametric Resonance