This book is addressed to graduate students and research workers in theoretical physics who want a thorough introduction to group theory and Hopf algebras. It is suitable for a one-semester course in group theory or a two-semester course which also treats advanced topics. Starting from basic definitions, it goes on to treat both finite and Lie groups as well as Hopf algebras. Because of the diversity in the choice of topics, which does not place undue emphasis on finite or Lie groups, it should be useful to physicists working in many branches.
A unique aspect of the book is its treatment of Hopf algebras in a form accessible to physicists. Hopf algebras are generalizations of groups and their concepts are acquiring importance in the treatment of conformal field theories, noncommutative spacetimes, topological quantum computation and other important domains of investigation. But there is a scarcity of treatments of Hopf algebras at a level and in a manner that physicists are comfortable with. This book addresses this need superbly.
There are illustrative examples from physics scattered throughout the book and in its set of problems. It also has a good bibliography. These features should enhance its value to readers.
General Notions
Finite Groups
Lie Groups
The Poincare Group
Hopf Algebras in Physics
Readership: Graduate students in theoretical physics and mathematical physics.
300pp (approx.) Pub. date: Jul 2010
ISBN: 978-981-4322-20-1
The book gives the practical means of finding asymptotic solutions to differential equations, and relates WKB methods, integral solutions, Kruskal-Newton diagrams, and boundary layer theory to one another. The construction of integral solutions and the use of analytic continuation are used in conjunction with the asymptotic analysis, to show the interrelatedness of these methods. Some of the functions of classical analysis are used as examples, to provide an introduction to their analytic and asymptotic properties, and to give derivations of some of the important identities satisfied by them. The emphasis is on the various techniques of analysis: obtaining asymptotic limits, connecting different asymptotic solutions, and obtaining integral representation.
The book gives the practical means of finding asymptotic solutions to differential equations, and relates WKB methods, integral solutions, Kruskal-Newton diagrams, and boundary layer theory to one another. The construction of integral solutions and the use of analytic continuation are used in conjunction with the asymptotic analysis, to show the interrelatedness of these methods. Some of the functions of classical analysis are used as examples, to provide an introduction to their analytic and asymptotic properties, and to give derivations of some of the important identities satisfied by them. The emphasis is on the various techniques of analysis: obtaining asymptotic limits, connecting different asymptotic solutions, and obtaining integral representation.
Dominant Balance
Exact Solutions
Complex Variables
Local Approximate Solutions
Phase Integral Methods I
Perturbation Theory
Asymptotic Evaluation of Integrals
The Euler Gamma Function
Integral Solutions
Expansion in Basis Functions
Airy
Phase Integral Methods II
Bessel
Weber?Hermite
Whittaker and Watson
Inhomogeneous Differential Equations
The Riemann Zeta Function
Boundary Layer Problems
Readership: Graduate students and researchers in mathematics, engineering and physics.
400pp (approx.) Pub. date: Aug 2010
ISBN: 978-1-84816-607-3
ISBN: 978-1-84816-608-0(pbk)
This book presents a complete theory of ordinary differential equations, with many illustrative examples and interesting exercises. A rigorous treatment is offered in this book with clear proofs for the theoretical results and with detailed solutions for the examples and problems.
This book is intended for undergraduate students who major in mathematics and have acquired a prerequisite knowledge of calculus and partly the knowledge of a complex variable, and are now reading advanced calculus and linear algebra. Additionally, the comprehensive coverage of the theory with a wide array of examples and detailed solutions, would appeal to mathematics graduate students and researchers as well as graduate students in majors of other disciplines.
As a handy reference, advanced knowledge is provided in this book with details developed beyond the basics; optional sections, where main results are extended, offer an understanding of further applications of ordinary differential equations.
Linear Equations
Systems of Linear First Order Equations
Power Series Solutions
Adjoint Operators and Nonhomogeneous Boundary Value Problems
Green Functions
Eigenfunction Expansions
Long Time Behavior of Systems of Differential Equations
Existence and Uniqueness Theorems
Readership: Advanced undergraduates in mathematics or physics; graduate students in engineering, science, economics, business or mathematics.
600pp (approx.) Pub. date: Scheduled Winter 2010
ISBN: 978-981-4307-12-3
This book provides a systematic treatment of the Volterra integral equation by means of a modern integration theory which extends considerably the field of differential equations. It contains many new concepts and results in the framework of a unifying theory. In particular, this new approach is suitable in situations where fast oscillations occur.
Kapitza's Pendulum and a Related Problem
Averaging
Internal Resonance
SR-Integration of Functions of a Pair of Coupled Variables
Generalized Ordinary Differential Equations
SR-Solutions
Continuous Dependence
SKH-Integration of Functions of a Pair of Coupled Variables
Differential Equations in Classical Form
Integration and Strong Integration
Integration by Parts
A Variant of Gronwall Formula
Linear Generalized Ordinary Differential Equations
Readership: Graduate students and mathematicians in differential equations; physicists and engineers interested in classical mechanics.
200pp (approx.) Pub. date: Scheduled Spring 2011
ISBN: 978-981-4324-02-1
Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.
Descent Theory
Etale Morphisms and Smooth Morphisms
Etale Fundamental Groups
Group Cohomology and Galois Cohomology
Etale Cohomology
Derived Categories and Derived Functors
Base Change Theorems
Duality
Finiteness Theorems
L-Adic Cohomology
Readership: Graduate students and researchers in pure mathematics.
550pp (approx.) Pub. date: Scheduled Winter 2010
ISBN: 978-981-4307-72-7
In this volume very simplified models are introduced to understand the random sequential packing models mathematically. The 1-dimensional model is sometimes called the Parking Problem, which is known by the pioneering works by Flory (1939), Renyi (1958), Dvoretzky and Robbins (1962). To obtain a 1-dimensional packing density, distribution of the minimum of gaps, etc., the classical analysis has to be studied. The packing density of the general multi-dimensional random sequential packing of cubes (hypercubes) makes a well-known unsolved problem. The experimental analysis is usually applied to the problem. This book introduces simplified multi-dimensional models of cubes and torus, which keep the character of the original general model, and introduces a combinatorial analysis for combinatorial modelings.
Random Interval Packing
The Speed of Convergence to the Renyi Constant
The Dvoretzky Robbins Central Limit Theorem
Gap Size
The Minimum of Gaps
Kakutani's Interval Splitting
Sequential Bisection and Binary Search Tree
Car Parking with Spin
Golay Code and Random Packing
Discrete Cube Packing
Torus Cube Packing
Continuous Random Cube Packing in Cube and Torus
Combinatorial Enumeration
Readership: Researchers in probability and statistics, combinatorics and graph theory, analysis & differential equations, coding theory and cryptography.
200pp (approx.) Pub. date: Scheduled Spring 2011
ISBN: 978-981-4307-83-3