This book primarily serves as a historical research monograph on the biographical sketch and career of Leonhard Euler and his major contributions to numerous areas in the mathematical and physical sciences. It contains fourteen chapters describing Euler's works on number theory, algebra, geometry, trigonometry, differential and integral calculus, analysis, infinite series and infinite products, ordinary and elliptic integrals and special functions, ordinary and partial differential equations, calculus of variations, graph theory and topology, mechanics and ballistic research, elasticity and fluid mechanics, physics and astronomy, probability and statistics.
The book is written to provide a definitive impression of Euler's personal and professional life as well as of the range, power, and depth of his unique contributions. This tricentennial tribute commemorates Euler the great man and Euler the universal mathematician of all time. Based on the author's historically motivated method of teaching, special attention is given to demonstrate that Euler's work had served as the basis of research and developments of mathematical and physical sciences for the last 300 years. An attempt is also made to examine his research and its relation to current mathematics and science. Based on a series of Euler's extraordinary contributions, the historical development of many different subjects of mathematical sciences is traced with a linking commentary so that it puts the reader at the forefront of current research.
Mathematics Before Leonhard Euler
Brief Biographical Sketch and Career of Leonhard Euler
Euler's Contributions to Number Theory and Algebra
Euler's Contributions to Geometry and Spherical Trigonometry
Euler's Formula for Polyhedra, Topology and Graph Theory
Euler's Contributions to Calculus and Analysis
Euler's Contributions to the Infinite Series and the Zeta Function
Euler's Beta and Gamma Functions and Infinite Products
Euler and Differential Equations
The Euler Equations of Motion in Fluid Mechanics
Euler's Contributions to Mechanics and Elasticity
Euler's Work on the Probability Theory
Euler's Contributions to Ballistics
Euler and His Work on Astronomy and Physics
Readership: Undergraduate and graduate students of mathematics, mathematics education, physics, engineering and science. As well as professionals and prospective mathematical scientists.
420pp Pub. date: Oct 2009
ISBN: 978-1-84816-525-0
This book is suitable for use in any graduate course on analytical methods and their application to representation theory. Each concept is developed with special emphasis on lucidity and clarity. The book also shows the direct link of Cauchy?Pochhammer theory with the Hadmard?Reisz?Schwartz?Gel'fand et al. regularization. The flaw in earlier works on the Plancheral formula for the universal covering group of SL(2,R) is pointed out and rectified. This topic appears here for the first time in the correct form.
Existing treatises are essentially magnum opus of the experts, intended for other experts in the field. This book, on the other hand, is unique insofar as every chapter deals with topics in a way that differs remarkably from traditional treatment. For example, Chapter 3 presents the Cauchy?Pochhammer theory of gamma, beta and zeta function in a form which has not been presented so far in any treatise of classical analysis.
Convergence, Analytic Functions, Complex Integration, Residue Theorem, Cauchy?Pochhammer Theory of Gamma, Beta and Zeta Function
Bargman?Segal Spaces, Elements of the Theory of Generalized Functions
Regularizations and Cauchy's Theory of Analytic Continuation
Gel'fand?Shilov Formulas for Gamma and Beta Function
Lie Group and Invariant Measure
Representations and Unitary Representation
Wigner?Eckart Theorem
SU(2) Group
Elements of SU(3)
Gell?Mann Basis and ă-Matrices
Gell?Mann Neeman Octet Model and Mass Formula
Locally Compact Groups: SL(2,R) (SU(1,1))
Principal Exceptional, Positive and Negative Discreet Series and Their Canonical Carrier Spaces
The Clebsch?Gordan Problem: D+ X D+,c
Dc X Dc,e
Group Ring and Invariant Definition of Character
Plancherel Formula as a Completeness Condition of Character
The Group SL(2,C) and Its Unitary Representations
Group Ring and Character
Plancherel Formula
SU(1,1) Content of SL(2,C)
Heisenberg?Weyl Group and Its Representations
Coherent-States and Bergman?Segal Spaces
Bargmann's Integral Transform
SU(1,1) Coherent States and Integral Transforms Connecting Well-Known Carrier Spaces of SU(1,1)
Readership: Academics, research scholars and graduate students of mathematical physics, mathematics and theoretical physics.
400pp (approx.) Pub. date: Scheduled Winter 2009
ISBN: 978-981-4273-29-9
ISBN: 978-981-4273-30-5(pbk)
This book is directed primarily at undergraduate and postgraduate students interested to get acquainted with the representation theory of Lie algebras. The book treats the case of the smallest simple Lie algebra, namely, the Lie algebra sl_2. It contains classical contents including the description of all finite-dimensional modules and an introduction to the universal enveloping algebras with its primitive ideals, alongside non-classical contents including the description of all simple weight modules, the category of all weight modules, a detailed description of the category O, and especially, a description of all simple modules. The book also contains an account of a new research direction: the categorification of simple finite-dimensional modules.
Finite-Dimensional Modules
The Universal Enveloping Algebra
Weight Modules
The Primitive Spectrum
Category O
Description of All Simple Modules
Categorification of Finite-Dimensional Modules
Readership: Researchers, graduate, undergraduate students and professionals in algebra.
272pp (approx.) Pub. date: Scheduled Winter 2009
ISBN: 978-1-84816-517-5
This textbook covers the mathematical foundations of the analysis of algorithms. The gist of the book is how to argue, without the burden of excessive formalism, that a given algorithm does what it is supposed to do. The two key ideas of the proof of correctness, induction and invariance, are employed in the framework of pre/post-conditions and loop invariants.
The algorithms considered are the basic and traditional algorithms of computer science, such as Greedy, Dynamic and Divide & Conquer. In addition, two classes of algorithms that rarely make it into introductory textbooks are discussed. Randomized algorithms, which are now ubiquitous because of their applications to cryptography; and Online algorithms, which are essential in fields as diverse as operating systems (caching, in particular) and stock-market predictions.
This self-contained book is intended for undergraduate students in computer science and mathematics.
Preliminaries
Greedy Algorithms
Divide and Conquer
Dynamic Programming
Randomized Algorithms
Online Algorithms
Readership: Undergraduate students in computer science, software engineers and mathematicians.
150pp (approx.) Pub. date: Scheduled Winter 2009
ISBN: 978-981-4271-40-0
This book is an indispensable guide for anyone seeking to familarize themselves with research in braid groups, configuration spaces and their applications. Starting at the beginning, and assuming only basic topology and group theory, the volume's noted expositors take the reader through the fundamental theory and on to current research and applications in fields as varied as astrophysics, cryptography and robotics. As leading researchers themselves, the authors write enthusiastically about their topics, and include many striking illustrations. The chapters have their origins in tutorials given at a Summer School on Braids, at the National University of Singapore's Institute for Mathematical Sciences in June 2007, to an audience of more than thirty international graduate students.
Tutorial on the Braid Groups
Simplicial Objects and Homotopy Groups
Introduction to Configuration Spaces and Their Applications
Braids and Magnetic Fields
Configuration Spaces, Braids, and Robotics
Braid Group Cryptography
Readership: Graduates and researchers in mathematics (low-dimensional topology, homotopy theory), applied mathematics (robotics and swarming, cryptography) and applications to magnetohydrodynamics and fluid flow.
420pp (approx.) Pub. date: Scheduled Winter 2009
ISBN: 978-981-4291-40-8