R Aldrovandi (State University of Sa~o Paulo, Brazil)

SPECIAL MATRICES OF MATHEMATICAL PHYSICS
Stochastic, Circulant and Bell Matrices

This book expounds three special kinds of matrices that are of physical interest, centering on physical examples. Stochastic matrices describe dynamical systems of many different types, involving (or not) phenomena like transience, dissipation, ergodicity, nonequilibrium, and hypersensitivity to initial conditions. The main characteristic is growth by agglomeration, as in glass formation. Circulants are the building blocks of elementary Fourier analysis and provide a natural gateway to quantum mechanics and noncommutative geometry. Bell polynomials offer closed expressions for many formulas concerning Lie algebra invariants, differential geometry and real gases, and their matrices are instrumental in the study of chaotic mappings.


Contents:
Basics: Some Fundamental Notions
Stochastic Matrices: Evolving Systems
Markov Chains
Glass Transition
The Kerner Model
Formal Developments
Equilibrium, Dissipation and Ergodicity
Circulant Matrices: Prelude
Definition and Main Properties
Discrete Quantum Mechanics
Quantum Symplectic Structure
Bell Matrices: An Organizing Tool
Bell Polynomials
Determinants and Traces
Projectors and Iterates
Gases: Real and Ideal


Readership: Mathematical physicists, statistical physicists and researchers in the field of combinatorics and graph theory.

340pp (approx.) Pub. date: Scheduled Fall 2001
ISBN 981-02-4708-7

Lars Bergstro"m & Ulf Lindstro"m (University of Stockholm, Sweden)

THE OSKAR KLEIN MEMORIAL LECTURES
(Volume 3)

This is an invaluable collection of colloquium-type lectures given by some of the most prominent theoretical physicists of today. In a form accessible to the interested general physicist, it covers topics ranging from the use of field-theoretical methods in different contexts via duality symmetries between various field theories, to the Ads/CFT correspondence and cosmology.


Contents:
The Weak Interaction: Its History and Impact on Physics (T D Lee)
Electron Orbits and Superconductivity of Carbon 60 (T D Lee)
The Power of Duality -- Exact Results in 4D SUSY Field Theory (N Seiberg)
String Theory as a Universal Language (A M Polyakov)
The Cosmological Tests (P J E Peebles)
Anti-de-Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories (E Witten)
Can There Be Physics Without Experiments, Challenges, and Pitfalls? (G 't Hooft)


Readership: Researchers and graduate students in high energy physics, theoretical physics, astrophysics, cosmology and the history of physics.

200pp (approx.) Pub. date: Scheduled Fall 2001
ISBN 981-02-4691-9
ISBN 981-02-4692-7(pbk)

Jay Kappraff (New Jersey Institute of Technology, USA)

BEYOND MEASURE
A Guided Tour Through Nature, Myth, and Numbers

Series on Knots and Everything - Vol. 28

This book consists of essays that stand on their own but are also loosely connected. Part 1 documents how numbers and geometry arise in several cultural contexts and in nature: the ancient musical scale, proportion in architecture, ancient geometry, megalithic stone circles, the hidden pavements of the Laurentian library, the shapes of the Hebrew letters, and the shapes of biological forms. The focus is on how certain numbers, such as the golden and silver means, present themselves within these systems. Part 2 shows how many of the same numbers and number sequences are related to the modern mathematical study of numbers, dynamical systems, chaos, and fractals.

Contents:
Spirals in Nature
The Relationship Between Projective Geometry and Biological Form
Harmonic Law of the Ancient Musical Scales
The Projective Nature of the Musical Scale
Kepler, Bode's Law and the Harmony of the Spheres
Tiling with Zonogons and Zonohedra and Their Applications to Design
A Secret of Ancient Geometry: An Eight-Pointed Star Created by Tons Brunes with Many Interesting Properties
Proportion in Architecture with Applications to the Roman System of Architecture and the Medici Chapel in Florence
The Hidden Pavements of Michelangelo's Laurentian Library
The Geometry of the British Megalithic Stone Circles
Flame Letters of the Hebrew Alphabet
and other essays

Readership: Polytechnic or college students, designers, mathematicians and general readers.

520pp Pub. date: Scheduled Spring 2002
ISBN 981-02-4701-X
ISBN 981-02-4702-8(pbk)

Kai Lai Chung (Stanford University, USA)

GREEN, BROWN, AND PROBABILITY & BROWNIAN MOTION ON THE LINE

This invaluable book consists of two parts. Part 1 is the second edition of the author's widely acclaimed publication Green, Brown, and Probability , which first appeared in 1995. In this exposition the author reveals, from a historical perspective, the beautiful relations between the Brownian motion process in probability theory and two important aspects of the theory of partial differential equations initiated from the problems in electricity -- Green's formula for solving the boundary value problem of Laplace equations and the Newton-Coulomb potential.

Part 2 of the book comprises lecture notes based on a short course on "Brownian Motion on the Line" which the author has given to graduate students at Stanford University. It emphasizes the methodology of Brownian motion in the relatively simple case of one-dimensional space. Numerous exercises are included.


Contents:
Part 1: Green's Ideas
Probability and Potential
Process
Random Time
Markov Property
Brownian Construct
The Trouble with Boundary
Return to Green
Strong Markov Property
Transience
Last but Not Least
Least Energy
Part 2: Brownian Motion on the Line
Stopped Feynman-Kac Functionals


Readership: Graduate students and researchers in probability and statistics.

200pp (approx.) Pub. date: Scheduled Winter 2001
ISBN 981-02-4689-7
ISBN 981-02-4690-0(pbk)

Li Guo, W Keigher (Rutgers University, Newark, USA),
P Cassidy (Smith College, USA) & W Sit (City University of New York, USA)

DIFFERENTIAL ALGEBRA AND RELATED TOPICS
Rutgers University at Newark, USA 2 - 3 November 2000

Differential algebra explores properties of solutions to systems of (ordinary or partial, linear or nonlinear) differential equations from an algebraic point of view. It includes as special cases algebraic systems as well as differential systems with algebraic constraints. This algebraic theory of Joseph F Ritt and Ellis R Kolchin is further enriched by its interactions with algebraic geometry, Diophantine geometry, differential geometry, model theory, control theory, automatic theorem proving, combinatorics, and difference equations. Differential algebra now plays an important role in computational methods such as symbolic integration, and symmetry analysis of differential equations. This proceedings volume includes tutorial and survey papers presented at an international workshop held at Rutgers University, Newark.


Contents:
The Evolution of Differential Algebraic Geometry
Differential Galois Theory and the Inverse Problem
Galois Theory of Difference Equations
Applications of Differential Galois Theory in Hamiltonian Mechanics
Differential Polynomial Algebra
Symmetries of Differential Equations
Symbolic Integration (Integration in Finite Terms)
Differential Schemes
Model Theory and Differential Algebra
Baxter Algebra and Differential Algebra


Readership: Graduate students, pure mathematicians, logicians, algebraic geometers, applied mathematicians and physicists.

250pp (approx.) Pub. date: Scheduled Winter 2001
ISBN 981-02-4703-6