This volume contains 27 refereed research articles and survey
papers written by experts in the field of stochastic analysis and
related topics. Most contributors are well known leading
mathematicians worldwide and prominent young scientists. The
volume reflects a review of the recent developments in stochastic
analysis and related topics. It puts in evidence the strong
interconnection of stochastic analysis with other areas of
mathematics, as well as with applications of mathematics in
natural and social economic sciences. The volume also provides
some possible future directions for the field.
Contents:
Invariant Gibbs Measures for the 2D Vortex Motion of Fluids (S
Albeverio & B Ferrario)
Limit Laws for Sums of Random Exponentials (G B Arous et al.)
Stochastic Models of Economic Optimization (M-F Chen)
Essential Spectrum on Riemannian Manifolds (K D Elworthy & F-Y
Wang)
Levy Process on Real Lie Algebras (U Franz)
Wick Rotation for Holomorphic Random Fields (H Gottschalk)
Stochastic Mollifier and Nash Inequality (R Leandre)
Precise Estimations Related to Large Deviations (S Liang)
Stochastic Holonomy (I Mitoma)
Independence and Product Systems (M Skeide)
and other papers
Readership: Graduate students, teachers and researchers in
stochastic analysis.
468pp Pub. date: Oct 2004
ISBN 981-256-104-8
This book provides a challenging and stimulating introduction
to the contemporary topics of complexity and criticality, and
explores their common basis.
Criticality refers to the behaviour of extended systems at a
phase transition where scale invariance prevails. The many
constituent microscopic parts bringing about macroscopic
phenomena that cannot be understood by considering a single part
alone. The phenomenology of phase transitions is introduced by
considering percolation, a simple model with a purely geometrical
phase transition, thus enabling the reader to become intuitively
familiar with concepts such as scale invariance and
renormalisation. The Ising model is then introduced, which
captures a thermodynamic phase transition from a disordered to an
ordered system as the temperature is lowered in zero external
field. By emphasising analogies between percolation and the Ising
model, the reader's intuition of phase transitions is developed
so that the underlying theoretical formalism may be appreciated
fully. These equilibrium systems undergo a phase transition only
if an external agent finely tunes certain external parameters to
particular values.
Besides fractals and phase transitions, there are many examples
in Nature of the emergence of such complex behaviour in slowly
driven non-equilibrium systems: earthquakes in seismic systems,
avalanches in granular media and rainfall in the atmosphere. A
class of non-equilibrium systems, not constrained by having to
tune external parameters to obtain critical behaviour, is
addressed in the framework of simple models, revealing that the
repeated application of simple rules may spontaneously give rise
to emergent complex behaviour not encoded in the rules themselves.
The common basis of complexity and criticality is identified and
applied to a range of non-equilibrium systems. Finally, the
reader is invited to speculate whether self-organisation in non-equilibrium
systems might be a unifying concept for disparate fields such as
statistical mechanics, geophysics and atmospheric physics.
Contents:
Percolation:
Percolating Phase Transition
In One and Two Dimensions, and in the Bethe Lattice
Geometric Properties of Clusters
Scaling Ansatz, Scaling Functions and Scaling Relations
Universality
Real-Space Renormalisation Group
Ising Model:
Review of Thermodynamics and Statistical Mechanics
Symmetry Breaking
Ferromagnetic Phase Transition
In One and Two Dimensions, and in the Mean-Field
Landau Theory of Continuous Phase Transitions
Scaling Ansatz, Scaling Functions and Scaling Relations
Universality
Real-Space Renormalisation Group
Self-Organised Criticality:
BTW Model in One and Two Dimensions, and in the Mean-Field
A Rice Pile Experiment and the Oslo Model
Earthquakes and the OFC Model
Rainfall
Self-Organised Criticality as a Unifying Principle
Readership: Students at all levels, researchers and instructors
interested in complexity and criticality.
400pp (approx.) Pub. date: Scheduled Summer 2005
ISBN 1-86094-504-X
ISBN 1-86094-517-1(pbk)
This proceedings volume is a collection of papers presented at
the International Conference on SPT2004 focusing on symmetry,
perturbation theory, and integrability.
The book provides an updated overview of the recent developments
in the various different fields of nonlinear dynamics, covering
both theory and applications. Special emphasis is given to
algebraic and geometric integrability, solutions to the N-body
problem of the “choreography” type, geometry and symmetry of
dynamical systems, integrable evolution equations, various
different perturbation theories, and bifurcation analysis.
The contributors to this volume include some of the leading
scientists in the field, among them: I Anderson, D Bambusi, S
Benenti, S Bolotin, M Fels, W Y Hsiang, V Matveev, A V Mikhailov,
P J Olver, G Pucacco, G Sartori, M A Teixeira, S Terracini, F
Verhulst and I Yehorchenko.
Contents:
Parametric Excitation in Nonlinear Dynamics (T Bakri)
Similarity Reductions of an Optical Model (M S Bruzon & M L
Gandarias)
A Regularity Theory for Optimal Partition Problems (M Conti et al.)
Periodic Solutions for Zero Mass Nonlinear Wave Equations (G
Gentile)
Geometric Reduction of Poisson Operators (K Marciniak & M
Biaszak)
Renormalization Group Symmetry and Gas Dynamics (S Murata)
Refined Computation of Hypernormal Forms (J Murdock)
Regularity of Pseudogroup Orbits (P J Olver & J Pohjanpelto)
On Birkhoff Conditional Integrals (G Pucacco)
and other papers
Readership: Researchers and academics.
400pp (approx.) Pub. date: Scheduled Spring 2005
ISBN 981-256-136-6
This book provides a broad introduction to gauge field
theories formulated on a space-time lattice, and in particular of
QCD. It serves as a textbook for advanced graduate students, and
also provides the reader with the necessary analytical and
numerical techniques to carry out research on his own. Although
the analytic calculations are sometimes quite demanding and go
beyond an introduction, they are discussed in sufficient detail,
so that the reader can fill in the missing steps. The book also
introduces the reader to interesting problems which are currently
under intensive investigation. Whenever possible, the main ideas
are exemplified in simple models, before extending them to
realistic theories. Special emphasis is placed on numerical
results obtained from pioneering work. These are displayed in
numerous figures.
Contents:
The Path Integral Approach to Quantization
The Free Scalar on the Lattice
Fermions on the Lattice
Abelian Gauge Fields on the Lattice and Compact QED
Non-Abelian Gauge Fields on the Lattice Compact QCD
The Wilson Loop and the Static Quark?Antiquak Potential
The QQ Potential in Some Simple Models
The Continuum Limit of Lattice QCD
Lattice Sum Rules
The Strong Coupling Expansion
The Hopping Parameter Expansion
Weak Coupling Expansion (I). The Φ3-Theory
Weak Coupling Expansion (II) & (III). Lattice QED
Monte Carlo Methods
Some Results of Monte Carlo Calculations
Path-Integral Representation of the Thermodynamical Partition
Function for Some Solvable Bosonic and Fermionic Systems
Finite Temperature Perturbation Theory Off and On the Lattice
Non-Perturbative QCD at Finite Temperature.
Readership: Graduate students and theoretical elementary particle
physicists in the field of lattice field theories.
580pp (approx.) Pub. date: Scheduled Summer 2005
ISBN 981-256-062-9
ISBN 981-256-168-4(pbk)
This comprehensive volume contains the state of the art on ODE’s
and PDE’s of different nature, functional differential
equations, delay equations, and others, mostly from the dynamical
systems point of view.
A broad range of topics are treated through contributions by
leading experts of their fields, presenting the most recent
developments. A large variety of techniques are being used,
stressing geometric, topological, ergodic and numerical aspects.
The scope of the book is wide, ranging from pure mathematics to
various applied fields. Examples of the latter are provided by
subjects from earth and life sciences, classical mechanics and
quantum-mechanics, among others.
Contents:
Computational Aspects of Differential Equations and Applications
Water Waves
Topological and Variational Methods
Qualitative Theory of Nonlinear Parabolic and Elliptic Equations
Around Hilbert’s 16th Problem
Navier?Stokes Equations and Reaction Diffusion Equations
Hyperbolic Dynamics and Beyond
Symmetry and Mechanics
Shock Waves and Conservation Laws
Nonlinear Elliptic Partial Differential Equations
Algebraic Aspects and Optimisation in Dynamical Systems
Case Studies in Theoretical Interpretation of Numerical
Experiments
Infinite-Dimensional Dynamics
Quasiperiodicity
Delay Equations
Wave Stability and Pattern Formation
Nonautonomous Dynamics
Normal Forms and Invariant Manifolds
Singular Perturbations
Differential Geometric Foliations and Flows
Homoclinic and Heteroclinic Dynamics
Mathematical Aspects of Celestical Mechanics
Readership: Graduate students and researchers in mathematics,
especially in ODE and PDE areas.
1180pp (approx.) Pub. date: Scheduled Summer 2005
ISBN 981-256-169-2
In the last decade, the development of new ideas in quantum
theory, including geometric and deformation quantization, the non-Abelian
Berry's geometric factor, super- and BRST symmetries, non-commutativity,
has called into play the geometric techniques based on the deep
interplay between algebra, differential geometry and topology.
The book aims at being a guide to advanced differential geometric
and topological methods in quantum mechanics. Their main
peculiarity lies in the fact that geometry in quantum theory
speaks mainly the algebraic language of rings, modules, sheaves
and categories. Geometry is by no means the primary scope of the
book, but it underlies many ideas in modern quantum physics and
provides the most advanced schemes of quantization.
Contents:
Commutative Geometry
Classical Hamiltonian Systems
Algebraic Quantization
Geometry of Algebraic Quantization
Geometric Quantization
Supergeometry
Deformation Quantization
Non-Commutative Geometry
Geometry of Quantum Groups
Readership: Theoreticians and mathematicians of postgraduate and
research level.
720pp (approx.) Pub. date: Scheduled Summer 2005
ISBN 981-256-129-3
Based on new and original results obtained by the authors, in
collaboration with other leading researchers such as R Deeley, J
T Horwood, R Milson and Jin Yue, this book presents a new
invariant theory along with many interesting applications. A
central theme of the book is that ITKT (invariant theory of
Killing tensors) is a pseudo-Riemannian analogue of the classical
invariant theory. The book presents a self-contained review
obtained so far in the development of ITKT. It also provides a
new approach to the study of Hamiltonian systems of classical
mechanics.
Although the subject lies at the intersection of differential
geometry, Lie group theory, classical mechanics and invariant
theory, the book is self-contained and accessible to upper years
undergraduate and graduate students.
Contents:
Introduction to Pseudo?Riemannian Geometry
A Glimpse into the Lie Group Theory
Classical Invariant Theory
Invariant Theory of Killing Tensors
Applications
Readership: Graduate students and researchers in differential
geometry and mathematical physics.
220pp (approx.) Pub. date: Scheduled Winter 2005
ISBN 981-256-058-0