Neshveyev, Sergey, Stormer, Erling
Dynamical Entropy in Operator Algebras
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A
Series of Modern Surveys in Mathematics , Vol. 50
2006, Approx. 305 p., Hardcover
ISBN: 3-540-34670-8
About this book
During the last 30 years there have been several attempts at
extending the notion of entropy to noncommutative dynamical
systems. The authors present in the book the two most successful
approaches to the extensions of measure entropy and topological
entropy to the noncommutative setting and analyze in detail the
main models in the theory.
The book addresses mathematicians and physicists, including
graduate students, who are interested in quantum dynamical
systems and applications of operator algebras and ergodic theory.
Although the authors assume a basic knowledge of operator
algebras, they give precise definitions of the notions and in
most cases complete proofs of the results which are used.
Table of contents
Part I. General Theory.- 1 Classical Dynamical Systems.- 2
Relative Entropy.- 3 Dynamical Entropy.- 4 Maximality of Entropy
and Commutativity.- 5 Dynamical Abelian Models.- 6 Topological
Entropy.- 7 Dynamics on the State Space.- 8 Crossed Products.- 9
Variational Principle.- Part II. Special Topics.- 10 Relative
Entropy and Subfactors.- 11 Systems of Algebraic Origin.- 12
Binary Shifts.- 13 Bogoliubov Automorphisms.- 14 Free Products.-
A Completely Positive Maps.- B Operator Inequalities.- C Direct
Integrals.- References.- List of Symbols.- Index
Bojarski, B.; Mishchenko, A.S.; Troitsky, E.V.; Weber, A. (Eds.)
C*-algebras and Elliptic Theory
Series: Trends in Mathematics
2006, Approx. 350 p., Hardcover
ISBN: 3-7643-7686-4
About this book
This book consists of reviewed original research papers and
expository articles in index theory (especially on singular
manifolds), topology of manifolds, operator and equivariant K-theory,
Hopf cyclic cohomology, geometry of foliations, residue theory,
Fredholm pairs and others, and applications in mathematical
physics. The wide spectrum of subjects reflects the diverse
directions of research for which the starting point was the
Atiyah-Singer index theorem.
Table of contents
Preface.- Correspondences and Index.- Approximation Properties
for Discrete Groups.- A Riemannian Invariant.- Morse Inequalities
for Foliations.- Index Theory for Generalized Dirac Operators on
Open Manifolds.- Semiclassical Asymptotics and Spectral Gaps for
Periodic Magnetic Schrodinger Operators.- The Group of Unital C*-extensions.-
Lefschetz Theory on Manifolds with Singularities.- Residues and
Index for Bisingular Operators.- Hopf-type Cyclic Homology.- Thom
Isomorphism in Gauge-Invariant K-theory.- Pseudodifferential
Subspaces.- L2-invariants of Chain Complexes.- Bundles of C*-algebras.
Woodhouse, Nicholas
General Relativity
Series: Springer Undergraduate Mathematics Series
2006, VI, 232 p. 33 illus., Softcover
ISBN: 1-84628-486-4
About this textbook
This book follows on from Special Relativity (by the same author)
to provide a first course on general relativity. Rather than
treating the subject as a piece of abstract mathematics, the book
leads the reader into general relativity from the problem of
making distance and time measurements in the presence of gravity,
and presents the basic theory using mathematical techniques -
such as phase-plane analysis - that will already be very familiar
to mathematics undergraduates. Many of those studying general
relativity will also study modern differential geometry in
parallel and so the author keeps the geometric aspects of the
theory firmly in view, while keeping the emphasis on tools for
calculation rather than abstract structures.
Numerous problems, of varying levels of difficulty, are provided
and there are explicit links with recent developments to tempt
readers to further study, including descriptions of further
theoretical work, unresolved problems and up-to-date
observational evidence.
Table of contents
Introduction: Special Relativity and Newtonian Gravity.- Special
Relativity Revisited.- Tensors in Minkowski Space.- Curved Space-time.-
Tensors in Curved Space-time.- Curvature.- The Field Equations.-
The Schwarzschild Solution.- Black Holes
Samelson, Roger M., Wiggins, Stephen
Lagrangian Transport in Geophysical Jets and Waves
The Dynamical Systems Approach
Series: Interdisciplinary Applied Mathematics , Vol. 31
2006, Approx. 150 p. 29 illus., Hardcover
ISBN: 0-387-33269-3
About this book
This book provides an accessible introduction to a new set of
methods for the analysis of Lagrangian motion in geophysical
flows. These methods were originally developed in the abstract
mathematical setting of dynamical systems theory, through a
geometric approach to differential equations. Despite the recent
developments in this field and the existence of a substantial
body of work on geophysical fluid problems in the dynamical
systems and geophysical literature, this is the first
introductory text that presents these methods in the context of
geophysical fluid flow. The book is organized into seven
chapters; the first introduces the geophysical context and the
mathematical models of geophysical fluid flow that are explored
in subsequent chapters. The second and third cover the simplest
case of steady flow, develop basic mathematical concepts and
definitions, and touch on some important topics from the
classical theory of Hamiltonian systems. The fundamental elements
and methods of Lagrangian transport analysis in time-dependent
flows that are the main subject of the book are described in the
fourth, fifth, and sixth chapters. The seventh chapter gives a
brief survey of some of the rapidly evolving research in
geophysical fluid dynamics that makes use of this new approach.
Related supplementary material, including a glossary and an
introduction to numerical methods, is given in the appendices.
This book will prove useful to graduate students, research
scientists, and educators in any branch of geophysical fluid
science in which the motion and transport of fluid, and of
materials carried by the fluid, is of interest. It will also
prove interesting and useful to the applied mathematicians who
seek an introduction to an intriguing and rapidly developing area
of geophysical fluid dynamics. The book was jointly authored by a
geophysical fluid dynamicist, Roger M. Samelson of the College of
Oceanic and Atmospheric Sciences at Oregon State University, USA
and an applied mathematician, Stephen Wiggins of the School of
Mathematics, University of Bristol, UK.
Table of contents
Introduction.- Steadily Translating Waves and Meanders.-
Integrability of Lagrangian Motion.- Fluctuating Waves and
Meanders.- Material Manifolds, Flow Regimes, and Fluid Exchange.-
Lobe Transport and Flux.- Transport and Dynamics.- A Mathematical
Properties of Fluid Trajectories.- B Action-Angle Coordinates.- C
Numerical Methods.- D Finite-Time Material Manifolds: An Example.-
E Glossary.- Index.
Sinclair, Nathalie; Pimm, David; Higginson, William (Eds.)
Mathematics and the Aesthetic
New Approaches to an Ancient Affinity
Series: CMS Books in Mathematics
2007, Approx. 300 p. 79 illus., 14 in colour., Hardcover
ISBN: 0-387-30526-2
About this book
The essays in this book explore the ancient affinity between the
mathematical and the aesthetic, focusing on the fundamental
connections between these two modes of reasoning and
communicating. From historical, philosophical and psychological
perspectives, with particular attention to certain mathematical
areas such as geometry and analysis, the authors examine the ways
in which the aesthetic is ever present in mathematical thinking
and contributes to the growth and value of mathematical knowledge.
Written for:
Mathematicians, scientists, mathematics educators, and those
interested in psychological, sociological and philosophical
aspects of mathematics and mathematical thinking
Table of contents
Preface.- Acknowledgements.- Notes about Authors.- A Historical
Gaze at the Mathematical Aesthetic, by Nathalie Sinclair and
David Pimm.-Chapter 1 Aesthetics for the Working Mathematician,
by Jonathan M. Borwein.- Chapter 2 Beauty and Truth in
Mathematics, by Doris Schattschneider.- Chapter 3 Experiencing
Meanings in Geometry, by David W. Henderson and Daina Taimina.-
Chapter 4 The Aesthetic Sensibilities of Mathematicians, by
Nathalie Sinclair.- Chapter 5 The Meaning of Pattern, by Martin
Schiralli.- Chapter 6 Mathematics, Aesthetics and Being Human,
William Higginson.- Chapter 7 Mechanism and Magic in the
Psychology of Dynamic Geometry, by R. Nicholas Jackiw.- Chapter 8
Drawing on the Image in Mathematics and Art, by David Pimm.-
Chapter 9 Sensible Objects, by Dick Tahta.- Chapter 10 Aesthetics
and the eMathematical Mindf by David Pimm and Nathalie
Sinclair.- References.- Index of Names.- Index.-
Haase, Markus
The Functional Calculus for Sectorial Operators
Series: Operator Theory: Advances and Applications , Vol. 169
2006, Approx. 405 p., Hardcover
ISBN: 3-7643-7697-X
A Birkhauser book
About this book
The present monograph deals with the functional calculus for
unbounded operators in general and for sectorial operators in
particular. Sectorial operators abound in the theory of evolution
equations, especially those of parabolic type. They satisfy a
certain resolvent condition that leads to a holomorphic
functional calculus based on Cauchy-type integrals. Via an
abstract extension procedure, this elementary functional calculus
is then extended to a large class of (even meromorphic) functions.
With this functional calculus at hand, the book elegantly covers
holomorphic semigroups, fractional powers, and logarithms.
Special attention is given to perturbation results and the
connection with the theory of interpolation spaces. A chapter is
devoted to the exciting interplay between numerical range
conditions, similarity problems and functional calculus on
Hilbert spaces. Two chapters describe applications, for example
to elliptic operators, to numerical approximations of parabolic
equations, and to the maximal regularity problem.
This book is the first systematic account of a subject matter
which lies in the intersection of operator theory, evolution
equations, and harmonic analysis. It is an original and
comprehensive exposition of the theory as a whole. Written in a
clear style and optimally organised, it will prove useful for the
advanced graduate as well as for the experienced researcher.
Table of contents