Student Mathematical Library, Volume: 52
2009; 314 pp; softcover
ISBN-13: 978-0-8218-4889-0
Expected publication date is November 26, 2009.
Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular "chaotic" motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them.
The first half of the book introduces some of the key ideas in fractal geometry and dimension theory--Cantor sets, Hausdorff dimension, box dimension--using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents.
In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science--the FitzHugh-Nagumo model and the Lorenz system of differential equations.
This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed.
This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.
This volume is published in cooperation with the Mathematics Advanced Study Semesters.
Undergraduate and graduate students interested in dynamical systems and fractal geometry.
Contemporary Mathematics, Volume: 499
2009; 270 pp; softcover
ISBN-13: 978-0-8218-4771-8
This volume represents the proceedings of the conference on Groups, Rings and Group Rings, held July 28-August 2, 2008, in Ubatuba, Brazil.
Papers in this volume contain results in active research areas in the theory of groups, group rings and algebras (including noncommutative rings), polynomial identities, Lie algebras and superalgebras. In particular, topics such as growth functions on varieties, groups of units in group rings, representation theory of Lie algebras, Jordan, alternative and Leibniz algebras, graded identities, automorphisms of trees, and partial actions, are discussed.
Graduate students and research mathematicians interested in ring theory.
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Contemporary Mathematics, Volume: 500
2009; 186 pp; softcover
ISBN-13: 978-0-8218-4744-2
This volume contains the proceedings of the conference on Spectral and Scattering Theory for Quantum Magnetic Systems, which took place at CIRM, Luminy, France, in July 2008. The main purpose of this conference was to bring together a number of specialists in the mathematical modelling of magnetic phenomena in quantum mechanics, to mark the recent progress as well as to outline the future development in this area.
This volume contains original results presented by some of the invited speakers and surveys on recent advances in the mathematical theory of quantum magnetic Hamiltonians.
Most of the talks at the conference, as well as the articles in this volume, have been dedicated to one of the following topics:
Spectral and scattering theory for magnetic Schrodinger operators
Magnetic Pauli and Dirac operators
Magnetic operators on manifolds
Microlocal analysis of magnetic Hamiltonians
Random Schrodinger operators and quantum Hall effect
Ginsburg-Landau equation, supraconductivity, magnetic bottles
Bose-Einstein condensate, Gross-Pitaevski equation
Magnetic Lieb-Thirring inequalities, stability of matter
Graduate students and research mathematicians interested in mathematics and theoretical physics.
Contemporary Mathematics, Volume: 501
2009; approx. 156 pp; softcover
ISBN-13: 978-0-8218-4647-6
Expected publication date is November 27, 2009. S
This volume reports on research related to Discrete Groups and Geometric Structures, as presented during the International Workshop held May 26-30, 2008, in Kortrijk, Belgium.
Readers will benefit from impressive survey papers by John R. Parker on methods to construct and study lattices in complex hyperbolic space and by Ursula Hamenstadt on properties of group actions with a rank-one element on proper textrm{CAT}(0)-spaces.
This volume also contains research papers in the area of group actions and geometric structures, including work on loops on a twice punctured torus, the simplicial volume of products and fiber bundles, the homology of Hantzsche-Wendt groups, rigidity of real Bott towers, circles in groups of smooth circle homeomorphisms, and groups generated by spine reflections admitting crooked fundamental domains.
Graduate students and research mathematicians interested in group actions on geometric structures and their properties.
J. R. Parker -- Complex hyperbolic lattices
U. Hamenstadt -- Rank-one isometries of proper CAT(0)-spaces
R. Agueda -- Trace polynomial for simple loops on the twice punctured torus
M. Bucher -- Simplicial volume of products and fiber bundles
K. Dekimpe and N. Petrosyan -- Homology of Hantzsche-Wendt groups
Y. Kamishima and A. Nazra -- Seifert fibred structure and rigidity on real Bott towers
A. Adouani and H. Marzougui -- Exotic circles in groups of piecewise smooth circle homeomorphisms
V. Charette -- Groups generated by spine reflections admitting crooked fundamental domains
AMS/IP Studies in Advanced Mathematics, Volume: 47
2009; 482 pp; hardcover
ISBN-13: 978-0-8218-4935-4
Expected publication date is January 2, 2010.
The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace-Beltrami operator and the associated heat equation.
The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectral-theoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels.
Grigor'yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research.
Titles in this series are co-published with International Press, Cambridge, MA.
Graduate students and research mathematicians interested in geometric analysis; heat kernel methods in geometry and analysis.
Laplace operator and the heat equation in mathbb{R}^n
Function spaces in mathbb{R}^n
Laplace operator on a Riemannian manifold
Laplace operator and heat equation in L^{2}(M)
Weak maximum principle and related topics
Regularity theory in mathbb{R}^n
The heat kernel on a manifold
Positive solutions
Heat kernel as a fundamental solution
Spectral properties
Distance function and completeness
Gaussian estimates in the integrated form
Green function and Green operator
Ultracontractive estimates and eigenvalues
Pointwise Gaussian estimates I
Pointwise Gaussian estimates II
Reference material
Bibliography
Some notation
Index