Graduate Studies in Mathematics, Volume: 162
2015; approx. 314 pp; hardcover
ISBN-13: 978-0-8218-7578-0
Expected publication date is May 19, 2015.
This is an introductory course on the methods of computing asymptotics of probabilities of rare events: the theory of large deviations. The book combines large deviation theory with basic statistical mechanics, namely Gibbs measures with their variational characterization and the phase transition of the Ising model, in a text intended for a one semester or quarter course.
The book begins with a straightforward approach to the key ideas and results of large deviation theory in the context of independent identically distributed random variables. This includes Cramer's theorem, relative entropy, Sanov's theorem, process level large deviations, convex duality, and change of measure arguments.
Dependence is introduced through the interactions potentials of equilibrium statistical mechanics. The phase transition of the Ising model is proved in two different ways: first in the classical way with the Peierls argument, Dobrushin's uniqueness condition, and correlation inequalities and then a second time through the percolation approach.
Beyond the large deviations of independent variables and Gibbs measures, later parts of the book treat large deviations of Markov chains, the Gartner-Ellis theorem, and a large deviation theorem of Baxter and Jain that is then applied to a nonstationary process and a random walk in a dynamical random environment.
The book has been used with students from mathematics, statistics, engineering, and the sciences and has been written for a broad audience with advanced technical training. Appendixes review basic material from analysis and probability theory and also prove some of the technical results used in the text.
Graduate students interested in probability, the theory of large deviations, and statistical mechanics.
Large deviations: General theory and i.i.d. processes
Introductory discussion
The large deviation principle
Large deviations and asymptotics of integrals
Convex analysis in large deviation theory
Relative entropy and large deviations for empirical measures
Process level large deviations for i.i.d. fields
Statistical mechanics
Formalism for classical lattice systems
Large deviations and equilibrium statistical mechanics
Phase transition in the Ising model
Percolation approach to phase transition
Additional large deviation topics
Further asymptotics for i.i.d. random variables
Large deviations through the limiting generating function
Large deviations for Markov chains
Convexity criterion for large deviations
Nonstationary independent variables
Random walk in a dynamical random environment
Appendixes
Analysis
Probability
Inequalities from statistical mechanics
Nonnegative matrices
Bibliography
Notation index
Author index
General index
XYZ Series, Volume: 14
2014; 249 pp; hardcover
ISBN-13: 978-0-9885622-2-6
This book represents a collection of carefully selected geometry problems designed for passionate geometers and students preparing for the IMO. Assuming the theory and the techniques presented in the first two geometry books published by XYZ Press, 106 Geometry Problems from the AwesomeMath Summer Program and 107 Problems from the AwesomeMath Year-Round Program, this book presents a multitude of beautiful synthetic solutions that are meant to give a sense of how one should think about difficult geometry problems. On average, each problem comes with at least two such solutions and with additional remarks about the underlying configuration.
A publication of XYZ Press. Distributed in North America by the American Mathematical Society.
Middle and high school students interested in mathematics competition preparation.
Abbreviations and notation
Problems
Solutions
References and further reading
XYZ Series, Volume: 15
2014; 499 pp; hardcover
ISBN-13: 978-0-9885622-1-9
This book is a compilation and revision of the 2012 and 2013 volumes from the online journal of the same name. This book is aimed at high school students, participants in math competitions, undergraduates, and anyone who has a fire for mathematics. Passionate readers submitted many of the problems, solutions, and articles and all require creativity, experience, and comprehensive mathematical knowledge. This book is a great resource for students training for advanced national and international mathematics competitions such as USAMO and IMO.
A publication of XYZ Press. Distributed in North America by the American Mathematical Society.
Students interested in advanced mathematics competition preparation.
Problems
Solutions
Articles
Problem author index
Article author index
Contemporary Mathematics, Volume: 631
2015; 258 pp; softcover
ISBN-13: 978-1-4704-0931-9
Expected publication date is March 30, 2015.
This volume contains the proceedings of the International Conference on Recent Trends in Ergodic Theory and Dynamical Systems, in honor of S. G. Dani's 65th Birthday, held December 26-29, 2012, in Vadodara, India.
This volume covers many topics of ergodic theory, dynamical systems, number theory and probability measures on groups. Included are papers on Teichmuller dynamics, Diophantine approximation, iterated function systems, random walks and algebraic dynamical systems, as well as two surveys on the work of S. G. Dani.
Graduate students and research mathematicians interested in ergodic theory and dynamics systems and their applications.