Jin Feng, University of Kansas, Lawrence, KS,
and Thomas G. Kurtz, University of Wisconsin at Madison, WI

Large Deviations for Stochastic Processes
Supplementary Material

Mathematical Surveys and Monographs, Volume: 131
2006; approx. 404 pp; hardcover
ISBN-10: 0-8218-4145-9

The book is devoted to the results on large deviations for a class of stochastic processes. Following an introduction and overview, the material is presented in three parts. Part 1 gives necessary and sufficient conditions for exponential tightness that are analogous to conditions for tightness in the theory of weak convergence. Part 2 focuses on Markov processes in metric spaces. For a sequence of such processes, convergence of Fleming's logarithmically transformed nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence of linear semigroups in weak convergence. Viscosity solution methods provide applicable conditions for the necessary convergence. Part 3 discusses methods for verifying the comparison principle for viscosity solutions and applies the general theory to obtain a variety of new and known results on large deviations for Markov processes. In examples concerning infinite dimensional state spaces, new comparison principles are derived for a class of Hamilton-Jacobi equations in Hilbert spaces and in spaces of probability measures.

Readership

Graduate students and research mathematicians interested in stochastic processes.

Table of Contents

Introduction
Introduction
An overview
The general theory of large deviations
Large deviations and exponential tightness
Large deviations for stochastic processes
Large deviations for Markov processes and semigroup convergence
Large deviations for Markov processes and nonlinear semigroup convergence
Large deviations and nonlinear semigroup convergence using viscosity solutions
Extensions of viscosity solution methods
The Nisio semigroup and a control representation of the rate function
Examples of large deviations and the comparison principle
The comparison principle
Nearly deterministic processes in $R^d$
Random evolutions
Occupation measures
Stochastic equations in infinite dimensions
Appendix
Operators and convergence in function spaces
Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators
Spectral properties for discrete and continuous Laplacians
Results from mass transport theory
Bibliography
Index

Edited by: Abba Gumel, Editor-in-Chief, University of Manitoba, Winnipeg, MB, Canada, Carlos Castillo-Chavez, Arizona State University, Tempe, AZ, Ronald E. Mickens, Clark Atlanta University, GA, and Dominic P. Clemence, North Carolina A&T State University, Greensboro, NC

Mathematical Studies on Human Disease Dynamics:
Emerging Paradigms and Challenges

Contemporary Mathematics, Volume: 410
2006; approx. 386 pp; softcover
ISBN-10: 0-8218-3775-3

This volume contains the proceedings of the AMS-SIAM-IMS Joint Summer Research Conference on Modeling the Dynamics of Human Diseases: Emerging Paradigms and Challenges, held in Snowbird, Utah, July 17-21, 2005.

The goal of the conference was to bring together leading and upcoming researchers to discuss the latest advances and challenges associated with the modeling of the dynamics of emerging and re-emerging diseases, and to explore various control strategies. The articles included in this book are devoted to some of the significant recent advances, trends, and challenges associated with the mathematical modeling and analysis of the dynamics and control of some diseases of public health importance. In addition to illustrating many of the diverse prevailing epidemiological challenges, together with the diversity of mathematical approaches needed to address them, this book provides insights on a number of topical modeling issues such as the modeling and control of mosquito-borne diseases, respiratory diseases, animal diseases (such as foot-and-mouth disease), cancer and tumor growth modeling, influenza, HIV, HPV, rotavirus, etc. This book also touches upon other important topics such as the use of modeling in homeland security and some review and new results on various modeling paradigms including network, stochastic and deterministic formulations together with the use of optimal control and related methods for evaluating control strategies.

Readership

Graduate students and research mathematicians interested in mathematical models of disease dynamics.

Table of Contents

L. J. S. Allen, E. J. Allen, and C. B. Jonsson -- The impact of environmental variation on hantavirus infection in rodents
J. P. Aparicio and J. C. Hernandez -- Preventive treatment of tuberculosis through contact tracing
C. Bowman and A. Gumel -- Optimal vaccination strategies for an influenza-like illness in a hetergeneous population
G. Chowell, A. Cintron-Arias, S. Del Valle, F. Sanchez, B. Song, J. M. Hyman, H. Hethcote, and C. Castillo-Chavez -- Mathematical applications associated with the deliberate release of infectious agents
G. Chowell, A. L. Rivas, N. W. Hengartner, J. M. Hyman, and C. Castillo-Chavez -- Critical response to post-outbreak vaccination against foot-and-mouth disease
L. G. de Pillis and A. E. Radunskaya -- Some promising approaches to tumor-immune modeling
E. H. Elbasha -- Impact of prophylactic vaccination against human papillomavirus infection
T. J. Emerson -- Population dynamics of developmental deficits due to an environmental neurotoxicant
W. Gu and H. Moore -- Optimal therapy regimens for treatment-resistant mutations of HIV
T. L. Jackson, R. Ashkenazi, S. Heusel, and H. V. Jain -- Cancer modeling: A perspective on what's new and what's next
Y. Jiang -- Understanding a killer: A predictive model for tumor development
H. R. Joshi, S. Lenhart, M. Y. Li, and L. Wang -- Optimal control methods applied to disease models
A. L. Lloyd, S. Valeika, and A. Cintron-Arias -- Infection dynamics on small-world networks
J. O. Lloyd-Smith, S. J. Schreiber, and W. M. Getz -- Moving beyond averages: Individual-level variation in disease transmission
E. M. Lungu, M. Kgosimore, and F. Nyabadza -- Models for the spread of HIV/AIDS: Trends in Southern Africa
R. E. Mickens -- Application of NSFD methods to the numerical integration of bio-sciences differential equation models
K. Rios-Soto, C. Castillo-Chavez, M. G. Neubert, E. S. Titi, and A.-A. Yakubu -- Epidemic spread in populations at demographic equilibrium
F. Sanchez, M. Engman, L. Harrington, and C. Castillo-Chavez -- Models for Dengue transmission and control
E. Shim, H. T. Banks, and C. Castillo-Chavez -- Seasonality of rotavirus infection with its vaccination
C. J. Struchiner, P. M. Luz, C. T. Codeco, F. C. Coelho, and E. Massad -- Current research issues in mosquito-borne diseases modeling
S. Tennenbaum, T. G. Kassem, S. Roudenko, and C. Castillo-Chavez -- The role of transactional sex in spreading HIV in Nigeria

David Gay

Explorations in Topology
Map Coloring, Surfaces and Knots

- Students begin to solve substantial problems right from the start.
- Mathematical ideas unfold through the context of a storyline
- Engaging and well-illustrated.

Description

This book give students a rich experience with low-dimensional topology, enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that would help them make sense of a future, more formal topology course. The innovative story-line style of the text models the problems-solving process, presents the development of concepts in a natural way, and through its informality seduces the reader into engagement with the material. The end-of-chapter Investigations give the reader opportunities to work on a variety of open-ended, non-routine problems, and, through a modified “Moore method”, to make conjectures from which theorems emerge. The students themselves emerge from these experiences owning concepts and results. The end-of-chapter Notes provide historical background to the chapter’s ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides opportunities for continued involvement in “research” beyond the topics of the book.

Readership: Upper division, junior/senior mathematics majors and for high school mathematics teachers; individuals who are interested in innovative approaches to the teaching of advanced undergraduate mathematics; mathematicians/mathematics educators interested/specializing in curriculum development.

ISBN: 0-12-370858-3 Paperback
Measurements: 6 X 9 in
Pages: 352
Publication Date: 15 November 2006

Sheldon Ross
University of Southern California, Los Angeles, CA

Introduction to Probability Models
Ninth Edition

A new section (3.7) on COMPOUND RANDOM VARIABLES, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions.

A new section (4.11) on HIDDDEN MARKOV CHAINS, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states.

Simplified Approach for Analyzing Nonhomogeneous Poisson processes

Additional results on queues relating to the
(a) conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system,;
(b) inspection paradox for M/M/1 queues
(c) M/G/1 queue with server breakdown

Many new examples and exercises.

Reviews

Praise from Reviewers:
“This is a fascinating introduction to applications from a variety of disciplines. Any curious student will love this book."
- Jean LeMaire, University of Pennsylvania
“I think Ross has done an admirable job of covering the breadth of applied probability. Ross
writes fantastic problems which really force the students to think divergently...The examples, like the exercises are great.”
- Matt Carlton, Cal Polytechnic Institute
“This book may be a model in the organization of the education process. I would definitely rate
this text to be the best probability models book at its level of difficulty...far more sophisticated and deliberate than its competitors.”
- Kris Ostaszewski, University of Illinois

Readership: Professionals and students in actuarial science, engineering, operations research, and other fields in applied probability.

ISBN: 0-12-598062-0 Book/Hardback
Measurements: 6 X 9 in
Pages: 750
Publication Date: 15 November 2006

Pajitnov, Andrei

Circle-valued Morse Theory

24 x 17 cm. Approx. 460 pages. Cloth.
ISBN 3-11-015807-8
Series: de Gruyter Studies in Mathematics 32

In 1927 M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology.

Reformulated in modern terms the geometric essence of Morse theory is as follows. For a smooth function f on a closed manifold having only non-degenerate critical points (a Morse function) there is a chain complex M* (the Morse complex) freely generated by the set of all critical points of f, such that the homology of M* is isomorphic to the homology of the manifold. The boundary operators in this complex are related to the geometry of the gradient flow of the function.

It is natural to consider also circle-valued Morse functions, that is, smooth functions with values in S1 having only non-degenerate critical points. The study of such functions was initiated by S. P. Novikov in the early 1980s in relation to a problem in hydrodynamics. The formulation of the circle-valued Morse theory is a new branch of topology.

At present the Morse?Novikov theory is a large and actively developing domain of differential topology, with applications and connections to many geometrical problems like the Arnol'd conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in S3. The aim of the present book is to give a systematic treatment of the geometric foundations of the subject, and of some recent research results.

The book is accessible for first year graduate students specializing in geometry and topology.