Mark Gross, University of California, San Diego, CA

Tropical Geometry and Mirror Symmetry

CBMS Regional Conference Series in Mathematics, Number: 114
2011; 317 pp; softcover
ISBN-13: 978-0-8218-5232-3
Expected publication date is February 21, 2011.

Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry.

The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for "integral tropical manifolds." A complete version of the argument is given in two dimensions.

Readership

Graduate students and research mathematicians interested in mirror symmetry and tropical geometry.

Table of Contents

The three worlds
The tropics
The A- and B-models
Log geometry
Example: mathbb{P}^2
Mikhalkin's curve counting formula
Period integrals
The Gross-Siebert program
The program and two-dimensional results
Bibliography
Index of symbols
General index



************************************************************

Alexander Korostelev, Wayne State University, Detroit, MI, and Olga Korosteleva, California State University, Long Beach, CA

Mathematical Statistics: Asymptotic Minimax Theory

Graduate Studies in Mathematics, Volume: 119
2011; approx. 243 pp; hardcover
ISBN-13: 978-0-8218-5283-5
Expected publication date is February 20, 2011.

This book is designed to bridge the gap between traditional textbooks in statistics and more advanced books that include the sophisticated nonparametric techniques. It covers topics in parametric and nonparametric large-sample estimation theory. The exposition is based on a collection of relatively simple statistical models. It gives a thorough mathematical analysis for each of them with all the rigorous proofs and explanations. The book also includes a number of helpful exercises.

Readership

Graduate students and research mathematicians interested in mathematical statistics.

Table of Contents

Parametric models
The Fisher efficiency
The Bayes and minimax estimators
Asymptotic minimaxity
Some irregular statistical experiments
Change-point problem
Sequential estimators
Linear parametric regression
Nonparametric regression
Estimation in nonparametric regression
Local polynomial approximation of regression function
Estimation of regression in global norms
Estimation by splines
Asymptotic optimality in global norms
Estimation in nonparametric models
Estimation of functionals
Dimension and structure in nonparametric regression
Adaptive estimation
Testing of nonparametric hypotheses
Bibliography
Index of notation
Index



************************************************************

Michael E. Taylor, University of North Carolina, Chapel Hill, NC

Introduction to Differential Equations

Pure and Applied Undergraduate Texts, Volume: 14
2011; 409 pp; hardcover
ISBN-13: 978-0-8218-5271-2
Expected publication date is March 11, 2011.

The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis.

The first chapter treats single differential equations, emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton's laws. The first order linear theory starts with a self-contained presentation of the exponential and trigonometric functions, which plays a central role in the subsequent development of this chapter. Chapter 2 provides a mini-course on linear algebra, giving detailed treatments of linear transformations, determinants and invertibility, eigenvalues and eigenvectors, and generalized eigenvectors. This treatment is more detailed than that in most differential equations texts, and provides a solid foundation for the next two chapters. Chapter 3 studies linear systems of differential equations. It starts with the matrix exponential, melding material from Chapters 1 and 2, and uses this exponential as a key tool in the linear theory. Chapter 4 deals with nonlinear systems of differential equations. This uses all the material developed in the first three chapters and moves it to a deeper level. The chapter includes theoretical studies, such as the fundamental existence and uniqueness theorem, but also has numerous examples, arising from Newtonian physics, mathematical biology, electrical circuits, and geometrical problems. These studies bring in variational methods, a fertile source of nonlinear systems of differential equations. The reader who works through this book will be well prepared for advanced studies in dynamical systems, mathematical physics, and partial differential equations.

Readership

Undergraduate students interested in ordinary differential equations.

Table of Contents

Single differential equations
Linear algebra
Linear systems of differential equations
Nonlinear systems of differential equations
Bibliography
Index



**************************************************************************

I. Ya. Novikov, Voronezh State University, Russia, V. Yu. Protasov, Moscow State University, Russia, and M. A. Skopina, St. Petersburg University, Russia

Wavelet Theory

Translations of Mathematical Monographs, Volume: 239
2011; approx. 508 pp; hardcover
ISBN-13: 978-0-8218-4984-2
Expected publication date is April 2, 2011.

Wavelet theory lies on the crossroad of pure and computational mathematics, with connections to audio and video signal processing, data compression, and information transmission.

The present book is devoted to a systematic exposition of modern wavelet theory. It details the construction of orthogonal and biorthogonal systems of wavelets and studies their structural and approximation properties, starting with basic theory and ending with special topics and problems. The book also presents some applications of wavelets. Historical commentary is supplied for each chapter in the book, and most chapters contain exercises.

The book is intended for professional mathematicians and graduate students working in functional analysis and approximation theory. It is also useful for engineers applying wavelet theory in their work. Prerequisites for reading the book consist of graduate courses in real and functional analysis.

Readership

Graduate students and research mathematicians interested in wavelet theory.

Table of Contents

Wavelets on the line
Multivariate wavelets
Compactly supported refinable functions
Wavelets with compact support
Fractal properties of wavelets
Factorization of refinement equations
Smoothness of compactly supported wavelets
Nonstationary wavelets
Periodic wavelets
Approximation by periodic wavelets
Remarkable properties of wavelet bases
Auxiliary facts of the theory of functions and functional analysis
Historical comments
Bibliography
Index



******************************************************************

C.R. Rao, The Pennsylvania State University, PA, USA
Dipak Dey, University of Connecticut, CT, USA

Essential Bayesian Models

Hardbound, 586 pages
Published: NOV-2010
ISBN 13: 978-0-444-53732-4
Imprint: NORTH-HOLLAND

This accessible reference includes selected contributions from Bayesian Thinking - Modeling and Computation, Volume 25 in the Handbook of Statistics Series, with a focus on key methodologies and applications for Bayesian models and computation. It describes parametric and nonparametric Bayesian methods for modeling, and how to use modern computational methods to summarize inferences using simulation. The book covers a wide range of topics including objective and subjective Bayesian inferences, with a variety of applications in modeling categorical, survival, spatial, spatiotemporal, Epidemiological, small area and micro array data.

Contents

. Model Selection and Hypothesis Testing based on Objective Probabilities and Bayes Factors; 2. Bayesian Model Checking and Model Diagnostics; 3. Bayesian Nonparametric Modeling and Data Analysis: An Introduction; 4. Some Bayesian Nonparametric Models; 5. Bayesian Modeling in the Wavelet Domain; 6. Bayesian Methods for Function Estimation; 7. MCMC Methods to Estimate Bayesian Parametric Models; 8. Bayesian Computation: From Posterior Densities to Bayes Factors, Marginal Likelihoods, and Posterior Model Probabilities; 9. Bayesian Modelling and Inference on Mixtures of Distributions; 10. Variable Selection and Covariance Selection in Multivariate Regression Models; 11. Dynamic Models; 12. Elliptical Measurement Error Models - A Bayesian Approach; 13. Bayesian Sensitivity Analysis in Skew-elliptical Models; 14. Bayesian Methods for DNA Microarray Data Analysis; 15. Bayesian Biostatistics; 16. Innovative Bayesian Methods for Biostatistics and Epidemiology; 17. Modeling and Analysis for Categorical Response Data; 18. Bayesian Methods and Simulation-Based Computation for Contingency Tables; 19. Teaching Bayesian Thought to Nonstatisticians