by Huai-Dong Cao and Xi-Ping Zhu

A Complete Proof of the Poincare and Geometrization Conjectures
- Application of the Hamilton-Perelman theory of the Ricci flow

The Poincare conjecture is one hundred years old, and one of the seven "Millennium Prize Problems" in mathematics.

The Asian Journal of Mathematics, Volume 10, Number 2 (June 2006):

In the past two decades, Ricci flow and in particular, Richard Hamilton's work therein, has received much attention as both having a profound influence on geometric evolution equations and as a possible approach to studying Thurston's Geometrization Conjecture. In this paper, Huai-Dong Cao (Lehigh University) and Xi-Ping Zhu (Sun Yat-Sen University, China) provide an essentially self-contained description of both the fundamental works of Hamilton and Perelman's recent breakthrough, as well as the important contributions by many others to the subject of Ricci flow and its application to the geometrization of three-manifolds.

The paper offers a complete proof of the famous Poincare conjecture and the Thurston geometrization conjecture based on the Hamilton-Perelman theory of Ricci flow.

Contents

Feifang Hu, William F. Rosenberger

The Theory of Response-Adaptive Randomization in Clinical Trials

ISBN: 0-471-65396-9
Hardcover
232 pages
August 2006

The Theory of Response-Adaptive Randomization in Clinical Trials is the result of the authors' ten-year collaboration as well as their collaborations with other researchers in investigating the important questions regarding response-adaptive randomization in a rigorous mathematical framework. Response-adaptive allocation has a long history in biostatistics literature; however, largely due to the disastrous ECMO trial in the early 1980s, there is a general reluctance to use these procedures.

This timely book represents a mathematically rigorous subdiscipline of experimental design involving randomization and answers fundamental questions, including:

How does response-adaptive randomization affect power?
Can standard inferential tests be applied following response-adaptive randomization?
What is the effect of delayed response?
Which procedure is most appropriate and how can "most appropriate" be quantified?
How can heterogeneity of the patient population be incorporated?
Can response-adaptive randomization be performed with more than two treatments or with continuous responses?
The answers to these questions communicate a thorough understanding of the asymptotic properties of each procedure discussed, including asymptotic normality, consistency, and asymptotic variance of the induced allocation. Topical coverage includes:

The relationship between power and response-adaptive randomization
The general result for determining asymptotically best procedures
Procedures based on urn models
Procedures based on sequential estimation
Implications for the practice of clinical trials
Useful for graduate students in mathematics, statistics, and biostatistics as well as researchers and industrial and academic biostatisticians, this book offers a rigorous treatment of the subject in order to find the optimal procedure to use in practice.

Boichenko, Vladimir A. / Leonov, Genadij A. / Reitmann, Volker

Dimension Theory for Ordinary Differential Equations

Aus der Reihe: Teubner-Texte zur Mathematik Bd. 141

2005. 443 pp. With 7 Fig. and 3 Tab. Softc.
ISBN: 3-519-00437-2 - Sofort lieferbar

This book is devoted to the estimation of dimension-like characteristics (Hausdorff dimension, fractal dimension, Lyapunov dimension, topological entropy) for attractors
(mainly global B-attractors) of ordinary differential equations, time-discrete systems and dynamical systems on finite-dimensional manifolds. The contraction under flows of
parameter-dependent outer measures is shown by introducing varying Lyapunov functions or metric tensors in the calculation of singular values. For the attractors of the Henon and Lorenz systems, exact formulae for the Lyapunov dimension are derived.

Aus dem Inhalt

Basic facts from matrix theory - Attractors, stability and Lyapunov functions - Introduction to dimension theory - Dimension and Lyapunov functions - Dimension estimates for invariant sets of vector fields on manifolds


Dr. Vladimir A. Boichenko, Barrikada Company, St. Petersburg
Prof. Dr. Gennadij A. Leonov, St. Petersburg State University
Dr. Volker Reitmann, MPI for the Physics of Complex Systems, Dresden

Vapnik, Vladimir

Estimation of Dependences Based on Empirical Data
Empirical Inference Science Afterword of 2006

Series: Information Science and Statistics
1st ed. 1982. Reprint, 2006, XVIII, 510 p., 28 illus., Hardcover
ISBN: 0-387-30865-2

About this book

In 1982, Springer published the English translation of the Russian book Estimation of Dependencies Based on Empirical Data which became the foundation of the statistical theory of learning and generalization (the VC theory). A number of new principles and new technologies of learning, including SVM technology, have been developed based on this theory.

The second edition of this book contains two parts:

- A reprint of the first edition which provides the classical foundation of Statistical Learning Theory

- Four new chapters describing the latest ideas in the development of statistical inference methods. They form the second part of the book entitled Empirical Inference Science

The second part of the book discusses along with new models of inference the general philosophical principles of making inferences from observations. It includes new paradigms of inference that use non-inductive methods appropriate for a complex world, in contrast to inductive methods of inference developed in the classical philosophy of science for a simple world.

The two parts of the book cover a wide spectrum of ideas related to the essence of intelligence: from the rigorous statistical foundation of learning models to broad philosophical imperatives for generalization.

The book is intended for researchers who deal with a variety of problems in empirical inference: statisticians, mathematicians, physicists, computer scientists, and philosophers.

Table of contents

Sperlich, Stefan; Hardle, Wolfgang; Aydinli, Gokhan (Eds.)

The Art of Semiparametrics

Series: Contributions to Statistics
2006, VIII, 178 p., 33 illus., Softcover
ISBN: 3-7908-1700-7

About this book

This selection of articles has emerged from different works presented at the conference "The Art of Semiparametrics" celebrated in 2003 in Berlin. The idea was to bring together junior and senior researchers but also practitioners working on semiparametric statistics in rather different fields. The meeting succeeded in welcoming a group that presents a broad range of areas where research on, respectively with, semiparametric methods is going on. It contains mathematical statistics, econometrics, finance, business statistics, etc. and thus combines theoretical contributions with more applied and partly even empirical studies. Although each article represents an original contribution to its own field, they all are written in a self-contained way to be read also by non-experts of the particular topic. This volume therefore offers a collection of individual works that together show the actual large spectrum of semiparametric statistics.

Table of contents