Gu, C., Purdue University, West Lafayette, IN, USA

Smoothing Spline ANOVA Models

2002. XIII, 289 pp. 42 figs. Hardcover
0-387-95353-1

Nonparametric function estimation with stochastic data, otherwise known as smoothing, has been studied by several generations of statisticians. Assisted by the recent availability of ample desktop and laptop computing power, smoothing methods are now finding their ways into everyday data analysis by practitioners.
While scores of methods have proved successful for univariate smoothing, ones practical in multivariate settings number far less. Smoothing spline ANOVA models are a versatile family of smoothing methods derived through roughness penalties that are suitable for both univariate and multivariate problems.
In this book, the author presents a comprehensive treatment of penalty smoothing under a unified framework. Methods are developed for (i) regression with Gaussian and non-Gaussian responses as well as with censored life time data; (ii) density and conditional density estimation under a variety of sampling schemes; and (iii) hazard rate estimation with censored life time data and covariates. The unifying themes are the general penalized likelihood method and the construction of multivariate models with built-in ANOVA decompositions. Extensive discussions are devoted to model construction, smoothing parameter selection, computation, and asymptotic convergence. Most of the computational and data analytical tools discussed in the book are implemented in R, an open-source clone of the popular S/S- PLUS language. Code for regression has been distributed in the R package gss freely available through the Internet on CRAN, the Comprehensive R Archive Network. The use of gss facilities is illustrated in the book through simulated and real data examples.

Contents: Introduction.- Model Construction.- Regression with Gaussian Type Responses.- More Splines.- Regression with Exponential Families.- Probability Density Estimation.- Hazard Rate Estimation.- Asymptotic Convergence.

Series: Springer Series in Statistics.

Ryan, R.A., National University of Ireland, Galway, Ireland

Introduction to Tensor Products of Banach Spaces

2002. XIV, 225 pp. 31 figs. Hardcover
1-85233-437-1

This volume provides a self-contained introduction to the theory of tensor products of Banach spaces. It is written for graduate students in analysis or for researchers in other fields who wish to become acquainted with this area. The only prerequisites are a basic knowledge of functional analysis and measure theory.
Features of particular interest include:
- A full treatment of the Grothendieck theory of tensor norms;
- Coverage of the Chevet-Saphar norms and their duals, along with the associated classes of nuclear, integral and summing operators;
- Chapters on the approximation property and the Radon-Nikodym property;
- Topics such as the Bochner and Pettis integrals, the principle of local reflexivity and the Grothendieck inequality placed in a natural setting;
- The classes of operators generated by a tensor norm and connections with the theory of operator ideals.
Each chapter is accompanied by worked examples and a set of exercises, and two appendices provide essential material on summability in Banach spaces and properties of spaces of measures that may be new to the beginner.

Contents: Preface.- Notation and Terminology.- Tensor Products.- The Projective Tensor Product.- The Injective Tensor Product.- The Approximation Property.- The Radon-Nikodym Property.- The Chevet-Saphar Tensor Products.- Tensor Norms.- Operator Ideals.- Appendix A: Suggestions for Further Reading.- Appendix B: Summability in Banach Spaces.- Appendix C: Spaces of Measures.- References.- Index.

Series: Springer Monographs in Mathematics.

Haigh, J., University of Sussex, Brighton, UK

Probability Models

2002. VIII, 256 pp. 15 figs. Softcover
1-85233-431-2

Probability Models is designed to aid students studying probability as part of an undergraduate course on mathematics or mathematics and statistics. It describes how to set up and analyse models of real-life phenomena that involve elements of chance. Motivation comes from everyday experiences of probability via dice and cards, the idea of fairness in games of chance, and the random ways in which, say, birthdays are shared or particular events arise. Applications include branching processes, random walks, Markov chains, queues, renewal theory, and Brownian motion. No specific knowledge of the subject is assumed, only a familiarity with the notions of calculus, and the summation of series. Where the full story would call for a deeper mathematical background, the difficulties are noted and appropriate references given. The main topics arise naturally, with definitions and theorems supported by fully worked examples and some 200 set exercises, all with solutions.

Contents: Preface.- Probability Spaces.- Conditional Probability and Independence.- Common Probability Distributions.- Random Variables.- Sums of Random Variables.- Convergence and Limit Theorems.- Stochastic Processes in Discrete Time.- Stochastic Processes in Continuous Time.- Appendix: Common Distributions, Mathfacts.- Bibliography.- Solutions.- Index.

Series: Springer Undergraduate Mathematics Series

Bollobas, B., The University of Memphis, TN, USA (Ed.)

Contemporary Combinatorics

2002. Approx. 300 pp. Hardcover
3-540-42725-2

This volume is a collection of survey papers in combinatorics that have grown out of lectures given in the workshop on Probabilistic Combinatorics at the Paul Erdos Summer Research Center in Mathematics in Budapest. The papers, reflecting the many facets of modern-day combinatorics, will be appreciated by specialists and general mathematicians alike: assuming relatively little background, each paper gives a quick introduction to an active area, enabling the reader to learn about the fundamental results and appreciate some of the latest developments. An important feature of the articles, very much in the spirit of Erdos, is the abundance of open problems.

Keywords: Combinatorics, probabilistic combinatorics, random graphs, combinatorical game theory, polynomials of knots, combinatorial optimization, Gibbs measures,

Contents: 1. N. Alon: Graph Powers.- 2. J. Beck: TIC-TAC-TOE.- 3. R. Arratia: On the Amount of Dependence in the Prime Factorization of a Uniform Random Integer.- 4. B. Bollobas/L. Pebody/D. Weinreich: A State Space Definition of the HOMFLY Invariant.- 5. G. Brightwell/P. Winkler: Random Colorings of a Cayley Tree.- 6. S. Janson: On Concentration of Probability.- 7. B. Bollobas/A.D. Scott: Better Bounds for Max Cut.- 8. D. Coppersmith/ G.B. Sorkin: On the Expected Incremental Cost of a Minimum Assignment.

Series: Bolyai Society Mathematical Studies. VOL. 10

Halasz, G., Eotvos Lorand University, Budapest, Hungary; Lovasz, L., Microsoft Research, Redmond, WA, USA; Simonovits, M., Hungarian Academy of Sciences, Budapest, Hungary; Sos, V.T., Hungarian Academy of Sciences, Budapest, Hungary (Eds.)

Paul Erdos and His Mathematics

2002. Approx. 1300 pp. (2-volume-set, not available separately). Hardcover
3-540-42236-6

Since his death in 1996, many scientific meetings have been dedicated to the memory of Paul Erdos. From July 4 to 11, 1999, the conference "Paul Erdos and his Mathematics" was held in Budapest, with the pretentious goal of showing the whole range of Erdos' work - a difficult task in view of Erdos' versatility and his broad scope of interest in mathematics. According to this goal, the topics of plenary lectures and parallel sections included number theory, combinatorics, analysis, set theory, probability, geometry and areas connecting them, like ergodic theory. The conference has contributed to changing the common view that Erdos worked only in combinatorics and combinatorial number theory. In the present two volumes, the editors have collected, besides some personal reminiscences by Paul's old friends, mainly survey articles of a different nature: on his work, and on areas he initiated or worked in.

Keywords: number theory, combinatorics, analysis, set theory, probability, geometry .

Series: Bolyai Society Mathematical Studies. VOL. 11