Series: Springer Monographs in Mathematics
2009, XVIII, 242 p. 101 illus., 15 in color., Hardcover
ISBN: 978-0-387-85493-9
Due: January 2009
For metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods ? the classical (separable metric) and the modern (not necessarily separable metric). While the classical theory is now well documented in several books, this is the first book to unify the modern theory (1960 ? 2007). Like the classical theory, the modern theory fundamentally involves the unit interval.
By the 1970s, the author of this monograph generalized Cantor’s 1883 construction (identify adjacent-endpoints in Cantor’s set) of the unit interval, obtaining ? for any given weight ? a one-dimensional metric space that contains rationals and irrationals as counterparts to those in the unit interval.
Following the development of fractal geometry during the 1980s, these new spaces turned out to be the first examples of attractors of infinite iterated function systems ? “generalized fractals.” The use of graphics to illustrate the fractal view of these spaces is a unique feature of this monograph. In addition, this book provides historical context for related research that includes imbedding theorems, graph theory, and closed imbeddings.
This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. It can also serve as a text for graduate seminars or self-study ? the interested reader will find many relevant open problems that will motivate further research into these topics.
Preface.- Introduction.- Construction of JA=j alpha.- Self-Similarity and Jn+1 for Finite n.- No-Carry Property of wA.- Imbedding Ja in Hilbert Space.- Infinite IFS with Attractor wA.- Infinite IFS with Attractor wA.- Dimension Zero.- Decompositions.- The Jn+1 Imbedding Theorem.- Minimal-Exponent Question.- The JA Imbedding Theorem.- 1992-2007 Ja Related Research.- Isotopy Moves J5 Into 3-Space.- From 2-Web IFS to 2 Simplex IFS 2-Space and the 1-Sphere.- From 3-Web IFS to 3-Simplex 3-Space and the 2-Sphere.- Background Basics.- The Standard Simplex.- Measures and Fractal Dimension.- Bibliography.- Index.-
Series: Springer Series in Statistics
2009, Approx. 225 p., Hardcover
ISBN: 978-0-387-79051-0
Due: January 2009
Methods of nonparametric estimation are located at the core of modern statistical science. The aim of this book is to give a short but mathematically self-contained introduction to the theory of nonparametric estimation. The emphasis is on the construction of optimal estimators; therefore the concepts of minimax optimality and adaptivity, as well as the oracle approach, occupy the central place in the book.
This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs.
The book has three chapters. Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinsker's theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity.
Nonparametric estimators.- Lower bounds on the minimax risk.- Asymptotic efficiency and adaptation.- Appendix.- References.- Index.
Series: Graduate Texts in Mathematics , Vol. 253
2009, X, 208 p. 5 illus., Hardcover
ISBN: 978-0-387-85528-8
Due: January 2009
This text is intended for a one-semester introductory course in functional analysis for graduate students and well-prepared advanced undergraduates in mathematics and related fields. It is also suitable for self-study, and could be used for an independent reading course for undergraduates preparing to start graduate school.
While this book is relatively short, the author has not sacrificed detail. Arguments are presented in full, and many examples are discussed, making the book ideal for the reader who may be learning the material on his or her own, without the benefit of a formal course or instructor. Each chapter concludes with an extensive collection of exercises.
The choice of topics presented represents not only the author's preferences, but also her desire to start with the basics and still travel a lively path through some significant parts of modern functional analysis. The text includes some historical commentary, reflecting the author's belief that some understanding of the historical context of the development of any field in mathematics both deepens and enlivens one's appreciation of the subject.
The prerequisites for this book include undergraduate courses in real analysis and linear algebra, and some acquaintance with the basic notions of point set topology. An Appendix provides an expository discussion of the more advanced real analysis prerequisites, which play a role primarily in later sections of the book.
Barbara MacCluer is Professor of Mathematics at University of Virginia. She also co-authored a book with Carl Cowen, Composition Operators on Spaces of Analytic
Preface.- Hilbert Space Preliminaries.- Operator Theory Basics.- The Big Three.- Compact Operators.- Banach and C+- Algebras.- The Spectral Theorem.- Real Analysis Topics.- References.- Index.-
Series: Progress in Mathematical Physics , Preliminary entry 251
2009, Approx. 150 p., Hardcover
ISBN: 978-3-7643-8796-9
Due: February 2009
The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, informatics, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. In 1978+, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained.
Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also their intersection numbers.
The so called Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces.
In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).
The book intends to be self-contained and pedagogical, and will provide comprehensive proofs, several examples, and will give the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, will be discussed, and in particular we give a proof of the Witten-Kontsevich conjecture.
Series: Problem Books in Mathematics
2009, Approx. 360 p. 6 illus., Hardcover
ISBN: 978-0-387-87861-4
Due: July 2009
The book is a collection of problems in the theory of stochastic processes and its application to financial mathematics and risk theory. More than 1000 problems are divided into 23 chapters. Each chapter is supplied by the brief overview of the main theoretical results. Most of the problems are supplied by either solutions or hints to the solutions.
Preface.- The definition of a stochastic process.- The characteristics of a stochastic process.- Properties of trajectories of a stochastic processes.- Continuity, differentiability, integrability.- Processes with independent increments.- Gaussian processes.- Martingales and associated processes with discrete and continuous time.- Stationary processes with discrete and continuous time.- Prognosis, interpolation, filtration.- Markov chains with discrete and continuous time.- Regeneration theory, Random walks, Queuing theory.- Markov and diffusion processes.- Ito integral and Ito formula..- Stochastic differential equations.- Optimal control for Markov processes, Optimal stopping.- Measures in a metric spaces.- Weak convergence of measures, functional limit theorems.- The basic functionals of risk theory.- Index.-