Contemporary Mathematics, Volume: 532
2010; 242 pp; softcover
ISBN-13: 978-0-8218-4958-3
Expected publication date is December 29, 2010.
This volume contains papers from the special program and international conference on Dynamical Numbers which were held at the Max-Planck Institute in Bonn, Germany in 2009.
These papers reflect the extraordinary range and depth of the interactions between ergodic theory and dynamical systems and number theory. Topics covered in the book include stationary measures, systems of enumeration, geometrical methods, spectral methods, and algebraic dynamical systems.
Graduate students and research mathematicians interested in ergodic theory, dynamical systems, and number theory.
Collected Works, Volume: 19
2010; approx. 408 pp; hardcover
ISBN-13: 978-0-8218-4876-0
Expected publication date is January 26, 2011.
In addition to his seminal work in topology, John Milnor is also an accomplished algebraist, producing a spectacular agenda-setting body of work related to algebraic K-theory and quadratic forms during the five year period 1965-1970. These papers, together with other (some of them previously unpublished) works in algebra are assembled here in this fifth volume of Milnor's Collected Papers. They constitute not only an important historical archive, but also, thanks to the clarity and elegance of Milnor's mathematical exposition, a valuable resource for work in the fields treated. In addition, Milnor's papers are complemented by detailed surveys on the current state of the field in two areas. One is on the congruence subgroup problem, by Gopal Prasad and Andrei Rapinchuk. The other is on algebraic K-theory and quadratic forms, by Alexander Merkurjev.
Graduate students and research mathematicians interested in algebra and group theory.
Algebras and groups
Introduction
On the structure of Hopf algebras, preprint
On the structure of Hopf algebras
Remarks on infinite-dimensional Lie groups
The representation rings of some classical groups
Growth of finitely generated solvable groups
The congruence subgroup problem
Introduction
On unimodular groups over number fields (preprint, 1965)
Solution of the congruence subgroup problem for SL_n(ngeq3) and Sp_{2n}(ngeq2)
On a functorial property of power residue symbols
On polylogarithms, Hurwitz zeta functions, and the Kubert identities
Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre
Algebraic K-theory and quadratic forms
Introduction
On isometries of inner product spaces
Algebraic K-theory and quadratic forms
Symmetric inner product spaces over a Dedekind domain (preprint, 1970)
Symmetric inner products in characteristic 2
Developments in algebraic K-theory and quadratic forms after the work of Milnor
Index
Pure and Applied Undergraduate Texts, Volume: 12
2010; 163 pp; hardcover
ISBN-13: 978-0-8218-5274-3
Expected publication date is December 18, 2010.
An Introduction to Complex Analysis and Geometry provides the reader with a deep appreciation of complex analysis and how this subject fits into mathematics. The book developed from courses given in the Campus Honors Program at the University of Illinois Urbana-Champaign. These courses aimed to share with students the way many mathematics and physics problems magically simplify when viewed from the perspective of complex analysis. The book begins at an elementary level but also contains advanced material.
The first four chapters provide an introduction to complex analysis with many elementary and unusual applications. Chapters 5 through 7 develop the Cauchy theory and include some striking applications to calculus. Chapter 8 glimpses several appealing topics, simultaneously unifying the book and opening the door to further study.
The 280 exercises range from simple computations to difficult problems. Their variety makes the book especially attractive.
A reader of the first four chapters will be able to apply complex numbers in many elementary contexts. A reader of the full book will know basic one complex variable theory and will have seen it integrated into mathematics as a whole. Research mathematicians will discover several novel perspectives.
Undergraduate students interested in complex analysis.
Student Mathematical Library, Volume: 56
2010; approx. 161 pp; softcover
ISBN-13: 978-0-8218-5281-1
Expected publication date is January 7, 2011.
The Erdos problem asks, What is the smallest possible number of distinct distances between points of a large finite subset of the Euclidean space in dimensions two and higher. The main goal of this book is to introduce the reader to the techniques, ideas, and consequences related to the Erdos problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience--from motivated high school students interested in mathematics to graduate students specializing in combinatorics and geometry--to absorb the content and appreciate its far reaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.
The book is heavily problem oriented, following the authors' firm belief that most of the learning in mathematics is done by working through the exercises. Many of these problems are recently published results by mathematicians working in the area. The order of the exercises is designed both to reinforce the material presented in the text and, equally importantly, to entice the reader to leave all worldly concerns behind and launch head first into the multifaceted and rewarding world of Erdos combinatorics.
Undergraduates, graduate students, and research mathematicians interested in geometric combinatorics and various topics in general combinatorics.
Introduction
The sqrt{n} theory
The n^{2/3} theory
The Cauchy-Schwarz inequality
Graph theory and incidences
The n^{4/5} theory
The n^{6/7} theory
Beyond n^{6/7}
Information theory
Dot products
Vector spaces over finite fields
Distances in vector spaces over finite fields
Applications of the Erd?s distance problem
Hyperbolas in the plane
Basic probability theory
Jensen's inequality
Bibliography
Biographical information
Index of terminology
Graduate Studies in Mathematics,Volume: 118
2011; approx. 411 pp; hardcover
ISBN-13: 978-0-8218-4945-3
Expected publication date is January 14, 2011.
The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term. It applies to infinite-dimensional as well as to finite-dimensional dynamical systems, and to discrete-time as well as to continuous-time semiflows.
This monograph provides a self-contained treatment of persistence theory that is accessible to graduate students. The key results for deterministic autonomous systems are proved in full detail such as the acyclicity theorem and the tripartition of a global compact attractor. Suitable conditions are given for persistence to imply strong persistence even for nonautonomous semiflows, and time-heterogeneous persistence results are developed using so-called "average Lyapunov functions".
Applications play a large role in the monograph from the beginning. These include ODE models such as an SEIRS infectious disease in a meta-population and discrete-time nonlinear matrix models of demographic dynamics. Entire chapters are devoted to infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bio-reactor, and an age-structured model of cells growing in a chemostat.
Graduate students and research mathematicians interested in dynamical systems and mathematical biology.
Introduction
Semiflows on metric spaces
Compact attractors
Uniform weak persistence
Uniform persistence
The interplay of attractors, repellers, and persistence
Existence of nontrivial fixed points via persistence
Nonlinear matrix models: Main act
Topological approaches to persistence
An SI endemic model with variable infectivity
Semiflows induced by semilinear Cauchy problems
Microbial growth in a tubular bio-reactor
Dividing cells in a chemostat
Persistence for nonautonomous dynamical systems
Forced persistence in linear Cauchy problems
Persistence via average Lyapunov functions
Tools from analysis and differential equations
Tools from functional analysis and integral equations
Bibliography
Index
Pure and Applied Undergraduate Texts, Volume: 13
2011; approx. 615 pp; hardcover
ISBN-13: 978-0-8218-5233-0
Expected publication date is March 9, 2011.
Foundations and Applications of Statistics simultaneously emphasizes both the foundational and the computational aspects of modern statistics. Engaging and accessible, this book is useful to undergraduate students with a wide range of backgrounds and career goals.
The exposition immediately begins with statistics, presenting concepts and results from probability along the way. Hypothesis testing is introduced very early, and the motivation for several probability distributions comes from p-value computations. Pruim develops the students' practical statistical reasoning through explicit examples and through numerical and graphical summaries of data that allow intuitive inferences before introducing the formal machinery. The topics have been selected to reflect the current practice in statistics, where computation is an indispensible tool. In this vein, the statistical computing environment textsf{R} is used throughout the text and is integral to the exposition. Attention is paid to developing students' mathematical and computational skills as well as their statistical reasoning. Linear models, such as regression and ANOVA, are treated with explicit reference to the underlying linear algebra, which is motivated geometrically.
Foundations and Applications of Statistics discusses both the mathematical theory underlying statistics and practical applications that make it a powerful tool across disciplines. The book contains ample material for a two-semester course in undergraduate probability and statistics. A one-semester course based on the book will cover hypothesis testing and confidence intervals for the most common situations.
Undergraduate students interested in statistics.
Summarizing data
Probability and random variables
Continuous distributions
Parameter estimation and testing
Likelihood-based statistics
Introduction to linear models
More linear models
A brief introduction to R
Some mathematical preliminaries
Geometry and linear algebra review
Review of Chapters 1-4
Hints, answers, and solutions to selected exercises
Bibliography
Index to R functions, packages, and data sets
Index