Paper | July 2003 | ISBN: 0-691-09291-5
Cloth | July 2003 | ISBN: 0-691-09290-7
392 pp. | 6 x 9 | 28 line illus.
The formulation, analysis, and re-evaluation of mathematical
models in population biology has become a valuable source of
insight to mathematicians and biologists alike. This book
presents an overview and selected sample of these results and
ideas, organized by biological theme rather than mathematical
concept, with an emphasis on helping the reader develop
appropriate modeling skills through use of well-chosen and varied
examples.
Part I starts with unstructured single species population models,
particularly in the framework of continuous time models, then
adding the most rudimentary stage structure with variable stage
duration. The theme of stage structure in an age-dependent
context is developed in Part II, covering demographic concepts,
such as life expectation and variance of life length, and their
dynamic consequences. In Part III, the author considers the
dynamic interplay of host and parasite populations, i.e., the
epidemics and endemics of infectious diseases. The theme of stage
structure continues here in the analysis of different stages of
infection and of age-structure that is instrumental in optimizing
vaccination strategies.
Each section concludes with exercises, some with solutions, and
suggestions for further study. The level of mathematics is
relatively modest; a "toolbox" provides a summary of
required results in differential equations, integration, and
integral equations. In addition, a selection of Maple worksheets
is provided.
The book provides an authoritative tour through a dazzling
ensemble of topics and is both an ideal introduction to the
subject and reference for researchers.
Horst R. Thieme is Professor of Mathematics at Arizona State
University. He is the editor of Differential Equations and
Applications to Biology and to Industry and an editor of the
Journal of Mathematical Analysis and Applications.
Endorsement:
"This is an inspiring book, which is carefully written in
rich language. What I like in particular is the systematic use of
submodels for behavioral processes on short time scales to derive
model ingredients for population changes on longer time scales."--Odo
Diekmann, University of Utrecht
Series: Princeton Series in Theoretical and Computational Biology
Cloth | February 2003 | ISBN: 0-691-11298-3
180 pp. | 6 x 9 | 5 line illus.
Rabinowitz's classical global bifurcation theory, which concerns
the study in-the-large of parameter-dependent families of
nonlinear equations, uses topological methods that address the
problem of continuous parameter dependence of solutions by
showing that there are connected sets of solutions of global
extent. Even when the operators are infinitely differentiable in
all the variables and parameters, connectedness here cannot in
general be replaced by path-connectedness. However, in the
context of real-analyticity there is an alternative theory of
global bifurcation due to Dancer, which offers a much stronger
notion of parameter dependence.
This book aims to develop from first principles Dancer's global
bifurcation theory for one-parameter families of real-analytic
operators in Banach spaces. It shows that there are globally
defined continuous and locally real-analytic curves of solutions.
In particular, in the real-analytic setting, local analysis can
lead to global consequences--for example, as explained in detail
here, those resulting from bifurcation from a simple eigenvalue.
Included are accounts of analyticity and implicit function
theorems in Banach spaces, classical results from the theory of
finite-dimensional analytic varieties, and the links between
these two and global existence theory.
Laying the foundations for more extensive studies of real-analyticity
in infinite-dimensional problems and illustrating the theory with
examples, Analytic Theory of Global Bifurcation is intended for
graduate students and researchers in pure and applied analysis.
Boris Buffoni holds a Swiss National Science Foundation
Professorship in Mathematics at the Swiss Federal Institute of
Technology-Lausanne. John Toland is Professor of Mathematical
Sciences at the University of Bath and a Senior Research Fellow
of the UK's Engineering and Physical Sciences Research Council
Series: Princeton Series in Applied Mathematics
Paper | February 2003 | ISBN: 0-691-02772-2
Cloth | February 2003 | ISBN: 0-691-02771-4
120 pp. | 6 x 9 | 33 line illus.
This book is a spectacular introduction to the modern
mathematical discipline known as the Theory of Games. Harold Kuhn
first presented these lectures at Princeton University in 1952.
They succinctly convey the essence of the theory, in part through
the prism of the most exciting developments at its frontiers half
a century ago. Kuhn devotes considerable space to topics that,
while not strictly the subject matter of game theory, are firmly
bound to it. These are taken mainly from the geometry of convex
sets and the theory of probability distributions.
The book opens by addressing "matrix games," a name
first introduced in these lectures as an abbreviation for two-person,
zero-sum games in normal form with a finite number of pure
strategies. It continues with a treatment of games in extensive
form, using a model introduced by the author in 1950 that quickly
supplanted von Neumann and Morgenstern's cumbersome approach. A
final section deals with games that have an infinite number of
pure strategies for the two players.
Throughout, the theory is generously illustrated with examples,
and exercises test the reader's understanding. A historical note
caps off each chapter. For readers familiar with the calculus and
with elementary matrix theory or vector analysis, this book
offers an indispensable store of vital insights on a subject
whose importance has only grown with the years.
Harold W. Kuhn is Professor Emeritus of Mathematics at Princeton
University. Joint winner of the 1980 von Neumann Prize in Theory,
he is internationally known for co-authoring a paper that
initiated the theory of "nonlinear programming." Kuhn
is the editor or coeditor of several books (all Princeton),
including The Essential John Nash, Classics in Game Theory,
Linear Inequalities and Related Systems, and Contributions to the
Theory of Games, I and II.
Series: Annals of Mathematics Studies
Cloth | April 2003 |ISBN: 0-691-11384-X
320 pp. | 6 x 9 | 40 line illus.
This first volume, a three-part introduction to the subject, is
intended for students with a beginning knowledge of mathematical
analysis who are motivated to discover the ideas that shape
Fourier analysis. It begins with the simple conviction that
Fourier arrived at in the early nineteenth century when studying
problems in the physical sciences--that an arbitrary function can
be written as an infinite sum of the most basic trigonometric
functions.
The first part implements this idea in terms of notions of
convergence and summability of Fourier series, while highlighting
applications such as the isoperimetric inequality and
equidistribution. The second part deals with the Fourier
transform and its applications to classical partial differential
equations and the Radon transform; a clear introduction to the
subject serves to avoid technical difficulties. The book closes
with Fourier theory for finite abelian groups, which is applied
to prime numbers in arithmetic progression.
In organizing their exposition, the authors have carefully
balanced an emphasis on key conceptual insights against the need
to provide the technical underpinnings of rigorous analysis.
Students of mathematics, physics, engineering and other sciences
will find the theory and applications covered in this volume to
be of real interest.
The Princeton Lectures in Analysis represents a sustained effort
to introduce the core areas of mathematical analysis while also
illustrating the organic unity between them. Numerous examples
and applications throughout its four planned volumes, of which
Fourier Analysis is the first, highlight the far-reaching
consequences of certain ideas in analysis to other fields of
mathematics and a variety of sciences. Stein and Shakarchi move
from an introduction addressing Fourier series and integrals to
in-depth considerations of complex analysis; measure and
integration theory, and Hilbert spaces; and, finally, further
topics such as functional analysis, distributions and elements of
probability theory.
Cloth | May 2003 |ISBN: 0-691-09983-9
376 pp. | 6 x 9 | 2 halftones. 87 line illus. 20 tables.
Among the myriad of constants that appear in mathematics, p, e,
and i are the most familiar. Following closely behind is g, or
gamma, a constant that arises in many mathematical areas yet
maintains a profound sense of mystery.
In a tantalizing blend of history and mathematics, Julian Havil
takes the reader on a journey through logarithms and the harmonic
series, the two defining elements of gamma, toward the first
account of gamma's place in mathematics.
Introduced by the Swiss mathematician Leonhard Euler (1707-1783),
who figures prominently in this book, gamma is defined as the
limit of the sum of 1 + 1/2 + 1/3 + . . . up to 1/n, minus the
natural logarithm of n--the numerical value being 0.5772156. . ..
But unlike its more celebrated colleagues p and e, the exact
nature of gamma remains a mystery--we don't even know if gamma
can be expressed as a fraction.
Among the numerous topics that arise during this historical
odyssey into fundamental mathematical ideas are the Prime Number
Theorem and the most important open problem in mathematics today--the
Riemann Hypothesis (though no proof of either is offered!).
Sure to be popular with not only students and instructors but all
math aficionados, Gamma takes us through countries, centuries,
lives, and works, unfolding along the way the stories of some
remarkable mathematics from some remarkable mathematicians.
Julian Havil is a Master at Winchester College, England, where he
has taught mathematics for nearly thirty years. He received a Ph.D.
in mathematics from Oxford University. Freeman Dyson is Professor
Emeritus of Physics at the Institute for Advanced Study,
Princeton. He is the author of several books, including
Disturbing the Universe and Origins of Life.
Endorsements:
"I like this book very much. So much, in fact, that I found
myself muttering 'neat stuff!' all the way through. While it is
about an important topic, there isn't a single competitor. This
amazing oversight by past authors is presumably the result of the
topic requiring an author with a pretty sophisticated
mathematical personality. Havil clearly has that. His skillful
weaving of mathematics and history makes the book a 'fun' read.
Many instructors will surely find the book attractive."--Paul
J. Nahin, author of Duelling Idiots and Other Probability
Puzzlers and An Imaginary Tale
"This is an excellent book, mathematically as well as
historically. It represents a significant contribution to the
literature on mathematics and its history at the upper
undergraduate and graduate levels. Julian Havil injects genuine
excitement into the topic."--Eli Maor, author of e: The
Story of a Number