Borisyuk, A., Ermentrout, G.B., Friedman, A., Terman, D.
Tutorials in Mathematical Biosciences I
Mathematical Neuroscience
Series: Lecture Notes in Mathematics,
Subseries: Mathematical Biosciences, Vol. 1860
2005, IX,170p., Softcover
ISBN: 3-540-23858-1
About this textbook
This volume introduces some basic theories on computational
neuroscience. Chapter 1 is a brief introduction to neurons,
tailored to the subsequent chapters. Chapter 2 is a self-contained
introduction to dynamical systems and bifurcation theory,
oriented towards neuronal dynamics. The theory is illustrated
with a model of Parkinson's disease. Chapter 3 reviews the theory
of coupled neural oscillators observed throughout the nervous
systems at all levels; it describes how oscillations arise, what
pattern they take, and how they depend on excitory or inhibitory
synaptic connections. Chapter 4 specializes to one particular
neuronal system, namely, the auditory system. It includes a self-contained
introduction, from the anatomy and physiology of the inner ear to
the neuronal network that connects the hair cells to the cortex,
and describes various models of subsystems.
Table of contents
Preface.- A. Friedman: Introduction to Neurons.- D. Terman: An
Introduction to Dynamical Systems and Neuronal Dynamics.- B.
Ermentrout: Neural Oscillators.- A. Borisyuk: Physiology and
Mathematical Modeling of the Auditory System.
Everest, G., Ward, Thomas
An Introduction to Number Theory
Series: Graduate Texts in Mathematics, Preliminary entry 331
2005, Approx. 270 p. 20 illus., Hardcover
ISBN: 1-85233-917-9
About this textbook
The book aims to take readers to a deeper understanding of the
patterns of thought that have shaped the modern understanding of
number theory. It begins with the fundamental theorem of
arithmetic and shows how it echoes through much of number theory
over the last two hundred years.
One of the main strengths of this book is the narrative. Everest
and Ward present number theory as a living subject, showing how
various new developments have drawn upon older traditions.
The authors concentrate on the underlying ideas instead of
working out the most general and complete version of a result.
They select material from both the algebraic and analytic
disciplines and sometimes present several different proofs of a
single result to show the differing viewpoints and also to
capture the imagination of the reader and help them to discover
their own tastes. They also cover important topics of significant
interest, eg. elliptic functions and the new primality test,
which are often omitted from other books at this level.
Table of contents
A Brief History of Prime;-Diophantine Equations;-Elliptic Curves;-Elliptic
Functions and Mordellfs Theorem;-Arithmetical Functions;-The
Riemann Zeta Function;- The Functional Equation of the Riemann
Zeta Function;-Primes in Arithmetic Progression;-Computational
Number Theory;-Factorizing;-Proof of Theorem 7.12;-Abelfs Limit
Theorem;-Bibliography / Index
Borwein, Jonathan, Zhu, Q.J.
Techniques in Variational Analysis
Series: CMS Books in Mathematics,
2005, Approx. 390 p. 14 illus., Hardcover
ISBN: 0-387-24298-8
About this textbook
This book aims to provide a concise account of the essential
tools of infinite-dimensional first-order variational analysis
that are presently scattered throughout the literature. It also
illustrates applications in many parts of analysis, optimization
and approximation, control theory, dynamic systems, mathematical
economics, and elsewhere.
It is arranged in 3 parts: Part I discusses the fundamentals of
variational analysis, and Parts II and II deal with applications.
Part II focuses on traditional topics while Part Iii contains
problems of a more applied nature. More technical examples and
extensions are found at the end of each section. This book is
intended for researchers and graduate students who might benefit
from using the variational methods. A prerequisite is a working
knowledge of basic analysis and principles of functional analysis.
Table of contents
* Introduction * Variational Principles * Variational Techniques
in Subdifferential Theory * Variational Techniques in Convex
Analysis * Variational Techniques and Multifunctions *
Variational Principles in Nonlinear Functional Analysis *
Variational Techniques in the Presence of Symmetry
Crossley, Martin D.
Essential Topology
Series: Springer Undergraduate Mathematics Series,
2005, Approx. 200 p. 50 illus., Softcover
ISBN: 1-85233-782-6
About this textbook
"Essential Topology" brings the most exciting ? and
useful - aspects of modern topology within reach of the average
second-year undergraduate student. It contains all the essentials.
The first chapter provides a complete account of continuity
beginning at a level that a high school student could understand.
The algebraic notions are introduced slowly through the text,
leading the reader to the celebrated Hairy Ball theorem, and on
to homotopy and homology ? the cornerstones of contemporary
algebraic topology.
Each topic is introduced with a thorough explanation of why it is
being studied, and the focus throughout is on providing
interesting examples that will motivate the student. Emphasis is
placed on the basic objects that occur in research topology, and
in its applications to other areas of mathematics.
This book is designed to provide a "one-stop shop" for
undergraduate topology, providing enough material for two
semester-long courses, and leaving students motivated and
prepared for postgraduate study.
Table of contents
Introduction;-Continuous Functions;-Topological Spaces;-Interlude;-Topological
Properties;-econstructionist Topology;-Interlude;-Homotopy;-The
Euler Number;-Homotopy Groups;-Simplicial Homology;-Singular
Homology;-More Deconstructionism;-Solutions to Selected
Exercises;-Bibliography;-Index
Verhulst, Ferdinand
Methods and Applications of Singular Perturbations
Boundary Layers and Multiple Timescale Dynamics
Series: Texts in Applied Mathematics, Vol. 50
2005, 332 pages, Hardcover
ISBN: 0-387-22966-3
About this textbook
Perturbation theory is a fascinating and fundamental topic in
mathematics and its applications to the natural and engineering
sciences. In this workbook, each explicit example is studied and
methods introduced beginning without proof, a learning method
very suitable for singular perturbation problems. The text
includes an extensive discussion of timescales and apriori
knowledge of the presence of certain timescales. This
comprehensive introduction to singular perturbation covers a
broad range of topics, includes odes' and pde's, boundary value
problems, and problems with initial values.
Table of contents
Introduction - Basic material - Approximation of integrals -
Boundary layer behaviour - Two-point boundary value problems -
Nonlinear boundary-value problems - Elliptic boundary value
problems - Boundary layers in time - Evolution equations with
boundary layers - The continuation method - Averaging and
timescales - Advanced averaging - Averaging for evolution
equations - Wave equations on unbounded domains - Appendices
Alho, Juha, Spencer, Bruce
Statistical Demography and Forecasting
Series: Springer Series in Statistics,
2005, Approx. 360 p., Hardcover
ISBN: 0-387-23530-2
About this book
Sustainability of pension systems, intergeneration fiscal equity
under population aging, and accounting for health care benefits
for future retirees are examples of problems that cannot be
solved without understanding the nature of population forecasts
and their uncertainty. Similarly, the accuracy of population
estimates directly affects both the distributions of formula-based
government allocations to sub-national units and the
apportionment of political representation. The book develops the
statistical foundation for addressing such issues. Areas covered
include classical mathematical demography, event history methods,
multi-state methods, stochastic population forecasting, sampling
and census coverage, and decision theory. The methods are
illustrated with empirical applications from Europe and the U.S.
For statisticians the book provides a unique introduction to
demographic problems in a familiar language. For demographers,
actuaries, epidemiologists, and professionals in related fields,
the book presents a unified statistical outlook on both classical
methods of demography and recent developments. To facilitate its
classroom use, exercises are included. Over half of the book is
readily accessible to undergraduates, but more maturity may be
required to benefit fully from the complete text. Knowledge of
differential and integral calculus, matrix algebra, basic
probability theory, and regression analysis is assumed.
Juha M. Alho is Professor of Statistics, University of Joensuu,
Finland, and Bruce D. Spencer is Professor of Statistics and
Faculty Fellow at the Institute for Policy Research, Northwestern
University. Both have contributed extensively to statistical
demography and served in advisory roles and as statistical
consultants in the field.
Table of contents
Introduction.- Sources of Demographic Data.- Sampling Designs and
Interference.- Waiting Times and Their Statistical Estimation.-
Regression Models for Counts and Survival.- Multistate Models and
Cohort-Component Book-keeping.- Approaches to Forecasting
Demographic Rates.- Uncertainty in Demographic Forecasts:
Concepts, Issues, and Evidence.- Statistical Propagation of Error
in Forecasting.- Errors in Census Numbers.- Economic Applications.-
Decision Analysis and Small Area Estimates.
Bjorner, Anders, Brenti, Francesco
Combinatorics of Coxeter Groups
Series: Graduate Texts in Mathematics, Vol. 231
2005, Approx. 367 pp., Hardcover
ISBN: 3-540-44238-3
About this textbook
This book is a carefully written exposition of Coxeter groups, an area
of mathematics which appears in algebra, geometry, and combinatorics. In
this book, the combinatorics of Coxeter groups has mainly to do with reduced
expressions, partial order of group elements, enumeration, associated graphs
and combinatorial cell complexes, and connections with combinatorial representation
theory. While Coxeter groups have already been exposited from algebraic
and geometric perspectives, this book will be presenting the combinatorial
aspects of Coxeter groups. The authors have included an exposition of Coxeter
groups along with a rich variety of exercises, ranging from easy to very
difficult, giving the book the unique character of serving as both a textbook
and a monograph.
Table of contents
Foreword * The Basics * Bruhat order * Weak order and reduced
words * Roots, games and automata * Kazhdan-Lusztig and R-polynomials
* Kazhdan-Lusztig representations * Enumeration * Combinatorial
descriptions * Appendices * Bibliography * Index