2002. Approx. 670 pp. 845 figs. Hardcover
0-387-95419-8
First published in 1202, Fibonacci's "Liber abaci" was
one of the most important books on mathematics in the Middle
Ages, introducing Arabic numerals and methods throughout Europe.
Its author, Leonardo Pisano, known today as Fibonacci, was a
citizen of Pisa, an active maritime power, with trading outposts
on the Barbary Coast and other points in the Muslim Empire. As a
youth Fibonacci was instructed in mathematics in one of these
outposts; he continued his study of mathematics while traveling
extensively on business and developed contacts with scientists
throughout the Mediterranean world. A member of the academic
court around the Emperor Frederick II, Leonardo saw clearly the
advantages for both commerce and scholarship of the Hindu
positional number system and the algebraic methods developed by
al-Khwarizmi and other Muslim scientists.
Though it is known as an introduction to the Hindu number system
and the algorithms of arithmetic that children now learn in grade
school, "Liber abaci" is much more: an encyclopaedia of
thirteenth-century mathematics, both theoretical and practical.
It develops the tools rigorously, establishing them with
Euclidean geometric proofs, and then shows how to apply them to
all kinds of situations in business and trade - conversion of
measures and currency, allocations of profit, computation of
interest, alloying of currencies, and so forth. It is rigorous
mathematics, well applied, and vividly described.
As the first translation into a modern language of the "Liber
abaci", this book will be of interest not only to historians
of science, but to all mathematicians and mathematics teachers
interested in the origins of their methods.
Contents: Introduction.- Liber Abaci.- Notes for Liber Abaci.-
Bibliography.
Series: Sources and Studies in the History of Mathematics and
Physical Sciences.
2002. Approx. 750 pp. 39 figs. With CD-ROM. Hardcover
0-387-95452-X
This book contains a large amount of information not found in
standard textbooks. Written for the advanced undergraduate/beginning
graduate student, it combines the modern mathematical standards
of numerical analysis with an understanding of the needs of the
computer scientist working on practical applications.
Among its many particular features are:
- fully worked-out examples
- many carefully selected and formulated problems
- fast Fourier transform methods
- a thorough discussion of some important minimization methods
- solution of stiff or implicit ordinary differential equations
and of differential algebraic systems
- modern shooting techniques for solving two-point boundary value
problems
- basics of multigrid methods.
Included are numerous references to contemporary research
literature.
Contents: Prefaces.- Error Analysis.- Interpolation.- Topics of
Integration.- Systems of Linear Equations.- Finding Zeros and
Minimum Points by Iterative Methods.- Eigenvalue Problems.-
Ordinary Differential Equations.- Iterative Methods for the
Solution of Large Systems of Linear Equations.- General
Literature on Numerical Methods.- Index.
Series: Texts in Applied Mathematics. VOL. 12
2002. XV, 601 pp. Hardcover
3-540-43566-2
In algebraic topology some classical invariants - such as Betti
numbers and Reidemeister torsion - are defined for compact spaces
and finite group actions. They can be generalized using von
Neumann algebras and their traces, and applied also to non-compact
spaces and infinite groups. These new L2-invariants contain very
interesting and novel information and can be applied to problems
arising in topology, K-Theory, differential geometry, non-commutative
geometry and spectral theory. It is particularly these
interactions with different fields that make L2-invariants very
powerful and exciting. The book presents a comprehensive
introduction to this area of research, as well as its most recent
results and developments. It is written in a way which enables
the reader to pick out a favourite topic and to find the result
she or he is interested in quickly and without being forced to go
through other material.
Keywords: L2-Invariants, topology, K-Theory
Contents: Introduction.- L2-Betti Numbers.- Novikov-Shubin
Invariants.- L2-Torsion.- L2-Invariants of 3-Manifolds.- L2-Invariants
of Symmetric Spaces.- L2-Invariants for General Spaces with Group
Action.- Applications to Groups.- The Algebra of Affiliated
Operators.- Middle Algebraic K-Theory and L-Theory of von Neumann
Algebras.- The Atiyah Conjecture.- The Singer Conjecture.- The
Zero-in-the-Spectrum Conjecture.- The Approximation Conjecture
and the Determinant Conjecture.- L2-Invariants and the Simplicial
Volume.- Survey on Other Topics Related to L2-Invariants.-
Solutions of the Exercises.- References.- Notation.- Index.
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.
Folge / A Series of Modern Surveys in Mathematics. VOL. 44
2002. Approx. 300 pp. 15 figs. Hardcover
0-387-95463-5
This book presents aspects of a geometric theory of infinite
dimensional spaces with major emphasis on retarded functional
differential equations. It contains results on Morse-Smale
systems for semiflows, persistence of hyperbolicity under
perturbations, nonuniform hyperbolicity, monotone dynamical
systems, realization of vector fields on center manifolds and
normal forms.
Contents: Preface.- Introduction.- Invariant Sets and Attractors.-
Functional Differential Equations on Manifolds.- The Dimension of
the Attractor.- Stability and Bifurcation.- Stability of Morse-Smale
Maps and Semiflows.- One-to-oneness, Persistence and
Hyperbolicity.- Realization of Vector Fields and Normal Forms.-
Attractor Sets as C1-Manifolds.- Monotonicity.- The Kupka-Smale
Theorem.- Appendix A: Conley Index Theory in Noncompact Spaces.-
References.- Index.
Series: Applied Mathematical Sciences. VOL. 47
2002. Approx. 155 pp. 41 figs. Hardcover
0-387-95465-1
In the ideal world, major decisions would be made based on
complete and reliable information available to the decision maker.
We live in a world of uncertainties, and decisions must be made
from information which may be incomplete and may contain
uncertainty. The key mathematical question addressed in this
volume is "how to make decision in the presence of
quantifiable uncertainty." The volume contains articles on
model problems of decision making process in the energy and power
industry when the available information is noisy and/or
incomplete. The major tools used in studying these problems are
mathematical modeling and optimization techniques; especially
stochastic optimization. These articles are meant to provide an
insight into this rapidly developing field, which lies in the
intersection of applied statistics, probability, operations
research, and economic theory. It is hoped that the present
volume will provide entry to newcomers into the field, and
stimulation for further research.
Contents: Foreward.- Preface.- Stochastic programming models:
wait-and-see versus here-and-now.- Optimal stimulation of oil
production.- Power management in a hydro-thermal system under
uncertainty by Lagrangian relaxation.- Hedging electricity
portfolios via stochastic programming.- Opportunities for
stochastic and probabilistic modeling in the deregulated
electricity industry.- On supply function bidding in electricity
markets.- Qualitative implications of uncertainty in economic
equilibrium models.- List of workshop participants.
Series: The IMA Volumes in Mathematics and its Applications. VOL.
128
2002. XX, 450 pp. Hardcover
3-540-43466-6
The book presents in a mathematical clear way the fundamentals of
algorithmic information theory and a few selected applications.
This 2nd edition presents new and important results obtained in
recent years: the characterization of computable enumerable
random reals, the construction of an Omega Number for which ZFC
cannot determine any digits, and the first successful attempt to
compute the exact values of 64 bits of a specific Omega Number.
Finally, the book contains a discussion of some interesting
philosophical questions related to randomness and mathematical
knowledge. "Professor Calude has produced a first-rate
exposition of up-to-date work in information and randomness."
D.S. Bridges, Canterbury University, co-author, with Errett
Bishop, of Constructive Analysis "The second edition of this
classic work is highly recommended to anyone interested in
algorithmic information and randomness." G.J. Chaitin, IBM
Research Division, New York, author of Conversations with a
Mathematician "This book is a must for a comprehensive
introduction to algorithmic information theory and for anyone
interested in its applications in the natural sciences." K.
Svozil, Technical University of Vienna, author of Randomness
& Undecidability in Physics
Keywords: Kolmogorov complexity, Chaitin complexity, random
string / sequence, halting probability, discrete mathematics,
logic, philosophy, random algorithms, computability,
randomization
Contents: Editor Foreword by A. Salomaa, Foreword by G.J.
Chaitin; 1. Mathematical Background; 2. Noiseless Coding; 3.
Program Size; 4. Computably Enumerable Instantaneous Codes; 5.
Random Strings; 6. Random Sequences; 7. Computably Enumerable
Random Reals; 8. Randomness and Incompleteness; 9. Applications;
10. Open Problems; Bibliography; Index
Series: Texts in Theoretical Computer Science. An EATCS Series.