Series: Springer Monographs in Mathematics
2006, Approx. 480 p. 53 illus., Hardcover
ISBN: 0-387-33285-5
Due: August 2006
About this book
Number theory, spectral geometry, and fractal geometry are
interlinked in this in-depth study of the vibrations of fractal
strings, that is, one-dimensional drums with fractal boundary.
Key Features
The Riemann hypothesis is given a natural geometric reformulation
in the context of vibrating fractal strings
Complex dimensions of a fractal string, defined as the poles of
an associated zeta function, are studied in detail, then used to
understand the oscillations intrinsic to the corresponding
fractal geometries and frequency spectra
Explicit formulas are extended to apply to the geometric,
spectral, and dynamic zeta functions associated with a fractal
Examples of such formulas include Prime Orbit Theorem with error
term for self-similar flows, and a tube formula
The method of diophantine approximation is used to study self-similar
strings and flows
Analytical and geometric methods are used to obtain new results
about the vertical distribution of zeros of number-theoretic and
other zeta functions
Throughout new results are examined. The final chapter gives a
new definition of fractality as the presence of nonreal complex
dimensions with positive real parts.
The significant studies and problems illuminated in this work may
be used in a classroom setting at the graduate level. Fractal
Geometry, Complex Dimensions and Zeta Functions will appeal to
students and researchers in number theory, fractal geometry,
dynamical systems, spectral geometry, and mathematical physics.
Written for:
Graduate students, geometers, math physicists, number theorists
Table of contents
List of Figures.- Preface.- Overview.- Introduction.- Complex
Dimensions of Ordinary Fractal Strings.- Complex Dimensions of
Self- Similar Fractal Strings.- Complex Dimensions of Nonlattice
Self- Similar Strings: Quasiperiodic Patterns and Diophantin
Approximation.- Generalized Fractal Strings Viewed as Measures.-
Explicit Formulas for Generalized Fractal Strings.- The geometry
and the Spectrum of Fractal Strings.- Periodic Orbits of Self-Similar
Flows.- Tubular Neighborhoods and Minkowski Measurability.- The
Riemann Hypothesis and Inverse Spectral Problems.- Generalized
Cantor Strings and their Oscillations.- The Critical Zeros of
Zeta Functions.- Concluding Comments, Open Problems, and
Perspectives.- Appendices.- A. Zeta Functions in Number Theory.-
B. Zeta Functions of Laplacians and Spectral Asymptotics.- C. An
Application of Nevanlinna Theory.- Bibliography.-
Acknolwedgements.- Conventions.- Index of Symbols.- Author Index.-
Subject Index.-
Series: Springer Undergraduate Mathematics Series
2006, Approx. 200 p., Softcover
ISBN: 1-84628-423-6
Due: November 15, 2006
About this textbook
This book is an informal introduction to game theory intended as
a first course for undergraduate students of mathematics.
Uniquely, it covers optimal decisions, classical games and
evolutionary game theory in one volume. Optimal decisions are
treated as a special case of game theory in which the game is
played against nature ? an opponent who is indifferent about the
outcome. For evolutionary game theory, a subject which is often
presented as a minor adjustment to the classical theory, this
book aims to provide a better understanding by emphasising the
differences between the two types of game theory, in particular
the population context in which evolutionary games are embedded.
In contrast to many textbooks at this level, the subject is
studied from a mathematical perspective so the emphasis is on
presenting the mathematics without getting bogged down in
examples which mathematics students, without the relevant
background in economics or biology, would struggle to follow, an
approach that should also help researchers in biology and
economics to understand each other’s models.
While this book is written primarily for students of mathematics,
proofs and other technical discussion are restricted to special
cases so this book should be accessible to students and
researchers of economics and biology as a second course in the
subject or as supplementary reading.
Table of contents
Part I: One-player Games.- Simple Decision Models. Strategic
Behaviour. Finite-horizon Markov Decision Processes. Infinite-horizon
Markov Decision Processes. Part II: Two-player Games.- Static
Games. Finite-horizon Dynamic Games. Games with Continuous
Strategies. Markov Games. Part III: Population Games.-
Evolutionarily Stable Strategies. Replicator Dynamics. Part IV:
Appendices.- Introduction to Dynamical Systems. Solutions to
Exercises.
Series: Graduate Texts in Mathematics, Vol. 149
2007, Approx. 790 p. 170 illus., Hardcover
ISBN: 0-387-33197-2
Due: November 2006
About this textbook
This book is an exposition of the theoretical foundations of
hyperbolic manifolds. It is intended to be used both as a
textbook and as a reference. The reader is assumed to have a
basic knowledge of algebra and topology at the first year
graduate level of an American university. The book is divided
into three parts. The first part is concerned with hyperbolic
geometry and discrete groups. The second part is devoted to the
theory of hyperbolic manifolds. The third part integrates the
first two parts in a development of the theory of hyperbolic
orbifolds.
The second edition contains hundreds of changes and corrections,
and new additions include:
* a more thorough discussion of polytopes;
* discussion of Simplex Reflection groups has been expanded to
give a complete classification of the Gram matrices of spherical,
Euclidean and hyperbolic n-simplices;
* A new section on the volume of a simplex, in which a derivation
of Schlafli’s differential formula is presented;
* A new section with a proof of the n-dimensional Gauss-Bonnet
theorem.
The exercises have been thoroughly reworked, pruned, and
upgraded, and over 100 new exercises have been added. The author
has also prepared a solutions manual which is available to
professors who choose to adopt this text for their course.
Table of contents
Preface.- Euclidean Geometry.- Spherical Geometry.- Hyperbolic
Geometry.- Inversive Geometry.- Isometries of Hyperbolic Space.-
Geometry of Discrete Groups.- Classical Discrete Groups.-
Geometric Manifolds.- Geometric Surfaces.- Hyperbolic 3-Manifolds.-
Hyperbolic n-Manifolds.- Geometrically Finite n-Manifolds.-
Geometric Orbifolds.- Bibliography.- Index.-
Series: Texts in Applied Mathematics, Preliminary entry 300
2007, Approx. 700 p., Hardcover
ISBN: 0-387-00450-5
Due: February 2007
About this textbook
Computer simulation has become, alongside experimentation and
abstract reasoning, a third major tool of science. The student in
the computational sciences needs a background in numerical
analysis, and it is one of the objectives of this text to provide
that background material. This book covers many of the
traditional areas of numerical analysis as well as topics in
software development and methods of Monte Carlo simulations. It
is designed to serve both as a textbook for a course in numerical
methods for students in natural sciences and as a reference for
scientists whose research involves numerical computations and
simulations. Exercises comprise an important part of the text.
The mathematical prerequisites include real analysis and linear
algebra, and the text does assume some familiarity with computer
programming.
Table of contents
Numerical Computations and Algorithms * Random Number Generation
and onte Carlo Methods * Numerical Linear Algebra * Optimization
and Nonlinear Systems * Structure in Multivariate Data
Volume package: David Hilbert's Lectures. Foundation:1849-1933
2007, Hardcover
ISBN: 3-540-40549-6
Due: October 2007
About this book
The first part of this volume documents Hilbert’s efforts in
the period 1898-1910 to base all known physics (including
thermodynamics, hydrodynamics and electrodynamics) on classical
mechanics. This period closes with a lecture course ‘Mechanik
der Kontinua’ (1911), in which Hilbert considers the
consequences of the new principle of special relativity for our
understanding of physics. The second part starts with the lecture
course ‘Kinetische Gastheorie’ (1911/12), which introduces a
new approach to problems of statistical physics. The lecture
course ‘Molekulartheorie der Materie’ (1913) deals with a
topic that was of great importance to Hilbert, returning to it
repeatedly. The last lecture course contained in this volume, ‘Statistische
Mechanik’ (1922) presents a very perceptive comparison of the
different approaches of Maxwell, Boltzmann, Gibbs etc. to the
foundational problems of statistical physics. It is a paradigm of
logical analysis and conceptual clarity.
Table of contents
Introduction.- Part A: Classical Mechanics.- Overview of the
texts on mechanics.- Chapter 1: Hilbert's Lectures on Mechanics
from 1898/99.- Chapter 2: Hilbert's Lectures on Stability from
1903.- Chapter 3: Hilbert's Lectures on Continuum Mechanics from
1905/06.- Chapter 4: Hilbert's Lectures on Mechanics in a New
Perspective, 1911 and 1924.- Part B: Kinetic Theories of Matter.-
Overview of the texts.- Hilbert's Lectures on Kinetic Theory of
Gases from 1911/12.- Chapter 6: Hilbert's Lectures on Molecular
Theory of Matter from 1912/13.-Chapter 7: Hilbert's Lectures on
Statictical Mechanics from 1922
Volume package: David Hilbert's Lectures. Foundation:1849-1933
2007, Approx. 650 p., Hardcover
ISBN: 3-540-20606-X
Due: December 2007
About this book
This Volume has three sections, General Relativity,
Epistemological Issues, and Quantum Mechanics. The core of the
first section is Hilbert’s two semester lecture course on ‘The
Foundations of Physics’ (1916/17). This is framed by Hilbert’s
published ‘First and Second Communications’ on the ‘Grundlagen
der Physik’ (1915, 1917). The section closes with a lecture on
the new concepts of space and time held in Bucharest in 1918. The
epistemological issues concern the principle of causality in
physics (1916), the intricate relation between nature and
mathematical knowledge (1921), and the subtle question whether
what Hilbert calls the ‘world equations’ are physically
complete (1923). The last section deals with quantum theory in
its early, advanced and mature stages. Hilbert held lecture
courses on the mathematical foundations of quantum theory twice,
before and after the breakthrough in 1926. These documents bear
witness to one of the most dramatic changes in the foundations of
science.
Table of contents
Introduction.- Part C: General Relativity.- Overview of the texts
on General Relativity.- Hilbert's Lectures on The Foundations of
Physics from 1915.- Hilbert's Lectures on The Foundations of
Physics, Main Lecture 1916/17.- Hilbert's Lectures on FoP, Second
Communication 1916.- Hilbert's Lectures on Space and Time (Bukarest),
1918.- Epistemological Issues.- Overview of the texts.- Hilbert's
Lectures on The Causality Principle from 1916.- Hilbert's
Lectures on Nature and Mathematical Knowledge from 1921.-
Hilbert's Lectures on The World Equations from 1923.-Part E:
Quantum Mechanics.- Overview of the texts.- Hilbert's Lectures on
Radiation Theory from 1916.- Chapter 16: Hilbert's Lectures on
Mathematical Methods of Quantum Theory from 19222/23 and 1926/27.