ISBN: 0-13-143140-4
Publisher: Prentice Hall
Copyright: 2004
Format: Cloth; 672 pp
Published: 01/05/2004
Description
For one- or two-semester courses in Dynamical Systems in the
department of Advanced Mathematics.
This text gives an introduction into the ideas of dynamical
systems. It is divided into two parts which can be treated in
either order: the first part treats the aspects coming from
systems of nonlinear ordinary differential equations, and the
second part is comprised of those aspects dealing with iteration
of a function. Its main emphasis is on the types of behavior
which nonlinear systems of differential equations can exhibit.
The text assumes that students have taken courses on calculus
covering both a single variable and multivariables, a course on
linear algebra, and an introductory course on differential
equations.
Table of Contents
(NOTE: Each chapter concludes with Applications, Theory and
Proofs, and Exercises.)
Prologue: Historical Perspective.
I. SYSTEMS OF DIFFERENTIAL EQUATIONS.
1. Geometric Approach to Differential Equations.
2. Linear Systems.
3. The Flow: Solutions of Nonlinear Equations.
4. Phase Portraits with Emphasis on Fixed Points.
5. Phase Portraits Using Energy and Other Test Functions.
6. Periodic Orbits.
7. Chaotic Attractors.
II. ITERATION OF FUNCTIONS.
8. Iteration of Functions as Dynamics.
9. Periodic Points of One-Dimensional Maps.
10. Itineraries for One-Dimensional Maps.
11. Invariant Sets for One-Dimensional Maps.
12. Periodic Points of Higher Dimensional Maps.
13. Invariant Sets for Higher Dimensional Maps.
14. Fractals.
Appendix A: Calculus Background.
Appendix B: Analysis and Topology Terminology.
Appendix C: Linear Algebra Background.
Bibliography.
Index.
The synergism between the World Wide Web and fiber optics is a
familiar story to researchers of digital communications. Fibers
are the enablers of the rates of information flow that make the
Internet possible. Currently transoceanic optical fiber cables
transmit data at rates that could transfer the contents of a
respectable university library in a few minutes. No other medium
is capable of this rate of transmission at such distances.
With the maturing of mobile portable telephony and the emerging
broadband access market, greater fiber transmission capacity will
be essential in the early 21st century. Since the demand for more
capacity drives the development of new optics-based technologies,
fiber optics therefore remains a vibrant area for research.
Knowing that the basic fiber optic technology is mature means
that open questions are more sharply focused and permit deeper
mathematic content.
Mathematical Principles of Optical Fiber Communications is
intended to support and promote interdisciplinary research in
optical fiber communications by providing essential background in
both the physical and mathematical principles of the discipline.
Chapter topics include the basics of fibers and their
construction, fiber modes and the criterion of single mode
operation, the nonlinear Schrodinger equation, the variational
approach to the analysis of pulse propagation, and, finally,
solitons and some new results on soliton formation energy
thresholds. These chapters are written to be as independent as
possible while taking the reader to the frontiers of research on
fiber optics communications.
Contents
Preface; Chapter 1: Background and Introduction; Chapter 2: Fiber
Modes; Chapter 3: Fiber Dispersion and Nonlinearity; Chapter 4:
The Variational Approach; Chapter 5: Optical Solitons;
Bibliography,Index
2004 | Softcover | ISBN 0-89871-556-3
Series: Mathematiques et Applications, Vol. 45
2004, XI, 310 p., Softcover
ISBN: 3-540-21369-4
L'analyse numerique de deux types de discretisations
variationnelles est effectuee en detail pour des problemes
elliptiques: les methodes spectrales et les methodes d'elements
finis. Les avantages de chaque type sont mis en valeur, et leur
mise en oeuvre est decrite. L'originalite de cet ouvrage est
d'inserer ces deux types de discretisation dans un cadre abstrait
commun, ce qui permet au lecteur d'etendre l'approche a bien
d'autres methodes et problemes. Sont presentes egalement un
algorithme pour coupler ces methodes dans un cadre de
decomposition de domaine et une application aux ecoulements de
fluides incompressibles dans des milieux poreux. L'ouvrage
s'adresse aux etudiants de 3eme cycle en mathematiques appliquees
et mecanique, ainsi qu'a tous les ingenieurs interesses par la
simulation numerique.
Table of contents
Preface.- Introduction aux methodes variationnelles: I.
Formulations et discretisations variationnelles.- Methodes
spectrales: II. Espace de polynomes et formules de quadrature.-
III. Erreur d'approximation polynomiale.- IV. Erreur
d'interpolation polynomial.- V. Discretisation spectrale des
equations de Laplace.- VI. Traitement de geometries complexes.-
Methodes d'elements finis: VII. Construction des element finis.-
VIII. Construction des espaces d'elements finis.- IX. Erreur
d'appoximation par elements finis.- X. Discretisation par
elements finis des equations de Laplace.- XI. Analyse a
posteriori de la discretisation.- Couplage de methodes: XII. Un
exemple de couplage spectral/elements finis.- Une application:
XIII. Discretisations des equations de milieux poreux.- Et
quelques problemes...: XIV. Quelques problemes.- References.-
Index.
Series: Algorithms and Computation in Mathematics, Vol. 11
2004, Approx. 291 p., Hardcover
ISBN: 3-540-40714-6
The theory of polynomials constitutes an essential part of
university of algebra and calculus. Nevertheless, there are very
few books entirely devoted to this theory. This book provides an
exposition of the main results in the theory of polynomials, both
classical and modern. Many of the modern results have only been
published in journals so far. Considerable attention is given to
Hilbert's 17th problem on the representation of non-negative
polynomials by the sums of squares of rational functions and its
generalizations. Galois theory is discussed primarily from the
point of view of the theory of polynomials, not from that of the
general theory of fields and their extensions.
Table of contents
Foreword Notational conventions Chapter 1. Roots of polynomials 1.
Inequalities for roots 2. The roots of a polynomial and of its
derivative 3. The resultant and the discriminant 4. Separation of
roots 5. Lagrange's series and estimates of the roots of a
polynomial 6. Problems to Chapter 1 7. Solutions of selected
problems Chapter 2. Irreducible polynomials 1. Main properties of
irreducible polynomials 2. Irreducibility criteria 3.
Irreducibility of trinomials and fournomials 4. Hilbert's
irreducibility theorem 5. Algorithms for factorization into
irreducible factors 6. Problems to Chapter 2 7. Solutions of
selected problems Chapter 3. Polynomials of a particular form 1.
Symmetric polynomials 2. Integer-valued polynomials 3. Cyclotomic
polynomials 4. Chebyshev polynomials 5. Bernoulli's polynomials 6.
Problems to Chapter 3 7. Solutions of selected problems Chapter 4.
Certain properties of polynomials 1. Polynomials with prescribed
values 2. The height of a polynomial and other norms 3. Equations
for polynomials 4. Transformations of polynomials 5. Algebraic
numbers 6. Problems to Chapter 4 Chapter 5. Galois theory 1.
Lagrange's theorem and the Galois resolvent 2. Basic Galois
theory 3. How to solve equations by radicals 4. Calculations of
the Galois groups Chapter 6. Ideals in polynomial rings 1.
Hilbert's basis theorem and Hilbert's theorem on zeros 2. Grobner
bases Chapter 7. Hilbert's seventeenth problem 1. The sums of
squares: introduction 2. Artin's theory 3. Pfister's theory
Chapter 8. Appendix 1. The Lenstra-Lenstra-Lovasz algorithm
Bibliography
Series: Lecture Notes in Mathematics, Vol. 1842
2004, VIII, 164 p., Softcover
ISBN: 3-540-21505-0
The exposition studies projective models of K3 surfaces whose
hyperplane sections are non-Clifford general curves. These models
are contained in rational normal scrolls. The exposition
supplements standard descriptions of models of general K3
surfaces in projective spaces of low dimension, and leads to a
classification of K3 surfaces in projective spaces of dimension
at most 10. The authors bring further the ideas in Saint-Donat's
classical article from 1974, lifting results from canonical
curves to K3 surfaces and incorporating much of the Brill-Noether
theory of curves and theory of syzygies developed in the mean
time.
Table of contents