Ted Burton Ph.D., Professor Emeritus of Southern Illinois University

Volterra Integral and Differential Equations,Second Edition

(Mathematics in Science and Engineering, Volume 202)

1. Smooth transition from ordinary differential equations to integral and functional differential equations.
2. Unification of the theories, methods, and applications of ordinary and functional differential equations.
3. Large collection of examples of Liapunov functions.
4. Description of the history of stability theory leading up to unsolved problems.
5. Applications of the resolvent to stability and periodic problems.

Most mathematicians, engineers, and many other scientists are well-acquainted with theory and application of ordinary differential equations. This book seeks to present Volterra integral and functional differential equations in that same framwork, allowing the readers to parlay their knowledge of ordinary differential equations into theory and application of the more general problems. Thus, the presentation starts slowly with very familiar concepts and shows how these are generalized in a natural way to problems involving a memory. Liapunov's direct method is gently introduced and applied to many particular examples in ordinary differential equations, Volterra integro-differential equations, and functional differential equations.

By Chapter 7 the momentum has built until we are looking at problems on the frontier. Chapter 7 is entirely new, dealing with fundamental problems of the resolvent, Floquet theory, and total stability. Chapter 8 presents a solid foundation for the theory of functional differential equations. Many recent results on stability and periodic solutions of functional differential equations are given and unsolved problems are stated.

Key Features:

- Smooth transition from ordinary differential equations to integral and functional differential equations.

- Unification of the theories, methods, and applications of ordinary and functional differential equations.

- Large collection of examples of Liapunov functions.

- Description of the history of stability theory leading up to unsolved problems.

- Applications of the resolvent to stability and periodic problems.

Contents

Preface.
Preface to the second edition.
Contents.

0 - Introduction and Overview
1 - The General Problems
2 - Linear Equations
3 - Existence Properties
4 - History, Examples and Motivation
5 - Instability, Stability and Perturbations
6 - Stability and Boundedness
7 - The Resolvent
8 - Functional Differential Equations
References.
Author Index.
Subject Index.

ISBN: 0-444-51786-3 Book/Hardback
Measurements: 5 X 9 15/16 in
Pages: 368

W.H.A. SCHILDERS /Philips Research Laboratories, Eindhoven, The Netherlands
E.J.W. TER MATEN /Philips Research Laboratories, Eindhoven, The Netherlands

Numerical Methods in Electromagnetics

Special Volume(Handbook of Numerical Analysis, Volume 13)

Complete survey of numerical methods used in the electronic industry

Each chapter is selfcontained

Presents state-of-the-art applications and methods

Internationally recognised authors

Description

This special volume provides a broad overview and insight in the way numerical methods are being used to solve the wide variety of problems in the electronics industry. Furthermore its aim is to give researchers from other fields of application the opportunity to benefit from the results wich have been obtained in the electronics industry.

Contents

1. Introduction to electromagnetism
2. Discretization of electromagnetic problems
3. Finite-difference time-domaine methods
4. Discretization of semiconductor divice problems I
5. Discretization of semiconductor divice problems II
6. Modelling and discretization of circuit problems
7. Simulation of EMC behaviour
8. Solution of linear systems
9. Reduced-order modelling

Series: Handbook of Numerical Analysis
ISBN: 0-444-51375-2 Book/Hardback
Measurements: 9 1/2 X 6 7/16 in
Pages: 928
Publication Date: 23 May 2005

Jimmie Gilbert - University of South Carolina, Spartanburg
Linda Gilbert - University of South Carolina, Spartanburg

Linear Algebra and Matrix Theory
2nd Edition

0534405819
544 pages Case Bound 8 x 9 1/4

Intended for a serious first course or a second course in linear algebra, this book carries students beyond eigenvalues and eigenvectors to the classification of bilinear forms, normal matrices, spectral decompositions, the Jordan form, and sequences and series of matrices. The authors present the material from a structural point of view: fundamental algebraic properties of the entities involved are emphasized. The approach is particularly important because the mathematical systems encountered in linear algebra furnish a wealth of examples for the structures studied in more advanced courses. By taking a straight and smooth path to the heart of linear algebra, students will be able to make the transition from the intuitive developments of courses at a lower level to the more abstract treatments encountered later.

Table of Contents

1. REAL COORDINATE SPACES.
The Vector Spaces Rn. Linear Independence. Subspaces of Rn. Spanning Sets. Geometric Interpretations of R2 and R3. Bases and Dimension.
2. ELEMENTARY OPERATIONS ON VECTORS.
Elementary Operations and Their Inverses. Elementary Operations and Linear Independence. Standard Bases for Subspaces.
3. MATRIX MULTIPLICATION.
Matrices of Transition. Properties of Matrix Multiplication. Invertible Matrices. Column Operations and Column-Echelon Forms. Row Operations and Row-Echelon Forms. Row and Column Equivalence. Rank and Equivalence. LU Decompositions.
4. VECTOR SPACES, MATRICES, AND LINEAR EQUATIONS.
Vector Spaces. Subspaces and Related Concepts. Isomorphisms of Vector Spaces. Standard Bases for Subspaces. Matrices over an Arbitrary Field. Systems of Linear Equations. More on Systems of Linear Equations.
5. LINEAR TRANSFORMATIONS.
Linear Transformations. Linear Transformations and Matrices. Change of Basis. Composition of Linear Transformations.
6. DETERMINANTS.
Permutations and Indices. The Definition of a Determinant. Cofactor Expansions. Elementary Operations and Cramer's Rule. Determinants and Matrix Multiplication.
7. EIGENVALUES AND EIGENVECTORS.
Eigenvalues and Eigenvectors. Eigenspaces and Similarity. Representation by a Diagonal Matrix.
8. FUNCTIONS OF VECTORS.
Linear Functionals. Real Quadratic Forms. Orthogonal Matrices. Reduction of Real Quadratic Forms. Classification of Real Quadratic Forms. Binlinear Forms. Symmetric Bilinear Forms. Hermitian Forms.
9. INNER PRODUCT SPACES.
Inner Products. Norms and Distances. Orthonormal Bases. Orthogonal Complements. Isometrics. Normal Matrices. Normal Linear Operators.
10. SPECTRAL DECOMPOSITIONS.
Projections and Direct Sums. Spectral Decompositions. Minimal Polynomials and Spectral Decompositions. Nilpotent Transformations. The Jordan Canonical Form.
11. NUMERICAL METHODS.
Sequences and Series of Vectors. Sequences and Series of Matrices. The Standard Method of Iteration. Cimmino's Method. An Iterative Method for Determining Eigenvalues.

New to the Edition

Through rewritten material and increased examples and exercises, the pace of the early chapters, especially the first, has been made slower to help students gain confidence early.

Four new sections appear in Chapters 1 ?10, including one on LU decompositions.

A new Chapter 11, "Numerical Methods," enhances the usefulness of the text for two courses.

Manin, Yu.I., Panchishkin, Alexei A.
Parshin, A.N.; Shafarevich, I.R. (Eds.)

Introduction to Modern Number Theory, 2nd ed.
Fundamental Problems, Ideas and Theories

Series: Encyclopaedia of Mathematical Sciences, Vol. 49
Volume package: Enc.Mathematical Sciences Number Theory
2005, XV, 514 p. 17 illus., Hardcover
ISBN: 3-540-20364-8

About this book

Introduction to Modern Number Theory surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions.

This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Moreover, the authors have added a part dedicated to arithmetical cohomology and noncommutative geometry, a report on point counts on varieties with many rational points, the recent polynomial time algorithm for primality testing, and some others subjects.

Written for:

Researchers and graduate students in number theory, algebra, logic and algebraic geometry

Keywords:

Arakelov geometry
Arithmetic der algebraischen Zahlen
Elementare Zahlentheorie
Elementary number theory

Langlands program
Langlands-Programm
Modular forms
Non-commutative geometry
arithmetic of algebraic numbers
diophantine equations
diophantische Gleichungen
elliptic curves
elliptische Kurven
logic
public key Verschlusselungssysteme
public key cryptosystems
zeta-functions

Suzuki, Takashi

Free Energy and Self-Interacting Particles

Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 62
2005, XIV, 366 p. 7 illus., Hardcover
ISBN: 0-8176-4302-8

About this book

This book examines a nonlinear system of parabolic partial differential equations (PDEs) arising in mathematical biology and statistical mechanics. In the context of biology, the system typically describes the chemotactic feature of cellular slime molds. One way of deriving these equations is via the random motion of a particle in a cellular automaton. In statistical mechanics, on the other hand, the system is associated with the motion of the mean field of self-interacting particles under gravitational force.

Physically, such a system is related to Langevin, Fokker?Planck, Liouville and gradient flow equations, which involve the issues of free energy and the second law of thermodynamics. Mathematically, the mechanism can be referred to as a quantized blowup. Actually, it is regarded as a nonlinear theory of quantum mechanics, and it comes from the mass and location quantization of the singular limit for the associated nonlinear eigenvalue problems. This book describes the whole picture, i.e., the mathematical and physical principles: derivation of a series of equations, biological modeling based on biased random walks, the study of equilibrium states via the variational structure derived from the free energy, and the quantized blowup mechanism based on several PDE techniques.

Free Energy and Self-Interacting Particles is suitable for researchers and graduate students of mathematics and applied mathematics who are interested in nonlinear PDEs in stochastic processes, cellular automatons, variational methods, and their applications to natural sciences. It is also suitable for researchers in other fields such as physics, chemistry, biology, and engineering.

Table of contents

* Dedication
* Preface
* Summary
* Background
* Fundamental Theorem
* Trudinger?Moser Inequality
* The Green's Function
* Equilibrium States
* Blowup Analysis for Stationary Solutions
* Multiple Existence
* Dynamical Equivalence
* Formation of Collapses
* Finiteness of Blowup Points
* Concentration Lemma
* Weak Solution
* Hyperparabolicity
* Quantized Blowup Mechanism
* Theory of Dual Variation
* References
* Index

Vytautas Paskunas

Coefficient systems and supersingular representations of GL2(F)

Memoires de la SMF 99 (2004), vi+84 pages

Resume :

Systemes de coefficients et representations supersingulieres de
Soit F un corps local non archimedien de caracteristique residuelle p. Nous construisons le ≪ bon ≫ nombre de -representations lisses et irreductibles de qui sont supersingulieres au sens de Barthel et Livne. Si , les resultats de Breuil impliquent alors que notre construction donne toutes les representations supersingulieres a la torsion pres par un quasi-caractere non ramifie. Nous conjecturons que ceci reste vrai pour F quelconque.

Mots clefs : Supersinguliere, representation mod p

Abstract:
Let F be a non-Archimedean local field with the residual characteristic p. We construct a ``good'' number of smooth irreducible -representations of , which are supersingular in the sense of Barthel and Livne. If then results of Breuil imply that our construction gives all the supersingular representations up to the twist by an unramified quasi-character. We conjecture that this is true for an arbitrary F.

Key words: Supersingular, mod p-representations

ISBN : 2-85629-165-1

Jean-Marc Deshouillers - Koichi Kawada - Trevor D. Wooley

On Sums of Sixteen Biquadrates

Memoires de la SMF 100 (2005), vi+120 pages

Resume :

Sur les Sommes de Seize Bicarres
En 1939, on savait que 13792 ne peut pas etre represente comme somme de seize bicarres (folklore), qu'il existe une infinite d'entiers qui ne peuvent pas etre ecrits comme sommes de quinze bicarres (Kempner) et que tout entier assez grand est somme de seize bicarres (Davenport).
Dans ce memoire, on montre que tout entier superieur a 10216 et non divisible par 16 peut s'exprimer comme somme de seize bicarres. Combine a une etude numerique menee par Deshouillers, Hennecart et Landreau, ce resultat implique que tout entier superieur a 13792 est somme de seize bicarres.

Mots clefs : Probleme de Waring, methode du cercle, sommes de Weyl, fonctions multiplicatives, equations diphantiennes

Abstract:

By 1939 it was known that 13792 cannot be expressed as a sum of sixteen biquadrates (folklore), that there exist infinitely many natural numbers which cannot be written as sums of fifteen biquadrates (Kempner) and that every sufficiently large integer is a sum of sixteen biquadrates (Davenport).
In this memoir it is shown that every integer larger than 10216 and not divisible by 16 can be represented as a sum of sixteen biquadrates. Combined with a numerical study by Deshouillers, Hennecart and Landreau, this result implies that every integer larger than 13792 is a sum of sixteen biquadrates.

Key words: Waring's problem, circle method, Weyl sums, multiplicative functions, Diophantine equations

ISBN : 2-85629-171-6