T. A. Burton

Stability & Periodic Solutions of Ordinary & Functional Differential Equations

ISBN: 0-486-44254-3

This book's discussion of a broad class of differential equations will appeal to professionals as well as graduate students. Beginning with the structure of the solution space and the stability and periodic properties of linear ordinary and Volterra differential equations, the text proceeds to an extensive collection of applied problems. The background for and application to differential equations of the fixed-point theorems of Banach, Brouwer, Browder, Horn, Schauder, and Tychonov are examined, in addition to those of the asymptotic fixed-point theorems. The text concludes with a unified presentation of the basic stability and periodicity theory for nonlinear ordinary and functional differential equations.


Table of Contents

An Overview
1. Linear Differential and Integrodifferential Equations
2. History, Motivation, Examples
3. Fixed-Point Theory
4. Limit Sets, Periodicity, and Stability
References
Indexes

Leonard Eugene Dickson

History of the Theory of Numbers, Volume I: Divisibility and Primality

ISBN: 0-486-4423-2

This first volume in the series History of the Theory of Numbers presents in twenty chapters the material related to the subjects of divisibility and primality. At the end of the volume is a separate index of authors for each of the chapters. Volume II: Diophantine Analysis (0-486-44233-0)and Volume III: Quadratic and Higher Forms (0-486-44234-9) complete the three-volume set. Accessible and well-indexed, the three books survey the works of leading experts and can be used independently of each other.
Suitable for upper-level undergraduates and graduate students, this series is the work of a distinguished mathematician who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory.

Table of Contents for History of the Theory of Numbers, Volume I.

I. Perfect, multiply perfect, and amicable numbers
II. Formulas for the number and sum of divisors, problems of Fermat and Wallis
III. Fermat’s and Wilson’s theorems, generalizations and converses; symmetric functions of 1, 2, ..., p-1, modulo p
IV Residue of (up-1-1)/p modulo p
V. Euler’s function, generalizations; Farey series
VI. Periodic decimal fractions; periodic fractions; factors of 10n
VII. Primitive roots, exponents, indices, binomial congruences
VIII. Higher congruences
IX. Divisibility of factorials and multinomial coefficients
X. Sum and number of divisors
XI. Miscellaneous theorems on divisibility, greatest common divisor, least common multiple
XII. Criteria for divisibility by a given number
XIII. Factor tables, lists of primes
XIV. Methods of factoring
XV. Fermat numbers
XVI. Factors of an+bn
XVII. Recurring series; Lucas’ un, vn
XVIII. Theory of prime numbers
XIX. Inversion of functions; Mobius’ function; numerical integrals and derivatives
XX. Properties of the digits of numbers
Indexes

Leonard Eugene Dickson

History of the Theory of Numbers, Volume ll: Diophantine Analysis

ISBN: 0-486-44233-0

This second volume in the series History of the Theory of Numbers presents twenty-six chapters of material related to Diophantine Analysis. At the end of the volume is a separate subject and author index. Volume I: Divisibility and Primality (0-486-44232-2)and Volume III: Quadratic and Higher Forms (0-486-44234-9) complete the three-volume set. Accessible and well-indexed, the three books survey the works of leading experts and can be used independently of each other.
Suitable for upper-level undergraduates and graduate students, this series is the work of a distinguished mathematician who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory.

Table of Contents

I. Polygonal, Pyramidal, and Figurate Numbers
II. Linear Diophantine Equations and Congruences
III. Partitions
IV. Rational Right Triangles
V. Triangles, Quadrilaterals, and Tetrahedra
VI. Sum of Two Squares
VII. Sum of Three Squares
VIII. Sum of Four Squares
IX. Sum of n Squares
X. Number of Solutions of Quadratic Congruences in n Unknowns
XI. Liouville’s Series of Eighteen Articles
XII. Pell Equation
XIII. Further Single Equations of the Second Degree
XIV. Squares in Arithmetical or Geometrical Progression
XV. Two or More Linear Functions Made Squares
XVI. Two Quadratic Functions of One or Two Unknowns Made Squares
XVII. Systems of Two Equations of Degree Two
XVIII. Three or More Quadratic Functions of One or Two Unknowns Made Squares
XIX. Systems of Three or More Equations of Degree Two in Three or More Unknowns
XX. Quadratic Form Made an nth Power
XI. Equations of Degree Three
XXII. Equations of Degree Four
XXIII. Equations of Degree n
XXIV. Sets of Integers with Equal Sums of Like Powers
XXV. Waring’s Problem and Related Results
XXVI. Fermat’s Last Theorem
Indexes

Leonard Eugene Dickson

History of the Theory of Numbers, Volume III: Quadratic and Higher Forms

ISBN: 0-486-44234-9

This third volume in the series History of the Theory of Numbers presents nineteen chapters of material related to Quadratic and Higher Forms. Volume III is mainly concerned with general theories rather than with special problems and special theorems. The investigations deal with the most advanced parts of the theory of numbers. At the end of the volume is a separate subject and author index. Volume I: Divisibility and Primality (0-486-44232-2)and Volume II: Diophantine Analysis (0-486-44233-0) complete the three-volume set. Accessible and well-indexed, the three books survey the works of leading experts and can be used independently of each other.
Suitable for upper-level undergraduates and graduate students, this series is the work of a distinguished mathematician who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory.

Table of Contents for History of the Theory of Numbers, Volume III

I. Reduction and Equivalence of Binary Quadratic Forms, Representation of Integers
II. Explicit Values of x, y
III. Composition of Binary Quadratic Forms
IV. Orders and Genera; Their Composition
V. Irregular Determinants
VI. Number of Classes of Binary Quadratic Forms With Integral Coefficients
VII. Binary Quadratic Forms Whose Coefficients Are Complete Integers or Integers of a Field
VIII. Number of Classes of Binary Quadratic Forms with Complex Integral Coefficients
IX. Ternary Quadratic Forms
X. Quaternary Quadratic Forms
XI. Quadratic Forms in n Variables
XII. Binary Cubic Forms
XIII. Cubic Forms in Three or More Variables
XIV. Forms of Degree n>4
XV. Binary Hermitian Forms
XVI. Hermitian Forms in n Variables and Their Conjugates
XVII. Bilinear Forms, Matrices, Linear Substitutions
XVIII. Representation by Polynomials Modulo p
XIX. Congruencial Theory of Forms
Indexes

Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.

The Mathematics of the Bose Gas and its Condensation

Series: Oberwolfach Seminars, Vol. 34
2005, VIII, 208 p., Softcover
ISBN: 3-7643-7336-9

About this textbook

This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It is a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but it is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. It is an active subject of ongoing research, and this book provides a pedagogical entry into the field for graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.

Written for:

Graduate students and researchers in mathematical and theoretical physics interested in the quantum-mechanical many-body problem, statisticians

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