Bluman, George, Anco, Stephen

Advanced Symmetry Method Differential Equations

Series: Applied Mathematical Sciences, Preliminary entry 200
2005, Hardcover
ISBN: 0-387-98612-X
Due: April 2006

About this book

This is an acessible book on the advanced symmetry methods for differential equations, including such subjects as conservation laws, Lie-Backlund symmetries, contact transformations, adjoint symmetries, Nother's Theorem, mappings with some modification, potential symmetries, nonlocal symmetries, nonlocal mappings, and non-classical method. Of use to graduate students and researchers in mathematics and physics.

Written for:

Graduate and senior graduate students, researchers

Table of contents

Chapter 1 Introductory chapter where basic ideas are introduced in an elementary way as motivation for later chapters including summary of relevant material from Symmetry Methods of Differential Equations * Chapter 2 Conservation laws, Lie-Backlund symmetries, contact transformations, adjoint symmetries, Noether's Theorem, new work of Anco/Bluman introducing new conservation law formula * Chapter 3 Mappings with some modification * Chapter 4 Potential symmetries, nonlocal symmetries, nonlocal mappings, Chapter 5 Non-classical methods

Pitman, Jim / Picard, Jean (Ed.)

Combinatorial Stochastic Processes
Ecole d'Ete de Probabilites de Saint-Flour XXXII - 2002

Series: Lecture Notes in Mathematics, Vol. 1875
2006, VIII, 260 p. 16 illus., Softcover
ISBN: 3-540-30990-X
Due: April 5, 2006

About this book

The purpose of this text is to bring graduate students specializing in probability theory to current research topics at the interface of combinatorics and stochastic processes, in particular the theory of random combinatorial structures such as partitions, permutations, trees, forests, and mappings, and connections between the asymptotic theory of enumeration of such structures and the theory of stochastic processes like Brownian motion and Poisson processes. The course is a summary and review of the author's research over the last ten years, much of it joint work with coauthors David Aldous, Jean Bertoin, Steven Evans, and Marc Yor.

Written for:

Researchers and graduate students in probability and stochastic processes and in combinatorics

Table of contents

Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.-Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.- Bibliography.- Index.

Herrlich, Horst

Axiom of Choice

Series: Lecture Notes in Mathematics, Vol. 1876
2006, XIV, 200 p. 1 illus., Softcover
ISBN: 3-540-30989-6
Due: April 5, 2006

About this book

AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that:

- Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC).
- Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games).
- Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness).

Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

Written for:

Researchers and graduate students interested in set theory, topology, order theory, algebra, analysis, measure theory, graph theory, and game theory

Table of contents

Origins: Hilbert's First Problem.- Choice Principles: Some Equivalents to the Axiom of Choice, Some Concepts Related to the Axiom of Choice.- Elementary Observations: Hidden Choice, Unnecessary Choice, Concepts Split Up: Compactness.- Disasters without Choice: Finiteness, Disasters in Cardinal Arithmetic, Disasters in Order Theory, Disasters in Algebra I: Vector Spaces, Disasters in Algebra II: Categories, Disasters in Elementary Analysis: The Reals and Continuity, Disasters in Topology I: Countable Sums, Disasters in Topology II: Products (The Tychonoff and the Cech-Stone Theorem), Disasters in Topology III: Function Spaces (The Ascoli Theorem), Disasters in Topology IV: The Baire Category Theorem, Disasters in Graph Theory: Coloring Problems.- Disasters with Choice: Disasters in Elementary Analysis, Disasters in Geometry: Paradoxical Decompositions.- Disasters either way: Disasters in Game Theory.- Beauty without Choice: Lindelof = Compact, Measurability (The Axiom of Determinateness).

Dumortier, Freddy, Llibre, Jaume, Artes, Joan C.

Qualitative Theory of Planar Differential Systems

Series: Universitext
2006, Approx. 370 p., Softcover
ISBN: 3-540-32893-9
Due: June 19, 2006

About this textbook

The book deals essentially with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced.

From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems.

The book is very appropriate for a first course in dynamical systems. Not only does it provide simple and appropriate proofs, but it also contains a lot of exercises and presents a survey of interesting results with the necessary references to the literature.

Table of contents

Basic Results on the Qualitative Theory of Differential Equations.- Normal Forms and Elementary Singularities.- Desingularization of Non-elementary Singularities. Centers and Lyapunov Constants. - Poincare and Poincare-Lyapunov Compactification.- Indices of Planar Singular Points.- Limit Cycles and Structural Stability.- Integrability and Algebaric Solutions in Polynomial Vector Fields.- Polynomial Planar Phase Portraits.- Examples for Running P4. - Bibliography

Knoebel, A., Laubenbacher, R., Lodder, J., Pengelley, D.

Mathematical Masterpieces
Further Chronicles by the Explorers

Series: Undergraduate Texts in Mathematics
2006, Approx. 350 p.,
Hardcover ISBN: 0-387-33060-7
Softcover ISBN: 0-387-33061-5
Due: May 2006

About this textbook

This book traces the historical development of four different mathematical concepts by presenting readers with the original sources. Although primary sources can be more demanding, the investment yields the rewards of a deeper understanding of the subject, an appreciation of the details, and a glimpse into the direction research has taken.

Each chapter contains a different story, each anchored around a sequence of selected primary sources showcasing a masterpiece of mathematical achievement. The authors begin by studying the interplay between the discrete and continuous, with a focus on sums of powers. They proceed to the development of algorithms for finding numerical solutions of equations as developed by Newton, Simpson and Smale. Next they explore our modern understanding of curvature, with its roots in the emerging calculus of the 17th century, while the final chapter ends with an exploration of the elusive properties of prime numbers, and the patterns found therein.

This book emerged from a course taught at New Mexico State University to juniors and seniors majoring in mathematics. The intended audience is juniors and seniors majoring in mathematics, as well as anyone pursuing independent study. The authors have included exercises, numerous historical photographs, and an annotated bibliography.

Written for:

undergrad mathematics students mathematics teachersmathematicians

Table of contents

* The Bridge Between the Continuous and the Discrete * Numerical Solutions of Equations * The Quadratic Reciprocity Law: Secrets of Prime Numbers * Prime Decomposition of Ideals: The Maturation of Abstract Algebra * Curvature and the Notion of Space