Paperback (ISBN-13: 9780521091886)
The purpose of this book is to bridge the gap between differential geometry of Euclidean space of three dimensions and the more advanced work on differential geometry of generalized space. The subject is treated with the aid of the Tensor Calculus, which is associated with the names of Ricci and Levi-Civita; and the book provides an introduction both to this calculus and to Riemannian geometry. The geometry of subspaces has been considerably simplified by use of the generalized covariant differentiation introduced by Mayer in 1930, and successfully applied by other mathematicians.
1. Some Preliminaries; 2. Coordinates, Vectors , Tensors; 3. Riemannian Metric; 4. Christoffel’s Three-Index Symbols. Covariant Differentiation; 5. Curvature of a Curve. Geodeics, Parallelism of Vectors; 6. Congruences and Orthogonal Ennuples; 7. Riemann Symbols. Curvature of a Riemannian Space; 8. Hypersurfaces; 9. Hypersurfaces in Euclidean Space. Spaces of Constant Curvature; 10. Subspaces of a Riemannian Space.
Series: Encyclopedia of Mathematics and its Applications (No. 26)
Paperback (ISBN-13: 9780521092029)
The theory of matroids is unique in the extent to which it connects such disparate branches of combinatorial theory and algebra as graph theory, lattice theory, design theory, combinatorial optimization, linear algebra, group theory, ring theory and field theory. Furthermore, matroid theory is alone among mathematical theories because of the number and variety of its equivalent axiom systems. Indeed, matroids are amazingly versatile and the approaches to the subject are varied and numerous. This book is a primer in the basic axioms and constructions of matroids. The contributions by various leaders in the field include chapters on axiom systems, lattices, basis exchange properties, orthogonality, graphs and networks, constructions, maps, semi-modular functions and an appendix on cryptomorphisms. The authors have concentrated on giving a lucid exposition of the individual topics; explanations of theorems are preferred to complete proofs and original work is thoroughly referenced. In addition, exercises are included for each topic.
1. Examples and Basic Concepts Henry Crapo; 2. Axiom Systems Giorgio Nicoletti and Neil White; 3. Lattices Ulrich Faigle; 4. Basis-Exchange Properties Joseph P. S. Kung; 5. Orthogonality Henry Crapo; 6. Graphs and Series-Parallel Networks James Oxley; 7. Constructions Thomas Brylawski; 8. Strong Maps Joseph P. S. Kung; 9. Weak Maps Joseph P. S. Kung and Hein Q. Nguyen; 10. Semimodular Functions Hein Q. Nguyen.
Series: London Mathematical Society Lecture Note Series (No. 357)
Paperback (ISBN-13: 9780521720083)
Integration of differential equations is a central problem in mathematics and several approaches have been developed by studying analytic, algebraic, and algorithmic aspects of the subject. One of these is Differential Galois Theory, developed by Kolchin and his school, and another originates from the Soliton Theory and Inverse Spectral Transform method, which was born in the works of Kruskal, Zabusky, Gardner, Green and Miura. Many other approaches have also been developed, but there has so far been no intersection between them. This unique introduction to the subject finally brings them together, with the aim of initiating interaction and collaboration between these various mathematical communities. The collection includes a LMS Invited Lecture Course by Michael F. Singer, together with some shorter lecture courses and review articles, all based upon a mini-programme held at the International Centre for Mathematical Sciences (ICMS) in Edinburgh.
* Features the LMS Invited Lecture Course by Professor Michael F. Singer
* Brings together various different approaches to the problem of integrability
* An introduction suitable for graduate students and academic researchers
Preface; 1. Galois theory of linear differential equations Michael F. Singer; 2. Solving in closed form Felix Ulmer and Jacques-Arthur Weil; 3. Factorization of linear systems Sergey P. Tsarev; 4. Introduction to D-modules Anton Leykin; 5. Symbolic representation and classification of integrable systems A. V. Mikhailov, V. S. Novikov and Jing Ping Wang; 6. Searching for integrable (P)DEs Jarmo Hietarinta; 7. Around differential Galois theory Anand Pillay.
Series: Encyclopedia of Mathematics and its Applications (No. 125)
Hardback (ISBN-13: 9780521517201)
The Hilbert transform has many uses, including solving problems in aerodynamics, condensed matter physics, optics, fluids, and engineering. Written in a style that will suit a wide audience (including the physical sciences), this book will become the reference of choice on the topic, whatever the subject background of the reader. It explains all the common Hilbert transforms, mathematical techniques for evaluating them, and has detailed discussions of their application. Especially useful for researchers are the tabulation of analytically evaluated Hilbert transforms, and an atlas that immediately illustrates how the Hilbert transform alters a function. A collection of exercises helps the reader to test their understanding of the material in each chapter. The bibliography is a wide-ranging collection of references both to the classical mathematical papers, and to a diverse array of applications.
Series: CBMS-NSF Regional Conference Series in Applied Mathematics (No. 78)
Paperback (ISBN-13: 9780898716511)
A novel approach to analysing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, based on ideas of the inverse scattering transform that the author introduced in 1997. This method is unique in also yielding novel integral representations for linear PDEs. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated Radon transform and the Dirichlet-to-Neumann map for a moving boundary; integral representations for linear boundary value problems; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author’s new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions.
* Unifies the most extensively used techniques for solving boundary value
problems for linear PDEs * Includes a new approach to an important medical
imaging technique * Unique in presenting an extension of the inverse scattering
method from initial value problems to boundary value problems
Preface; Introduction; 1. Evolution equations on the half-line; 2. Evolution
equations on the finite interval; 3. Asymptotics and a novel numerical
technique; 4. From PDEs to classical transforms; 5. Riemann*Hilbert and
d-Bar problems; 6. The Fourier transform and its variations; 7. The inversion
of the attenuated Radon transform and medical imaging; 8. The Dirichlet
to Neumann map for a moving boundary; 9. Divergence formulation, the global
relation, and Lax pairs; 10. Rederivation of the integral representations
on the half-line and the finite interval; 11. The basic elliptic PDEs in
a polygonal domain; 12. The new transform method for elliptic PDEs in simple
polygonal domains; 13. Formulation of Riemann*Hilbert problems; 14. A collocation
method in the Fourier plane; 15. From linear to integrable nonlinear PDEs;
16. Nonlinear integrable PDEs on the half-line; 17. Linearizable boundary
conditions; 18. The generalized Dirichlet to Neumann map; 19. Asymptotics
of oscillatory Riemann*Hilbert problems; Epilogue; Bibliography; Index.
Series: Cambridge Mathematical Library
Hardback (ISBN-13: 9780521883894)
Paperback (ISBN-13: 9780521737944)
Gian-Carlo Rota was one of the most original and colourful mathematicians of the 20th century. His work on the foundations of combinatorics focused on the algebraic structures that lie behind diverse combinatorial areas, and created a new area of algebraic combinatorics. Written by two of his former students, this book is based on notes from his influential graduate courses and on face-to-face discussions. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed. A must-have for all students and researchers in combinatorics and related areas.
* Emphasizes unifying ideas in combinatorics * Shows connections with other
areas of mathematics * Gives elementary accounts of topics like entropy,
linear lattices, umbral calculus, and Baxter algebras, not available elsewhere
in book form
1. Sets, functions, and relations; 2. Matching theory; 3. Partially ordered sets and lattices; 4. Generating functions and the umbral calculus; 5. Symmetric functions and Baxter algebras; 6. Determinants, matrices, and polynomials; 7. Selected solutions.