Yu. P. Solovyov and E. V. Troitsky, Moscow State University, Russia
C^*-Algebras and Elliptic Operators in Differential Topology
Description
The aim of this book is to present some applications of functional analysis and the theory of differential operators to the investigation of topological invariants of manifolds.
The main topological application discussed in the book concerns the problem of the description of homotopy-invariant rational Pontryagin numbers of non-simply connected manifolds and the Novikov conjecture of homotopy invariance of higher signatures. The definition of higher signatures and the formulation of the Novikov conjecture are given in Chapter 3. In this chapter, the authors also give an overview of different approaches to the proof of the Novikov conjecture. First, there is the Mishchenko symmetric signature and the generalized Hirzebruch formulae and the Mishchenko theorem of homotopy invariance of higher signatures for manifolds whose fundamental groups have a classifying space, being a complete Riemannian non-positive curvature manifold. Then the authors present Solovyov's proof of the
Novikov conjecture for manifolds with fundamental group isomorphic to a discrete subgroup of a linear algebraic group over a local field, based on the notion of the Bruhat-Tits building. Finally, the authors discuss the approach due to Kasparov based on the operator KK-theory and another proof of the Mishchenko theorem. In Chapter 4, they outline the approach to the Novikov conjecture due to Connes and Moscovici involving cyclic homology. That allows one to prove the conjecture in the case when the fundamental group is a (Gromov) hyperbolic group.
The text provides a concise exposition of some topics from functional analysis (for instance, C^*-Hilbert modules, K-theory or C^*-bundles, Hermitian K-theory, Fredholm representations, KK-theory, and functional integration) from the theory of differential operators (pseudodifferential calculus and Sobolev
chains over C^*-algebras), and from differential topology (characteristic classes).
The book explains basic ideas of the subject and can serve as a course text for an introduction to the study of original works and special monographs.
Contents
$C^*-algebras and K-theory
Index theorems
The higher signatures
Noncommutative differential geometry
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs,
Publication Year: 2001
ISBN: 0-8218-1399-4
Paging: approximately 232 pp.
Binding: Hardcover
Thierry Aubin, University of Paris, France
A Course in Differential Geometry
Description
This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III
addresses integration of vector fields and p-plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative.
Chapter V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely. Chapter VI explores some problems in PDEs suggested by the geometry of manifolds.
The author is well-known for his significant contributions to the field of geometry and PDEs--particularly for his work on the Yamabe problem--and for his expository accounts on the subject.
The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.
Contents
Background material
Differentiable manifolds
Tangent space
Integration of vector fields and differential forms
Linear connections
Riemannian manifolds
The Yamabe problem-An introduction to research
Bibliography
Subject index
Notation
Details:
Series: Graduate Studies in Mathematics, Volume: 27
Publication Year: 2001
ISBN: 0-8218-2709-X
Paging: approximately 208 pp.
Binding: Hardcover
Kenneth H. Rosen, AT&T Laboratories
Elementary Number Theory and Its Applications, 4th edition
Copyright 2000, 544 pp.
Cloth format
ISBN 0-201-87073-8
Description
The fourth edition of Kenneth Rosen's widely used and successful text, Elementary Number Theory and Its Applications, preserves the strengths of the previous editions, while enhancing the book's flexibility and depth of content coverage.
The blending of classical theory with modern applications is a hallmark feature of the text. The Fourth Edition builds on this strength with new examples, additional applications and increased cryptology coverage. Up-to-date information on the latest discoveries is included.
Elementary Number Theory and Its Applications provides a diverse group of exercises, including basic exercises designed to help students develop skills, challenging exercises and computer projects. In addition to years of use and professor feedback, the fourth edition of this text has been thoroughly accuracy checked to ensure the quality of the mathematical content and the exercises.
Features
Exercises. Extensive and diverse exercise sets, including routine computational exercises to develop basic skills, exercises devoted to developing new concepts, and more difficult exercises to challenge students.
Applications. Lots and lots of applications of number theory are well integrated into the text, illustrating the usefulness of the theory.
Cryptography. The Fourth Edition has updated and increased the coverage of this popular topic.
New Content. Two new sections included in the Fourth Edition will cover Möbius Inversion and Solving Polynomial Congruences.
Proofs. Carefully motivated and fully explained proofs.
Computer Projects. In each section of the text, involving concepts or algorithms from that section. So students can tie together the mathematics in the text with their computer skills.
Accuracy. The text has been carefully checked to ensure accuracy of exercises and mathematical content.
Web Links. The text will include Website listings for relevant number theory sites.
Historical Content. Has been enhanced in the Fourth Edition, showing the old and rich history for this modern material.
Professor O C Zienkiewicz
UNESCO Professor of Numerical Methods in Engineering,
Professor R L Taylor
Professor in the Graduate School, University of California at Berkeley, USA.
Finite Element Method Set
3 Volume Set
New material on fields that have rapidly altered since the previous edition.
A 'must have' reference in this field.
"...the publication of the first edition was an epoch making event...it is written by...the greatest theorist of the subject. If you are serious about finite elements, this is a book that you simply cannot afford to be without." International Journal of Numerical Methods in Engineering.
"...the pre-eminent reference work on finite element analysis." Applied Mechanical Review.
"...a very good book...presentation is first class...will be of great assistance to all engineers and scientists interested in the method...a very commendable piece of work." Journal of the British Society for Strain Measurement.
In the years since the fourth edition of this seminal work was published, active research has developed the Finite Element Method into the pre-eminent tool for the modelling of physical systems. Written by the pre-eminent professors in their fields, this new edition of the Finite Element Method maintains the
comprehensive style of the earlier editions and authoritatively incorporates the latest developments of this dynamic field. Expanded to three volumes the book now covers the basis of the method and its application to advanced solid mechanics and also advanced fluid dynamics. Aimed at undergraduate and postgraduate students, and essential as a reference tool for professional engineers, it provides a complete introduction
to the method.
Volume 1 of The Finite Element Method provides a complete introduction to the method, and is essential reading for undergraduates, postgraduates and professional engineers.
Volume 2 concentrates on non-linear solid and structural mechanics and is ideal for postgraduate students and professional engineers working in this discipline.
Volume 3 covers the whole range of fluid dynamics and is ideal reading for postgraduate level students and professional engineers working in this discipline.
Contents: VOLUME 1: Some preliminaries: The standard discrete system * A direct approach to problems in elasticity * Generalisation of the finite element concepts - Galerkin-weighted residual and variational approaches * Plane stress and plane strain * Axisymmetric stress analysis * Three-dimensional stress analysis * Steady-state field problems - heat conduction, electric and magnetic potential,
fluid flow etc * 'Standard' and 'hierarchical' element shape functions: some general familiarities of C0 continuity * Mapped elements and numerical integration - 'infinite' and 'singularity' elements * The patch test, reduced integration and non-conforming elements * Mixed formulation and constraints - complete field methods * Incompressible problems, mixed methods and other procedures of solution * Mixed formulation and constraints - incomplete (hybrid) field methods, boundary*Trefftz methods * Errors, recovery
processes and error estimates * Adaptive finite element refinement * The time dimension semi-discretisation of field and dynamic problems and analytical solution procedures * The time dimension discrete approximation in time * Coupled systems * Computer procedures for finite element analysis * Appendices. VOLUME 2: Plate and shell bending approximation: thin (Kirchoff) plates and C1 continuity requirements * 'Thick' Reissner-Mindlin plates - irreducible and mixed formulations * Shells as an assembly of flat elements
* Axisymmetric shells * VOLUME 3: Introduction and the equations of fluid dynamics * Convection dominated problems - finite element * A general algorithm for compressible and incompressible flows - the characteristic based split (CBS) algorithm * Incompressible laminar flow - Newtonian and non-Newtonian fluids * Free surface, buoyancy and turbulent incompressible flows * Compressible high-speed gas flow * Shallow water problems * Waves * Computer implementation of the CBS algorithm
Readership: Undergraduate, postgraduate and professional civil, mechanical, aeronautical, automotive and marine engineers and material scientists.
ISBN: 0 7506 5160 1 Hardback
Colour plates: 12 Line Illustrations: 566 Halftones:
Measurements: 244 x 172 mm
Pages: 1440pp
Approximate Publication Date: 25 August 2000