Serafina Cuomo
Pappus of Alexandria and the Mathematics of Late Antiquity
This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. This is one of very few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus's text as part of a wider context and relates it to other contemporary cultural practices and opens new avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity.
Introduction; 1. The outside world; 2. Bees and philosophers; 3. Inclined planes and architects; 4. Altars and strange curves; 5. The inside story.
ENew topic - no other monograph on Pappus of Alexandria
ENew methodology - relates mathematical practices to other cultural practices in Late antiquity
ENew conclusions - concludes that Pappus had a precise agenda and used his sources strategically
S. R. Alpern and V. S. Prasad
Typical Dynamics of Volume-Preserving Homeomorphisms
This book provides a self-contained introduction to typical properties of volume preserving homeomorphisms, examples of which include transitivity, chaos and ergodicity. A key notion is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the underlying measure space. The authors make the first part of the book very concrete by focusing on volume preserving homeomorphisms of the unit n-dimensional cube. They also prove fixed point theorems (Conley-Zehnder-Franks). This is done in a number of short self-contained chapters which would be suitable for an undergraduate analysis seminar or a graduate lecture course. Parts II and III consider compact manifolds and sigma compact manifolds respectively. Much of this work describes the work of the two authors in extending to different settings and properties, the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.
Part I. Volume Preserving Homomorphisms of the Cube: 1. Introduction to Parts 1, 2 (Compact manifolds); 2. Measure preserving homeomorphisms; 3. Discrete approximations; 4. Transitive homeomorphisms of In and Rn; 5. Fixed points and area preservation; 6. Measure preserving Lusin theorem; 7. Ergodic homeomorphisms; 8. Uniform approximation in G[In, l]; Part II. Measure Preserving Homeomorphisms of a Compact Manifold: 9. Measures on compact manifolds; 10. Dynamics on compact manifolds; Part III. Measure Preserving Homeomorphisms of a Noncompact Manifold: 11. Introduction to Part 3; 12. Ergodic volume preserving homeomorphisms of Rn; 13. Manifolds where ergodic is not generic; 14. Noncompact manifolds and ends; 15. Ergodic homeomorphisms: the results; 16. Ergodic homeomorphisms: proof; Other properties typical in M[X, m]; Appendix 1. Multiple Rokhlin towers and conjugacy approximation; Appendix 2. Homeomorphic measures.
R. Iorio and V. Iorio
Fourier Analysis and Partial Differential Equations
An Introduction
Hardback 0 521 62116 X 2000/10/25
This modern introduction to Fourier analysis and partial differential equations is intended to be used with courses for beginning graduate students. With minimal prerequisites the authors take the reader from fundamentals to research topics in the area of nonlinear evolution equations. The first part of the book consists of some very classical material, followed by a discussion of the theory of periodic distributions and the periodic Sobolev spaces. The authors then turn to the study of linear and nonlinear equations in the setting provided by periodic distributions. They assume only some familiarity with Banach and Hilbert spaces and the elementary properties of bounded linear operators. After presenting a fairly complete discussion of local and global well-posedness for the nonlinear Schrödinger and the Korteweg-de Vries equations, they turn their attention, in the two final chapters, to the non-periodic setting, concentrating on problems that do not occur in the periodic case.
Part I. Fourier Series and Periodic Distributions: 1. Preliminaries; 2. Fourier series: basic theory; 3. Periodic distributions and Sobolev spaces; Part II. Applications to Partial Differential Equations: 4. Linear equations; 5. Nonlinear evolution equations; 6. The Korteweg-de Vries; Part III: 7. Distributions, Fourier transforms and linear equations; 8. KdV, BO and friends; Appendix A. Tools from the theory of ODEs; Appendix B. Commutator estimates; Bibliography; Index.
M. Hashimoto
Auslander-Buchweitz Approximations of Equivariant Modules
Paperback 0 521 79696 2 2000/11/02
London Mathematical Society Lecture Note Series, 282
This book focuses on homological aspects of equivariant modules. It presents a new homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of co-algebras over an arbitrary base. It aims to overcome the difficulty of generalising known homological results in representation theory. This book will be of interest to researchers and graduate students in algebra, specialising in commutative ring theory and representation theory.
Introduction; Conventions and terminology; Part I. Background Materials: 1. From homological algebra; 2. From Commutative ring theory; 3. Hopf algebras over an arbitrary base; 4. From representation theory; 5. Basics on equivariant modules; Part II. Equivariant Modules: 1. Homological aspects of (G, A)-modules; 2. Matijevic-Roberts type theorem; Part III. Highest Weight Theory: 1. Highest weight theory over a field; 2. Donkin systems; 3. Ringel's theory over a field; 4. Ringel's theory over a commutative ring; Part IV. Approximations of Equivariant Modules; 1. Approximations of (G, A)-modules; 2. An application to determinantal rings; Bibliography; Index; Glossary.
C.J. Ash ,
J. Knight, University of Notre Dame, Department of Mathematics, Notre Dame, IN, USA
Computable Structures and the Hyperarithmetical Hierarchy
Included in series
Studies in Logic and the Foundations of Mathematics, 144
Description
This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties).
Contents
Preface.
Computability.
The arithmetical hierarchy.
Languages and structures.
Ordinals.
The hyperarithmetical hierarchy.
Infinitary formulas.
Computable infinitary formulas.
The Barwise-Kreisel Compactness Theorem.
Existence of computable structures.
Completeness and forcing.
The Ash-Nerode Theorem.
Computable categoricity and stability.
n-systems.
-systems.
Back-and forth relations.
Theorems of Barker and Davey.
Pairs of computable structures.
Models of arithmetic.
Special classes of structures.
Year 2000
Hardbound
ISBN: 0-444-50072-3
358 pages