Denis R. Bell

The Malliavin Calculus

ISBN: 0486449947

This introduction to Malliavin's stochastic calculus of variations is suitable for graduate students and professional mathematicians. Author Denis R. Bell particularly emphasizes the problem that motivated the subject's development, with detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and descriptions of a variety of applications. The text includes enough technical background to make the subsequent material accessible to readers without specialized knowledge of stochastic analysis.1987 edition.

Table of Contents

Introduction
1. Background material
2. The functional analytic approach
3. The variational approach
4. An elementary derivation of Malliavinfs inequalities
5. A discussion of the different forms of the theory
6. Non-degeneracy of the covariance matrix under Hormanderfs condition
7. Some further applications of the Malliavin calculus
References
Index

David B. Gauld

Differential Topology: An Introduction

ISBN: 048645021X

Offering classroom-proven results, Differential Topology presents an introduction to point set topology via a naive version of nearness space. Its treatment encompasses a general study of surgery, laying a solid foundation for further study and greatly simplifying the classification of surfaces. This self-contained treatment features 88 helpful illustrations. Its subjects include topological spaces and properties, some advanced calculus, differentiable manifolds, orientability, submanifolds and an embedding theorem, and tangent spaces. Additional topics comprise vector fields and integral curves, surgery, classification of orientable surfaces, and Whitney's embedding theorem. 1982 ed.

Table of Contents

1. What Is Topology?
2. Topological Spaces
3. Some Topological Properties
4. Some Advanced Calculus
5. Differentiable Manifolds
6. Orientability
7. Submanifolds and an Embedding Theorem
8. Tangent Spaces
9. Critical Points Again
10. Vector Fields and Integral Curves
11. Surgery
12. The Trace of a Surgery
13. Surgery on a Surface
14. Classification of Orientable Surfaces
15. Whitneyfs Embedding Theorem
Appendix A. The Unproved Theorems
Appendix B. Further Topics
Notation
Bibliography
Index

Robert Goldblatt

Topoi: The Categorial Analysis of Logic

ISBN: 0486450260

A classic exposition of the branch of mathematical logic known as category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Beginning with a survey of set theory and its role in mathematics, it proceeds to definitions and examples of categories and explains the use of arrows in the place of epsilon. Subsequent topics include topos logic, algebra of subobjects, intuitionism and its logic, functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry. 1983 ed.

Table of Contents

1. Mathematics = Set Theory
2. What Categories Are
3. Arrows Instead of Epsilon
4. Introducing Topoi
5. Topos Structure: First Steps
6. Logic Classically Conceived
7. Algebra of Subobjects
8. Institutionism and Its Logic
9. Functors
10. Set Concepts and Validity
11. Elementary Truth
12. Categorial Set Theory
13. Arithmetic
14. Local Truth
15. Adjointness and Quantifiers
16. Logical Geometry
References
Catalogue of Notation
Index of Definitions

William C. Graustein

Differential Geometry

ISBN: 0486450112

This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions, using vector notation and technique. It also introduces Riemannian geometry. Written by a noted mathematician, and suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of surfaces, and the absolute geometry of a surface; the final chapter considers the applications of the theory to selected classes of surfaces. Nearly 200 problems appear throughout the text. 1935 ed.

Table of Contents

1. Introduction
2. Space Curves
3. Curves and Surfaces Associated with a Space Curve
4. Fundamentals of the Theory of Surfaces
5. Curvature. Important Systems of Curves
6. The Fundamental Theorem
7. Geodesic Curvature. Geodesics
8. Mapping of Surfaces
9. The Absolute Geometry of a Surface
10. Surfaces of Special Type
Index

Wilfrid Hodges

Building Models by Games

ISBN: 0486450171

This volume presents research by algebraists and model theorists in accessible form for advanced undergraduates or beginning graduate students studying algebra, logic, or model theory. It introduces a general method for building infinite mathematical structures and surveys applications in algebra and model theory. Starting with an overview of basic model theory, the text examines a variety of algebraic applications, including completeness for Magidor-Malitz quantifiers, Shelah's recent and sophisticated omitting types theorem for L(Q), and applications to Boolean algebras and models of arithmetic. More than 160 exercises range from elementary drills to research-related items. 1985 ed.

Table of Contents

1. Preliminaries
2. Games and Forcing
3. Existential Closure
4. Chaos or Regimentation
5. Classical Languages
6. Proper Extensions
7. Generalised Quantifiers
8. L(Q) in Higher Cardinalities
List of types of forcing
List of open questions
Bibiliography
Index

M. A. Lavrentfev

Variational Methods for Boundary Value Problems:
for Systems of Elliptic Equations

ISBN: 0486450783

In this famous work, a distinguished Russian mathematical scholar presents an innovative approach to classical boundary value problems -- one that may be used by mathematicians as well as by theoreticians in mechanics. The first two chapters cover variational principles of the theory of conformal mapping and behavior of a conformal transformation on the boundary. Succeeding chapters address hydrodynamic applications and quasi-conformal mappings, as well as linear systems and the simplest classes of non-linear systems.

Table of Contents

Introduction
1. Variational principles
2. Sufficient conditions
3. Generalizations
Chapter 1. Variational principles of the theory of conformal mapping
1.1 The principles of Lindelof and Montel
1.1.1 The case of the circle
1.1.2 Mapping on to a strip
1.2 Mechanical interpretation
1.3 Quantitative estimates
Chapter 2. Behaviour of a conformal transformation on the boundary
2.1 Derivatives at the boundary
2.2 Narrow strips
2.3 Behaviour of the extension at points of maximum inclination and extreme curvature
Chapter 3. Hydrodynamic applications
3.1 Stream line flow
3.2 Generalizations
3.3 Stream line flow with detachment
3.4 Wave motions of a fluid
3.5 The linear theory of waves
3.6 Rayleigh waves
3.7 The exact theory
3.8 Generalizations
3.8.1 Motion of a fluid over a submarine trench
3.8.2 Motion over a bottom with a ridge
3.8.3 Spillway with singularities
Chapter 4. Quasi-conformal mappings
4.1 The concept of the quasi-conformal map
4.2 Derivative systems
4.3 Strong ellipticity
Chapter 5. Linear systems
5.1 Transformations with bounded distortion
5.1.1 Equi-graded continuity
5.1.2 Almost conformal mappings
5.2 The simplest class of linear systems
5.2.1 Invariance with respect to conformal mappings
5.2.2. Stability of conformal mappings
5.2.3 Condition of smoothness of a transformation
5.2.4 Application to arbitrary linear systems
5.2.5 Existence theorem
Chapter 6. The simplest classes of non-linear systems
6.1 Maximum principle
6.2 The principle of Schwarz-Lindelof
6.3 Quantitative estimates
6.4 Inductive proof of Lindelof's principle
6.5 The existence theorem
6.6 Generalizations
6.7 Hydrodynamic applications
References; Index