Cutland, N.J., University of Hull, UK
Loeb Measures in Practice: Recent Advances
EMS Lectures 1997
2000. XI, 111 pp. Softcover
3-540-41384-7
This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the
construction of Loeb Measures, which are rich measures discovered in 1975 by Peter Loeb, using
techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb
measures due to the author and his collaborators, in such diverse fields as (stochastic) fluid
mechanics, stochastic calculus of variations ("Malliavin" calculus) and the mathematical finance
theory. The exposition is designed for a general audience, and no previous knowledge of either NSA
or the various fields of applications is assumed.
Keywords: Nonstandard analysis, Loeb measures, stochastic fluid mechanics, stochastic
analysis, mathematical finance theory
Contents: 1. Loeb Measures
1.1 Introduction
1.2 Nonstandard Analysis
1.2.1 The hyperreals
1.2.2 The nonstandard universe
1.2.3 N1-saturation
1.2.4 Nonstandard topology
1.3 Construction of Loeb Measures
1.3.1 Example: Lebesgue measure
1.3.2 Example: Haar measure
1.3.3 Example: Wiener measure
1.3.4 Loeb measurable functions 1.4 Loeb Integration Theory
1.5 Elementary Applications
1.5.1 Lebesgue integration
1.5.2 Peano's Existence Theorem
1.5.3 Ito integration and stochastic differential equations 2 Stochastic Fluid Mechanics
2.1 Introduction
2.1.1 Function spaces
2.1.2 Functional formulation of the Navier-Stokes equations
2.1.3 Definition of solutions to the stochastic Navier-Stokes equations
2.1.4 Nonstandard topology in Hilbert spaces
2.2 Solution of the Deterministic Navier-Stokes Equations
2.2.1 Uniqueness
2.3 Solution of the Stochastic Navier-Stokes Equations
2.3.1 Stochastic Flow
2.3.2 Nonhomogeneous stochastic Navier-Stokes equations
2.4 Stochastic Euler Equations
2.5 Statistical Solutions
2.5.1 The Foias equation
2.5.2 Construction of statistical solutions using Loeb measures
2.5.3 Measures by nonstandard densities
2.5.4 Construction of statistical solutions using nonstandard densities
2.5.5 Statistical solutions for stochastic Navier-Stokes equations
2.6 Attractors for the Navier-Stokes Equations
2.6.1 Introduction
2.6.2 Nonstandard attractors and standard attractors
2.6.3 Attractors for 3-dimensional Navier-Stokes equations
2.7 Measure Attractors for Stochastic Navier-Stokes Equations
2.8 Stochastic Attractors for Navier-Stokes Equations
2.8.1 Stochastic attractors
2.8.2 Existence of a stochastic attractor for the Navier-Stokes equations
2.9 Attractors for the 3-dimensional Stochastic Navier-Stokes Equations
3. Stochastic Calculus of Variations
3.1 Introduction
3.1.1 Notation
3.2 Flat Integral Representation of Wiener Measure
3.3 The Wiener Sphere
3.4 Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck
Process
3.5 Malliavin Calculus
3.5.1 Notation and preliminaries
3.5.2 The Wiener-Ito chaos decomposition
3.5.3 The derivation operator
3.5.4 The Skorohod integral
3.5.5 The Malliavin operator
4. Mathematical Finance Theory
4.1 Introduction
4.2 The Cox-Ross-Rubinstein Models
4.3 Options and Contingent Claims
4.3.1 Pricing a claim
4.4 The Black-Scholes Model
4.5 The Black-Scholes Model and Hyperfinite CRR Models
4.5.1 The Black-Scholes formula
4.5.2 General claims
4.6 Convergence of Market Models
4.7 Discretisation Schemes
4.8 Further Developments
4.8.1 Poisson pricing models
4.8.2 American options 4.8.3 Incomplete markets
4.8.4 Fractional Brownian motion
4.8.5 Interest rates
Index
Series: Lecture Notes in Mathematics.VOL. 1751
Bobenko, A.I., Technische Universitat Berlin, Germany
Eitner, U., Technische Universitat Berlin, GermanyPainleve Equations in the Differential Geometry of Surfaces
2000. VI, 120 pp. Softcover
3-540-41414-2
This book brings together two different branches of mathematics: the theory of Painlev・and the
theory of surfaces. Self-contained introductions to both these fields are presented. It is shown how some classical problems in surface theory can be solved using the modern theory of Painlev・
equations. In particular, an essential part of the book is devoted to Bonnet surfaces, i.e. to surfaces possessing families of isometries preserving the mean curvature function. A global classification of Bonnet surfaces is given using both ingredients of the theory of Painlev・equations: the theory of isomonodromic deformation and the Painlev・property. The book is illustrated by plots of surfaces. It is intended to be used by mathematicians and graduate students interested in differential geometry and Painlev・equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.
Keywords: Painleve equations, Bonnet surfaces, integrable systems . MSC : 53-02, 34-02, 53C42,
35Q53, 30D05, 53A05, 53A15
Contents: 1. Introduction
2. Basics of Painlev・Equations and Quaternionic Description of Surfaces
2.1. Painlev・Property and Painlev・Equations
2.2. Isomonodromic Deformations 2.3. Conformally Parametrized Surfaces 2.4. Quaternionic
Description of Surfaces 3. Bonnet Surfaces in Euclidean three-space 3.1. Definition of Bonnet
Surfaces and Simplest Properties 3.2. Local Theory away from Critical Points 3.3. Local Theory at
Critical Points 3.4. Bonnet Surfaces via Painlev Transcendents 3.5. Global Properties of Bonnet
Surfaces 3.6. Examples of Bonnet Surfaces 3.7. Schlesinger Transformations for Bonnet Surfaces
4. Bonnet Surfaces in S and H and Surfaces with Harmonic Inverse Mean Curvature 4.1. Surfaces
in S3 and H3 4.2. Definition and Simplest Properties 4.3. Bonnet Surfaces in S3 and H3 away from
Critical Points 4.4. Local Theory of Bonnet Surfaces in S and H at Critical Points 4.5. Bonnet
Surfaces in S3 and H3 in Terms of Painlev Transcendents 4.6. Global Properties of Bonnet
Surfaces in Space Forms 4.7. Surfaces with Harmonic Inverse Mean Curvature 4.8. Bonnet Pairs of
HIMC Surfaces 4.9. HIMC Bonnet Pairs in Painlev Transcendents 4.10. Examples of HIMC
Surfaces 5. Surfaces with Constant Curvature 5.1. Surfaces with Constant Negative Gaussian
Curvature and Two Straight Asymptotic Lines 5.2. Smyth Surfaces 5.3. Affine Spheres with Affine
Straight Lines 6. Appendices
Series: Lecture Notes in Mathematics.VOL. 1753
Bertram, W., Universite Henri Poincare, Vandoeuvre-les-Nancy, France
The Geometry of Jordan and Lie Structures
2000. XVI, 269 pp. Softcover
3-540-41426-6
The geometry of Jordan and Lie structures tries to answer the following question: what is the
integrated, or geometric, version of real Jordan algebras, - triple systems and - pairs? Lie theory
shows the way one has to go: Lie groups and symmetric spaces are the geometric version of Lie
algebras and Lie triple systems. It turns out that both geometries are closely related via a functor
between them, called the Jordan-Lie functor, which is constructed in this book.
The reader is not assumed to have any knowledge of Jordan theory; the text can serve as a
self-contained introduction to (real finite-dimensional) Jordan theory.
Keywords: Symmetric space, complexification, Jordan triple system, Lie triple system, Liouville
theorem . MSC : 17C36, 17C30, 17C37, 53B35, 53C15, 53C35, 53C55, 22E15, 53A40, 53B05
Contents: First Part: The Jordan-Lie functor
I.Symetric spaces and the Lie-functor
1. Lie functor: group theoretic version
2. Lie functor:differential geometric version
3. Symmetries and group of displacements
4. The multiplication map
5. Representations os symmetric spaces
6. Examples
Appendix A: Tangent objects and their extensions
Appendix B: Affine Connections
II. Prehomogeneous symmetric spaces and Jordan algebras
1. Prehomogeneous symmetric spaces
2. Quadratic prehomogeneous symmetric spaces
3. Examples
4. Symmetric submanifolds and Helwig spaces
III. The Jordan-Lie functor
1. Complexifications of symmetric spaces
2. Twisted complex symmetric spaces and Hermitian JTS
3. Polarizations, graded Lie algebras and Jordan pairs
4. Jordan extensions and the geometric Jordan-Lie functor
IV. The classical spaces
1. Examples
2. Principles of the classification
V. Non.degenerate spaces
1. Pseudo-Riemannian symmetric spaces
2. Pseudo-Hermitian and para-Hermitian symmetric spaces
3. Pseudo-Riemannian symmetric spaces with twist
4. Semisimple Jordan algebras
5. Compact spaces and duality
Second Part: Conformal group and global theory
VI. Integration of Jordan structures
1. Circled spaces
2. Ruled spaces
3. Integrated version of Jordan triple systems
Appendix A: Integrability of almost complex structures
VII. The conformal Lie algebra
1. Euler operators and conformal Lie algebra
2. The Kantor-Koecher-Tits construction
3. General structure of the conformal Lie algebra
VIII. Conformal group and conformal completion
1. Conformal group: general properties
2. Conformal group: fine structure
3. The conformal completion and its dual
4. Conformal completion of the classical spaces
Appendix A: Some identities for Jordan triple systems
Appendix B: Equivariant bundles over homogeneous spaces
IX. Liouville theorem and fundamental theorem
1. Liouville theorem and and fundamental theorem
2. Application to the classical spaces
X. Algebraic structures of symmetric spaces with twist
1. Open symmetric orbits in the conformal completion
2. Harish-Chandra realization
3. Jordan analog of the Campbell-Hausdorff formula
4. The exponential map
5. One-parameter subspaces and Peirce-decomposition
6. Non-degenerate spaces
Appendix A: Power associativity
XI. Spaces of the first and of the second kind
1. Spaces of the first kind and Jordan algebras
2. Cayley transform and tube realizations
3. Causal symmetric spaces
4. Helwig-spaces and the extension problem
5. Examples
XII.Tables
1. Simple Jordan algebras
2. Simple Jordan systems
3. Conformal groups and conformal completions
4. Classification of simple symmetric spaces with twist
XIII. Further topics
Series: Lecture Notes in Mathematics.VOL. 1754
Ben-Ari, M., Weizmann Institute of Science, Rehovot, Israel
Mathematical Logic for Computer Science 2nd rev. ed. 2001.
Approx. 320 pp. Softcover
1-85233-319-7
Mathematical Logic for Computer Science is a mathematics textbook with theorems and
proofs, but the choice of topics has been guided by the needs of computer science students. The
method of semantic tableaux provides an elegant way to teach logic that is both theoretically sound and yet sufficiently elementary for undergraduates. To provide a balanced treatment of logic, tableaux are related to deductive proof systems.
The logical systems presented are:
- Propositional calculus (including binary decision diagrams);
- Predicate calculus;
- Resolution;
- Hoare logic;
- Z;
- Temporal logic.
Answers to exercises as well as Prolog source code for algorithms may be found via the Springer
London web site: http://www.springer.co.uk/com pubs/ct mlcs.htm
Mordechai Ben-Ari is Associate Professor at the Department of Science Teaching of the Weizmann Institute of Science. He has published textbooks on concurrent programming and programming languages.
Contents: Preface.- Chapter 1 Introduction.- Chapter 2 Propositional Calculus: Formulas, Models,
Tableaux.- Chapter 3 Propositional Calculus: Deductive Systems.- Chapter 4 Propositional
Calculus: Resolution and BDDs.- Chapter 5 Predicate Calculus: Formulas, Models, Tableau.-
Chapter 6 Predicate Calculus: Deductive Systems.- Chapter 7 Predicate Calculus: Resolution.-
Chapter 8 Logic Programming.- Chapter 9 Programs: Semantics and Verification.- Chapter 10
Programs: Formal Specification with Z.- Chapter 11 Temporal Logic: Formulas, Models, Tableaux.-
Chapter 12 Temporal Logic: Deduction and Applications.- Appendix A Set Theory.- Further
Reading.- Index.
Babitsky, V.I., Loughborough University, UK
Krupenin, V.L., Russian Academy of Science, Moscow, RussiaVibrations of Strongly Nonlinear Systems
2001. Approx. 330 pp. Hardcover
3-540-41447-9
Among the wide variety of nonlinear mechanical systems, it appears possible to distinguish a
representative class which may be characterized by the presence of threshold nonlinear positional
forces. Such systems demonstrate a sudden change in the behavior of elastic and dissipative
forces. This monograph addresses the systematic representation of the new methods of analysis
developed by the authors recently as applied to such systems. Specific features of dynamic
processes of these systems are studied. Special attention is given to an analysis of different
resonant phenomena taking unusual and diverse forms. These methods are applied to the analysis
of mechanical systems designed for the generation and transformation of intensive processing of an impulsive nature. These are machines for rock fragmentation, impact processing, special types of shock testing machines and many other types of machinery.
Keywords: Applied mechanics
Contents: Operators of Linear Systems.- Strongly Nonlinear Single Degrees of Freedom Systems.-
Multiple Degrees of Freedom Systems.
Series: Foundations of Engineering Mechanics.