Hersh, Reuben (Ed.)

18 Unconventional Essays on the Nature of Mathematics

2006, XXII, 330 p. 10 illus., Softcover
ISBN: 0-387-25717-9

About this book

This book collects some of the most interesting recent writings that are tackling, from various points of view, the problem of giving an accounting of the nature, purpose, and justification of real mathematical practice--mathematics as actually done by real live mathematicians. What is the nature of the objects being studied? What determines the directions and styles in which mathematics progresses (or, perhaps, degenerates)? What certifies its claim to certainty, or to a priori status, to independence of experience? Why is mathematics the same for all times and places, or is it really the same, or in what senses is it the same and in what senses different? Many of these writings were read at conferences in Europe and America under the heading of "history" or "cultural studies" as well as "philosophy." It is the editor’s hope to help foster healthy interdisciplinary mutual aid in this young and fertile area.

REUBEN HERSH is professor emeritus at the University of New Mexico, Albuquerque. He is the recipient (with Martin Davis) of the Chauvenet Prize and (with Edgar Lorch) the Ford Prize. Hersh is the author (with Philip J. Davis) of The Mathematical Experience and Descartes' Dream, which won the National Book Award in l983, and What is Mathematics, Really?

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Mills, Bruce

Theoretical Introduction to Programming

2006, X, 358 p. 29 illus., Softcover
ISBN: 1-84628-021-4

About this textbook

Including well-digested information about fundamental techniques and concepts in software construction, this book is distinct in unifying pure theory with pragmatic details. Driven by generic problems and concepts, with brief and complete illustrations from languages including C, Prolog, Java, Scheme, Haskell and HTML.

This book is intended to be both a how-to handbook and easy reference guide. Discussions of principle, worked examples and exercises are presented. All concepts outside introductory programming are explained with clear demarcation and dependencies so the experienced programmer can quickly locate material. Readable in a linear manner, with short mono-thematic to encourage dipping and reference. Also included are sections on open problems in software theory and practice.

While little other than a novice programmer's knowledge is explicitly assumed, a certain conceptual maturity, either through commercial programming or academic training is required ? each language is introduced and explained briefly as needed.

Table of contents

The Abstract Rational Outlook.- A Grab-bag of Computational Models.- Some Formal Technology.- Limitations on Exact Knowledge.- Some Orthodox Languages.- Arithmetic Computation.- Repetitive Computation.- Temporal Interaction.- Container Datatypes.- End-notes and Candle-wax.

Sirmakessis, Spiros (Ed.)

Knowledge Mining
Proceedings of the NEMIS 2004 Final Conference

Series: Studies in Fuzziness and Soft Computing, Vol. 185
2005, VIII, 290 p. 94 illus., Hardcover
ISBN: 3-540-25070-0

About this textbook

This book contains the papers presented during the 3rd International Workshop on Text Mining and its Applications held in Athens, Greece. This workshop is the final event of the activities of a network of excellence in the area of text mining and its applications. Knowledge Mining includes contributions in the areas of: Document processing, visualization techniques, Web mining, Text mining, or knowledge management.

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Fengler, Matthias R.

Semiparametric Modeling of Implied Volatility

Series: Springer Finance
2005, XV, 224 p. 61 illus., Softcover
ISBN: 3-540-26234-2

About this book

The implied volatility surface is a key financial variable for the pricing and the risk management of plain vanilla and exotic options portfolios alike. Consequently, statistical models of the implied volatility surface are of immediate importance in practice: they may appear as estimates of the current surface or as fully specified dynamic models describing its propagation through space and time.

This book fills a gap in the financial literature by bringing together both recent advances in the theory of implied volatility and refined semiparametric estimation strategies and dimension reduction methods for functional surfaces: the first part of the book is devoted to smile-consistent pricing appoaches. The theory of implied and local volatility is presented concisely, and vital smile-consistent modeling approaches such as implied trees, mixture diffusion, or stochastic implied volatility models are discussed in detail. The second part of the book familiarizes the reader with estimation techniques that are natural candidates to meet the challenges in implied volatility modeling, such as the rich functional structure of observed implied volatility surfaces and the necessity for dimension reduction: non- and semiparametric smoothing techniques.

The book introduces Nadaraya-Watson, local polynomial and least squares kernel smoothing, and dimension reduction methods such as common principle components, functional principle components models and dynamic semiparametric factor models. Throughout, most methods are illustrated with empirical investigations, simulations and pictures.

Table of contents

Introduction.- The Implied Volatility Surface.- Smile Consistent Volatility Models.- Smoothing Techniques.- Dimension-Reduced Modeling.

Krantz, Steven G.

Geometric Function Theory
Explorations in Complex Analysis

Series: Cornerstones
2006, XIV, 314 p. 41 illus., Hardcover
ISBN: 0-8176-4339-7

About this textbook

Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy?Riemann equations, and the corona problem.

The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.

This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis?and also to spark the interest of seasoned workers in the field?the book imparts a solid education both in complex analysis and in how modern mathematics works.

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Malliavin, Paul, Thalmaier, Anton

Stochastic Calculus of Variations in Mathematical Finance

Series: Springer Finance
2006, XII, 142 p., Hardcover
ISBN: 3-540-43431-3

About this book

Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous paths stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. The discretization error of the Euler scheme for a stochastic differential equation is expressed as a generalized Watanabe distribution on the Wiener space. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear.

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Calvin C. Moore / University of California, Berkeley
Claude L. Schochet / Wayne State University, Detroit

Global Analysis on Foliated Spaces, 2nd Edition

Series: Mathematical Sciences Research Institute Publications
Paperback (ISBN-10: 0521613051 | ISBN-13: 9780521613057)

Foliated spaces look locally like products, but their global structure is generally not a product, and tangential differential operators are correspondingly more complex. In the 1980s, Alain Connes founded what is now known as noncommutative geometry and topology. One of the first results was his generalization of the Atiyah-Singer index theorem to compute the analytic index associated with a tangential (pseudo) - differential operator and an invariant transverse measure on a foliated manifold, in terms of topological data on the manifold and the operator. This second edition presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds). It includes the necessary background from analysis, geometry, and topology. The present edition has improved exposition, an updated bibliography, an index, and additional material covering developments and applications since the first edition came out, including the confirmation of the Gap Labeling Conjecture of Jean Bellissard.

* Background information in all fields necessary to understand the proof of the Index Theorem; fundamental text in non-commutative topology

* Contains new chapter updates, an appendix discussing the Gap Labelling Theorem, an updated bibliography, and completely redone illustrations

* The first edition of the book was widely and positively reviewed

Contents

Introduction; 1. Locally traceable operators; 2. Foliated spaces; 3. Tangential cohomology; 4. Transverse measures; 5. Characteristic classes; 6. Operator algebra; 7. Pseudodifferential operators; 8. The index theorem; Appendices.

Review
Praise for the first edition … ‘The quest for the proof leads through functional analysis, C^* and von Neumann algebras, topological groupoids, characteristic classes and K-theory along a foliation, and the theory of pseudodifferential operators. It is a long but very rewarding journey and Moore and Schochet have performed a valuable service in putting all this material in one place in an easily readable form … The book contains a wealth of information. It is not for those who wish an overview...However, for those wishing a comprehensive proof … this book is indispensable.’ AMS Bulletin