Fernholz, E.R., INTECH, Princeton, NJ, USA

Stochastic Portfolio Theory

2002. XIV, 177 pp. 38 figs. Hardcover
0-387-95405-8

Stochastic portfolio theory is a mathematical methodology for constructing stock portfolios and for analyzing the effects induced on the behavior of these portfolios by changes in the distribution of capital in the market.
Stochastic portfolio theory has both theoretical and practical applications: as a theoretical tool it can be used to construct examples of theoretical portfolios with specified characteristics and to determine the distributional component of portfolio return. On a practical level, stochastic portfolio theory has been the basis for strategies used for over a decade by the institutional equity manager INTECH, where the author has served as chief investment officer.
This book is an introduction to stochastic portfolio theory for investment professionals and for students of mathematical finance. Each chapter includes a number of problems of varying levels of difficulty and a brief summary of the principal results of the chapter, without proofs.

Contents: Stochastic Portfolio Theory.- Stock Market Diversity.- Portfolio Generating Functions.- Functions of Ranked Market Weights.- Stable Model for the Distribution of Capital.- Behavior of Functionally Generated Portfolios.- Applications of Stochastic Portfolio Theory.

Series: Applications of Mathematics. VOL. 48

Bialynicki-Birula, A., University of Warsaw, Poland; Carrell, J., The University of British Columbia, Vancouver, BC, Canada; McGovern, W.M., University of Washington, Seattle, WA, USA

Algebraic Quotients. Torus Actions and Cohomology.
The Adjoint Representation and the Adjoint Action

2002. V, 242 pp. Hardcover
3-540-43211-6

This is the second volume of the new subseries "Invariant Theory and Algebraic Transformation Groups". The aim of the survey by A. Bialynicki-Birula is to present the main trends and achievements of research in the theory of quotients by actions of algebraic groups. This theory contains geometric invariant theory with various applications to problems of moduli theory. The contribution by J. Carrell treats the subject of torus actions on algebraic varieties, giving a detailed exposition of many of the cohomological results one obtains from having a torus action with fixed points. Many examples, such as toric varieties and flag varieties, are discussed in detail. W.M. McGovern studies the actions of a semisimple Lie or algebraic group on its Lie algebra via the adjoint action and on itself via conjugation. His contribution focuses primarily on nilpotent orbits that have found the widest application to representation theory in the last thirty-five years.

Keywords: invariants, quotients, adjoint representation, transformation group

Series: Encyclopaedia of Mathematical Sciences. VOL. 131

Castillo-Chavez, C., Cornell University, Ithaca, NY, USA; Blower, S., UCLA School of Medicine, Los Angeles, CA, USA; Driessche, P.vande, University of Victoria, BC, Canada; Kirschner, D., University of Michigan Medical Scholl, Ann Arbor, MI,

Mathematical Approaches for Emerging and Reemerging Infectious Diseases Part I: An Introduction to Models, Methods, and Theory

2002. Approx. 390 pp. Hardcover
0-387-95354-X

This book grew out of the discussions and presentations that began during the Workshop on Emerging and Reemerging Diseases (May 17-21, 1999) sponsored by the Institute for Mathematics and its Application (IMA) at the University of Minnesota with the support of NIH and NSF. The workshop started with a two-day tutorial session directed at ecologists, epidemiologists, immunologists, mathematicians, and scientists interested in the study of disease dynamics. The core of this first volume, Volume 125, covers tutorial and research contributions on the use of dynamical systems (deterministic discrete, delay, PDEs, and ODEs models) and stochastic models in disease dynamics. The volume includes the study of cancer, HIV, pertussis, and tuberculosis.
Beginning graduate students in applied mathematics, scientists in the natural, social, or health sciences or mathematicians who want to enter the fields of mathematical and theoretical epidemiology will find this book useful.

Contents: From the contents: New directions in the mathematics of infectious disease.- Fred Brauer: The man and his mathematics.- Kenneth L. Cooke: Researcher, educator par excellance.- Basic ideas of mathematical epidemiology.- Extensions of the basic models.- New vaccination strategies for pertussis.- Time delay in epidemic models.- Nonlocal response in a simple epidemiological model.- Discrete-time S-I-S models with simple and complex population dynamics.- Intraspecific competition, dispersal, and disease dynamics in discrete-time patchy environments.- The impact of long-range dispersal on the rate of spread in population and epidemic models.- Endemicity, persistence, and quasi-stationarity.- On the computation of Ro and its role in global stability.- Nonlinear mating models for populations with discrete generations.- Center manifolds and normal forms in epidemic models.- Remarks on modeling host-viral dynamics and treatment.- A multiple compartment model for the evolution of HIV-1 after highly active antiretroviral therapy.- Modeling cancer as an infectious disease.- Frequency dependent risk of infection and the spread of infectious diseases.- Long-term dynamics and emergence of tuberculosis.

Series: The IMA Volumes in Mathematics and its Applications. VOL. 125

Castillo-Chavez, C., Cornell University, Ithaca, NY, USA; Blower, S., UCLA School of Medicine, Los Angeles, CA, USA; Driessche, P.vande, University of Victoria, BC, Canada (Eds.)

Mathematical Approaches for Emerging and Reemerging Infectious Diseases Part II: Models, Methods, and Theory

2002. Approx. 390 pp. Hardcover
0-387-95355-8

This book grew out of the discussions and presentations that began during the Workshop on Emerging and Reemerging Diseases (May 17-21, 1999) sponsored by the Institute for Mathematics and its Application (IMA) at the University of Minnesota with the support of NIH and NSF. The workshop started with a two-day tutorial session directed at ecologists, epidemiologists, immunologists, mathematicians, and scientists interested in the study of disease dynamics. The core of this second volume, Volume 126, covers research contributions on the use of dynamical systems (deterministic discrete, delay, PDEs, and ODEs models) and stochastic models in disease dynamics. Contributions motivated by the study of diseases like influenza, HIV, tuberculosis, and macroparasitic like schistosomiasis are also included.
This second volume requires additional mathematical sophistication, and graduate students in applied mathematics, scientists in the natural, social, and health sciences, or mathematicians who want to enter the field of mathematical and theoretical epidemiology will find it useful. The collection of contributors includes many who have been in the forefront of the development of the subject.

Contents: From the contents: Maximal prevalence and the basic reproduction number in simple epidemics.- The transition through stages with arbitrary length distributions, and applications in epidemics.- Measles outbreaks are not chaotic.- Epidemics among a population of households.- Infection transmission dynamics and vaccination program effectiveness as a function of vaccine effects in individuals.- The influence of different forms of cross-protective immunity on the population dynamics of antigenetically diverse pathogens.- Dynamics of multiple strains of infectious agents coupled by cross-immunity.- Virulence evolution in macro-parasites.- Mathematical models for schistosomiasis with delays and multiple definitive hosts.- Infectious disease models with chronological age structure and epidemiological age structure.- Effects of genetic heterogeneity on HIV transmission in homosexual populations.- Age-structured core group model and its impact on STD dynamics.- Global dynamics of tuberculosis models with density dependent demography.- Global stability in some SEIR epidemic models.

Series: The IMA Volumes in Mathematics and its Applications. VOL. 126


Chu, C.-H., University of London, UK;
Lau, A.T.-M., University of Alberta, Edmonton, AB, Canada

Harmonic Functions on Groups and Fourier Algebras

2002. VII, 100 pp. Softcover
3-540-43595-6

This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals.

Keywords: Harmonic function . Locally compact group . Jordan algebra . Fourier algebra . Semigroup . Mathematics subject classification ( 2000 ): 43A05, 43A20, 43A35 ,31C05, 45E10, 22D25, 46L70, 32M15 .

Series: Lecture Notes in Mathematics. VOL. 1782


Aguilar, M.A., Universidad Nacional Autonoma de Mexico, DF, Mexico; Gitler, S., University of Rochester, NY, USA; Prieto, C., Universidad Nacional Autonoma de Mexico, DF, Mexico

Algebraic Topology from a Homotopical Viewpoint

2002. Approx. 595 pp. 37 figs. Hardcover
0-387-95450-3

The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the text, no knowledge about category theory is expected from the reader. This book is intended for advanced undergraduates and graduate students with a basic knowledge of point set topology as well as group theory and can be used in a two semester course.
Marcelo Aguilar and Carlos Prieto are Professors at the Instituto de Matemticas, Universidad Nacional Autonoma de Mexico, and Samuel Gitler is a member of El Colegio Nacional and professor at the Centro de Investigacion y Estudios Avanzados del IPN.

Keywords: Algebraic topology

Contents: Introduction.- Basic Concepts and Notation.- Function Spaces.- Connectedness and Algebraic Invariants.- Homotopy Groups.- Homotopy Extension and Lifting Properties.- CW-Complexes Homology.- Homotopy Properties of CW-Complexes.- Cohomology Groups and Related Topics.- Vector Bundles.- K-Theory.- Adams Operations and Applications.- Relations Between Cohomology and Vector Bundles.- Cohomology Theories and Brown Representability.- Appendix A: Proof of the Dold-Thom Theorem.- Appendix B: Proof of the Bott Periodicity Theorem.- References.- Index.- Glossary.

Series: Universitext.