Heyde, C.C., Columbia University, New York, NY, USA
Seneta, E., The University of Sydney, NSW, Australia (Eds.)

Statisticians of the Centuries

2001. XII, 500 pp.
Hardcover 0-387-95329-9
Softcover 0-387-95283-7

Statisticians of the Centuries aims to demonstrate the achievements of statistics to a broad audience, and to commemorate the work of celebrated statisticians. This is done through short biographies that put the statistical work in its historical and sociological context, emphasizing contributions to science and society in the broadest terms rather than narrow technical achievement. The discipline is treated from its earliest times and only individuals born prior to the 20th Century are included. The volume arose through the initiative of the International Statistical Institute (ISI), the principal representative association for international statistics (founded in 1885). Extensive consultations within the statistical community, and with prominent members of ISI in particular, led to the names of the 104 individuals who are included in the volume. The biographies were contributed by 73 authors from across the world.
The editors are the well-known statisticians Chris Heyde and Eugene Seneta. Chris Heyde is Professor of Statistics at both Columbia University in New York and the Australian National University in Canberra. Eugene Seneta is Professor of Mathematical Statistics at the University of Sydney and a Member of the ISI. His historical writings focus on 19th Century France and the Russian Empire. Both editors are Fellows of the Australian Academy of Science and have, at various times, been awarded the Pitman Medal of the Statistical Society of Australia for their distinguished research contributions.

Contents: From the contents: Probability Prior to Pascal.- 17th Century: Pierre de Fermat (1601-1665). John Graunt (1620-1674). Blaise Pascal (1623-1662). Christian Huygens (1629-1695). Caspar Neumann (1648-1715). Jakob Bernoulli (1655-1705). John Arbuthnot (1667-1735). Abraham de Moivre (1667-1754). Pierre Remond de Montmort (1678-1719). Nicolaus Bernoulli (1687-1759). Daniel Bernoulli (1700-1782).- 18th Century: Thomas Bayes (1701-1761). Johann Peter Sussmilch (1707-1767). Georges-Louis Leclerc, Comte de Buffon (1707-1788). Rogerius Josephus Boscovich (1711-1787). D'Alembert (1717-1783). Marquis de Condorcet (1743-1794). Pierre-Simon Marquis de Laplace (1749-1827). Adrien-Marie Legendre (1752-1833). William Playfair (1759-1823). Thomas Robert Malthus (1766-1834). Sir Frederick Morton Eden (1765-1809). Carl Friederich Gauss (1777-1855). Simeon-Denis Poisson (1781-1840). Adolphe Quetelet (1796-1874). Irenee-Jules Bienayme (1796-1878). Stefano Franscini (1796-1857).- 19th Century: Gustav Theodor Fechner (1801-1887). Anton Meyer (1801-1857). Antoine Augustin Cournot (1801-1877). Augustus De Morgn (1806-1871). William Farr (1807-1883). George Boole (1815-1864). Florence Nightingale (1820-1910).

Kerler, T., Ohio State University, Columbus, OH, USA
Lyubashenko, V.V., Institute of Mathematics, Kiev, Ukraine

Non-Semisimple Topological Quantum Field Theories
for 3-Manifolds with Corners

2001. VI, 379 pp. Softcover
3-540-42416-4

This book presents the (to date) most general approach to combinatorial constructions of topological quantum field theories (TQFTs) in three dimensions. The authors describe extended TQFTs as double functors between two naturally defined double categories: one of topological nature, made of 3-manifolds with corners, the other of algebraic nature, made of linear categories, functors, vector spaces and maps. Atiyah's conventional notion of TQFTs as well as the notion of modular functor from axiomatic conformal field theory are unified in this concept. A large class of such extended modular catergory is constructed, assigning a double functor to every abelian modular category, which does not have to be semisimple.

Contents: Introduction and summary of results.- The double category of framed, relative 3-cobordisms.- Tangle-categories and presentation of cobordisms.- Isomorphism between tangle and cobordism categories.- Monoidal categories and monoidal 2-categories.- Coends and construction of Hopf algebras.- Construction of TQFT-Double Functors.- Generalization of modular functor.- A: From quantum field theory of axiomatics.- B: Double categories and double functors.- C: Thick tangles.

Series: Lecture Notes in Mathematics. VOL. 1765

Hennion, H., University of Rennes, France/ Herve, L., University of Rennes, France

Limit Theorems for Markov Chains and Stochastic Properties
of Dynamical Systems by Quasi-Compactness

2001. VIII, 145 pp. Softcover
3-540-42415-6

This book shows how techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel, may be used to obtain limit theorems for Markov chains or to describe stochastic properties of dynamical systems.
A general framework for this method is given and then applied to treat several specific cases. An essential element of this work is the description of the peripheral spectra of a quasi-compact Markov kernel and of its Fourier-Laplace perturbations. This is first done in the ergodic but non-mixing case. This work is extended by the second author to the non-ergodic case.
The only prerequisites for this book are a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis.

Contents: General facts about the method, purpose of the paper.- The central limit theorems for Markov chains.- Quasi-compact operators of diagonal type and perturbations.- First properties of Fourier kernels, application.- Peripheral eigenvalues of Fourier kernels.- Proofs of theorems A, B, C.- Renewal theorem for Markov chains (theorem D).- Large deviations for Markov chains (theorem E).- Ergodic properties for Markov chains.- Stochastic properties of dynamical systems.- Expanding maps.- Proofs of some statements in probability theory.- Functional analysis results on quasi-compactness.- Generalization to the non-ergodic case (by L. Herve).

Series: Lecture Notes in Mathematics. VOL. 1766

Harville, D.A., IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

Matrix Algebra: Exercises and Solutions

2001. Approx. 295 pp. Softcover
0-387-95318-3
Publication date: October 2001

This book contains over 300 exercises and solutions that together cover a wide variety of topics in matrix algebra. They can be used for independent study or in creating a challenging and stimulating environment that encourages active engagement in the learning process. The requisite background is some previous exposure to matrix algebra of the kind obtained in a first course. The exercises are those from an earlier book by the same author entitled Matrix Algebra From a Statistician's Perspective. They have been restated (as necessary) to stand alone, and the book includes extensive and detailed summaries of all relevant terminology and notation. The coverage includes topics of special interest and relevance in statistics and related disciplines, as well as standard topics. The overlap with exercises available from other sources is relatively small. This collection of exercises and their solutions will be a useful reference for students and researchers in matrix algebra. It will be of interest to mathematicians and statisticians.

Bourbaki, N.

Elements of Mathematics
Topological Vector Spaces. Chapters 1-5

1st ed. 1987. 2nd printing 2001. VII, 364 pp. Softcover
3-540-42338-9

This book is the English translation of the new and expanded version of Bourbaki's "Espaces vectoriels topologiques". Chapters 1 and 2 contain the general definitions and a thorough study of convexity; they are organized around the basic theorems (closed graph, Hahn-Banach and Krein-Milman) and differ only by minor changes from those of older editions. Chapters 3 and 4 have been substantially rewritten; the order of exposition has been modified and a number of notions and results have been inserted, whose importance emerged in the last twenty years. Bornological spaces are introduced together with barrelled ones; almost every space of practical use today belongs in fact to these two categories, which have good stability properties, and in which the basic theorems (the Banach-Steinhaus theorem for example) apply. Recent results on the completion of a dual space (Grothendieck theorem) or on the continuity of linear maps with measurable graphs are treated. An important place is devoted to properties of Frechet spaces and of their dual spaces, to compactness criteria (Eberlein-Smullian) and to the existence of fixed points for groups of linear maps. Chapter 5 is devoted to Hilbert spaces; it includes in particular the spectral decomposition of Hilbert-Schmidt operators and the construction of symmetric and exterior powers of Hilbert spaces, whose applications are of growing importance. At the end, an appendix restates the principal results obtained in the case of normed spaces, providing convenient references.
The book addresses all mathematicians and physicists interested in a structural presentation of contemporary mathematics.

Keywords: Functional analysis, operators MSC ( 2000 ): 46Axx, 46-02, 46Exx

Contents: Topological Vector Spaces over a Valued Division Ring.- Convex Sets and Locally Convex Spaces.- Spaces of Continuous Linear Mappings.- Duality in Topological Vector Spaces.- Hilbertian Spaces (Elementary Theory).