Five recent book reviews
Fifteen papers presented at the workshop Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications (held in May 2003 at the Rutgers University) are collected in this book. The recent development of data depth and its applications are presented by leading researchers in the field. Contributions may be divided into three main groups. Statistical theory and applications of data depth comprise the first part of the book. Among the main topics the reader will find are a general study of depth functions, tests based on data depth, development of zonoid, simplicial and spherical depth, classification and discrimination based on the depth function, regression depth and depth for functional data. The second section of presentations covers computational problems related to data depth. The main problem is to find fast algorithms for computing the depth, a process which is typically very slow, in particular in higher dimensions. The last section of contributions is devoted to geometric aspects of statistical data depth. The book can be recommended to researchers or students interested in multivariate statistical analysis and its applications.
This monograph is based on ideas by A. Dempster (1967) and G. Shafer (1976). They developed probabilistic argumentation systems combining logic and probability with a unified theory of inference under uncertainty. It has numerous applications in various fields (artificial intelligence, diagnostics and reliability, among others). The book applies this new theory to statistical inference. In particular, a new principle, called by the authors assumption-based inference in statistics reasoning, is introduced and worked out. Both the Bayesian approach and the Fisher fiducial probabilities are seen as special cases of a more general theory. The authors show the possibility of a new approach to discrete probability models, continuous probability models and linear models. Basic principles are introduced and explained in part concerning simple discrete probability models. A number of illustrative examples help in understanding the theory. The book is intended to open a new view on statistical inference.
This is the first book collecting the English translations of the historical mathematical texts from the five ancient and medieval non-Western mathematical cultures. It is the biggest sourcebook containing the newest fruit of historical research and that is why the book can replace older sources for the history of mathematics. The book is divided into five sections: Egypt (written by A. Imhausen), Mesopotamia (by E. Robson), China (by J. W. Dauben), India (by K. Plofker) and Islam (by J. L. Berggen). Each chapter starts with a detailed introduction that gives an overview of each culture, historical and social contexts, events and heritages, discovery of writing, origin of mathematics, numerical systems, etc. Then a deeper analysis of mathematical sources is presented. The authors of each part are renowned experts in their fields; they have carefully selected key mathematical texts, they provide new translations and in many cases they give new interpretations, commentaries and notes. They explain the mathematical skills and knowledge of the ancient and medieval non-Western mathematicians and they analyse the role of mathematics in these civilizations and its impact on their developments. The book can be recommended to students, teachers, historians and mathematicians as well as anyone who wants to understand the depth and power of ancient mathematics and who wants to learn about mathematical ideas and their use in the daily life of non-Western cultures.
This book contains 24 lectures by a group of authors on the topic of local cohomology. The notion of local cohomology was originally introduced by A. Grothendieck in the realm of algebraic geometry. Nowadays, the subject has a lot of relations to various other fields of mathematics. The book contains a revised set of lectures notes on the theme of local cohomology presented at the summer school organised at Snowbird, Utah, in 2005. Quite a few first lectures in the series cover the basic prerequisites needed later from geometry, sheaf theory and homological algebra (the Krull dimension of a ring, the dimension of an algebraic set and the dimension of a module; sheaves and Čech cohomology; complexes, resolutions and derived functors, and projective dimension; gradings, filtrations and Gröbner bases; and the Koszul complex and depth). It makes it possible to introduce the local cohomology functors and its first properties (depth and cohomological dimension). A further two lectures discuss properties of Cohen-Macaulay and Gorenstein rings. Further lectures treat relations to commutative algebra, algebraic geometry, topology and combinatorics. A few lectures also cover computational aspects (algorithms related to Gröbner bases, Weyl algebras and D-modules).
This book is devoted to the analytic theory of ordinary differential equations with complex time. Methods for the investigation of local and global properties of solutions are deeply examined so that the current state of typical problems like the 16th Hilbert problem (the number of limit cycles of polynomial planar vector fields) and the Riemann-Hilbert problem (i.e. the 21st Hilbert problem on the existence of a linear system with prescribed monodromy group and position of all singularities) can be presented. The first two chapters are devoted to an analysis of singular points of holomorphic vector fields by holomorphic normal forms. A local analysis of singularities is based on the notion of algebraic and analytic solvability. Chapter 3 deals with the local and global theory of linear systems. This chapter also contains positive and negative results on the solvability of the Riemann-Hilbert problem. Chapter 4 is concerned with analytic classification of resonant singularities. The main working tool here is an almost complex structure and quasiconformal maps. In the last chapter, the global theory of polynomial differential equations on the real and complex plane is investigated. Tools for the study of limit cycles (the 16th Hilbert problem) near poly-cycles or those which bifurcate from non-isolated periodic orbits (Abelian integrals) are presented. It is also shown how generic properties of complex foliations differ from real ones.
The book is carefully written and important notions are motivated and explained in detail. All sections end with exercises and problems often lying at the frontier of current research. A good knowledge of various parts of analysis (e.g. complex analysis in several variables) and topology is required. The book is aimed at senior graduate students in differential equations. Professionals can also find here an initiation into the present-day level of research and interesting applications of algebraic geometry to differential equations.