Five recent book reviews
The object of this book is to study elliptic boundary value problems with data, which may have singularities (e.g. mixed boundary value problems, boundary value problems with transmission property and singular crack problems, a prominent example being the Zaremba problem for the Laplace equation with mixed Dirichlet and Neumann boundary conditions). The authors’ aim is to study such problems based on the general calculus of operators on a manifold with edges or conical singularities and boundary. General theory then allows the construction of parametrix for considered problems and the proof of the regularity of the solutions. In the last chapter, an applied approach with operator algebras and symbolic structures on manifolds with singularities is discussed, together with motivations and branches of research over the past years and new challenges and open problems for the future. The mathematicians and physicists interested in elliptic differential equations with singularities will definitely appreciate the unified approach presented in the book. The authors also address the text to advanced students and specialists working in the field of analysis on manifolds with geometric singularities, applications of index theory and spectral theory, operator algebras with symbolic structures, quantization and asymptotic analysis.
The presented book is, as are all V. Arnold's books, full of geometric insight. Its aim is to cover most basic parts of the subject, in particular the Cauchy and Neumann problems for the classical linear equations of mathematical physics. In the preface to the second Russian edition (which is included in the book) one reads: “The author...has attempted to make students of mathematics with minimal knowledge...acquainted with a kaleidoscope of fundamental ideas of mathematics and physics. Instead of the principle of maximal generality, which is usual in mathematical books, the author has attempted to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation; only then can the developed method be extended to more complicated cases....” The book follows a series of lectures, which were delivered to the third year students in the Mathematical College of the Independent University of Moscow during the fall semester of 1994/95. The examination problems at the end of the book form an essential part of it.
This book is the fourth extended edition of the handbook designed to provide researchers, teachers and students with a comprehensive reference in the area of parametric and nonparametric statistical methods. The author focuses on hypothesis testing. The structure of the handbook is the following: introduction, 40 tests (with a separate chapter for each test) and appendix containing tables. The third edition has only 32 tests. The fourth one contains eight additional chapters (tests) concerning multivariate statistical methods and some medical statistics. Their titles are: multivariate regression, Hotelling’s T2, multivariate analysis of variance, multivariate analysis of covariance, discriminant function analysis, canonical correlational, logistic regression, and principal components analysis and factor analysis. The chapters (tests) all have the same structure: basic information on the particular problem, illustrative example, formulation of the null and alternative hypotheses, test computation (statistical software package SPSS serves as the basic software), interpretation of the test results, related tests, additional discussion, additional examples and references. The same structure in each chapter ensures that chapters are self-contained. The handbook can serve as a reference book for classical statistical procedures concerning hypothesis testing.
The main theme of this book is a study of properties of a certain class of infinite-dimensional groups. It is very much an interdisciplinary topic having relations to many parts of mathematics, including representations theory, descriptive set theory, infinite dimensional Lie groups, topological transformation groups, operator algebras, topological and ergodic dynamics and Ramsey theory. An approach presented in the book describes relations among dynamical properties of these groups, asymptotic geometric analysis, Ramsey type theorems in combinatorics and concentration properties. A presentation of a circle of problems relating the mentioned topics is based on a detailed study of representative examples (e.g. the unitary group of a Hilbert space, the group of homeomorphisms of a manifold, the infinite symmetric group, the group of transformations of measure spaces) and it covers many recent results in the field.
This book brings a modern exposition of complex topological K-theory. It is designed for beginners in the field of K-theory, primarily for graduate students. The exposition is quite self-contained and the author has reduced the number of prerequisites for reading to a minimum. On the other hand, the text is not very long, consisting of only 208 pages. This was possible because the author has limited the exposition to the most central and classical part of K-theory, namely the above mentioned complex topological K-theory. No other parts of K-theory are included. Nevertheless, the reader can find here hints for reading about other parts of K-theory. Vector bundles are often studied here in terms of idempotents and invertible matrices over Banach algebras of continuous complex-valued functions. We should also remark that the last quarter of the book deals with characteristic classes of vector bundles in the Chern-Weil style. Each chapter is followed by exercises. Generally we can say that the presentation is very nice and the book can be strongly recommended.