Five recent book reviews
The main topic of this comprehensive monograph is a detailed study of Lie algebras over an algebraically closed field of zero characteristic. The first ten chapters summarize basic results from commutative algebra, topology, sheaf theory, Jordan decomposition and basic facts on groups and their representations. The following seven chapters review required facts from algebraic geometry. The next part of the book (nine chapters) contains a detailed study of the relationship between algebraic groups and corresponding Lie algebras. The next two chapters contain the theory of representations of semisimple Lie algebras and the Chevalley theorem on invariants. Then the author introduces S-triples and describes properties of nilpotent orbits in semisimple Lie algebras. The final chapters are devoted to symmetric Lie algebras, semisimple symmetric Lie algebras, sheets of Lie algebras and a study of properties of the coadjoint representation. The main advantage of the book is a systematic treatment of the field, including detailed proofs.
The central object of this book is the nonlinear partial differential equation, ut – div (|u|m-1grad u); x Rn, t > 0, equipped with the initial value condition u = u0; x Rn, t = 0. The author is concerned with the smoothing effect of the equation and the time decay of positive solutions, i.e. whether the fact that u0 belongs to some function space X implies that the solution u(t) in time t > 0 is a member of some "better" function space Y and if it is possible to get estimates of the form |u(t)|Y < C(t, X, n, m,|u0|X). Well-posedness of the problem and some other substantial results such as the comparison theorem are mentioned in the preliminary part of the book and references are given for the proofs. Smoothing is carefully studied for all n N, m R (if m = 1 the classical results for the heat equation are reconstructed), X and Y being Lebesgue or weak Lebesgue spaces, which naturally appear as the correct spaces for studies of smoothing. It is very interesting that depending on m, n, X, Y, the solutions of the equation exhibit qualitatively very different properties, which are sometimes very surprising. The last chapter is devoted to the question of whether the results for the equation introduced at the beginning of the review also remain valid for the p-Laplacian equation. The book is very nicely written, well ordered and gives a rather complete overview of known results in the chosen field. At the beginning of each chapter there is a summary of the whole chapter with remarks of which sections of the chapter are essential for the following sections. The text is equipped with historical notes, remarks and a number of exercises. These properties make the book useful as a graduate textbook or a source of information for graduate students and researchers.
This book deals with the Euclidean geometry of plane and space, not only over the field of real numbers but over a general field, in particular over the field Fp, p 2 and prime. The reader will become acquainted with many theorems of elementary geometry such as Menelaus’ theorem, Ceva’s theorem, Ptolemy’s theorem and Stewart’s theorem. Feurbach’s nine point circle and the Euler line are presented. Conics are given by equations in linear coordinates and then parabolas, quadrolas and grammolas are defined. In three-dimensional space, Platonic solids are investigated. Besides linear coordinates the book also contains an explanation of polar and spherical coordinates, but not in the usual way (angles are not used). There is an interesting study of circles and other conics and their tangents in geometries over finite fields.
The book differs from current textbooks; it gives a new foundation for Euclidean geometry and trigonometry. The author suggests working with quadrance, i.e. using the square of distance and “spread” and “gross” instead of sine and cosine, which means the square of sine and the square of cosine in the case of real numbers. The author calls this technique “Rational Trigonometry” and adopts a purely algebraic approach, which is in his opinion a conceptually simpler framework for students. The reviewer does not share his optimistic point of view. For example, it is not clear in his modification of the cosine law which of two angles he is dealing with (acute or obtuse). The method used by the author has a lot of negative effects, nevertheless the book presents a very interesting exposition of many facts and theorems of elementary geometry.
There are not too many books devoted entirely to generating functions (GF). Even less of them have appeared in this century. One of them is this book. It is the third edition of a very popular book based on the author’s lectures at the University of Pennsylvania. GF are an indispensable tool for discrete mathematics. Roots of the use of GF are deep. Let us recall that the Binet formula for Fibonacci numbers was discovered using a generating function method by Moivre in 1718. At the end of the eighteenth century GF were installed as a basic method of probabilistic computations by Laplace.
The book gently introduces the reader to the way that GF are used for solving problems. It also underlines another role of GF: they form a natural bridge between two seemingly distant areas of continuous and discrete mathematics. Despite the fact that the author says that he tried only to communicate some of the main ideas on the subject, the book gives the beginner a surprisingly broad view of many different uses of GF. The author presents applications of GF ranging from set partitions, the money changing problem and graph theory to relations of GF to unimodality, convexity and proofs of some congruences. For about the first seventy pages basic notions and notations are introduced and then various problems are solved. Each of the five chapters contains exercises (all solutions are provided at the end of the book). The book is very readable and it should not be missing from any university library. In particular I would like to quote that the author nicely compares a GF to a “clothesline on which we hang up a sequence of numbers to display”.
This book is concerned with differentiability of nonlinear operators on Function Space. An extensive analysis of p-variation of functions is done with many applications to stochastic processes and nonlinear integral equations.
It is divided into twelve chapters on different topics on Functional calculus and an appendix on non-atomic measure spaces. Its many historical comments and remarks clarify the development of the theory.