Five recent book reviews
This volume contains contributions arising from lectures and related discussions held at the meeting on ‘Differential Equations: Inverse and Direct Problems’, which took place at Cortona, 21st-25th June 2004. Almost all of fourteen contributions contain original results; they do not just survey and explain results already published elsewhere. They cover a wide scope of up-to-date topics from the field of differential equations. Thus there are papers dealing with abstract differential equations in Banach spaces, linear and nonlinear semigroup theory, direct and inverse problems for non-degenerate and degenerate equations, differential equations with Wentzell boundary conditions and a deep study of equations of mathematical physics (e.g. from superconductivity and phase transition problems). The book will be an interesting and stimulating read for research workers in the field.
This text is based on a series of graduate lectures given by V. Markovic at the University of Warwick. The book is intended as an introduction to quasiconformal mappings in dimension 2 and Teichmüller theory. The reader is assumed to have a background in complex analysis and to be familiar with Riemann surfaces and hyperbolic geometry. The book starts with analytic and geometric definitions of quasiconformal mappings and continues with a study of their basic analytical properties. The connection with quasisymmetric maps and the Beltrami equation is explained in the following chapters, before a presentation of the definitions and basic properties of holomorphic motions and Teichmüller spaces. Extremal quasiconformal mappings are studied together with conditions that guarantee the unique extremality of a mapping. The book is well-written and illustrated with a number of examples. Some proofs are omitted but references to other sources are always given.
The main topic of this book is an analysis and numerical simulation of a mathematical model consisting of magnetohydrodynamic equations describing the motion of two immiscible incompressible Newtonian fluids. The prototypical industrial application of this model is a simulation of reduction cells for the production of aluminium. However, the authors do not restrict themselves only to this specific application and cover a wide area of related topics. The book starts with a brief explanation of principles leading to the governing equations of magnetohydrodynamics. The resulting model consists of the incompressible Navier-Stokes equations with the Lorentz force on the right-hand side, a parabolic equation for the magnetic induction depending on the fluid velocity and appropriate boundary conditions. Chapters 2 and 3 are devoted to the analysis of the one-fluid problem and to its numerical solution by the finite element method. The emphasis is on the coupling between hydrodynamic and electromagnetic phenomena. The material is rather standard but very useful for readers not familiar with the respective techniques. Moreover, it allows the authors to concentrate only on key issues, with a special emphasis on nonlinear phenomena.
The next two chapters are the central chapters of the book and are mainly based on original research by the authors. These chapters deal with the mathematical analysis and numerical solution of multifluid magnetohydrodynamic problems, where the basic additional difficulty is a geometric nonlinearity due to the presence of free interfaces separating the immiscible fluids. The discretization is based on the popular arbitrary Lagrangian Eulerian method but other approaches are also briefly mentioned. The final chapter is devoted to the industrial application that motivated the preceding five chapters and it serves as an illustration of the efficiency of the approach presented in the book. The comprehensive text is well written and it contains many examples of numerical simulations and many references to relevant literature. It is intended for mathematicians, engineers and physicists and it will also be valuable for experts in mathematical and numerical analysis of magnetohydrodynamics as well as for those who want to learn the basic issues in this fascinating area.
This book contains a wealth of information about Vito Volterra and his life. A large part of the personal history of Volterra and his family is based on the letters sent to his mother and wife; numerous excerpts presented in the book have been translated into English. The scientific career of Volterra's is traced, starting with his university studies and early professorship at Pisa, followed by a shorter period in Turin and finally his appointment at the University of Rome. Volterra was also a passionate traveller and we learn about his impressions of people and places throughout Europe and America. Volterra's scientific achievements are discussed in a reprint of E. Whittaker's obituary from 1941. The book provides a very readable account of the Italian scientific community and Italian history and politics in the 19th and 20th century.
This is a nicely readable textbook on differential geometry. It offers an outstanding, comprehensive presentation of both theoretical and computational aspects. All 27 chapters are accompanied by Mathematica code in the form of Mathematica notebooks in the appendix (the code can be downloaded from the publisher’s web site). This approach enables the reader to better understand how to define and compute standard geometric functions and how to construct new curves and surfaces from existing ones. Moreover, work with Mathematica notebooks may serve as a natural way of acquiring the basic and intermediate knowledge of Mathematica by example. There are hundreds of illustrations that help the reader visualize the concepts. Throughout the book the reader will find biographical information about 75 scientists, most of them with small portraits. It is a nicely written book, strongly recommended to all with an interest in differential geometry, its computational aspects and related fields.