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The standard starting point in cosmology is the cosmological principle; the assumption that the universe is spatially homogeneous and isotropic. After imposing this assumption, the only freedom left, as far as the geometry is concerned, is the choice of one out of three permissible spatial geometries, and one scalar function of time. Combining the cosmological principle with an appropriate description of the matter leads to the standard models. It is worth noting that these models yield quite a successful description of our universe. However, even though the universe may, or may not, be almost spatially homogeneous and isotropic, it is clear that the cosmological principle is not exactly satisfied. This leads to several questions. The most natural one concerns stability: given initial data corresponding to an expanding model of the standard type, do small perturbations give rise to solutions that are similar to the future? Another question concerns the shape of the universe: what are the restrictions if we only assume the universe to appear almost spatially homogeneous and isotropic to every observer? The main purpose of the book is to address these questions. However, to begin with, it is necessary to develop the general theory of the Cauchy problem for the Einstein-Vlasov equations. In order to to make the results accessible to researchers who are not mathematicians, but who are familiar with general relativity, the book contains an extensive prologue putting the results into a more general context.
This book is an updated version of the classic 1987 monograph "Spectral Theory and Differential Operators".The original book was a cutting edge account of the theory of bounded and closed linear operators in Banach and Hilbert spaces relevant to spectral problems involving differential equations. It is accessible to a graduate student as well as meeting the needs of seasoned researchers in mathematics and mathematical physics. This revised edition corrects various errors, and adds extensive notes to the end of each chapter which describe the considerable progress that has been made on the topic in the last 30 years.
"Basic Concepts in Physics: From the Cosmos to Quarks" is the outcome of the authors' long and varied teaching experience in different countries and for different audiences, and gives an accessible and eminently readable introduction to all the main ideas of modern physics. The book’s fresh approach, using a novel combination of historical and conceptual viewpoints, makes it ideal complementary reading to more standard textbooks. The first five chapters are devoted to classical physics, from planetary motion to special relativity, always keeping in mind its relevance to questions of contemporary interest. The next six chapters deal mainly with newer developments in physics, from quantum theory and general relativity to grand unified theories, and the book concludes by discussing the role of physics in living systems. A basic grounding in mathematics is required of the reader, but technicalities are avoided as far as possible; thus complex calculations are omitted so long as the essential ideas remain clear. The book is addressed to undergraduate and graduate students in physics and will also be appreciated by many professional physicists. It will likewise be of interest to students, researchers and teachers of other natural sciences, as well as to engineers, high-school teachers and the curious general reader, who will come to understand what physics is about and how it describes the different phenomena of Nature. Not only will readers of this book learn much about physics, they will also learn to love it.
The topic of special functions, normally presented as a mere collection of functions exhibiting particular properties, is treated from a fresh and unusual perspective in this book. The authors have based the special functions on the theory of second-order ordinary differential equations in the complex domain. Several physical applications are presented. Numerous tables and figures will help the reader find his way through the subject.
Complex hyperbolic geometry is a particularly rich area of study, enhanced by the confluence of several areas of research including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis. The boundary of complex hyperbolic geometry, known as spherical CR or Heisenberg geometry, is equally rich, and although there exist accounts of analysis in such spaces there is currently no account of their geometry. This book redresses the balance and provides an overview of the geometry of both the complex hyperbolic space and its boundary. Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.
Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier-type heat conduction, yet this book, which aims to provide a point of reference in the field, is the first monograph on the subject since the 1970s. It focuses on dynamic thermoelasticity governed by hyperbolic equations, and, in particular, on theories with one or two relaxation times. While the resulting field equations are linear partial differential ones, the complexity of the theories is due to the coupling of mechanical with thermal fields. The mathematical aspects of the theories - existence and uniqueness theorems, domain of influence theorems, convolutional variationalprinciples - as well as the methods for various initial/boundary value problems are explained and illustrated in detail and several applications of generalized thermoelasticity are reviewed.
Describing the applications found for the Wiles and Taylor technique, this book generalizes the deformation theoretic techniques of Wiles-Taylor to Hilbert modular forms (following Fujiwara's treatment), and also discusses applications found by the author.

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