Series: Progress in Mathematical Physics , Vol. 59
2010, Approx. 225 p., Hardcover
ISBN: 978-3-0346-0303-4
Due: April 2010
The aim of this book is to establish the precise mathematical framework in which linearization stability of Einstein equation with matter makes sense, and once this is achieved, to find conditions for this type of stability when a Robertson-Walker model for the universe is considered. The concept of linearization stability arises when one wonders whether the solutions of a linearized equation do really approximate solutions of the corresponding true equation. This requires a new definition of linearization stability adapted to Einstein equation, a bit different from the classical one. However, this new definition can not be applied directly to the Einstein equation because energy conditions ties both deformations of the metric and of the stress-energy tensor. Therefore, one must look for a background where the variables representing the geometry and the energy-matter are independent. This representation is given by a well-posed Cauchy problem for the Einsteinfs field equations.
Preface // I Pseudo-Riemannian Manifolds: I.1 Connections / I.2 Firsts results on pseudo-Riemannian manifolds / I.3 Laplacians / I.4 Sobolev spaces of tensors on Riemannian manifolds / I.5 Lorentzian manifolds // II Introduction to Relativity: II.1 Classical fluid mechanics / II.2 Kinematics of the special relativity / II.3 Dynamics of special relativity / II.4 General relativity / II.5 Cosmological models / II. 6 Appendix: a theorem in affine geometry // III. Approximation of Einstein's Equation by the Wave Equation: III.1 Perturbations of Ricci tensor / III.2 Einstein's equation for small perturbations of the Minkowski metric / III.3 Action on metrics of diffeomorphisms close to identity / III.4 Continuing the calculation of Section 2 / III.5 Comparison with the classical gravitation // IV. Cauchy Problem for Einstein's Equation with Matter: IV.1 1. Differential operators in an open set of Rn+1 / IV.2 Differential operators in vector bundles / IV.3 Harmonic maps / IV.4 Admissible classes of stress-energy tensors / IV.5 Differential operator associated to Einstein's equation / IV.6 Constraint equations / IV.7 Hyperbolic reduction / IV.8 Fundamental theorem / IV.9 An example: the stress-energy tensor of holonomic media / IV.10 The Cauchy problem in the vacuum // V. Stability by Linearization of Einstein's Equation, General Concepts: V.1 Classical concept of stability by linearization of Einstein's equation in the vacuum / V.2 A new concept of stability by linearization of Einstein's equation in the presence of matter / V.3 How to apply the definition of stability by linearization of Einstein's equation in the presence of matter / V.4 Change of notation / V.5 Technical details concerning the map f / V.6 Tangent linear map of f // VI. General Results on Stability by Linearization when the Submanifold M of V is Compact: IV.1 1. Adjoint of D(g,k) f / VI.2 Results by A. Fischer and J. E. Marsden / VI.3 A result by V. Moncrief / VI.4 Appendix: general results on elliptic operators in compact manifolds // VII. Stability by Linearization of Einstein's Equation at Minkowski's Initial Metric: VII.1 A further expression of D(g,k) f / VII.2 The relation between Euclidean Laplacian and stability by linearization at the initial Minkowski's metric / VII.3 Some proofs on topological isomorphisms in Rn / VII.4 Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Deser's result / VII.5 The Euclidean asymptotic case: generalization of a result by Y. Choquet-Bruhat, A. Fischer and J. E. Marsden // VIII. Stability by Linearization of Einstein's Equation in Robertson-Walker Cosmological Models: VIII.1 Euclidean model / VIII.2 Hyperbolic model / VIII.3 Sobolev spaces and hyperbolic Laplacian / VIII.4 Spherical model / VIII.5 Universes that are not simply connected // References
Series: Operator Theory: Advances and Applications , Vol. 203
2010, Approx. 500 p., Hardcover
ISBN: 978-3-0346-0160-3
Due: April 2010
This is the second volume of a collection of original and review articles on recent advances and new directions in a multifaceted and interconnected area of mathematics and its applications. It encompasses many topics in theoretical developments in operator theory and its diverse applications in applied mathematics, physics, engineering, and other disciplines. The purpose is to bring in one volume many important original results of cutting edge research as well as authoritative review of recent achievements, challenges, and future directions in the area of operator theory and its applications.
Researchers in operator theory and related fields
integral equation
operator theory
systems theory
2010, Approx. 1500 p., Hardcover
ISBN: 978-3-642-04897-5
Due: May 2010
This is the first attempt in Statistics to engage the most recognized international authors - including the most prominent authors from many developing countries - to write relatively brief papers on topics from their respective fields of expertise.
The goal of this book is multifaceted:
a) to help revive Statistics education in many parts in the world where it is in crisis. For the first time authors from many developing countries have an opportunity to write together with the most prominent world authorities. The editor has spent several years searching for the most reputable statisticians all over the world. The international contributors are either presidents of their local statistical societies, heads of the Statistics department at the main university, or the most distinguished statisticians in their respective countries.
b) to enable any non-statistician to obtain quick and yet comprehensive and highly accessible information on specific statistical terms, methods and applications.
c) to enable researchers, managers and practicioners to refresh their knowledge of Statistics, especially in certain controversial fields.
d) to revive interest in statistics among students, since they will see its usefulness and relevance in nearly all disciplines.
Overview of Statistical Sciences - Past, Recent trends and Advances, Present and Future perspective.
Series: Springer Monographs in Mathematics
2010, Approx. 260 p., Hardcover
ISBN: 978-1-4419-1731-7
Due: May 2010
Examines algebraic number theory and the theory of semisimple algebras
Covers classification over an algebraic number field and classification over the ring of algebraic integers
Discusses local class field theory
This book is divided in two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic diophantine equations. The first topic is further divided into two types of theories: classification over an algebraic number field and classification over the ring of algebraic integers.
To ensure the book would be concise, the author proved basic theorems in local class field theory only in some special cases, and the Hilbert reciprocity law only over the rational number field. However, the author stated the main theorems with an arbitrary number field as the base field, so the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references.
Goro Shimura is Professor Emeritus at Princeton University.
Graduate students; professionals
Preface.- Notation and Terminology.- The Quadratic Reciprocity Law.- Arithmetic in an Algebraic Number Field.- Various Basic Theorems.- Algebras Over a Field.- Quadratic Forms Over a Field.- Deeper Arithmetic of Quadratic Forms.- Quadratic Diophantine Equations.- References.- Index.-
Series: Trends in Mathematics
2010, Approx. 370 p., Hardcover
ISBN: 978-3-0346-0285-3
Due: May 2010
This volume consists of refereed research and expository articles by both plenary and other speakers at the International Conference on Algebra and Applications held at Ohio University in June 2008, to honor S.K. Jain on his 70th birthday. The articles are on a wide variety of areas in classical ring theory and module theory, such as rings satisfying polynomial identities, rings of quotients, group rings, homological algebra, injectivity and its generalizations, etc. Included are also applications of ring theory to problems in coding theory and in linear algebra.
Postgraduates and researchers in algebra
Foreward.- Biographical sketch of S.K. Jain.- Applications of Cogalois theory to elementary field arithmetic.- On big lattices of classes of R-modules defined by closure properties.- Reversible and duo group rings.- Principally quasi-Baer ring hulls.- Strongly prime ideals of near-rings of continuous functions.- Elements of minimal prime ideals in general rings.- The Camps-Dicks theorem.- Applications of the stone duality in the theory of precompact Boolean rings.- Over rings and functors.- On some classes of repeated-root constacyclic codes of length a power of 2 over Galois Rings.- Couniformly presented modules and dualities.- Semiclassical limits of quantized coordinate rings.- On unit-central rings.- Symplectic modules and Von Neumann regular matrices over commutative rings.- Extensions of simple modules and the converse of Schur's lemma.- Report on exchange rings.- Filtrations in semisimple Lie algebras, III.- On the blowing-up rings, Arf rings and type sequences.- A guide to supertropical algebra.- Projective modules, idempotent ideals and intersection theorems.- On ef-extending modules and rings with chain conditions.- On clean group rings.