Metric Affine Geometry

Author: Ernst Snapper,Robert J. Troyer

Publisher: Elsevier

ISBN: 1483269337

Category: Mathematics

Page: 456

View: 3413

Metric Affine Geometry focuses on linear algebra, which is the source for the axiom systems of all affine and projective geometries, both metric and nonmetric. This book is organized into three chapters. Chapter 1 discusses nonmetric affine geometry, while Chapter 2 reviews inner products of vector spaces. The metric affine geometry is treated in Chapter 3. This text specifically discusses the concrete model for affine space, dilations in terms of coordinates, parallelograms, and theorem of Desargues. The inner products in terms of coordinates and similarities of affine spaces are also elaborated. The prerequisites for this publication are a course in linear algebra and an elementary course in modern algebra that includes the concepts of group, normal subgroup, and quotient group. This monograph is suitable for students and aspiring geometry high school teachers.

Wormholes, Warp Drives and Energy Conditions

Author: Francisco S. N. Lobo

Publisher: Springer

ISBN: 3319551825

Category: Science

Page: 303

View: 6745

Top researchers in the field of gravitation present the state-of-the-art topics outlined in this book, ranging from the stability of rotating wormholes solutions supported by ghost scalar fields, modified gravity applied to wormholes, the study of novel semi-classical and nonlinear energy conditions, to the applications of quantum effects and the superluminal version of the warp drive in modified spacetime. Based on Einstein's field equations, this cutting-edge research area explores the more far-fetched theoretical outcomes of General Relativity and relates them to quantum field theory. This includes quantum energy inequalities, flux energy conditions, and wormhole curvature, and sheds light on not just the theoretical physics but also on the possible applications to warp drives and time travel. This book extensively explores the physical properties and characteristics of these 'exotic spacetimes,' describing in detail the general relativistic geometries that generate closed timelike curves.

The Twelfth Marcel Grossmann Meeting

On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories(In 3 Volumes)

Author: Thibault Damour,Robert Jantzen,Remo Ruffini

Publisher: World Scientific

ISBN: 9814458031

Category: Science

Page: 2660

View: 9837

Marcel Grossmann Meetings are formed to further the development of General Relativity by promoting theoretical understanding in the fields of physics, mathematics, astronomy and astrophysics and to direct future technological, observational, and experimental efforts. In these meetings are discussed recent developments in classical and quantum gravity, general relativity and relativistic astrophysics, with major emphasis on mathematical foundations and physical predictions, with the main objective of gathering scientists from diverse backgrounds for deepening the understanding of spacetime structure and reviewing the status of test-experiments for Einstein's theory of gravitation. The range of topics is broad, going from the more abstract classical theory, quantum gravity and strings, to the more concrete relativistic astrophysics observations and modeling. The three volumes of the proceedings of MG12 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting includes 29 plenary talks stretched over 6 mornings, and 74 parallel sessions over 5 afternoons. Volume A contains plenary and review talks ranging from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theories, to precision tests of general relativity including progress towards the detection of gravitational waves, to relativistic astrophysics including such topics as gamma ray bursts, black hole physics both in our galaxy, in active galactic nuclei and in other galaxies, neutron stars, pulsar astrophysics, gravitational lensing effects, neutrino physics and ultra high energy cosmic rays. The rest of the volumes include parallel sessions on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, binary systems, radiative transfer, accretion disks, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, cosmic background radiation & observational cosmology, numerical relativity & algebraic computing, gravitational lensing, variable “constants” of nature, large scale structure, topology of the universe, brane-world cosmology, early universe models & cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, gamma ray burst modeling, supernovas, global structure, singularities, cosmic censorship, chaos, Einstein–Maxwell systems, inertial forces, gravitomagnetism, wormholes & time machines, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors & data analysis, precision gravitational measurements, history of relativity, quantum gravity & loop quantum gravity, Casimir effect, quantum cosmology, strings & branes, self-gravitating systems, gamma ray astronomy, cosmic rays, gamma ray bursts and quasars. Sample Chapter(s) Space-Time from the Spectral Point of View (467k) Contents: Space-Time from the Spectral Point of View (Ali H Chamseddine and Alain Connes)The Formation of Black Holes in General Relativity (Demetrios Christodoulou)Matching Conditions in Relativistic Astrophysics (Hernando Quevedo)Black Holes as a Source of Information (Juan Maldacena)Black Hole Microstate Counting and Its Macroscopic Counterpart (Ipsita Mandal and Ashoke Sen)Transplanckian String Collisions: An Update (Gabriele Veneziano)Ultraviolet Divergences and Scale-Dependent Gravitational Couplings (Herbert W Hamber)The Black Hole Stability Problem for Linear Scalar Perturbations (Mihalis Dafermos and Igor Rodnianski)The Global Network of Laser Interferometer Gravitational Wave Detectors (David H Reitze)Analytical Relativity of Black Holes (Thibault Damour)Detection of Gravitational Waves Using Pulsar Timing (Richard N Manchester)Relativistic Spin-Precession in Binary Pulsars (Michael Kramer)Supernovae and Gamma-Ray Bursts: 10 Years of Observations (Massimo Della Valle)Gamma-Ray Bursts as Relativistic Objects (Tsvi Piran)Fundamental Physics from Black Holes, Neutron Stars and Gamma-Ray Bursts (Remo Ruffini)The Fascinating TeV Sky (Felix Aharonian)Galaxy Clusters and Their Central Supermassive Black Holes: Case of M87 (Eugene Churazov, Sergey Sazonov, Rashid Sunyaev, William Forman, Christine Jones and Hans Böhringer)Intergalactic Shock Fronts (Maxim Markevitch)Studies of Dark Energy with X-Ray Observations of Galaxy Clusters (Alexey Vikhlinin)and other papers Keywords:General Relativity;Gravitation;Astrophysics;Quantum Gravity;Particle Physics;Cosmology;Theoretical Physics

General Relativity and Gravitation

Invited Papers and Discussion Reports of the 10th International Conference on General Relativity and Gravitation, Padua, July 3–8, 1983

Author: B. Bertotti,F. de Felice,Alessandro Pascolini

Publisher: Springer Science & Business Media

ISBN: 9789027718198

Category: Science

Page: 536

View: 6600

The Tenth International Conference on General Relativity and Gravitation (GR10) was held from July 3 to July 8, 1983, in Padova, Italy. These Conferences take place every three years, under the auspices of the International Society on General Relativity and Gravitation, with the purpose of assessing the current research in the field, critically discussing the prog ress made and disclosing the points of paramount im portance which deserve further investigations. The Conference was attended by about 750 scientists active in the various subfields in which the current research on gravitation and general relativity is ar ticulated, and more than 450 communications were sub mitted. In order to fully exploit this great occur rence of experience and creative capacity, and to pro mote individual contributions to the collective know ledge, the Conference was given a structure of work shops on the most active topics and of general sessions in which the Conference was addressed by invited speakers on general reviews or recent major advance ments of the field. The individual communications were collected in a two-volume publication made available to the participants upon their arrival and widely distributed to Scientific Institutions and Research Centres.

Affine Differential Geometry

Geometry of Affine Immersions

Author: Katsumi Nomizu,Nomizu Katsumi,Takeshi Sasaki

Publisher: Cambridge University Press

ISBN: 9780521441773

Category: Mathematics

Page: 263

View: 4650

This is a self-contained and systematic account of affine differential geometry from a contemporary viewpoint, not only covering the classical theory, but also introducing the modern developments that have happened over the last decade. In order both to cover as much as possible and to keep the text of a reasonable size, the authors have concentrated on the significant features of the subject and their relationship and application to such areas as Riemannian, Euclidean, Lorentzian and projective differential geometry. In so doing, they also provide a modern introduction to the last. Some of the important geometric surfaces considered are illustrated by computer graphics, making this a physically and mathematically attractive book for all researchers in differential geometry, and for mathematical physicists seeking a quick entry into the subject.

Projective Geometry and Projective Metrics

Author: Herbert Busemann,Paul J. Kelly

Publisher: Courier Corporation

ISBN: 0486154696

Category: Mathematics

Page: 352

View: 3569

This text examines the 3 classical geometries and their relationship to general geometric structures, with particular focus on affine geometry, projective metrics, non-Euclidean geometry, and spatial geometry. 1953 edition.

Encyclopedia of Distances

Author: Michel Marie Deza,Elena Deza

Publisher: Springer Science & Business Media

ISBN: 364200234X

Category: Mathematics

Page: 590

View: 6023

Distance metrics and distances have become an essential tool in many areas of pure and applied Mathematics, and this encyclopedia is the first one to treat the subject in full. The book appears just as research intensifies into metric spaces and especially, distance design for applications. These distances are particularly crucial, for example, in computational biology, image analysis, speech recognition, and information retrieval. Here, an assessment of the practical questions arising during selection of a "good'' distance function has been left aside in favor of a comprehensive listing of the main available distances, a useful tool for the distance design community. This reader-friendly reference offers both independent introductions and definitions, while at the same time making cross-referencing easy through hyperlink-like boldfaced references to original definitions. This high-quality publication is a mix of reference resource and coffee-table book.

Lie Groups

A Problem Oriented Introduction Via Matrix Groups

Author: Harriet Pollatsek

Publisher: MAA

ISBN: 9780883857595

Category: Mathematics

Page: 177

View: 2670

The work of the Norwegian mathematician Sophus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires extensive mathematical prerequisites beyond the reach of the typical undergraduate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and important material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world.Lie Groups is an active learning text that can be used by students with a range of backgrounds and interests. The material is developed through 200 carefully chosen problems. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers.

Orthogonality and Spacetime Geometry

Author: Robert Goldblatt

Publisher: Springer Science & Business Media

ISBN: 1468463454

Category: Mathematics

Page: 194

View: 2823

This book examines the geometrical notion of orthogonality, and shows how to use it as the primitive concept on which to base a metric structure in affine geometry. The subject has a long history, and an extensive literature, but whatever novelty there may be in the study presented here comes from its focus on geometries hav ing lines that are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern four-dimensional special-relativistic spacetime (Minkowskian geometry), and its var ious sub-geometries, and these will be prominent throughout. But the project is intended as an exercise in the foundations of geome try that does not presume a knowledge of physics, and so, in order to provide the appropriate intuitive background, an initial chapter has been included that gives a description of the different types of line (timelike, spacelike, lightlike) that occur in spacetime, and the physical meaning of the orthogonality relations that hold between them. The coordinatisation of affine spaces makes use of constructions from projective geometry, including standard results about the ma trix represent ability of certain projective transformations (involu tions, polarities). I have tried to make the work sufficiently self contained that it may be used as the basis for a course at the ad vanced undergraduate level, assuming only an elementary knowledge of linear and abstract algebra.

Applications of Affine and Weyl Geometry

Author: Eduardo García-Río,Peter Gilkey,Stana Nikčević,Ramón Vázquez-Lorenzo

Publisher: Morgan & Claypool Publishers

ISBN: 1608457605

Category: Mathematics

Page: 168

View: 5266

Pseudo-Riemannian geometry is, to a large extent, the study of the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric structure. There are, however, other affine connections which arise in different contexts, such as conformal geometry, contact structures, Weyl structures, and almost Hermitian geometry. In this book, we reverse this point of view and instead associate an auxiliary pseudo-Riemannian structure of neutral signature to certain affine connections and use this correspondence to study both geometries. We examine Walker structures, Riemannian extensions, and Kähler--Weyl geometry from this viewpoint. This book is intended to be accessible to mathematicians who are not expert in the subject and to students with a basic grounding in differential geometry. Consequently, the first chapter contains a comprehensive introduction to the basic results and definitions we shall need---proofs are included of many of these results to make it as self-contained as possible. Para-complex geometry plays an important role throughout the book and consequently is treated carefully in various chapters, as is the representation theory underlying various results. It is a feature of this book that, rather than as regarding para-complex geometry as an adjunct to complex geometry, instead, we shall often introduce the para-complex concepts first and only later pass to the complex setting. The second and third chapters are devoted to the study of various kinds of Riemannian extensions that associate to an affine structure on a manifold a corresponding metric of neutral signature on its cotangent bundle. These play a role in various questions involving the spectral geometry of the curvature operator and homogeneous connections on surfaces. The fourth chapter deals with Kähler--Weyl geometry, which lies, in a certain sense, midway between affine geometry and Kähler geometry. Another feature of the book is that we have tried wherever possible to find the original references in the subject for possible historical interest. Thus, we have cited the seminal papers of Levi-Civita, Ricci, Schouten, and Weyl, to name but a few exemplars. We have also given different proofs of various results than those that are given in the literature, to take advantage of the unified treatment of the area given herein.

Mathematically Speaking

A Dictionary of Quotations

Author: C.C. Gaither,Alma E Cavazos-Gaither

Publisher: CRC Press

ISBN: 9781420050301

Category: Mathematics

Page: 484

View: 7005

For the first time, a book has brought together in one easily accessible form the best expressed thoughts that are especially illuminating and pertinent to the discipline of mathematics. Mathematically Speaking: A Dictionary of Quotations provides profound, wise, and witty quotes from the most famous to the unknown. You may not find all the quoted "jewels" that exist, but you will definitely a great many of them here. The extensive author and subject indexes provide you with the perfect tools for locating quotations for practical use or pleasure, and you will soon enjoy discovering what others have said on topics ranging from addition to zero. This book will be a handy reference for the mathematician or scientific reader and the wider public interested in who has said what on mathematics.

Conformal Geometry of Surfaces in S4 and Quaternions

Author: Francis E. Burstall,Dirk Ferus,Katrin Leschke,Franz Pedit,Ulrich Pinkall

Publisher: Springer Science & Business Media

ISBN: 9783540430087

Category: Mathematics

Page: 86

View: 7565

The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

Introduction to Mathematical Thinking

The Formation of Concepts in Modern Mathematics

Author: Friedrich Waismann

Publisher: Courier Corporation

ISBN: 0486167429

Category: Mathematics

Page: 272

View: 3964

Examinations of arithmetic, geometry, and theory of integers; rational and natural numbers; complete induction; limit and point of accumulation; remarkable curves; complex and hypercomplex numbers; more. Includes 27 figures. 1959 edition.

Machine Proofs in Geometry

Automated Production of Readable Proofs for Geometry Theorems

Author: Shang-Ching Chou,Xiao-Shan Gao,Jingzhong Zhang

Publisher: World Scientific

ISBN: 9789810215842

Category: Mathematics

Page: 461

View: 2460

This book reports recent major advances in automated reasoning in geometry. The authors have developed a method and implemented a computer program which, for the first time, produces short and readable proofs for hundreds of geometry theorems.The book begins with chapters introducing the method at an elementary level, which are accessible to high school students; latter chapters concentrate on the main theme: the algorithms and computer implementation of the method.This book brings researchers in artificial intelligence, computer science and mathematics to a new research frontier of automated geometry reasoning. In addition, it can be used as a supplementary geometry textbook for students, teachers and geometers. By presenting a systematic way of proving geometry theorems, it makes the learning and teaching of geometry easier and may change the way of geometry education.

Applicable Differential Geometry

Author: Mike Crampin,M. Crampin,N. Crampin,F. A. E. Pirani,Pirani F A E

Publisher: Cambridge University Press

ISBN: 9780521231909

Category: Mathematics

Page: 394

View: 5504

An introduction to geometrical topics used in applied mathematics and theoretical physics.

Emergent Evolution

Qualitative Novelty and the Levels of Reality

Author: David Blitz

Publisher: Springer Science & Business Media

ISBN: 9401580421

Category: Science

Page: 241

View: 6399

Emergent evolution combines three separate but related claims, whose background, origin, and development I trace in this work: firstly, that evolution is a universal process of change, one which is productive of qualitative novelties; secondly, that qualitative novelty is the emergence in a system of a property not possessed by any of its parts; and thirdly, that reality can be analyzed into levels, each consisting of systems characterized by significant emergent properties. In part one I consider the background to emergence in the 19th century discussion of the philosophy of evolution among its leading exponents in England - Charles Darwin, Herbert Spencer, T. H. Huxley, Alfred Russel Wallace, and G. J. Romanes. Unlike the scientific aspect of the debate which aimed to determine the factors and causal mechanism of biological evolution, this aspect of the debate centered on more general problems which form what I call the "philosophical framework for evolutionary theory." This considers the status of continuity and discontinuity in evolution, the role of qualitative and quantitative factors in change, the relation between the organic and the inorganic, the relation between the natural and the supernatural, the mind-body problem, and the scope of evolution, including its extension to ethics and morals.