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OverviewThis is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the resulting ""gyrolanguage"" of the book one attaches the prefix ""gyro"" to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book.The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and non-gyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Moebius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Moebius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented. Full Product DetailsAuthor: Abraham Albert Ungar (North Dakota State Univ, Usa)Publisher: World Scientific Publishing Co Pte Ltd Imprint: World Scientific Publishing Co Pte Ltd Dimensions: Width: 15.40cm , Height: 3.10cm , Length: 21.10cm Weight: 0.821kg ISBN: 9789812564573ISBN 10: 9812564578 Pages: 484 Publication Date: 07 September 2005 Audience: College/higher education , Professional and scholarly , Postgraduate, Research & Scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: Awaiting stock  The supplier is currently out of stock of this item. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out for you. Table of ContentsReviewsThis new book by Ungar is very well-written, with plenty of references and explanatory pictures. Almost all chapters include exercises which ensure that the book will reach a large audience from undergraduate and graduate students to researchers and academics in different areas of mathematics and mathematical physics. In this book, the author sets out his improved gyrotheory, capturing the curiosity of the reader with discernment, elegance and simplicity. Mathematical Reviews This book under review provides an efficient algebraic formalism for studying the hyperbolic geometry of Bolyai and Lobachevsky, which underlies Einstein special relativity ... It is of interest both to mathematicians, working in the field of geometry, and the physicists specialized in relativity or quantum computation theory ... It is recommended to graduate students and researchers interested in the interrelations among non-associative algebra, hyperbolic and differential geometry, Einstein relativity theory and the quantum computation theory. Journal of Geometry and Symmetry in Physics This book represents an exposition of the author's single-handed creation, over the past 17 years, of an algebraic language in which both hyperbolic geometry and special relativity find an aesthetically pleasing formulation, very much like Euclidean geometry and Newtonian mechanics find them in the language of vector spaces. Zentralblatt MATH """This new book by Ungar is very well-written, with plenty of references and explanatory pictures. Almost all chapters include exercises which ensure that the book will reach a large audience from undergraduate and graduate students to researchers and academics in different areas of mathematics and mathematical physics. In this book, the author sets out his improved gyrotheory, capturing the curiosity of the reader with discernment, elegance and simplicity."" Mathematical Reviews ""This book under review provides an efficient algebraic formalism for studying the hyperbolic geometry of Bolyai and Lobachevsky, which underlies Einstein special relativity ... It is of interest both to mathematicians, working in the field of geometry, and the physicists specialized in relativity or quantum computation theory ... It is recommended to graduate students and researchers interested in the interrelations among non-associative algebra, hyperbolic and differential geometry, Einstein relativity theory and the quantum computation theory."" Journal of Geometry and Symmetry in Physics ""This book represents an exposition of the author's single-handed creation, over the past 17 years, of an algebraic language in which both hyperbolic geometry and special relativity find an aesthetically pleasing formulation, very much like Euclidean geometry and Newtonian mechanics find them in the language of vector spaces."" Zentralblatt MATH" Author InformationTab Content 6Author Website:Countries AvailableAll regions |
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