Analytic hyperbolic geometry: mathematical foundations and applications
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| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New Jersey
World Scientific
©2005
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| Schlagworte: | |
| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | Includes bibliographical references (pages 445-456) and index This 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. 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 book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and nongyrocommutative, 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 (Mobius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Mobius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented Introduction -- Gyrogroups -- Gyrocommutative gyrogroups -- Gyrogroup extension -- Gyrovectors and cogyrovectors -- Gyrovector spaces -- Rudiments of differential geometry -- Gyrotrigonometry -- Bloch gyrovector of quantum computation -- Special theory of relativity: the analytic hyperbolic geometric viewpoint |
| Beschreibung: | 1 Online-Ressource (xvii, 463 pages) |
| ISBN: | 1281899224 9781281899224 9789812564573 9789812703279 9812564578 9812703276 |
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| dewey-full | 516.9 |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516.9 |
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| id | DE-604.BV043091989 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:17:10Z |
| institution | BVB |
| isbn | 1281899224 9781281899224 9789812564573 9789812703279 9812564578 9812703276 |
| language | English |
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| physical | 1 Online-Ressource (xvii, 463 pages) |
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| spelling | Ungar, Abraham A. Verfasser aut Analytic hyperbolic geometry mathematical foundations and applications Abraham A. Ungar New Jersey World Scientific ©2005 1 Online-Ressource (xvii, 463 pages) txt rdacontent c rdamedia cr rdacarrier Includes bibliographical references (pages 445-456) and index This 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. 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 book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and nongyrocommutative, 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 (Mobius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Mobius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented Introduction -- Gyrogroups -- Gyrocommutative gyrogroups -- Gyrogroup extension -- Gyrovectors and cogyrovectors -- Gyrovector spaces -- Rudiments of differential geometry -- Gyrotrigonometry -- Bloch gyrovector of quantum computation -- Special theory of relativity: the analytic hyperbolic geometric viewpoint Geometry, Hyperbolic Vector algebra Géométrie hyperbolique / Manuels d'enseignement supérieur Algèbre vectorielle / Manuels d'enseignement supérieur MATHEMATICS / Geometry / Non-Euclidean bisacsh Geometry, Hyperbolic Textbooks Vector algebra Textbooks Hyperbolische Geometrie (DE-588)4161041-6 gnd rswk-swf Relativitätstheorie (DE-588)4049363-5 gnd rswk-swf Hyperbolischer Raum (DE-588)4161046-5 gnd rswk-swf Analytische Geometrie (DE-588)4001867-2 gnd rswk-swf Analytische Geometrie (DE-588)4001867-2 s Hyperbolischer Raum (DE-588)4161046-5 s Relativitätstheorie (DE-588)4049363-5 s 1\p DE-604 Hyperbolische Geometrie (DE-588)4161041-6 s 2\p DE-604 http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=174551 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ungar, Abraham A. Analytic hyperbolic geometry mathematical foundations and applications Geometry, Hyperbolic Vector algebra Géométrie hyperbolique / Manuels d'enseignement supérieur Algèbre vectorielle / Manuels d'enseignement supérieur MATHEMATICS / Geometry / Non-Euclidean bisacsh Geometry, Hyperbolic Textbooks Vector algebra Textbooks Hyperbolische Geometrie (DE-588)4161041-6 gnd Relativitätstheorie (DE-588)4049363-5 gnd Hyperbolischer Raum (DE-588)4161046-5 gnd Analytische Geometrie (DE-588)4001867-2 gnd |
| subject_GND | (DE-588)4161041-6 (DE-588)4049363-5 (DE-588)4161046-5 (DE-588)4001867-2 |
| title | Analytic hyperbolic geometry mathematical foundations and applications |
| title_auth | Analytic hyperbolic geometry mathematical foundations and applications |
| title_exact_search | Analytic hyperbolic geometry mathematical foundations and applications |
| title_full | Analytic hyperbolic geometry mathematical foundations and applications Abraham A. Ungar |
| title_fullStr | Analytic hyperbolic geometry mathematical foundations and applications Abraham A. Ungar |
| title_full_unstemmed | Analytic hyperbolic geometry mathematical foundations and applications Abraham A. Ungar |
| title_short | Analytic hyperbolic geometry |
| title_sort | analytic hyperbolic geometry mathematical foundations and applications |
| title_sub | mathematical foundations and applications |
| topic | Geometry, Hyperbolic Vector algebra Géométrie hyperbolique / Manuels d'enseignement supérieur Algèbre vectorielle / Manuels d'enseignement supérieur MATHEMATICS / Geometry / Non-Euclidean bisacsh Geometry, Hyperbolic Textbooks Vector algebra Textbooks Hyperbolische Geometrie (DE-588)4161041-6 gnd Relativitätstheorie (DE-588)4049363-5 gnd Hyperbolischer Raum (DE-588)4161046-5 gnd Analytische Geometrie (DE-588)4001867-2 gnd |
| topic_facet | Geometry, Hyperbolic Vector algebra Géométrie hyperbolique / Manuels d'enseignement supérieur Algèbre vectorielle / Manuels d'enseignement supérieur MATHEMATICS / Geometry / Non-Euclidean Geometry, Hyperbolic Textbooks Vector algebra Textbooks Hyperbolische Geometrie Relativitätstheorie Hyperbolischer Raum Analytische Geometrie |
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