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Abraham A. Ungar - One of the best experts on this subject based on the ideXlab platform.
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THE Hyperbolic SSQUARE AND MÖBIUS TRANSFORMATIONS
2014Co-Authors: Abraham A. UngarAbstract:Abstract. Professor Themistocles M. Rassias ’ special predilection and con-tribution to the study of Möbius transformations is well known. Möbius trans-formations of the open unit disc of the complex plane and, more generally, of the open unit ball of any real inner product space, give rise to Möbius addition in the ball. The latter, in turn, gives rise to Möbius gyrovector spaces that enable the Poincare ́ ball model of Hyperbolic Geometry to be approached by gyrovector spaces, in full analogy with the common vector space approach to the standard model of Euclidean Geometry. The purpose of this paper, dedi-cated to Professor Themistocles M. Rassias, is to employ the Möbius gyrovector spaces for the introduction of the Hyperbolic square in the Poincare ́ ball model of Hyperbolic Geometry. We will find that the Hyperbolic square is richer i
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Analytic Hyperbolic Geometry in N Dimensions: An Introduction
2014Co-Authors: Abraham A. UngarAbstract:List of Figures Preface Author's Biography Introduction Gyrovector Spaces in the Service of Abalytic Hyperbolic Geometry When Two Counterintuitive Theories Meet The Fascinating Rich Mathematical Life of Einstein's Velocity Addition Law Parts of the Book Einstein Gyrogroups and Gyrovector Spaces Einstein Gyrogroups Introduction Einstein Velocity Addition Einstein Addition for Computer Algebra Thomas Precession Angle Einstein Addition With Respect to Cartesian Coordinates Einstein Addition Vs. Vector Addition Gyrations Gyration Angles From Einstein Velocity Addition to Gyrogroups Gyrogroup Cooperation (Coaddition) First Gyrogroup Properties Elements of Gyrogroup Theory The Two Basic Gyrogroup Equations The Basic Gyrogroup Cancellation Laws Automorphisms and Gyroautomorphisms Gyrosemidirect Product Basic Gyration Properties An Advanced Gyrogroup Equation Gyrocommutative Gyrogroups Problems Einstein Gyrovector Spaces 65 The Abstract Gyrovector Space Einstein Gyrovector Spaces Einstein Addition and Differential Geometry Euclidean Lines Gyrolines - The Hyperbolic Lines Gyroangles - The Hyperbolic Angles Euclidean Isometries The Group of Euclidean Motions Gyroisometries - The Hyperbolic Isometries Gyromotions - The Motions of Hyperbolic Geometry Problems Relativistic Mass Meets Hyperbolic Geometry Lorentz Transformation and Einstein Addition Mass of Particle Systems Resultant Relativistically Invariant Mass Problems Mathematical Tools for Hyperbolic Geometry Barycentric and Gyrobarycentric Coordinates Barycentric Coordinates Segments Gyrobarycentric Coordinates Uniqueness of Gyrobarycentric Representations Gyrovector Gyroconvex Span Gyrosegments Triangle Centroid Gyromidpoint Gyroline Boundary points Gyrotriangle Gyrocentroid Gyrodistance in Gyrobarycentric Coordinates Gyrolines in Gyrobarycentric Coordinates Problems Gyroparallelograms and Gyroparallelotopes The Parallelogram Law Einstein Gyroparallelograms The Gyroparallelogram Law The Higher-Dimensional Gyroparallelotope Law Gyroparallelotopes Gyroparallelotope Gyrocentroid Gyroparallelotope Formal Definition and Theorem Low Dimensional Gyroparallelotopes Hyperbolic Plane Separation GPSA for the Einstein Gyroplane Problems Gyrotrigonometry Gyroangles Gyroangle - Angle Relationship The Law of Gyrocosines The SSS to AAA Conversion Law Inequalities for Gyrotriangles The AAA to SSS Conversion Law The Law of Sines/Gyrosines The Law of Gyrosines The ASA to SAS Conversion Law Gyrotriangle Defect Right Gyrotriangles Gyrotrigonometry Gyroangle of Parallelism Useful Gyrotriangle Gyrotrigonometric Identities A Determinantal Pattern Problems Hyperbolic Triangles and Circles Gyrotriangles and Gyrocircles Gyrocircles Gyrotriangle Circumgyrocenter Triangle Circumcenter Gyrotriangle Circumgyroradius Triangle Circumradius The Gyrocircle Through Three Points The Inscribed Gyroangle Theorem I The Inscribed Gyroangle Theorem II Gyrocircle Gyrotangent Gyrolines Semi-Gyrocircle Gyrotriangles Problems Gyrocircle Theorems The Gyrotangent-Gyrosecant Theorem The Intersecting Gyrosecants Theorem Gyrocircle Gyrobarycentric Representation Gyrocircle Interior and Exterior Points Circle Barycentric Representation Gyrocircle Gyroline Intersection Gyrocircle-Gyroline Tangency Points Gyrocircle Gyrotangent Gyrolength Circle-Line Tangency Points Circumgyrocevians Gyrodistances Related to the Gyrocevian A Gyrodistance Related to the Circumgyrocevian Circumgyrocevian Gyrolength The Intersecting Gyrochords Theorem Problems Hyperbolic Simplices, Hyperplanes and Hyperspheres in N Dimensions Gyrosimplices Gyrotetrahedron Circumgyrocenter Gyrotetrahedron Circumgyroradius Gyrosimplex Gyrocentroid Gamma Matrices Gyrosimplex Gyroaltitudes Gyrosimplex Circumhypergyrosphere The Gyrosimplex Constant Point to Gyrosimplex Gyrodistance Cramer's Rule Point to Gyrosimplex Perpendicular Projection Gyrosimplex In-Exgyrocenters and In-Exgyroradii Gyrotriangle In-Exgyrocenters Gyrosimplex Gyrosymmedian Problems Gyrosimplex Gyrovolume Gyrovolume Problems Hyperbolic Ellipses and Hyperbolas Gyroellipses and Gyrohyperbolas Gyroellipses - A Gyrobarycentric Representation Gyroellipses - Gyrotrigonometric Gyrobarycentric Representation Gyroellipse Major Vertices Gyroellipse Minor Vertices Canonical Gyroellipses Gyrobarycentric Representation of Canonical Gyroellipses Barycentric Representation of Canonical Ellipses Some Properties of Canonical Gyroellipses Canonical Gyroellipses and Ellipses Canonical Gyroellipse Equation A Gyrotrigonometric Constant of the Gyroellipse Ellipse Eccentricity Gyroellipse Gyroeccentricity Gyrohyperbolas - A Gyrobarycentric Representation Problems Thomas Precession Thomas Precession Introduction The Gyrotriangle Defect and Thomas Precession Thomas Precession Thomas Precession Matrix Thomas Precession Graphical Presentation Thomas Precession Angle Thomas Precession Frequency Thomas Precession and Boost Composition Thomas Precession Angle and Generating Angle have Opposite Signs Problems Bibliography Index
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gyrogroups the grouplike loops in the service of Hyperbolic Geometry and einstein s special theory of relativity
arXiv: Mathematical Physics, 2013Co-Authors: Abraham A. UngarAbstract:In this era of an increased interest in loop theory, the Einstein velocity addition law has fresh resonance. One of the most fascinating aspects of recent work in Einstein’s special theory of relativity is the emergence of special grouplike loops. The special grouplike loops, known as gyrocommutative gyrogroups, have thrust the Einstein velocity addition law, which previously has operated mostly in the shadows, into the spotlight. We will find that Einstein (Mobius) addition is a gyrocommutative gyrogroup operation that forms the setting for the Beltrami-Klein (Poincare) ball model of Hyperbolic Geometry just as the common vector addition is a commutative group operation that forms the setting for the standard model of Euclidean Geometry. The resulting analogies to which the grouplike loops give rise lead us to new results in (i) Hyperbolic Geometry; (ii) relativistic physics; and (iii) quantum information and computation.
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mobius transformation and einstein velocity addition in the Hyperbolic Geometry of bolyai and lobachevsky
arXiv: Mathematical Physics, 2012Co-Authors: Abraham A. UngarAbstract:In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, Mobius transformation and Einstein velocity addition meet in the Hyperbolic Geometry of Bolyai and Lobachevsky. It turns out that Mobius addition that is extracted from Mobius transformation of the complex disc and Einstein addition from his special theory of relativity enable the introduction of Cartesian coordinates and vector algebra as novel tools in the study of Hyperbolic Geometry.
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The Relativistic Hyperbolic Parallelogram Law
2012Co-Authors: Abraham A. UngarAbstract: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 “gyrolanguage” of this paper 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 paper share. The aim of this article is to employ recent developments in analytic Hyperbolic Geometry for the presentation of the relativistic Hyperbolic parallelogram law, and the relativistic particle aberration.
Victor Pambuccian - One of the best experts on this subject based on the ideXlab platform.
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the simplest axiom system for plane Hyperbolic Geometry revisited
Studia Logica, 2011Co-Authors: Victor PambuccianAbstract:Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane Hyperbolic Geometry (in Tarski's language L B?), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely of at most 4-variable statements.
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the complexity of plane Hyperbolic incidence Geometry is
Mathematical Logic Quarterly, 2005Co-Authors: Victor PambuccianAbstract:We show that plane Hyperbolic Geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized Hyperbolic planes, with arbitrary ordered fields as coordinate fields. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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the simplest axiom system for plane Hyperbolic Geometry
Studia Logica, 2004Co-Authors: Victor PambuccianAbstract:We provide a quantifier-free axiom system for plane Hyperbolic Geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean Geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane Hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean Geometry, expressed in the same language, all of whose axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane Hyperbolic Geometry in Tarski's language LB≡ which might be the simplest possible one in that language.
M Chaichian - One of the best experts on this subject based on the ideXlab platform.
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vector generation functions q spectral functions of Hyperbolic Geometry and vertex operators for quantum affine algebras
Journal of Mathematical Physics, 2017Co-Authors: A A Bytsenko, M Chaichian, R LunaAbstract:We investigate the concept of q-replicated argument in symmetric functions with its connection to spectral functions of Hyperbolic Geometry. This construction suffices for vector generation functions in the form of q-series and string theory. We hope that the mathematical side of the construction can be enriched by ideas coming from physics.
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s functions spectral functions of Hyperbolic Geometry and vertex operators with applications to structure for weyl and orthogonal group invariants
Nuclear Physics, 2016Co-Authors: A A Bytsenko, M ChaichianAbstract:In this paper we analyze the quantum homological invariants (the Poincare polynomials of the slN link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the procedure of the calculation of the Kovanov–Rozansky type homology, based on the Euler–Poincare formula can be appreciably simplified. We express the formal character of the irreducible tensor representation of the classical groups in terms of the symmetric and spectral functions of Hyperbolic Geometry. On the basis of Labastida–Marino–Ooguri–Vafa conjecture, we derive a representation of the Chern–Simons partition function in the form of an infinite product in terms of the Ruelle spectral functions (the cases of a knot, unknot, and links have been considered). We also derive an infinite-product formula for the orthogonal Chern–Simons partition functions and analyze the singularities and the symmetry properties of the infinite-product structures. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
Misha Verbitsky - One of the best experts on this subject based on the ideXlab platform.
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Hyperbolic Geometry of the ample cone of a hyperkahler manifold
Research in the Mathematical Sciences, 2016Co-Authors: Ekaterina Amerik, Misha VerbitskyAbstract:Let M be a compact hyperkahler manifold with maximal holonomy (IHS). The group $$H^2(M, {\mathbb {R}})$$ H 2 ( M , R ) is equipped with a quadratic form of signature $$(3, b_2-3)$$ ( 3 , b 2 - 3 ) , called Bogomolov–Beauville–Fujiki form. This form restricted to the rational Hodge lattice $$H^{1,1}(M,{\mathbb {Q}})$$ H 1 , 1 ( M , Q ) has signature (1, k). This gives a Hyperbolic Riemannian metric on the projectivization H of the positive cone in $$H^{1,1}(M,{\mathbb {Q}})$$ H 1 , 1 ( M , Q ) . Torelli theorem implies that the Hodge monodromy group $$\varGamma $$ Γ acts on H with finite covolume, giving a Hyperbolic orbifold $$X=H/\varGamma $$ X = H / Γ . We show that there are finitely many geodesic hypersurfaces, which cut X into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient $$P(M')/{\text {Aut}}(M')$$ P ( M ′ ) / Aut ( M ′ ) , where $$P(M')$$ P ( M ′ ) is the projectivization of the ample cone of a birational model $$M'$$ M ′ of M, and $${\text {Aut}}(M')$$ Aut ( M ′ ) the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkahler birational model of a simple hyperkahler manifold of Picard number at least 5 and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism (cf. Markman and Yoshioka in Int. Math. Res. Not. 2015(24), 13563–13574, 2015).
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Hyperbolic Geometry of the ample cone of a hyperkahler manifold
arXiv: Algebraic Geometry, 2015Co-Authors: Ekaterina Amerik, Misha VerbitskyAbstract:Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a Hyperbolic Riemannian metric on the projectivisation of the positive cone in $H^{1,1}(M,Q)$, denoted by $H$. Torelli theorem implies that the Hodge monodromy group $\Gamma$ acts on $H$ with finite covolume, giving a Hyperbolic orbifold $X=H/\Gamma$. We show that there are finitely many geodesic hypersurfaces which cut $X$ into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient $P(M')/Aut(M')$, where $P(M')$ is the projectivization of the ample cone of a birational model $M'$ of $M$, and $Aut(M')$ the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkahler birational model of a simple hyperkahler manifold of Picard number at least 5, and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism, originally deduced by Markman and Yoshioka from the Morrison-Kawamata cone conjecture.
Howard S Cohl - One of the best experts on this subject based on the ideXlab platform.
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fourier and gegenbauer expansions for a fundamental solution of the laplacian in the hyperboloid model of Hyperbolic Geometry
Journal of Physics A, 2012Co-Authors: Howard S Cohl, E G KalninsAbstract:Due to the isotropy of d-dimensional Hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hyperboloid model of Hyperbolic Geometry with R > 0 represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the Hyperbolic sine, finite summation expressions over Hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace’s equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace’s equation on the R-radius hyperboloid. For d ⩾ 2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace’s equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace’s equation is obtained through comparison with its corresponding Gegenbauer expansion.
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fourier and gegenbauer expansions for a fundamental solution of the laplacian in the hyperboloid model of Hyperbolic Geometry
arXiv: Mathematical Physics, 2011Co-Authors: Howard S Cohl, E G KalninsAbstract:Due to the isotropy $d$-dimensional Hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional Hyperbolic Geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental solution of Laplace's equation. For $d\ge 2$, we compute a Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant sectional curvature Riemannian manifold. In three-dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion.
