Academic literature on the topic 'Hyperbolic Geometry'

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Journal articles on the topic "Hyperbolic Geometry"

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Reynolds, William F. "Hyperbolic Geometry on a Hyperboloid." American Mathematical Monthly 100, no. 5 (May 1993): 442. http://dx.doi.org/10.2307/2324297.

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Reynolds, William F. "Hyperbolic Geometry on a Hyperboloid." American Mathematical Monthly 100, no. 5 (May 1993): 442–55. http://dx.doi.org/10.1080/00029890.1993.11990430.

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Kostin, A. V., and I. Kh Sabitov. "Smarandache Theorem in Hyperbolic Geometry." Zurnal matematiceskoj fiziki, analiza, geometrii 10, no. 2 (June 25, 2014): 221–32. http://dx.doi.org/10.15407/mag10.02.221.

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Li, Hongbo. "Hyperbolic geometry with geometric algebra." Chinese Science Bulletin 42, no. 3 (February 1997): 262–63. http://dx.doi.org/10.1007/bf02882454.

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Lord, Nick, and B. Iversen. "Hyperbolic Geometry." Mathematical Gazette 79, no. 486 (November 1995): 622. http://dx.doi.org/10.2307/3618121.

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Baseilhac, Stephane, and Riccardo Benedetti. "Quantum hyperbolic geometry." Algebraic & Geometric Topology 7, no. 2 (June 20, 2007): 845–917. http://dx.doi.org/10.2140/agt.2007.7.845.

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Ungar, Abraham A. "Ptolemy’s Theorem in the Relativistic Model of Analytic Hyperbolic Geometry." Symmetry 15, no. 3 (March 4, 2023): 649. http://dx.doi.org/10.3390/sym15030649.

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Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of hyperbolic geometry is based on the Einstein addition of relativistically admissible velocities and, as such, it coincides with the well-known Beltrami–Klein ball model of hyperbolic geometry. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the relativistic model of analytic hyperbolic geometry gives rise.
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Eppstein, David, and Michael T. Goodrich. "Succinct Greedy Geometric Routing Using Hyperbolic Geometry." IEEE Transactions on Computers 60, no. 11 (November 2011): 1571–80. http://dx.doi.org/10.1109/tc.2010.257.

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Eskew, Russell Clark. "Suspending the Principle of Relativity." Applied Physics Research 8, no. 2 (March 29, 2016): 82. http://dx.doi.org/10.5539/apr.v8n2p82.

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A unique hyperbolic geometry paradigm requires suspending the Relativistic principle that absolute velocity is unmeasurable. The idea that of two observers each sees the same constant velocity, therefore there is no absolute velocity, is true only because Relativity uses a particular Lorentz geometry. Our mathematical geometry constructs circle and hyperbola vectors with hyperbolic terms in an original formulation of complex numbers. We use a point on a hyperbola as a frame of reference. A theory of time is given. The physical laws of motion by Galileo, Newton and Einstein are forged using the absolute velocity and the precondition to electromagnetic velocity. The field of real and fictitious force accelerations is established. We utilize Galilean Invariance to measure absolute velocity. An experiment exemplifies the math from the Earth’s frame of reference. But Relativity is based on local Lorentz geometry. We discover a possible dark energy and gravitational accelerations and a geometry of gravitational collapse.
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Lohkamp, Joachim. "Hyperbolic Unfoldings of Minimal Hypersurfaces." Analysis and Geometry in Metric Spaces 6, no. 1 (August 1, 2018): 96–128. http://dx.doi.org/10.1515/agms-2018-0006.

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Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.
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Dissertations / Theses on the topic "Hyperbolic Geometry"

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Markham, Sarah. "Hypercomplex hyperbolic geometry." Thesis, Durham University, 2003. http://etheses.dur.ac.uk/3698/.

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The rank one symmetric spaces of non-compact type are the real, complex, quaternionic and octonionic hyperbolic spaces. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still in its infancy. The purpose of this thesis is to investigate the conditions for discrete group action in quaternionic and octonionic hyperbolic 2-spaces and their geometric consequences, in the octonionic case, in terms of lower bounds on the volumes of non-compact manifolds. We will also explore the eigenvalue problem for the 3 x 3 octonionic matrices germane to the Jordan algebra model of the octonionic hyperbolic plane. In Chapters One and Two we concentrate on discreteness conditions in quaternionic hyperbolic 2-space. In Chapter One we develop a quaternionic Jørgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. In Chapter Two we give a generalisation of Shimizu's Lemma to groups of isometries of quaternionic hyperbolic 2-space containing a screw-parabolic element. In Chapter Three we present the Jordan algebra model of the octonionic hyperbolic plane and develop a generalisation of Shimizu's Lemma to groups of isometries of octonionic hyperbolic 2-space containing a parabolic map. We use this result to determine estimates of lower bounds on the volumes of non-compact closed octonionic 2-manifolds. In Chapter Four we construct an octonionic Jørgensen's inequality for non-elementary groups of isometries of octonionic hyperbolic 2-space generated by two elements, one of which is loxodromic. In Chapter Five we solve the real eigenvalue problem Xv = λv, for the 3 x 3 ɸ-Hermitian matrices, X, of the Jordan algebra model of the octonionic hyperbolic plane. Finally, in Chapter Six we consider the embedding of collars about real geodesies in complex hyperbohc 2-space, quaternionic hyperbolic 2-space and octonionic hyperbolic 2-space.
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Marshall, T. H. (Timothy Hamilton). "Hyperbolic Geometry and Reflection Groups." Thesis, University of Auckland, 1994. http://hdl.handle.net/2292/2140.

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The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ≤ k ≤ n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length γ we then derive power series for the volume, both in u = sinγ/2, and in γ itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and pεC). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all π/2, and the remaining dihedral angles are all sub-multiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g≥2, which we conjecture to be of least volume among such manifolds.
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Murray, Marilee Anne. "Hyperbolic Geometry and Coxeter Groups." Bowling Green State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1343040882.

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Walker, Mairi. "Continued fractions and hyperbolic geometry." Thesis, Open University, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.700134.

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This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. This intuitive approach gives insight into the theory of continued fractions that is not so easy to obtain from traditional algebraic methods. Using it, we provide a more extensive study of both Rosen continued fractions and even-integer continued fractions than many previous works, yielding new results, and revisiting classical theorems. We also study two types of complex continued fractions, namely Gaussian integer continued fractions and Bianchi continued fractions. As well as providing a more elegant and simple theory of continued fractions, our approach leads to a natural generalisation of continued fractions that has not been explored, before.
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Thorgeirsson, Sverrir. "Hyperbolic geometry: history, models, and axioms." Thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503.

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Gharamti, Moustafa. "Supersymmetry and geometry of hyperbolic monopoles." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/10479.

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This thesis studies the geometry of hyperbolic monopoles using supersymmetry in four and six dimensions. On the one hand, we show that starting with a four dimensional supersymmetric Yang-Mills theory provides the necessary information to study the geometry of the complex moduli space of hyperbolic monopoles. On the other hand, we require to start with a six dimensional supersymmetric Yang-Mills theory to study the geometry of the real moduli space of hyperbolic monopoles. In chapter two, we construct an off-shell supersymmetric Yang-Mills-Higgs theory with complex fields on three-dimensional hyperbolic space starting from an on-shell supersymmetric Yang-Mills theory on four-dimensional Euclidean space. We, then, show that hyperbolic monopoles coincide precisely with the configurations that preserve one half of the supersymmetry. In chapter three, we explore the geometry of the moduli space of hyperbolic monopoles using the low energy linearization of the field equations. We find that the complexified tangent bundle to the hyperbolic moduli space has a 2-sphere worth of integrable structures that act complex linearly and behave like unit imaginary quaternions. Moreover, we show that these complex structures are parallel with respect to the Obata connection, which implies that the geometry of the complexified moduli space of hyperbolic monopoles is hypercomplex. We also show, as a requirement of analysing the geometry, that there is a one-to-one correspondence between the number of solutions of the linearized Bogomol’nyi equation on hyperbolic space and the number of solutions of the Dirac equation in the presence of hyperbolic monopole. In chapter four and five, we shift the focus to supersymmetric Yang-Mills theories in six dimensional Minkowskian spacetime. Via dimensional reduction we construct a supersymmetric Yang-Mills Higgs theory on R3 with real fields which we then promote to H3. Under certain supersymmetric constraints, we show that hyperbolic monopoles configurations of this theory preserve, again, one half of the supersymmetry. Then, through investigating the geometry of the moduli space we showthat the moduli space is described by real coordinate functions (zero modes), and we construct two sets of 2-sphere of real complex structures that act linearly on the tangent bundle of the moduli space, but don’t behave like unit quaternions. This result coincides with the result of Bielawski and Schwachhöfer, who called this new type of geometry pluricomplex geometry. Finally, we show that in the limiting case, when the radius of curvature H3 is set to infinity, the geometry becomes hyperkähler which is the geometry of the moduli space of Euclidian monopoles.
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Rippy, Scott Randall. "Applications of hyperbolic geometry in physics." CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/1099.

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Naeve, Trent Phillip. "Conics in the hyperbolic plane." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3075.

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An affine transformation such as T(P)=Q is a locus of an affine conic. Any affine conic can be produced from this incidence construction. The affine type of conic (ellipse, parabola, hyperbola) is determined by the invariants of T, the determinant and trace of its linear part. The purpose of this thesis is to obtain a corresponding classification in the hyperbolic plane of conics defined by this construction.
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Bowen, Lewis Phylip. "Density in hyperbolic spaces." Access restricted to users with UT Austin EID Full text (PDF) from UMI/Dissertation Abstracts International, 2002. http://wwwlib.umi.com/cr/utexas/fullcit?p3077409.

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Saratchandran, Hemanth. "Four dimensional hyperbolic link complements via Kirby calculus." Thesis, University of Oxford, 2015. http://ora.ox.ac.uk/objects/uuid:ba72ee75-c22f-4800-a38c-76e5cf411ad9.

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The primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S4 and a smooth standard S2 x S2. We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples.
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Books on the topic "Hyperbolic Geometry"

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Anderson, James W. Hyperbolic Geometry. London: Springer London, 1999. http://dx.doi.org/10.1007/978-1-4471-3987-4.

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Anderson, James W. Hyperbolic geometry. London: Springer, 1999.

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Ramsay, Arlan. Introduction to hyperbolic geometry. New York: Springer-Verlag, 1995.

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Benedetti, Riccardo, and Carlo Petronio. Lectures on Hyperbolic Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/978-3-642-58158-8.

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Ramsay, Arlan, and Robert D. Richtmyer. Introduction to Hyperbolic Geometry. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4757-5585-5.

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Stillwell, John. Sources of hyperbolic geometry. Providence, R.I: American Mathematical Society, 1996.

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Benedetti, R. Lectures on hyperbolic geometry. Berlin: Springer-Verlag, 1992.

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Boskoff, Wladimir-Georges. Hyperbolic geometry and barbilian spaces. Palm Harbor, FL, USA: Hadronic Press, 1996.

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Ratcliffe, John G. Foundations of hyperbolic manifolds. New York: Springer-Verlag, 1994.

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1864-, Schlesinger Ludwig, ed. Libellus post saeculum quam Ioannes Bolyai: De Bolya anno MDCCCII A.D. XVIII kalendas ianuarias Claudiopoli natus est ad celebrandam memoriam eius immortalem. Claudiopoli: [typis Societatis franklinianae budapestinensis], 1991.

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Book chapters on the topic "Hyperbolic Geometry"

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Millman, Richard S., and George D. Parker. "Hyperbolic Geometry." In Geometry, 196–223. New York, NY: Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-4436-3_8.

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Trudeau, Richard J. "Hyperbolic Geometry." In The Non-Euclidean Revolution, 173–231. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-2102-9_6.

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Pressley, Andrew. "Hyperbolic geometry." In Elementary Differential Geometry, 269–304. London: Springer London, 2010. http://dx.doi.org/10.1007/978-1-84882-891-9_11.

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Ratcliffe, John G. "Hyperbolic Geometry." In Foundations of Hyperbolic Manifolds, 56–104. New York, NY: Springer New York, 1994. http://dx.doi.org/10.1007/978-1-4757-4013-4_3.

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Richter-Gebert, Jürgen. "Hyperbolic Geometry." In Perspectives on Projective Geometry, 483–503. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-17286-1_25.

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Li, Hongbo. "Hyperbolic Geometry." In Geometric Algebra with Applications in Science and Engineering, 61–85. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0159-5_4.

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Hariri, Parisa, Riku Klén, and Matti Vuorinen. "Hyperbolic Geometry." In Springer Monographs in Mathematics, 49–66. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-32068-3_4.

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Ratcliffe, John G. "Hyperbolic Geometry." In Foundations of Hyperbolic Manifolds, 52–96. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-31597-9_3.

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Beals, Richard, and Roderick S. C. Wong. "Hyperbolic geometry." In Explorations in Complex Functions, 33–40. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-54533-8_3.

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Agricola, Ilka, and Thomas Friedrich. "Hyperbolic geometry." In Elementary Geometry, 167–207. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/stml/043/04.

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Conference papers on the topic "Hyperbolic Geometry"

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Costa, S. I. R., S. A. Santos, and J. E. Strapasson. "Fisher information matrix and hyperbolic geometry." In IEEE Information Theory Workshop, 2005. IEEE, 2005. http://dx.doi.org/10.1109/itw.2005.1531851.

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Sleator, D. D., R. E. Tarjan, and W. P. Thurston. "Rotation distance, triangulations, and hyperbolic geometry." In the eighteenth annual ACM symposium. New York, New York, USA: ACM Press, 1986. http://dx.doi.org/10.1145/12130.12143.

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You, Di, Thanh Tran, and Kyumin Lee. "Multi-Behavior Recommendation with Hyperbolic Geometry." In 2022 IEEE International Conference on Big Data (Big Data). IEEE, 2022. http://dx.doi.org/10.1109/bigdata55660.2022.10020616.

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Zhu, Yudong, Di Zhou, Jinghui Xiao, Xin Jiang, Xiao Chen, and Qun Liu. "HyperText: Endowing FastText with Hyperbolic Geometry." In Findings of the Association for Computational Linguistics: EMNLP 2020. Stroudsburg, PA, USA: Association for Computational Linguistics, 2020. http://dx.doi.org/10.18653/v1/2020.findings-emnlp.104.

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Saaty, Thomas L. "Reciprocal Comparisons, Inversion, and Hyperbolic Geometry (From Psychology to Geometry)." In The International Symposium on the Analytic Hierarchy Process. Creative Decisions Foundation, 1994. http://dx.doi.org/10.13033/isahp.y1994.040.

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SERIES, CAROLINE. "WHY IS THERE HYPERBOLIC GEOMETRY IN DYNAMICS?" In Proceedings of the Tenth General Meeting. WORLD SCIENTIFIC, 2003. http://dx.doi.org/10.1142/9789812704276_0010.

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MATSUEDA, HIROAKI. "SCALING OF ENTANGLEMENT ENTROPY AND HYPERBOLIC GEOMETRY." In Symposium on Interface between Quantum Information and Statistical Physics. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814425285_0007.

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Shahid, Simra, Tanay Anand, Nikitha Srikanth, Sumit Bhatia, Balaji Krishnamurthy, and Nikaash Puri. "HyHTM: Hyperbolic Geometry-based Hierarchical Topic Model." In Findings of the Association for Computational Linguistics: ACL 2023. Stroudsburg, PA, USA: Association for Computational Linguistics, 2023. http://dx.doi.org/10.18653/v1/2023.findings-acl.742.

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Tan, Xingwei, Gabriele Pergola, and Yulan He. "Extracting Event Temporal Relations via Hyperbolic Geometry." In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. Stroudsburg, PA, USA: Association for Computational Linguistics, 2021. http://dx.doi.org/10.18653/v1/2021.emnlp-main.636.

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Assylbekov, Zhenisbek, Sultan Nurmukhamedov, Arsen Sheverdin, and Thomas Mach. "From Hyperbolic Geometry Back to Word Embeddings." In Proceedings of the 7th Workshop on Representation Learning for NLP. Stroudsburg, PA, USA: Association for Computational Linguistics, 2022. http://dx.doi.org/10.18653/v1/2022.repl4nlp-1.5.

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Reports on the topic "Hyperbolic Geometry"

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Ungar, Abraham A. Hyperbolic Geometry. GIQ, 2014. http://dx.doi.org/10.7546/giq-15-2014-259-282.

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Ungar, Abraham A. Hyperbolic Geometry. Jgsp, 2013. http://dx.doi.org/10.7546/jgsp-32-2013-61-86.

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Brezov, Danail S., Clementina D. Mladenova, and Ivaïlo M. Mladenov. Vector Parameters in Classical Hyperbolic Geometry. GIQ, 2014. http://dx.doi.org/10.7546/giq-15-2014-79-105.

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Brezov, Danail, Clementina Mladenova, and Ivaïlo Mladenov. Vector Parameters in Classical Hyperbolic Geometry. Journal of Geometry and Symmetry in Physics, 2013. http://dx.doi.org/10.7546/jgsp-30-2013-19-.

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Brezov, Danail, Clementina Mladenova, and Ivaïlo Mladenov. Vector Parameters in Classical Hyperbolic Geometry. Journal of Geometry and Symmetry in Physics, 2013. http://dx.doi.org/10.7546/jgsp-30-2013-19-48.

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Hasslacher, B., and R. Mainieri. Geometry in the large and hyperbolic chaos. Office of Scientific and Technical Information (OSTI), November 1998. http://dx.doi.org/10.2172/674860.

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Holzapfel, Rolf-Peter. Enumerative Geometry on Quasi-Hyperbolic 4-Spaces with Cusps. GIQ, 2012. http://dx.doi.org/10.7546/giq-4-2003-42-87.

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