Relevant bibliographies by topics / Riemannian and barycentric geometry

Academic literature on the topic 'Riemannian and barycentric geometry'

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Published: 13 February 2024

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Journal articles on the topic "Riemannian and barycentric geometry"

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Pihajoki, Pauli, Matias Mannerkoski, and Peter H. Johansson. "Barycentric interpolation on Riemannian and semi-Riemannian spaces." Monthly Notices of the Royal Astronomical Society 489, no. 3 (September 2, 2019): 4161–69. http://dx.doi.org/10.1093/mnras/stz2447.

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ABSTRACT Interpolation of data represented in curvilinear coordinates and possibly having some non-trivial, typically Riemannian or semi-Riemannian geometry is a ubiquitous task in all of physics. In this work, we present a covariant generalization of the barycentric coordinates and the barycentric interpolation method for Riemannian and semi-Riemannian spaces of arbitrary dimension. We show that our new method preserves the linear accuracy property of barycentric interpolation in a coordinate-invariant sense. In addition, we show how the method can be used to interpolate constrained quantities so that the given constraint is automatically respected. We showcase the method with two astrophysics related examples situated in the curved Kerr space–time. The first problem is interpolating a locally constant vector field, in which case curvature effects are expected to be maximally important. The second example is a general relativistic magnetohydrodynamics simulation of a turbulent accretion flow around a black hole, wherein high intrinsic variability is expected to be at least as important as curvature effects.
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Miranda Jr., Gastão F., Gilson Giraldi, Carlos E. Thomaz, and Daniel Millàn. "Composition of Local Normal Coordinates and Polyhedral Geometry in Riemannian Manifold Learning." International Journal of Natural Computing Research 5, no. 2 (April 2015): 37–68. http://dx.doi.org/10.4018/ijncr.2015040103.

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The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.
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Sabatini, Luca. "Volume Comparison in the presence of a Gromov-Hausdorff ε−approximation II." Annals of West University of Timisoara - Mathematics and Computer Science 56, no. 1 (July 1, 2018): 99–135. http://dx.doi.org/10.2478/awutm-2018-0008.

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Abstract Let (M, g) be any compact, connected, Riemannian manifold of dimension n. We use a transport of measures and the barycentre to construct a map from (M, g) onto a Hyperbolic manifold (ℍn/Λ, g0) (Λ is a torsionless subgroup of Isom(ℍn,g0)), in such a way that its jacobian is sharply bounded from above. We make no assumptions on the topology of (M, g) and on its curvature and geometry, but we only assume the existence of a measurable Gromov-Hausdorff ε-approximation between (ℍn/Λ, g0) and (M, g). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison, if $\varepsilon < {1 \over {64\,{n^2}}}\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)$ , then $$\matrix{{Vol\left( {{M^n},g} \right) \ge }\cr {{{\left( {1 + 160n\left( {n + 1} \right)\sqrt {{\varepsilon \over {\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)}}} } \right)}^{{n \over 2}}}\left| {\deg \,h} \right| \cdot Vol\left( {{X^n},{g_0}} \right).} \cr }$$
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Wu, H., and Wilhelm Klingenberg. "Riemannian Geometry." American Mathematical Monthly 92, no. 7 (August 1985): 519. http://dx.doi.org/10.2307/2322529.

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Lord, Nick, M. P. do Carmo, S. Gallot, D. Hulin, J. Lafontaine, I. Chavel, and D. Martin. "Riemannian Geometry." Mathematical Gazette 79, no. 486 (November 1995): 623. http://dx.doi.org/10.2307/3618122.

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Mrugała, R. "Riemannian geometry." Reports on Mathematical Physics 27, no. 2 (April 1989): 283–85. http://dx.doi.org/10.1016/0034-4877(89)90011-6.

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M.Osman, Mohamed. "Differentiable Riemannian Geometry." International Journal of Mathematics Trends and Technology 29, no. 1 (January 25, 2016): 45–55. http://dx.doi.org/10.14445/22315373/ijmtt-v29p508.

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Dimakis, Aristophanes, and Folkert Müller-Hoissen. "Discrete Riemannian geometry." Journal of Mathematical Physics 40, no. 3 (March 1999): 1518–48. http://dx.doi.org/10.1063/1.532819.

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Beggs, Edwin J., and Shahn Majid. "Poisson–Riemannian geometry." Journal of Geometry and Physics 114 (April 2017): 450–91. http://dx.doi.org/10.1016/j.geomphys.2016.12.012.

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Strichartz, Robert S. "Sub-Riemannian geometry." Journal of Differential Geometry 24, no. 2 (1986): 221–63. http://dx.doi.org/10.4310/jdg/1214440436.

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Dissertations / Theses on the topic "Riemannian and barycentric geometry"

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Farina, Sofia. "Barycentric Subspace Analysis on the Sphere and Image Manifolds." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/15797/.

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In this dissertation we present a generalization of Principal Component Analysis (PCA) to Riemannian manifolds called Barycentric Subspace Analysis and show some applications. The notion of barycentric subspaces has been first introduced first by X. Pennec. Since they lead to hierarchy of properly embedded linear subspaces of increasing dimension, they define a generalization of PCA on manifolds called Barycentric Subspace Analysis (BSA). We present a detailed study of the method on the sphere since it can be considered as the finite dimensional projection of a set of probability densities that have many practical applications. We also show an application of the barycentric subspace method for the study of cardiac motion in the problem of image registration, following the work of M.M. Rohé.
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Lord, Steven. "Riemannian non-commutative geometry /." Title page, abstract and table of contents only, 2002. http://web4.library.adelaide.edu.au/theses/09PH/09phl8661.pdf.

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Maignant, Elodie. "Plongements barycentriques pour l'apprentissage géométrique de variétés : application aux formes et graphes." Electronic Thesis or Diss., Université Côte d'Azur, 2023. http://www.theses.fr/2023COAZ4096.

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Une image obtenue par IRM, c'est plus de 60 000 pixels. La plus grosse protéine connue chez l'être humain est constituée d'environ 30 000 acides aminés. On parle de données en grande dimension. En réalité, la plupart des données en grande dimension ne le sont qu'en apparence. Par exemple, de toutes les images que l'on pourrait générer aléatoirement en coloriant 256 x 256 pixels, seule une infime proportion ressemblerait à l'image IRM d'un cerveau humain. C'est ce qu'on appelle la dimension intrinsèque des données. En grande dimension, apprentissage rime donc souvent avec réduction de dimension. Il existe de nombreuses méthodes de réduction de dimension, les plus récentes pouvant être classées selon deux approches.Une première approche, connue sous le nom d'apprentissage de variétés (manifold learning) ou réduction de dimension non linéaire, part du constat que certaines lois physiques derrière les données que l'on observe ne sont pas linéaires. Ainsi, espérer expliquer la dimension intrinsèque des données par un modèle linéaire est donc parfois irréaliste. Au lieu de cela, les méthodes qui relèvent du manifold learning supposent un modèle localement linéaire.D'autre part, avec l'émergence du domaine de l'analyse statistique de formes, il y eu une prise de conscience que de nombreuses données sont naturellement invariantes à certaines symétries (rotations, permutations, reparamétrisations...), invariances qui se reflètent directement sur la dimension intrinsèque des données. Ces invariances, la géométrie euclidienne ne peut pas les retranscrire fidèlement. Ainsi, on observe un intérêt croissant pour la modélisation des données par des structures plus fines telles que les variétés riemanniennes. Une deuxième approche en réduction de dimension consiste donc à généraliser les méthodes existantes à des données à valeurs dans des espaces non-euclidiens. On parle alors d'apprentissage géométrique. Jusqu'à présent, la plupart des travaux en apprentissage géométrique se sont focalisés sur l'analyse en composantes principales.Dans la perspective de proposer une approche qui combine à la fois apprentissage géométrique et manifold learning, nous nous sommes intéressés à la méthode appelée locally linear embedding, qui a la particularité de reposer sur la notion de barycentre, notion a priori définie dans les espaces euclidiens mais qui se généralise aux variétés riemanniennes. C'est d'ailleurs sur cette même notion que repose une autre méthode appelée barycentric subspace analysis, et qui fait justement partie des méthodes qui généralisent l'analyse en composantes principales aux variétés riemanniennes. Ici, nous introduisons la notion nouvelle de plongement barycentrique, qui regroupe les deux méthodes. Essentiellement, cette notion englobe un ensemble de méthodes dont la structure rappelle celle des méthodes de réduction de dimension linéaires et non linéaires, mais où le modèle (localement) linéaire est remplacé par un modèle barycentrique -- affine.Le cœur de notre travail consiste en l'analyse de ces méthodes, tant sur le plan théorique que pratique. Du côté des applications, nous nous intéressons à deux exemples importants en apprentissage géométrique : les formes et les graphes. En particulier, on démontre que par rapport aux méthodes standard de réduction de dimension en analyse statistique des graphes, les plongements barycentriques se distinguent par leur meilleure interprétabilité. En plus des questions pratiques liées à l'implémentation, chacun de ces exemples soulève ses propres questions théoriques, principalement autour de la géométrie des espaces quotients. Parallèlement, nous nous attachons à caractériser géométriquement les plongements localement barycentriques, qui généralisent la projection calculée par locally linear embedding. Enfin, de nouveaux algorithmes d'apprentissage géométrique, novateurs dans leur approche, complètent ce travail
An MRI image has over 60,000 pixels. The largest known human protein consists of around 30,000 amino acids. We call such data high-dimensional. In practice, most high-dimensional data is high-dimensional only artificially. For example, of all the images that could be randomly generated by coloring 256 x 256 pixels, only a very small subset would resemble an MRI image of a human brain. This is known as the intrinsic dimension of such data. Therefore, learning high-dimensional data is often synonymous with dimensionality reduction. There are numerous methods for reducing the dimension of a dataset, the most recent of which can be classified according to two approaches.A first approach known as manifold learning or non-linear dimensionality reduction is based on the observation that some of the physical laws behind the data we observe are non-linear. In this case, trying to explain the intrinsic dimension of a dataset with a linear model is sometimes unrealistic. Instead, manifold learning methods assume a locally linear model.Moreover, with the emergence of statistical shape analysis, there has been a growing awareness that many types of data are naturally invariant to certain symmetries (rotations, reparametrizations, permutations...). Such properties are directly mirrored in the intrinsic dimension of such data. These invariances cannot be faithfully transcribed by Euclidean geometry. There is therefore a growing interest in modeling such data using finer structures such as Riemannian manifolds. A second recent approach to dimension reduction consists then in generalizing existing methods to non-Euclidean data. This is known as geometric learning.In order to combine both geometric learning and manifold learning, we investigated the method called locally linear embedding, which has the specificity of being based on the notion of barycenter, a notion a priori defined in Euclidean spaces but which generalizes to Riemannian manifolds. In fact, the method called barycentric subspace analysis, which is one of those generalizing principal component analysis to Riemannian manifolds, is based on this notion as well. Here we rephrase both methods under the new notion of barycentric embeddings. Essentially, barycentric embeddings inherit the structure of most linear and non-linear dimension reduction methods, but rely on a (locally) barycentric -- affine -- model rather than a linear one.The core of our work lies in the analysis of these methods, both on a theoretical and practical level. In particular, we address the application of barycentric embeddings to two important examples in geometric learning: shapes and graphs. In addition to practical implementation issues, each of these examples raises its own theoretical questions, mostly related to the geometry of quotient spaces. In particular, we highlight that compared to standard dimension reduction methods in graph analysis, barycentric embeddings stand out for their better interpretability. In parallel with these examples, we characterize the geometry of locally barycentric embeddings, which generalize the projection computed by locally linear embedding. Finally, algorithms for geometric manifold learning, novel in their approach, complete this work
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Lidberg, Petter. "Barycentric and harmonic coordinates." Thesis, Uppsala universitet, Algebra och geometri, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-179487.

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Hall, Stuart James. "Numerical methods and Riemannian geometry." Thesis, Imperial College London, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.538692.

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Ferreira, Ana Cristina Castro. "Riemannian geometry with skew torsion." Thesis, University of Oxford, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526550.

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Wu, Bao Qiang. "Geometry of complete Riemannian Submanifolds." Lyon 1, 1998. http://www.theses.fr/1998LYO10064.

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La géométrie rienmannienne des sous-variétés a connu ces cinquante dernières années un essor considérable, essentiellement dans le cas compact. Cette thèse a pour but de développer des outils consacrés à l'étude des sous-variétés riemanniennes complètes. Ces outils sont proches de ceux développés par Bochner et Lichnérowicz. Ils sont particulièrement adaptés aux problèmes de rigidité de certains types de sous-variétés complètes : celles qui sont à courbure moyenne constante dans un espace hyperbolique. Il est ainsi possible d'obtenir un théorème de classification de ces sous-variétés. D'autres applications sont données pour des sous-variétés totalement réelles des espaces projectifs complexes
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Boarotto, Francesco. "Topics in sub-Riemannian geometry." Doctoral thesis, SISSA, 2016. http://hdl.handle.net/20.500.11767/4881.

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This thesis is concerned with three different problems in sub-Riemannian geometry faced during my PhD. The first one is a problem in differential geometry and is about the local conformal classification of a certain class of sub-Riemannian structures. In the second one we deal with topology, and our main result establish some path-fibration properties for the Endpoint map. In the third and last problem, we begin the development of some variational calculus around critical points of the endpoint map, called abnormal controls, and we estabilish a counterpart of the classical Morse deformation techniques and of the Min-Max variational principle.
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Palmer, Ian Christian. "Riemannian geometry of compact metric spaces." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34744.

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A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
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Raineri, Emanuele. "Quantum Riemannian geometry of finite sets." Thesis, Queen Mary, University of London, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.414738.

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Books on the topic "Riemannian and barycentric geometry"

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Gallot, Sylvestre, Dominique Hulin, and Jacques Lafontaine. Riemannian Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-97242-3.

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Petersen, Peter. Riemannian Geometry. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4757-6434-5.

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Carmo, Manfredo Perdigão do. Riemannian Geometry. Boston, MA: Birkhäuser Boston, 1992. http://dx.doi.org/10.1007/978-1-4757-2201-7.

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Gallot, Sylvestre, Dominique Hulin, and Jacques Lafontaine. Riemannian Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-642-18855-8.

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Petersen, Peter. Riemannian Geometry. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-26654-1.

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Gallot, Sylvestre, Dominique Hulin, and Jacques Lafontaine. Riemannian Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/978-3-642-97026-9.

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Gallot, S. Riemannian geometry. Berlin: Springer-Verlag, 1987.

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Sakai, T. Riemannian geometry. Providence, R.I: American Mathematical Society, 1996.

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Klingenberg, Wilhelm. Riemannian geometry. 2nd ed. Berlin: W. de Gruyter, 1995.

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Carmo, Manfredo Perdigão do. Riemannian geometry. Boston: Birkhäuser, 1992.

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Book chapters on the topic "Riemannian and barycentric geometry"

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Bambi, Cosimo. "Riemannian Geometry." In Introduction to General Relativity, 85–105. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-1090-4_5.

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Conlon, Lawrence. "Riemannian Geometry." In Differentiable Manifolds, 293–348. Boston, MA: Birkhäuser Boston, 1993. http://dx.doi.org/10.1007/978-1-4757-2284-0_10.

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Aubin, Thierry. "Riemannian Geometry." In Some Nonlinear Problems in Riemannian Geometry, 1–31. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/978-3-662-13006-3_1.

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Kumaresan, S. "Riemannian Geometry." In A Course in Differential Geometry and Lie Groups, 232–80. Gurgaon: Hindustan Book Agency, 2002. http://dx.doi.org/10.1007/978-93-86279-08-8_5.

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Gadea, P. M., and J. Muñoz Masqué. "Riemannian Geometry." In Analysis and Algebra on Differentiable Manifolds, 233–349. Dordrecht: Springer Netherlands, 2009. http://dx.doi.org/10.1007/978-90-481-3564-6_6.

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Koch, Helmut. "Riemannian geometry." In Introduction to Classical Mathematics I, 182–209. Dordrecht: Springer Netherlands, 1991. http://dx.doi.org/10.1007/978-94-011-3218-3_14.

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McInerney, Andrew. "Riemannian Geometry." In Undergraduate Texts in Mathematics, 195–270. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-7732-7_5.

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Chow, Bennett, Peng Lu, and Lei Ni. "Riemannian geometry." In Hamilton’s Ricci Flow, 1–93. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/gsm/077/01.

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Gadea, Pedro M., Jaime Muñoz Masqué, and Ihor V. Mykytyuk. "Riemannian Geometry." In Analysis and Algebra on Differentiable Manifolds, 343–546. Dordrecht: Springer Netherlands, 2013. http://dx.doi.org/10.1007/978-94-007-5952-7_6.

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Hassani, Sadri. "Riemannian Geometry." In Mathematical Physics, 1143–77. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-01195-0_37.

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Conference papers on the topic "Riemannian and barycentric geometry"

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Moran, William, Stephen D. Howard, Douglas Cochran, and Sofia Suvorova. "Sensor management via riemannian geometry." In 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE, 2012. http://dx.doi.org/10.1109/allerton.2012.6483240.

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Hadwiger, Markus, Thomas Theußl, and Peter Rautek. "Riemannian Geometry for Scientific Visualization." In SA '22: SIGGRAPH Asia 2022. New York, NY, USA: ACM, 2022. http://dx.doi.org/10.1145/3550495.3558227.

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GMIRA, B., and L. VERSTRAELEN. "A CURVATURE INEQUALITY FOR RIEMANNIAN SUBMANIFOLDS IN A SEMI–RIEMANNIAN SPACE FORM." In Geometry and Topology of Submanifolds IX. WORLD SCIENTIFIC, 1999. http://dx.doi.org/10.1142/9789812817976_0016.

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Lenz, Reiner, Rika Mochizuki, and Jinhui Chao. "Iwasawa Decomposition and Computational Riemannian Geometry." In 2010 20th International Conference on Pattern Recognition (ICPR). IEEE, 2010. http://dx.doi.org/10.1109/icpr.2010.1086.

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Bejancu, Aurel. "Sub-Riemannian geometry and nonholonomic mechanics." In ALEXANDRU MYLLER MATHEMATICAL SEMINAR CENTENNIAL CONFERENCE. AIP, 2011. http://dx.doi.org/10.1063/1.3546072.

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Chen, Guohua. "Digital Riemannian Geometry and Its Application." In International Conference on Advances in Computer Science and Engineering. Paris, France: Atlantis Press, 2013. http://dx.doi.org/10.2991/cse.2013.63.

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Barachant, Alexandre, Stphane Bon, Marco Congedo, and Christian Jutten. "Common Spatial Pattern revisited by Riemannian geometry." In 2010 IEEE 12th International Workshop on Multimedia Signal Processing (MMSP). IEEE, 2010. http://dx.doi.org/10.1109/mmsp.2010.5662067.

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Zeestraten, Martijn J. A., Ioannis Havoutis, Sylvain Calinon, and Darwin G. Caldwell. "Learning task-space synergies using Riemannian geometry." In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2017. http://dx.doi.org/10.1109/iros.2017.8202140.

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Shao, Hang, Abhishek Kumar, and P. Thomas Fletcher. "The Riemannian Geometry of Deep Generative Models." In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW). IEEE, 2018. http://dx.doi.org/10.1109/cvprw.2018.00071.

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Gordina, Maria. "Riemannian geometry of Diff(S1)/S1 revisited." In Proceedings of a Satellite Conference of ICM 2006. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812791559_0002.

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