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Добірка наукової літератури з теми "Convergence of Riemannian manifolds"

Автор: Grafiati

Опубліковано: 10 березня 2023

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Статті в журналах з теми "Convergence of Riemannian manifolds"

1

Perales, Raquel. "Convergence of manifolds and metric spaces with boundary." Journal of Topology and Analysis 12, no. 03 (November 28, 2018): 735–74. http://dx.doi.org/10.1142/s1793525319500638.

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We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov–Hausdorff (GH) and Sormani–Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably [Formula: see text] rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require non-negative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.

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2

Kasue, Atsushi, and Hironori Kumura. "Spectral convergence of Riemannian manifolds." Tohoku Mathematical Journal 46, no. 2 (1994): 147–79. http://dx.doi.org/10.2748/tmj/1178225756.

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3

Greene, Robert, and Hung-Hsi Wu. "Lipschitz convergence of Riemannian manifolds." Pacific Journal of Mathematics 131, no. 1 (January 1, 1988): 119–41. http://dx.doi.org/10.2140/pjm.1988.131.119.

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4

Argyros, Ioannis K., and Santhosh George. "ON THE SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SECTIONS ON RIEMANNIAN MANIFOLDS." Asian-European Journal of Mathematics 07, no. 01 (March 2014): 1450007. http://dx.doi.org/10.1142/s1793557114500077.

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We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] in combination with the weaker center 2-piece L1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal.22 (2002) 359–390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math.8 (2008) 197–226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant α-theory, IMA J. Numer. Anal.23 (2003) 395–419].

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5

Yang, Le. "Riemannian median and its estimation." LMS Journal of Computation and Mathematics 13 (December 2010): 461–79. http://dx.doi.org/10.1112/s1461157020090531.

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AbstractIn this paper, we define the geometric median for a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to compute the geometric median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also improves a recent result of P. T. Fletcheret al. [NeuroImage45 (2009) S143–S152].

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6

Kasue, Atsushi. "A convergence theorem for Riemannian manifolds and some applications." Nagoya Mathematical Journal 114 (June 1989): 21–51. http://dx.doi.org/10.1017/s0027763000001380.

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The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

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7

Kasue, Atsushi, and Hironori Kumura. "Spectral convergence of Riemannian manifolds, II." Tohoku Mathematical Journal 48, no. 1 (1996): 71–120. http://dx.doi.org/10.2748/tmj/1178225413.

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8

Boumal, Nicolas, P.-A. Absil, and Coralia Cartis. "Global rates of convergence for nonconvex optimization on manifolds." IMA Journal of Numerical Analysis 39, no. 1 (February 7, 2018): 1–33. http://dx.doi.org/10.1093/imanum/drx080.

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Abstract We consider the minimization of a cost function f on a manifold $\mathcal{M}$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance ε. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of f to the tangent spaces of $\mathcal{M}$, both of these algorithms produce points with Riemannian gradient smaller than ε in $\mathcal{O}\big(1/\varepsilon ^{2}\big)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian’s least eigenvalue is larger than −ε in $\mathcal{O} \big(1/\varepsilon ^{3}\big)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy ε (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of ${\mathbb{R}^{n}}$, under simpler assumptions.

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Katsuda, Atsushi. "Gromov’s convergence theorem and its application." Nagoya Mathematical Journal 100 (December 1985): 11–48. http://dx.doi.org/10.1017/s0027763000000209.

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One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

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10

Greene, Robert, and Hung-Hsi Wu. "Addendum to: “Lipschitz convergence of Riemannian manifolds”." Pacific Journal of Mathematics 140, no. 2 (December 1, 1989): 398. http://dx.doi.org/10.2140/pjm.1989.140.398.

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Дисертації з теми "Convergence of Riemannian manifolds"

1

Zergänge, Norman [Verfasser]. "Convergence of Riemannian manifolds with critical curvature bounds / Norman Zergänge." Magdeburg : Universitätsbibliothek, 2017. http://d-nb.info/1141230488/34.

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Martins, Tiberio Bittencourt de Oliveira. "Newton's methods under the majorant principle on Riemannian manifolds." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4847.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
Apresentamos, nesta tese, uma an álise da convergência do m étodo de Newton inexato com tolerância de erro residual relativa e uma an alise semi-local de m etodos de Newton robustos exato e inexato, objetivando encontrar uma singularidade de um campo de vetores diferenci avel de nido em uma variedade Riemanniana completa, baseados no princ pio majorante a m invariante. Sob hip oteses locais e considerando uma fun ção majorante geral, a Q-convergância linear do m etodo de Newton inexato com uma tolerância de erro residual relativa xa e provada. Na ausência dos erros, a an alise apresentada reobtem o teorema local cl assico sobre o m etodo de Newton no contexto Riemanniano. Na an alise semi-local dos m etodos exato e inexato de Newton apresentada, a cl assica condi ção de Lipschitz tamb em e relaxada usando uma fun ção majorante geral, permitindo estabelecer existência e unicidade local da solu ção, uni cando previamente resultados pertencentes ao m etodo de Newton. A an alise enfatiza a robustez, a saber, e dada uma bola prescrita em torno do ponto inicial que satifaz as hip oteses de Kantorovich, garantindo a convergência do m etodo para qualquer ponto inicial nesta bola. Al em disso, limitantes que dependem da função majorante para a taxa de convergência Q-quadr atica do m étodo exato e para a taxa de convergência Q-linear para o m etodo inexato são obtidos.
A local convergence analysis with relative residual error tolerance of inexact Newton method and a semi-local analysis of a robust exact and inexact Newton methods are presented in this thesis, objecting to nd a singularity of a di erentiable vector eld de ned on a complete Riemannian manifold, based on a ne invariant majorant principle. Considering local assumptions and a general majorant function, the Q-linear convergence of inexact Newton method with a xed relative residual error tolerance is proved. In the absence of errors, the analysis presented retrieves the classical local theorem on Newton's method in Riemannian context. In the semi-local analysis of exact and inexact Newton methods presented, the classical Lipschitz condition is also relaxed by using a general majorant function, allowing to establish the existence and also local uniqueness of the solution, unifying previous results pertaining Newton's method. The analysis emphasizes robustness, being more speci c, is given a prescribed ball around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Furthermore, the bounds depending on the majorant function for Q-quadratic convergence rate of the exact method and Q-linear convergence rate of the inexact method are obtained.

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3

Luckhardt, Daniel [Verfasser], Thomas [Akademischer Betreuer] Schick, Thomas [Gutachter] Schick, Ralf [Gutachter] Meyer, Stephan [Gutachter] Huckemann, Russell [Gutachter] Luke, Viktor [Gutachter] Pidstrygach, and Ingo [Gutachter] Witt. "Benjamini-Schramm Convergence of Normalized Characteristic Numbers of Riemannian Manifolds / Daniel Luckhardt ; Gutachter: Thomas Schick, Ralf Meyer, Stephan Huckemann, Russell Luke, Viktor Pidstrygach, Ingo Witt ; Betreuer: Thomas Schick." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2020. http://d-nb.info/1209358239/34.

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Guevara, Stefan Alberto Gómez. "Unificando o análise local do método de Newton em variedades Riemannianas." Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/6951.

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In this work we consider the problem of finding a singularity of a field of differentiable vectors X on a Riemannian manifold. We present a local analysis of the convergence of Newton's method to find a singularity of field X on an increasing condition. The analysis shows a relationship between the major function and the vector field X. We also present a semi-local Kantorovich type analysis in the Riemannian context under a major condition. The two results allow to unify some previously unrelated results.
Neste trabalho consideramos o problema de encontrar uma singularidade de um campo de vetores diferenciável X sobre uma variedade Riemanniana. Apresentamos uma análise local da convergência do método de Newton para encontrar uma singularidade do Campo X sobre uma condição majorante. A análise mostra uma relação entre a função majorante e o campo de vetores X. Também apresentamos uma análise semi-local do tipo Kantorovich no contexto Riemanniana sob uma condição majorante. Os dois resultados permitem unificar alguns resultados não previamente.

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5

Erb, Wolfgang. "Uncertainty principles on Riemannian manifolds." kostenfrei, 2010. https://mediatum2.ub.tum.de/node?id=976465.

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6

Dunn, Corey. "Curvature homogeneous pseudo-Riemannian manifolds /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1188874491&sid=3&Fmt=2&clientId=11238&RQT=309&VName=PQD.

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Thesis (Ph. D.)--University of Oregon, 2006.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 146-147). Also available for download via the World Wide Web; free to University of Oregon users.

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7

Longa, Eduardo Rosinato. "Hypersurfaces of paralellisable Riemannian manifolds." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2017. http://hdl.handle.net/10183/158755.

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Introduzimos uma aplicação de Gauss para hipersuperfícies de variedades Riemannianas paralelizáveis e definimos uma curvatura associada. Após, provamos um teorema de Gauss-Bonnet. Como exemplo, estudamos cuidadosamente o caso no qual o espaço ambiente é uma esfera Euclidiana menos um ponto e obtemos um teorema de rigidez topológica. Ele é utilizado para dar uma prova alternativa para um teorema de Qiaoling Wang and Changyu Xia, o qual afirma que se uma hipersuperfície orientável imersa na esfera está contida em um hemisfério aberto e tem curvatura de Gauss-Kronecker nãonula então ela é difeomorfa a uma esfera. Depois, obtemos alguns invariantes topol_ogicos para hipersuperfícies de variedades translacionais que dependem da geometria da variedade e do espaço ambiente. Finalmente, encontramos obstruções para a existência de certas folheações de codimensão um.
We introduce a Gauss map for hypersurfaces of paralellisable Riemannian manifolds and de ne an associated curvature. Next, we prove a Gauss- Bonnet theorem. As an example, we carefully study the case where the ambient space is an Euclidean sphere minus a point and obtain a topological rigidity theorem. We use it to provide an alternative proof for a theorem of Qiaoling Wang and Changyu Xia, which asserts that if an orientable immersed hypersurface of the sphere is contained in an open hemisphere and has nowhere zero Gauss-Kronecker curvature, then it is di eomorphic to a sphere. Later, we obtain some topological invariants for hypersurfaces of translational manifolds that depend on the geometry of the manifold and the ambient space. Finally, we nd obstructions to the existence of certain codimension-one foliations.

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8

Catalano, Domenico Antonino. "Concircular diffeomorphisms of pseudo-Riemannian manifolds /." [S.l.] : [s.n.], 1999. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=13064.

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9

Afsari, Bijan. "Means and averaging on riemannian manifolds." College Park, Md. : University of Maryland, 2009. http://hdl.handle.net/1903/9978.

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Thesis (Ph.D.) -- University of Maryland, College Park, 2009.
Thesis research directed by: Dept. of Mathematics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

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10

Popiel, Tomasz. "Geometrically-defined curves in Riemannian manifolds." University of Western Australia. School of Mathematics and Statistics, 2007. http://theses.library.uwa.edu.au/adt-WU2007.0119.

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[Truncated abstract] This thesis is concerned with geometrically-defined curves that can be used for interpolation in Riemannian or, more generally, semi-Riemannian manifolds. As in much of the existing literature on such curves, emphasis is placed on manifolds which are important in computer graphics and engineering applications, namely the unit 3-sphere S3 and the closely related rotation group SO(3), as well as other Lie groups and spheres of arbitrary dimension. All geometrically-defined curves investigated in the thesis are either higher order variational curves, namely critical points of cost functionals depending on (covariant) derivatives of order greater than 1, or defined by geometrical algorithms, namely generalisations to manifolds of algorithms from the field of computer aided geometric design. Such curves are needed, especially in the aforementioned applications, since interpolation methods based on applying techniques of classical approximation theory in coordinate charts often produce unnatural interpolants. However, mathematical properties of higher order variational curves and curves defined by geometrical algorithms are in need of substantial further investigation: higher order variational curves are solutions of complicated nonlinear differential equations whose properties are not well-understood; it is usually unclear how to impose endpoint derivative conditions on, or smoothly piece together, curves defined by geometrical algorithms. This thesis addresses these difficulties for several classes of curves. ... The geometrical algorithms investigated in this thesis are generalisations of the de Casteljau and Cox-de Boor algorithms, which define, respectively, polynomial B'ezier and piecewise-polynomial B-spline curves by dividing, in certain ratios and for a finite number of iterations, piecewise-linear control polygons corresponding to finite sequences of control points. We show how the control points of curves produced by the generalised de Casteljau algorithm in an (almost) arbitrary connected finite-dimensional Riemannian manifold M should be chosen in order to impose desired endpoint velocities and (covariant) accelerations and, thereby, piece the curves together in a C2 fashion. A special case of the latter construction simplifies when M is a symmetric space. For the generalised Cox-de Boor algorithm, we analyse in detail the failure of a fundamental property of B-spline curves, namely C2 continuity at (certain) knots, to carry over to M.

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Книги з теми "Convergence of Riemannian manifolds"

1

Lee, John M. Riemannian Manifolds. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/b98852.

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Lee, John M. Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91755-9.

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3

Tondeur, Philippe. Foliations on Riemannian Manifolds. New York, NY: Springer New York, 1988. http://dx.doi.org/10.1007/978-1-4613-8780-0.

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4

Lang, Serge, ed. Differential and Riemannian Manifolds. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4182-9.

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5

Lang, Serge. Differential and Riemannian manifolds. New York: Springer-Verlag, 1995.

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6

Tondeur, Philippe. Foliations on Riemannian manifolds. New York: Springer-Verlag, 1988.

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7

Riemannian foliations. Boston: Birkhäuser, 1988.

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8

Hebey, Emmanuel. Sobolev Spaces on Riemannian Manifolds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/bfb0092907.

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9

Berestovskii, Valerii, and Yurii Nikonorov. Riemannian Manifolds and Homogeneous Geodesics. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56658-6.

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10

C, Wood John, ed. Harmonic morphisms between Riemannian manifolds. Oxford: Clarendon Press, 2003.

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Частини книг з теми "Convergence of Riemannian manifolds"

1

Godinho, Leonor, and José Natário. "Riemannian Manifolds." In Universitext, 95–122. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08666-8_3.

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2

DeWitt, Bryce, and Steven M. Christensen. "Riemannian Manifolds." In Bryce DeWitt's Lectures on Gravitation, 51–62. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-540-36911-0_4.

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3

Saller, Heinrich. "Riemannian Manifolds." In Operational Spacetime, 29–80. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-0898-8_3.

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4

Wells, Raymond O. "Riemannian Manifolds." In Differential and Complex Geometry: Origins, Abstractions and Embeddings, 187–210. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-58184-2_13.

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5

Torres del Castillo, Gerardo F. "Riemannian Manifolds." In Differentiable Manifolds, 115–60. Boston: Birkhäuser Boston, 2012. http://dx.doi.org/10.1007/978-0-8176-8271-2_6.

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6

Burago, Yuriĭ Dmitrievich, and Viktor Abramovich Zalgaller. "Riemannian Manifolds." In Geometric Inequalities, 232–99. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/978-3-662-07441-1_6.

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7

Berestovskii, Valerii, and Yurii Nikonorov. "Riemannian Manifolds." In Springer Monographs in Mathematics, 1–74. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56658-6_1.

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Torres del Castillo, Gerardo F. "Riemannian Manifolds." In Differentiable Manifolds, 141–202. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-45193-6_6.

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9

Kühnel, Wolfgang. "Riemannian manifolds." In The Student Mathematical Library, 189–224. Providence, Rhode Island: American Mathematical Society, 2005. http://dx.doi.org/10.1090/stml/016/05.

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10

Aubin, Thierry. "Riemannian manifolds." In Graduate Studies in Mathematics, 111–67. Providence, Rhode Island: American Mathematical Society, 2000. http://dx.doi.org/10.1090/gsm/027/06.

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Тези доповідей конференцій з теми "Convergence of Riemannian manifolds"

1

OU, YE-LIN. "BIHARMONIC MORPHISMS BETWEEN RIEMANNIAN MANIFOLDS." In Differential Geometry in Honor of Professor S S Chern. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812792051_0018.

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2

Snoussi, Hichem, and Ali Mohammad-Djafari. "Particle Filtering on Riemannian Manifolds." In Bayesian Inference and Maximum Entropy Methods In Science and Engineering. AIP, 2006. http://dx.doi.org/10.1063/1.2423278.

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3

KASHANI, S. M. B. "ON COHOMOGENEITY ONE RIEMANNIAN MANIFOLDS." In Proceedings of the Summer School. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812810571_0010.

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4

Brendle, Simon, and Richard Schoen. "Riemannian Manifolds of Positive Curvature." In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India, 2011. http://dx.doi.org/10.1142/9789814324359_0021.

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5

Elworthy, K. D., and Feng-Yu Wang. "Essential spectrum on Riemannian manifolds." In Proceedings of the First Sino-German Conference on Stochastic Analysis (A Satellite Conference of ICM 2002). WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812702241_0010.

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6

Jacobs, H., S. Nair, and J. Marsden. "Multiscale surveillance of Riemannian manifolds." In 2010 American Control Conference (ACC 2010). IEEE, 2010. http://dx.doi.org/10.1109/acc.2010.5531152.

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7

Yi Wu, Bo Wu, Jia Liu, and Hanqing Lu. "Probabilistic tracking on Riemannian manifolds." In 2008 19th International Conference on Pattern Recognition (ICPR). IEEE, 2008. http://dx.doi.org/10.1109/icpr.2008.4761046.

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8

Yang, Hyun Seok. "Riemannian Manifolds and Gauge Theory." In Proceedings of the Corfu Summer Institute 2011. Trieste, Italy: Sissa Medialab, 2012. http://dx.doi.org/10.22323/1.155.0063.

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9

Lee, Sangyul, and Hee-Seok Oh. "Robust Multivariate Regression on Riemannian Manifolds." In 2020 IEEE 7th International Conference on Data Science and Advanced Analytics (DSAA). IEEE, 2020. http://dx.doi.org/10.1109/dsaa49011.2020.00099.

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10

Chazal, Frédéric, Leonidas J. Guibas, Steve Y. Oudot, and Primoz Skraba. "Persistence-based clustering in riemannian manifolds." In the 27th annual ACM symposium. New York, New York, USA: ACM Press, 2011. http://dx.doi.org/10.1145/1998196.1998212.

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Звіти організацій з теми "Convergence of Riemannian manifolds"

1

Bozok, Hülya Gün. Bi-slant Submersions from Kenmotsu Manifolds onto Riemannian Manifolds. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, March 2020. http://dx.doi.org/10.7546/crabs.2020.03.05.

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2

Chiang, Yuan-Jen. f-biharmonic Maps Between Riemannian Manifolds. GIQ, 2013. http://dx.doi.org/10.7546/giq-14-2013-74-86.

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3

Dušek, Zdenek. Examples of Pseudo-Riemannian G.O. Manifolds. GIQ, 2012. http://dx.doi.org/10.7546/giq-8-2007-144-155.

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4

Chiang, Yuan-Jen. f-biharmonic Maps Between Riemannian Manifolds. Journal of Geometry and Symmetry in Physics, 2012. http://dx.doi.org/10.7546/jgsp-27-2012-45-58.

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5

Mirzaei, Reza. Cohomogeneity Two Riemannian Manifolds of Non-Positive Curvature. GIQ, 2012. http://dx.doi.org/10.7546/giq-13-2012-233-244.

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6

Iyer, R. V., R. Holsapple, and D. Doman. Optimal Control Problems on Parallelizable Riemannian Manifolds: Theory and Applications. Fort Belvoir, VA: Defense Technical Information Center, January 2002. http://dx.doi.org/10.21236/ada455175.

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7

R. Mirzaie. Topological Properties of Some Cohomogeneity on Riemannian Manifolds of Nonpositive Curvature. GIQ, 2012. http://dx.doi.org/10.7546/giq-3-2002-351-359.

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8

Tanimura, Shogo. Path Integrals on Riemannian Manifolds with Symmetry and Stratified Gauge Structure. GIQ, 2012. http://dx.doi.org/10.7546/giq-3-2002-431-441.

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