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Academic literature on the topic 'Curvature singularities'

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Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Curvature singularities"

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Van-Brunt, B., and K. Grant. "Hyperbolic Weingarten surfaces." Mathematical Proceedings of the Cambridge Philosophical Society 116, no. 3 (November 1994): 489–504. http://dx.doi.org/10.1017/s0305004100072765.

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AbstractWeingarten surfaces which can be represented locally as solutions to second order hyperbolic partial differential equations are examined in this paper. In particular, the geometry of the families of curves corresponding to characteristics on these surfaces is investigated and the relationships of these curves with other curves on the surface such as asymptotic lines and lines of curvature are explored. It is shown that singularities in the lines of curvature, i.e. umbilic points, correspond to singularities in the families of characteristics, and that lines of curvature are non-characteristic curves. If there is a linear relation between the Gaussian and mean curvatures and real characteristics exist, then the characteristics form a Tchebychef net on the corresponding Weingarten surface.
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Sáez, Mariel, and Oliver C. Schnürer. "Mean curvature flow without singularities." Journal of Differential Geometry 97, no. 3 (July 2014): 545–70. http://dx.doi.org/10.4310/jdg/1406033979.

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Andrews, Ben. "Singularities in crystalline curvature flows." Asian Journal of Mathematics 6, no. 1 (2002): 101–22. http://dx.doi.org/10.4310/ajm.2002.v6.n1.a6.

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Xin, Yuanlong. "Singularities of mean curvature flow." Science China Mathematics 64, no. 7 (April 26, 2021): 1349–56. http://dx.doi.org/10.1007/s11425-020-1840-1.

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RUDNICKI, WIESŁAW, ROBERT J. BUDZYŃSKI, and WITOLD KONDRACKI. "GENERALIZED STRONG CURVATURE SINGULARITIES AND COSMIC CENSORSHIP." Modern Physics Letters A 17, no. 07 (March 7, 2002): 387–97. http://dx.doi.org/10.1142/s021773230200659x.

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A new definition of a strong curvature singularity is proposed. This definition is motivated by the definitions given by Tipler and Królak, but is significantly different and more general. All causal geodesics terminating at these new singularities, which we call generalized strong curvature singularities, are classified into three possible types; the classification is based on certain relations between the causal structure and the curvature strength of the singularities. A cosmic censorship theorem is formulated and proved which shows that only one class of generalized strong curvature singularities, corresponding to a single type of geodesics according to our classification, can be naked. Implications of this result for the cosmic censorship hypothesis are indicated.
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Królak, Andrzej. "Strong curvature singularities and causal simplicity." Journal of Mathematical Physics 33, no. 2 (February 1992): 701–4. http://dx.doi.org/10.1063/1.529804.

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S Martila, Dmitri. "On Naked Singularities of spacetime Curvature." Journal of Contradiciting Results in Science 1, no. 1 (July 4, 2012): 09–13. http://dx.doi.org/10.5530/jcrsci.2012.1.4.

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Mantz, Christiaan L. M., and Tomislav Prokopec. "Resolving Curvature Singularities in Holomorphic Gravity." Foundations of Physics 41, no. 10 (June 4, 2011): 1597–633. http://dx.doi.org/10.1007/s10701-011-9570-3.

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Li, Chao, and Christos Mantoulidis. "Positive scalar curvature with skeleton singularities." Mathematische Annalen 374, no. 1-2 (September 15, 2018): 99–131. http://dx.doi.org/10.1007/s00208-018-1753-1.

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Benedini Riul, P., and R. Oset Sinha. "A relation between the curvature ellipse and the curvature parabola." Advances in Geometry 19, no. 3 (July 26, 2019): 389–99. http://dx.doi.org/10.1515/advgeom-2019-0002.

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Abstract At each point in an immersed surface in ℝ4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. Recently, at the singular point of a corank 1 singular surface in ℝ3, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in ℝ4 to ℝ3 in a tangent direction, corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in ℝ4 at a certain point to the geometry of the projection of the surface to ℝ3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.
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Dissertations / Theses on the topic "Curvature singularities"

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Höffer, v. Loewenfeld Philipp. "Resolution of Curvature Singularities in Black Holes and the Early Universe." Diss., lmu, 2010. http://nbn-resolving.de/urn:nbn:de:bvb:19-118659.

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Maurer, Wolfgang [Verfasser]. "Beauty and the Beast in Mean Curvature Flow Without Singularities / Wolfgang Maurer." Konstanz : KOPS Universität Konstanz, 2021. http://d-nb.info/1230755888/34.

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Schlichting, Arthur [Verfasser], and Miles [Akademischer Betreuer] Simon. "Smoothing singularities of Riemannian metrics while preserving lower curvature bounds / Arthur Schlichting. Betreuer: Miles Simon." Magdeburg : Universitätsbibliothek, 2014. http://d-nb.info/1054638039/34.

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Behrndt, Tapio. "Generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:f8a490d4-5b7c-4709-96e5-65ad3fefe922.

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In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.
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Wells-Day, Benjamin Michael. "Structure of singular sets local to cylindrical singularities for stationary harmonic maps and mean curvature flows." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/290409.

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In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.
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Binotto, Rosane Rossato. "Projetivos de curvatura." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306624.

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Orientadores: Sueli Irene Rodrigues Costa, Maria del Carmen Romero-Fuster
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: O projetivo de curvatura em um ponto de uma 3-variedade M de classe 'C POT. 2' imersa em 'IR POT. ?' , n >-4, é o lugar geométrico de todos os extremos dos vetores curvatura de secções normais ao longo de todas as direções tangentes a M em p. Mostramos que o projetivo de curvatura em p é isomorfo (difeomorfo) à superfície de Veronese clássica de ordem 2, composta com uma transformação linear. Conforme o posto desta transformação linear, o projetivo de curvatura será dado por projeções da superfície de Veronese em subespaços do espaço normal da variedade M. Quanto menor o posto, maior será a umbilicidade da variedade no ponto em questão. Também estudamos a natureza geométrica e singularidades para os diferentes casos de projetivos de curvatura em pontos de M, os quais incluem a superfície Romana de Steiner, a Cross-Cap, a superfície de Steiner de Tipo 5 e a Cross-Cup. Além disso, analisamos os pontos singulares de segunda ordem da imersão, no sentido de Feldman e estabelecemos condições relacionadas à natureza do projetivo de curvatura, para que uma 3-variedade imersa em 'IR POT. ?', n >_ 9, tenha contato de ordem _ 2 com k-planos e k-esferas de IRn, 3 _ k _ 8
Abstract: The curvature projective plane at each point p of three-manifolds M immersed in 'IR POT. ?', n _ 4, is the geometric locus of all end points of the curvature vectors of normal sections along of all tangent directions of M at p. In this study, we show that the curvature projective plane is isomorphic (diffeomorphic) to the classical Veronese surface of order two, composed with a linear transformation, and that according to the rank of this mapping, the curvature projective plane will be given by projections of the Veronese surface into subspaces of the normal space of M at p. Thus, the smaller the rank the greater the umbilicity of the manifold at this point. We also study the geometric nature and singularities of the curvature projective planes considering different possibilities, which include the Roman Steiner surface, the Cross-Cap, the Steiner surface of five-type, and the Cross-Cup. In addition, we analyze the order-two singularities of the immersion in the Feldman¿s sense and establish conditions related to the nature of the curvature projective plane for the existence of contacts of the three-manifolds in 'IR POT. ?', n _ 9, with k-planes and k-spheres, 3 _ k _ 8
Doutorado
Geometria
Doutor em Matemática
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Ashley, Michael John Siew Leung, and ashley@gravity psu edu. "Singularity theorems and the abstract boundary construction." The Australian National University. Faculty of Science, 2002. http://thesis.anu.edu.au./public/adt-ANU20050209.165310.

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The abstract boundary construction of Scott and Szekeres has proven a practical classification scheme for boundary points of pseudo-Riemannian manifolds. It has also proved its utility in problems associated with the re-embedding of exact solutions containing directional singularities in space-time. Moreover it provides a model for singularities in space-time - essential singularities. However the literature has been devoid of abstract boundary results which have results of direct physical applicability.¶ This thesis presents several theorems on the existence of essential singularities in space-time and on how the abstract boundary allows definition of optimal em- beddings for depicting space-time. Firstly, a review of other boundary constructions for space-time is made with particular emphasis on the deficiencies they possess for describing singularities. The abstract boundary construction is then pedagogically defined and an overview of previous research provided.¶ We prove that strongly causal, maximally extended space-times possess essential singularities if and only if they possess incomplete causal geodesics. This result creates a link between the Hawking-Penrose incompleteness theorems and the existence of essential singularities. Using this result again together with the work of Beem on the stability of geodesic incompleteness it is possible to prove the stability of existence for essential singularities.¶ Invariant topological contact properties of abstract boundary points are presented for the first time and used to define partial cross sections, which are an generalization of the notion of embedding for boundary points. Partial cross sections are then used to define a model for an optimal embedding of space-time.¶ Finally we end with a presentation of the current research into the relationship between curvature singularities and the abstract boundary. This work proposes that the abstract boundary may provide the correct framework to prove curvature singularity theorems for General Relativity. This exciting development would culminate over 30 years of research into the physical conditions required for curvature singularities in space-time.
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LEE, Fang Chou. "Par de Curvas no Plano: Geometria da Bicicleta." Universidade Federal de Goiás, 2011. http://repositorio.bc.ufg.br/tede/handle/tde/1939.

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The main objective is to study the curves generated by the front and rear wheels of a bicycle from the standpoint of differential geometry.
O principal objetivo deste trabalho é estudar as curvas geradas pelas rodas traseira e dianteira de uma bicicleta do ponto de vista da Geometria diferencial.
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Silva, Paulo do Nascimento. "Superfícies em R4 do ponto de vista da teoria das singularidades." Universidade Federal da Paraí­ba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/7447.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
We study the geometry of surfaces immersed in R4 through the singularities of their families of height functions. Inflection points on the surfaces are shown to be umbilic points from their families of height functions. Furthermore, we see that inflection points of imaginary type are isolated points of the curve --1(0). As a consequence we prove that any dive generic convexly embedded S2 in R4 has inflexion points.
Neste trabalho estudamos a geometria das superfícies em R4 através da variedade canal e das singularidades das famílias de funções altura das superfícies. Provaremos que os pontos de inflexão das superfície são os pontos umbílicos das famílias de funções altura. Além disso, veremos que pontos de inflexão do tipo imaginário serão pontos isolados da curva --1(0). Como uma consequência deste estudo provaremos que qualquer mergulho genérico convexo de S2 em R4 tem pelo menos um ponto de inflexão.
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Miranda, Gláucia Aparecida Soares. "Configurações das linhas de curvatura principal sobre superfícies seccionalmente suaves." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-03102014-112150/.

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Nesta tese apresentamos uma contribuição para o estudo da transição do retrato de fase de uma equação diferencial descontínua específica ao longo de uma linha de descontinuidade. A equação diferencial que tratamos neste trabalho é a das linhas de curvatura principal de uma superfície S contendo uma curva distinguida B e imersa em R^3. A linha de descontinuidade é a curva B, a qual é o bordo comum de duas superfícies suaves justapostas que formam S. Na primeira parte do trabalho consideramos a superfície seccionalmente suave, S = S+ U B U S-, obtida pela justaposição de S+ e S- ao longo do bordo comum B. O estudo da configuração principal de S nos casos em que as linhas de curvatura principal das superfícies S+ e S- tem contato quadrático ou cruzam transversalmente B foi feito por comparação com a configuração principal de uma superfície suave, obtida de S pelo processo da \"regularização\" ao longo da curva de descontinuidade B. Na segunda parte do trabalho estudamos as linhas de curvatura principal de uma superfície S em R^3 com bordo B e da superfície suave obtida de S através dos processos de engrossamento e regularização definidos por Garcia e Sotomayor em [5], onde os autores consideraram o caso genérico, sem pontos umbílicos e contato quadrático de uma linha de curvatura principal com B. Damos aqui continuidade ao estudo feito em [5] analisando o caso de contato cúbico com o bordo B. Obtivemos que dos pontos da curva bordo comum B de contato quadrático e de cruzamento transversal emergem, sobre a superfície regularizada, pontos umbílicos Darbouxianos dos tipos D1 e D3, enquanto que, para o ponto sobre B de contato cúbico obtivemos pontos umbílicos Darbouxianos dos tipos D1, D2 e D3 e também pontos umbílicos não Darbouxianos dos tipos D12 e D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117.
In this work we present a contribution to the study of the transition of the phase portrait of a specific discontinuous differential equation along a line of discontinuity. The differential equations under consideration will be that of the principal curvature lines of a surface S with a distinguished curve B immersed in R^3, where the line of discontinuity is the curve B which is the common border of two smooth surfaces attached to make up S. In the first part of the work we consider a piecewise smooth surface S = S+ U B U S-, obtained by the juxtaposition of two smooth surfaces S+ and S- along their common border B. The analysis of the principal configuration of S in the cases where the principal curvature lines of the surfaces S+ and S- have quadratic contact or cross transversally B was carried out by comparison with a smooth surface, obtained from S by the \"regularization\" along the discontinuity curve B. In the second part of the work we study the principal curvature lines of a surface S in R^3 with boundary B and of the smooth surface obtained from S by thickening and smoothing introduced by Garcia and Sotomayor in [5], where they considered the generic case of no umbilic points and at most quadratic contact of principal lines with B. Here we pursue the study in [5] and analyze the case of cubic contact with the border B. We established that while from quadratic contact points with B emerge on the smoothed surface Darbouxian umbilics of D1 and D3 types, from the cubic contact points appear Darbouxian umbilics of types D1, D2 and D3 as well as non Darbouxian points of types D12 and D23. [5] Garcia, R., and Sotomayor, J. Umbilic and tangential singularities on configurations of principal curvature lines. Anais da Academia Brasileira de Ciências 74, 1 (2002), 117.
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Books on the topic "Curvature singularities"

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Izumiya, Shyuichi. Differential geometry from singularity theory viewpoint. New Jersey: World Scientific, 2015.

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Book chapters on the topic "Curvature singularities"

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Mantegazza, Carlo. "Type II Singularities." In Lecture Notes on Mean Curvature Flow, 85–114. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0145-4_4.

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Mantegazza, Carlo. "Monotonicity Formula and Type I Singularities." In Lecture Notes on Mean Curvature Flow, 49–84. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0145-4_3.

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Ritoré, Manuel, and Carlo Sinestrari. "Local existence and formation of singularities." In Mean Curvature Flow and Isoperimetric Inequalities, 10–16. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0213-6_4.

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Hopf, Heinz. "Singularities of Surfaces with Constant Negative Gauss Curvature." In Lecture Notes in Mathematics, 174–84. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/3-540-39482-6_14.

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Asada, Akira. "Curvature forms with singularities and non-integral characteristic classes." In Lecture Notes in Mathematics, 152–68. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0074582.

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Troyanov, Marc. "Metrics of constant curvature on a sphere with two conical singularities." In Lecture Notes in Mathematics, 296–306. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/bfb0086431.

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Hopf, Heinz. "The Total Curvature (Curvatura Inteqra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements." In Lecture Notes in Mathematics, 107–18. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/3-540-39482-6_8.

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"Geometric singularities under the Gigli-Mantegazza flow." In Mean Curvature Flow, 109–15. De Gruyter, 2020. http://dx.doi.org/10.1515/9783110618365-011.

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Pansonato, Claudia C., and Sueli I. R. Costa. "Vertices of Curves on Constant Curvature Manifolds." In Real and Complex Singularities, 267–82. CRC Press, 2003. http://dx.doi.org/10.1201/9780203912089-14.

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Pansonato, Claudia, and Sueli Costa. "Vertices of Curves on Constant Curvature Manifolds." In Real And Complex Singularities. CRC Press, 2003. http://dx.doi.org/10.1201/9780203912089.ch14.

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Conference papers on the topic "Curvature singularities"

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KONKOWSKI, D. A., and T. M. HELLIWELL. "“SINGULARITIES” IN SPACETIMES WITH DIVERGING HIGHER-ORDER CURVATURE INVARIANTS." In Proceedings of the MG12 Meeting on General Relativity. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814374552_0360.

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Wang, Yu-Xin, Yu-Tong Li, Zheng Huang, and Shuang-Xia Pian. "Singular Assembly Configurations and Configuration Bifurcation Characteristics of the SRHGSMP." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49316.

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In this paper, the configuration bifurcation characteristics at the vicinities of singular points going with different input parameters are investigated. Then, with the aid of the assembly configurations at the theoretical singular points, the reasons to cause the singularities are analyzed. We find that the dimensional-utmost singularities, line vectors correlation singularities and Jacobian matrix correlation singularities can occur individually or jointly while choosing different number of input parameters. The number and the combination form of the input parameters have great influences on the complexity of the singularities and the curvature radiuses of the configuration curves. Selecting a group of adjacent input parameters, the simple configuration bifurcation and the large singularity-free input parameters zones can be obtained. And adopting multi-input parameters, the self-motion regions and the singularity avoidance errors can be reduced. These new discoveries are valuable and of significance for the trajectory design, the singularity avoidance, and the self-motion control for the parallel manipulator, as well as the parallel tools.
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Ghosal, Ashitava, and Bahram Ravani. "Differential Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/mech-5967.

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Abstract In this paper, we present a differential-geometric analysis of singularities of point trajectories of two and three-degree-of-freedom serial and parallel manipulators. At non-singular configurations, the first order local properties are characterized by the metric coefficients, and, geometrically, by the shape and size of a velocity ellipse and ellipsoid for two and three-degree-of-freedom motions respectively. At singular configurations, the definition of a metric is no longer valid and the velocity ellipsoid degenerates to an ellipse, a line or a point, and the area or the volume of the velocity ellipse or ellipsoid becomes zero. The second and higher order properties, such as curvature, are also not defined at a singularity. In this paper, we use the rate of change of the area or volume to characterize the singularities of the point trajectory. For parallel manipulators, singularities may lead to either loss or gain of one or more degrees-of-freedom. For loss of degree of freedom, the ellipsoid degenerates to an ellipse, a line, or a point as in serial manipulators. For a gain of degree-of-freedom the singularities can be pictured as growth to lines, ellipses, and ellipsoids. The method presented gives a clear geometric picture as to the possible directions and magnitude of motion at a singularity and the local geometry near a singularity. The theoretical results are illustrated with the help of a general spatial 2R manipulator and a three-degree-of-freedom RPSSPR-SPR parallel manipulator.
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Giorelli, Michele, Federico Renda, Gabriele Ferri, and Cecilia Laschi. "A Feed Forward Neural Network for Solving the Inverse Kinetics of Non-Constant Curvature Soft Manipulators Driven by Cables." In ASME 2013 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/dscc2013-3740.

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The solution of the inverse kinetics problem of soft manipulators is essential to generate paths in the task space to perform grasping operations. To address this issue, researchers have proposed different iterative methods based on Jacobian matrix. However, although these methods guarantee a good degree of accuracy, they suffer from singularities, long-term convergence, parametric uncertainties and high computational cost. To overcome intrinsic problems of iterative algorithms, we propose here a neural network learning of the inverse kinetics of a soft manipulator. To our best knowledge, this represents the first attempt in this direction. A preliminary work on the feasibility of the neural network solution has been proposed for a conical shape manipulator driven by cables. After the training, a feed-forward neural network (FNN) is able to represent the relation between the manipulator tip position and the forces applied to the cables. The results show that a desired tip position can be achieved quickly with a degree of accuracy of 0.73% relative average error with respect to total length of arm.
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Diaconescu, Emanuel. "A Correlation Between Pressure Distribution and Bounding Surfaces in Elastic Contacts." In STLE/ASME 2003 International Joint Tribology Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/2003-trib-0275.

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Many authors attempted to establish a direct correlation between the equation of the surface of an equivalent rigid punch and the pressure distribution arising when this punch is pressed against a corresponding elastic half-space. The results are encouraging, but they have a limited applicability and the involved calculations are still complicated. This paper advances a simple correlation based on a new interpretation of integral condition of deformation of a contact and on the dependence of pressure or pressure gradient on singularities in surface gradient or curvature. Based on this correlation, a method to find the pressure distribution for contacts bound by surfaces described by up to second order polynomials is advanced. The new method is applied in several classical contact problems and a very good agreement is found with existing solutions.
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6

Bonner, David L., Mark J. Jakiela, and Masaki Watanabe. "Pseudoedge: A Hierarchical Skeletal Modeler for the Design of Structural Components." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0118.

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Abstract A new design model for the creation of mechanical components has been developed. In this model, the shape is expressed by its areas of prominence or maximum curvature, for which we use the term pseudoedges. In terms of traditional design, these represent both fillet, chamfer and intersection lines, and more general shape features. The pseudoedges of the model combine with a skeletal shape that is used as a starting form, thereby creating a hierarchy of geometric dependencies that affords both global and local control. The surface is represented by a quilt of parametric Bezier patches, with tangent plane continuity everywhere and only certain isolated singularities. Considerable degrees of deformation are possible, with predictable control and at small computational expense; there is no need for computation of intersections or parameter space trimming of patches.
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7

Bauchau, Olivier A., and Minghe Shan. "Finite Element Models for Flexible Cosserat Solids." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22134.

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Abstract The application of the finite element method to the modeling of Cosserat solids is investigated in detail. In two- and three-dimensional elasticity problems, the nodal unknowns are the components of the displacement vector, which form a linear field. In contrast, when dealing with Cosserat solids, the nodal unknowns form the special Euclidean group SE(3), a nonlinear manifold. This observation has numerous implications on the implementation of the finite element method and raises numerous questions: (1) What is the most suitable representation of this nonlinear manifold? (2) How is it interpolated over one element? (3) How is the associated strain field interpolated? (4) What is the most efficient way to obtain the discrete equations of motion? All these questions are, of course intertwined. This paper shows that reliable schemes are available for the interpolation of the motion and curvature fields. The interpolated fields depend on relative nodal motions only, and hence, are both objective and tensorial. Because these schemes depend on relative nodal motions only, only local parameterization is required, thereby avoiding the occurrence of singularities. For Cosserat solids, it is preferable to perform the discretization operation first, followed by the variation operation. This approach leads to considerable computation efficiency and simplicity.
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