Relevant bibliographies by topics / Elliptic manifold

Academic literature on the topic 'Elliptic manifold'

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Published: 10 December 2022
Last updated: 28 January 2023

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Journal articles on the topic "Elliptic manifold"

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Bridges, Thomas J. "Canonical multi-symplectic structure on the total exterior algebra bundle." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2069 (February 8, 2006): 1531–51. http://dx.doi.org/10.1098/rspa.2005.1629.

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The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.
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Paternain, Gabriel P. "On the topology of manifolds with completely integrable geodesic flows." Ergodic Theory and Dynamical Systems 12, no. 1 (March 1992): 109–21. http://dx.doi.org/10.1017/s0143385700006623.

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AbstractWe show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy of M is finite dimensional. We also show that rational ellipticity is shared by simply connected compact manifolds whose cotangent bundle admits a multiplicity free compact action that leaves invariant the Hamiltonian associated with some Riemannian metric. In particular it follows that if M is a Riemannian manifold whose geodesic flow is completely integrable by the Thimm method, then M is rationally elliptic. Other questions concerning the global behaviour of geodesics on homogeneous spaces are discussed.
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Rogov, Vasily. "Complex Geometry of Iwasawa Manifolds." International Mathematics Research Notices 2020, no. 23 (November 7, 2018): 9420–39. http://dx.doi.org/10.1093/imrn/rny230.

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Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.
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Terzic, Svjetlana. "On geometric formality of rationally elliptic manifolds in dimensions 6 and 7." Publications de l'Institut Math?matique (Belgrade) 103, no. 117 (2018): 211–22. http://dx.doi.org/10.2298/pim1817211t.

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We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2 = 3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2 ? 2 and b3 = 0 can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well.
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EZHOV, V. V., A. V. ISAEV, and G. SCHMALZ. "INVARIANTS OF ELLIPTIC AND HYPERBOLIC CR-STRUCTURES OF CODIMENSION 2." International Journal of Mathematics 10, no. 01 (February 1999): 1–52. http://dx.doi.org/10.1142/s0129167x99000021.

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We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves "almost" like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of "semi-flat" manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.
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Fan, Haining. "Existence Theorems for a Class of Edge-Degenerate Elliptic Equations on Singular Manifolds." Proceedings of the Edinburgh Mathematical Society 58, no. 2 (February 17, 2015): 355–77. http://dx.doi.org/10.1017/s0013091514000145.

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AbstractIn this paper we establish the Nehari manifold on edge Sobolev spaces and study some of their properties. Furthermore, we use these results and the mountain pass theorem to get non-negative solutions of a class of edge-degenerate elliptic equations on singular manifolds under different conditions.
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Fegan, H. D., and B. Steer. "First Order Operators on Manifolds With a Group Action." Canadian Journal of Mathematics 48, no. 4 (August 1, 1996): 758–76. http://dx.doi.org/10.4153/cjm-1996-039-6.

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AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.
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Park, Jiewon. "Matrix Inequality for the Laplace Equation." International Mathematics Research Notices 2019, no. 11 (September 27, 2017): 3485–97. http://dx.doi.org/10.1093/imrn/rnx226.

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Abstract Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci flow, Kähler–Ricci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold with nonnegative Ricci curvature. In this article, we prove a related matrix inequality on manifolds with suitable curvature and volume growth assumptions.
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De Filippis, Cristiana, and Giuseppe Mingione. "Manifold Constrained Non-uniformly Elliptic Problems." Journal of Geometric Analysis 30, no. 2 (September 20, 2019): 1661–723. http://dx.doi.org/10.1007/s12220-019-00275-3.

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Ashyralyev, A., Y. Sozen, and F. Hezenci. "A remark on elliptic differential equations on manifold." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 99, no. 3 (September 30, 2020): 75–85. http://dx.doi.org/10.31489/2020m3/75-85.

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For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.
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Dissertations / Theses on the topic "Elliptic manifold"

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Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : V. Index formulas for elliptic problems on manifolds with edges." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2650/.

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For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.
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Nazaikinskii, Vladimir, and Boris Sternin. "On surgery in elliptic theory." Universität Potsdam, 2000. http://opus.kobv.de/ubp/volltexte/2008/2587/.

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We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
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Nazaikinskii, Vladimir, Bert-Wolfgang Schulze, and Boris Sternin. "Localization problem in index theory of elliptic operators." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2617/.

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This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.
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Lu, Nan. "Normally elliptic singular perturbation problems: local invariant manifolds and applications." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41090.

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In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative per- turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external forcing and other potential fields.
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Egorov, Yu, V. Kondratiev, and Bert-Wolfgang Schulze. "On completeness of eigenfunctions of an elliptic operator on a manifold with conical points." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2593/.

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Schulze, Bert-Wolfgang, Vladimir E. Nazaikinskii, and Boris Yu Sternin. "On the homotopy classification of elliptic operators on manifolds with singularities." Universität Potsdam, 1999. http://opus.kobv.de/ubp/volltexte/2008/2557/.

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We study the homotopy classification of elliptic operators on manifolds with singularities and establish necessary and sufficient conditions under which the classification splits into terms corresponding to the principal symbol and the conormal symbol.
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Nazaikinskii, Vladimir E., and Boris Yu Sternin. "Surgery and the relative index in elliptic theory." Universität Potsdam, 1999. http://opus.kobv.de/ubp/volltexte/2008/2553/.

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We prove a general theorem on the local property of the relative index for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions) this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities as well as for elliptic boundary value problems with a symmetry condition for the conormal symbol.
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Delengov, Vladimir. "Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions." Scholarship @ Claremont, 2018. https://scholarship.claremont.edu/cgu_etd/113.

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In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.
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Guillermou, Stéphane. "Classe de Lefschetz des paires elliptiques." Paris 6, 1995. http://www.theses.fr/1995PA066339.

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A la donnee d'une variete analytique complexe, d'une paire elliptique sur cette variete (i. E. Un d-module et un faisceau constructible dont les varietes caracteristiques n'ont pas d'intersection hors de la section nulle du fibre cotangent), d'un morphisme de cette variete dans elle-meme et d'un releve de ce morphisme pour la paire elliptique nous associons une classe de cohomologie microlocale. Nous montrons une formule d'image directe relative et une formule de produit pour cette classe, nous la calculons dans le cas transverse et en etudions des deformations. Ceci permet en particulier de retrouver la formule de lefschetz des complexes elliptiques de m. Atiyah et r. Bott
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Lekaus, Silke. "Vector bundles of degree zero over an elliptic curve, flat bundles and Higgs bundles over a compact Kähler manifold." [S.l. : s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=964273802.

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Books on the topic "Elliptic manifold"

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Kirk, P. Analytic deformations of the spectrum of a family of Dirac operators on an odd-dimensional manifold with boundary. Providence, RI: American Mathematical Society, 1996.

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Elliptic structures on 3-manifolds. Cambridge [Cambridgeshire]: Cambridge University Press, 1986.

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Hong, Sungbok. Diffeomorphisms of Elliptic 3-Manifolds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.

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Hong, Sungbok, John Kalliongis, Darryl McCullough, and J. Hyam Rubinstein. Diffeomorphisms of Elliptic 3-Manifolds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-31564-0.

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Nazaĭkinskiĭ, V. E. Elliptic theory on singular manifolds. Boca Raton: Chapman & Hall/CRC, 2006.

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1953-, Pinchover Yehuda, ed. Manifolds with group actions and elliptic operators. Providence, R.I: American Mathematical Society, 1994.

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Schapira, Pierre. Index theorem for elliptic pairs. Paris: Société mathématique de France, 1994.

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1922-, Markus L., ed. Elliptic partial differential operators and symplectic algebra. Providence, RI: American Mathematical Society, 2003.

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Compactness and stability for nonlinear elliptic equations. Zürich, Switzerland: European Mathematical Society, 2014.

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Werner, Müller. L²-index of elliptic operators on manifolds with cusps of rank one. Berlin: Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut für Mathematik, 1985.

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Book chapters on the topic "Elliptic manifold"

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Levendorskii, Serge. "Some Classes of Hypoelliptic Pseudodifferential Operators on Closed Manifold." In Degenerate Elliptic Equations, 203–44. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_8.

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Hörmander, Lars. "Elliptic Operators on a Compact Manifold Without Boundary." In The Analysis of Linear Partial Differential Operators III, 180–230. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-49938-1_4.

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Ghimenti, Marco, and Anna Maria Micheletti. "Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary." In Contributions to Nonlinear Elliptic Equations and Systems, 299–323. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19902-3_19.

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Léandre, R. "Brownian Motion Over a Kahler Manifold and Elliptic Genera of Level N." In Stochastic Analysis and Applications in Physics, 193–217. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0219-3_8.

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Habal, N., W. Rungrottheera, and B. W. Schulze. "A Class of Elliptic Operators on a Manifold with Edge and Boundary." In Fourier Analysis, 187–209. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02550-6_9.

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Hirzebruch, Friedrich, Thomas Berger, and Rainer Jung. "Elliptic genera." In Manifolds and Modular Forms, 23–33. Wiesbaden: Vieweg+Teubner Verlag, 1992. http://dx.doi.org/10.1007/978-3-663-14045-0_2.

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Hirzebruch, Friedrich, Thomas Berger, and Rainer Jung. "Elliptic genera." In Manifolds and Modular Forms, 23–33. Wiesbaden: Vieweg+Teubner Verlag, 1994. http://dx.doi.org/10.1007/978-3-663-10726-2_2.

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Taylor, J. "The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold 𝑀." In CRM Proceedings and Lecture Notes, 153–202. Providence, Rhode Island: American Mathematical Society, 2001. http://dx.doi.org/10.1090/crmp/028/06.

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Singer, I. M. "Elliptic Operators on Manifolds." In Pseudo-differential Operators, 333–75. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11074-0_9.

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Amann, Herbert. "Uniformly Regular and Singular Riemannian Manifolds." In Elliptic and Parabolic Equations, 1–43. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-12547-3_1.

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Conference papers on the topic "Elliptic manifold"

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Klimenko, Elena, and Natalia Kopteva. "Kleinian orbifolds uniformized by RP groups with an elliptic and a hyperbolic generators." In Heegaard splittings of 3--manifolds. Mathematical Sciences Publishers, 2007. http://dx.doi.org/10.2140/gtm.2007.12.121.

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Ulucakli, M. Erol. "Chaotic Mixing of Highly Viscous Liquids With Rectangular or Elliptical Rotors." In ASME 2005 International Mechanical Engineering Congress and Exposition. ASMEDC, 2005. http://dx.doi.org/10.1115/imece2005-81036.

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The objective of this research is to experimentally investigate various mixing regions in a two-dimensional Stokes flow driven by a rectangular or elliptical rotor. Flow occurs in a rectangular cell filled with a very viscous fluid. The Reynolds number based on rotor size is in the order of 0.5. The flow is time-periodic and can be analyzed, both theoretically and experimentally, by considering the Poincare map that maps the position of a fluid particle to its position one period later. The mixing regions of the flow are determined, theoretically, by the fixed points of this map, either hyperbolic or degenerate, and their stable and unstable manifolds. Experimentally, the mixing regions are visualized by releasing a blob of a passive dye at one of these fixed points: as the flow evolves, the blob stretches to form a streak line that lies on the unstable manifold of the fixed point.
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Yejun Gu, Rong Li, and Hanyang You. "Some remarks on nonlinear elliptic operators on Riemannian manifolds." In 2011 International Conference on Multimedia Technology (ICMT). IEEE, 2011. http://dx.doi.org/10.1109/icmt.2011.6002632.

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Duduchava, R., T. Buchukuri, O. Chkadua, and D. Natroshvili. "Interface Cracks Problems in Composites With Piezoelectric and Thermal Effects." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-13352.

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We investigate three–dimensional interface crack problems (ICP) for metallic-piezoelectric composite bodies with regard to thermal effects. We give a mathematical formulation of the physical problem when the metallic and piezoelectric bodies are bonded along some proper parts of their boundaries where interface cracks occur. By potential methods the ICP is reduced to an equivalent strongly elliptic system of pseudodifferential equations (ψDEs) on overlapping manifolds with boundary, which have no analogues in mathematical literature. We study the solvability of obtained ψDEs on overlapping manifolds with boundary by reduction to ψDEs on non-overlapping manifolds with boundary in different function spaces. These general results are applied to prove the uniqueness and the existence theorems for the original ICP-Problem.
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