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Full-Text Articles in Physical Sciences and Mathematics
Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern
Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern
Mathematics Department Faculty Scholarship
A hypersurface Mn-1 in Euclidean space En is proper Dupin if the number of distinct principal curvatures is constant on Mn-1, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together …
The Poincaré Duality Theorem And Its Applications, Natanael Alpay, Melissa Sugimoto, Mihaela Vajiac
The Poincaré Duality Theorem And Its Applications, Natanael Alpay, Melissa Sugimoto, Mihaela Vajiac
SURF Posters and Papers
In this talk I will explain the duality between the deRham cohomology of a manifold M and the compactly supported cohomology on the same space. This phenomenon is entitled “Poincaré duality” and it describes a general occurrence in differential topology, a duality between spaces of closed, exact differentiable forms on a manifold and their compactly supported counterparts. In order to define and prove this duality I will start with the simple definition of the dual space of a vector space, with the definition of a positive definite inner product on a vector space, then define the concept of a manifold. …
A Robust Hyperviscosity Formulation For Stable Rbf-Fd Discretizations Of Advection-Diffusion-Reaction Equations On Manifolds, Varun Shankar, Grady B. Wright, Akil Narayan
A Robust Hyperviscosity Formulation For Stable Rbf-Fd Discretizations Of Advection-Diffusion-Reaction Equations On Manifolds, Varun Shankar, Grady B. Wright, Akil Narayan
Mathematics Faculty Publications and Presentations
We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds �� ⊂ ℝ3 of codimension 1. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and …