Math 500 - Topology and Geometry

Math 500 - Topology and Geometry

A note on approximation conditions, standard triangularizability and a power set topology

The main result of this article is that for collections of entry-wise non-negative matrices the property of possessing a standard triangularization is stable under approximation. The methodology introduced to prove this result allows us to offer quick proofs of the corresponding results of [B. R. Yahaghi, Near triangularizability implies triangularizability, Canad. Math. Bull. 47, (2004), no. 2...

A Geometry Preserving Kernel over Riemannian Manifolds

Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...

Math 631 Notes Algebraic Geometry Lectures

1 Algebraic sets, affine varieties, and the Zariski topology 4 1.1 Algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Hilbert basis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Proof that affine algebraic sets form closed sets on a t...

Construction of Hexahedral Block Topology and its Decomposition to Generate Initial Tetrahedral Grids for Aerodynamic Applications

Making an initial tetrahedral grid for complex geometry can be a tedious and time consuming task. This paper describes a novel procedure for generation of starting tetrahedral cells using hexahedral block topology. Hexahedral blocks are arranged around an aerodynamic body to form a flow domain. Each of the hexahedral blocks is then decomposed into six tetrahedral elements to obtain an initial t...