arXiv subjects

Emil Saucan

Publications and source records attributed to Emil Saucan.

40 records · Page 3Linked to original sources

arXiv2004

Note on a Theorem of Munkres

We prove that given a $\mathcal{C^\infty}$ Riemannian manifold with boundary, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result holds extends to $\mathcal{C}^1$ manifolds, and that in dimensions $2,3$ and 4 it also holds for $PL$ manifolds. We employ the main result to prove that given any orientable $\mathcal{C^\infty}$ Riemannian manifold with boundary admits quasimeromorphic mappings onto $\hat{\mathbb{R}^n}$. In addition some generalizations are given.

math.GT↗

arXiv2004

The Existence of Quasimeromorphic Mappings in Dimension 3

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{3}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (i.e torsion) elements are uniformly bounded. This is accomplished by developing a technique for meshing distinct fat triangulations while preserving fatness. We further show how to adapt the proof to higher dimensions.

math.CV↗

arXiv2004

Surface Triangulation -- The Metric Approach

We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their Haussdorff-Gromov limit. This allows us to define in a more natural way the relevant elements, constants and invariants s.a. principal directions and principal values, Gaussian and Mean curvature, etc. By a "natural way" we mean an intrinsic, discrete, metric definitions as opposed to approximating or paraphrasing the differentiable notions. In this way we hope to circumvent computational errors and, indeed, conceptual ones, that are often inherent to the classical, "numerical" approach. In this first study we consider the problem of determining the Gaussian curvature of a polyhedral surface, by using the {\em embedding curvature} in the sense of Wald (and Menger). We present two modalities of employing these definitions for the computation of Gaussian curvature.

cs.GR↗

arXiv2003

Digital Version of Green`s Theorem and its Application to The Coverage Problem in Formal Verification

We present a novel scheme to the coverage problem, introducing a quantitative way to estimate the interaction between a block and its enviroment.This is achieved by setting a discrete version of Green`s theorem, specially adapted for Model Checking based verification of integrated circuits.This method is best suited for the coverage problem since it enables one to quantify the incompleteness or, on the other hand, the redundancy of a set of rules, describing the model under verification.Moreover this can be done continuously throughout the verification process, thus enabling the user to pinpoint the stages at which incompleteness/redundancy occurs. Although the method is presented locally on a small hardware example, we additionally show its possibility to provide precise coverage estimation also for large scale systems. We compare this method to others by checking it on the same test-cases.

cs.SC↗