Accession Number : ADA619846


Title :   Discrete Ricci Flow in Higher Dimensions


Descriptive Note : Final rept. Sep 2011-Sep 2014


Corporate Author : FLORIDA ATLANTIC UNIV BOCA RATON DEPT OF PHYSICS


Personal Author(s) : Miller, Warner A


Full Text : https://apps.dtic.mil/dtic/tr/fulltext/u2/a619846.pdf


Report Date : Feb 2015


Pagination or Media Count : 38


Abstract : The objective of this research to develop an efficient and justifiable algorithm to geometrize a given closed 3-manifold, and to show how its topologic characterization can be applied to complex networks. Hamilton's Ricci flow (RF) was developed in order to geometrize such 3-manifolds. The geometrization theorem (GT) states that each prime 3-manifold is either geometric or its simple pieces are geometric. The continuum approach is not numerically practical. Accordingly, we developed a discrete piecewise linear (PL) version of Hamilton's RF. It is the first dimensionally agnostic generalization of RF for PL geometries. We refer to our approach as simplicial Ricci flow (SRF). For a broad class of examples, the SRF equations reproduced the continuum RF. SRF provides an efficient approach to the 3-manifold recognition problem. Student S. Ray applied SRF to implement the 1916 Weyl isometric embedding problem. His results are being used to develop a discrete quasi-local measure of congestion in networks -- a possible filtration parameter to guide network reconfiguration.


Descriptors :   *ALGORITHMS , *MANIFOLDS(MATHEMATICS) , GEOMETRIC FORMS


Subject Categories : Numerical Mathematics
      Theoretical Mathematics


Distribution Statement : APPROVED FOR PUBLIC RELEASE