Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure--a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

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In this follow-up of our earlier two works D (11.1)(arXiv: 1406.0929 [math. DG]) and D (11.2)(arXiv: 1412.0771 [hep-th]) in the D-project, we study further the notion of adifferentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture is made that the notion of differentiable maps from Azumaya/matrix manifolds as defined in D (11.1) is equivalent to one defined through the contravariant ring-homomorphisms alone. A proof of this conjecture for the smooth (ie C^{\infty} ) case is given in this note. Thus, at least in the smooth case, our setting for D-branes in the realm of differential geometry is completely parallel to that in the realm of algebraic geometry, cf.\arXiv: 0709.1515 [math. AG] and arXiv: 0809.2121 [math. AG]. A related conjecture on such maps to C^{\infty} , as a C^{\infty} -manifold, and its proof in the C^{\infty} case is also given. As a by-product, a conjecture on a division lemma in the finitely differentiable case that generalizes the division lemma in the smooth case from Malgrange is given in the end, as well as other comments on the conjectures in the general C^{\infty} case. We remark that there are similar conjectures in general and theorems in the smooth case for the fermionic/super generalization of the notion.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature dimension inequality C D E(n, K), which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are the first kind of results in this direction for unbounded Laplacian on graphs.