In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in  for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wilking have for dimensions twelve and above,… (More)
— According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cup will start lifting up. In contrast, physicists believe… (More)
Many problems in computation and data analysis can be framed by graphs and solved with graph algorithms. A graph, which is defined as a set of vertices connected by edges, as shown on the left in Figure 1, adapts well to presenting data and relationships. Graphs take two forms: a directed graph has edges with orientation as shown in Figure 1, and an… (More)
We present a novel architecture for sparse pattern processing, using flash storage with embedded accelerators. Sparse pattern processing on large data sets is the essence of applications such as document search, natural language processing, bioinformatics, subgraph matching, machine learning, and graph processing. One slice of our prototype accelerator is… (More)
This paper discusses the initial inverse heat problem (backward heat problem) with time-dependent coefficient. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. Two regularization solutions of the backward heat problem will be given by a modified quasi-boundary value method. The Hölder type… (More)
Opinions, interpretations, conclusions d d ti th f th th d t il d d b th U it d St t G t 999999-1 XYZ 11/3/2010 MIT Lincoln Laboratory and recommendations are those of the authors and are not necessarily endorsed by the United States Government.