arXiv subjects

Erin Pearse

Publications and source records attributed to Erin Pearse.

2 recordsLinked to original sources

arXiv2019

Condensed Ricci Curvature of Complete and Strongly Regular Graphs

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive exact Ricci curvature formulas for strongly regular graphs of girth 4 and 5 using elementary methods. An example is provided that shows there is no exact formula for the Ricci curvature for strongly regular graphs of girth $3$ that is purely in terms of graph parameters.

math.CO↗

arXiv2017

Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise linking between such two Hilbert spaces when it is assumed that D is dense in one of the two; but generally not in the other. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and operator theory of reflection positivity.

math.FA↗