Living in to its Name and Living up to its Legacy: The public practical theology of KAIROS

KAIROS Canada looks hopefully towards the emerging challenges of continuing public practical theology in a pluralistic, post-apology time and place

An LpL^p theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.Comment: 44 page

On generating a diminimal set of polyhedral maps on the torus

We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus