Maximal Spanning Time for Neighborhood Growth on the Hamming Plane

Abstract

We consider a long-range growth dynamics on the two-dimensional integer lattice, initialized by a finite set of occupied points. Subsequently, a site x becomes occupied if the pair consisting of the counts of occupied sites along the entire horizontal and vertical lines through x lies outside a fixed Young diagram Z. We study the extremal quantity µ(Z), the maximal finite time at which the lattice is fully occupied. We give an upper bound on µ(Z) that is linear in the area of the bounding rectangle of Z, and a lower bound √ s − 1, where s is the side length of the largest square contained in Z. We give more precise results for a restricted family of initial sets, and for a simplified version of the dynamics.

Modeling Supply Chains by Critical Paths and Leontief Input-Output Table

“…[6], [7]), which bears the common construct of a partially ordered set. In addition, we extend the scope of a directed tree of uni-directional edges/paths connecting all the vertices/nodes [8], [9], to a network of mutually dependent economic agents, which then readily leads to Leontief's inputoutput table for the gross domestic product (GDP) [10].As such, Section 2 below will connect CPM to a supply-chain problem, and Section 3 will show how an input-output analysis can address a global disruption over an economy, where we will incorporate the apparatus of elasticities of substitution, proportional changes in the ratio of two factors due to a change in the ratio of their prices. Section 4 will draw a summary.…”