In this paper we characterise the structural transition in random mappings with in-degree restrictions. Specifically, for integers 0 6 r 6 n, we consider a random mapping model T̂ r n from [n] = {1, 2, . . . , n} into [n] such that Ĝn, the directed graph on n labelled vertices which represents the mapping T̂ r n , has r vertices that are constrained to have in-degree at most 1 and the remaining v...

We present dimensionless effective transport coefficients (formation factors) for arrays of cylinders of various orientation distributions, namely, cylinders randomly positioned and oriented with their axes parallel to a line, parallel to a plane, or in the three-dimensional space with no preferred orientation. Both cases of conducting cylinders dispersed in a nonconducting matrix and nonconduc...

The problem of compact tables is to maximise the overlap when building a word that is to include permutations of every given words (all the words being the same length). This problem is shown to be NP-complete in the general case, and some specific restrictions are studied. 1. Presentation of the problem 1.1. Informal description and examples In several fields, it can be helpful to give a compa...

Intestinal enteroendocrine cells secrete hormones that are vital for the regulation of glucose metabolism but their differentiation from intestinal stem cells is not fully understood. Asymmetric stem cell divisions have been linked to intestinal stem cell homeostasis and secretory fate commitment. We monitored cell divisions using 4D live cell imaging of cultured intestinal crypts to characteri...

We illustrate and discuss several general issues associated with the random component of utility, or more generally ‘‘unobserved variability’’. We posit a general conceptual framework that suggests a variance components view as an appropriate structure for unobserved variability. This framework suggests that ‘‘unobserved heterogeneity’’ is only one component of unobserved variability; hence, a ...

We study Doob’s martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, D...