We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting tra...

The skyline operator returns records in a dataset that provide optimal trade-offs of multiple dimensions. State-of-theart skyline computation involves complex tree traversals, data-ordering, and conditional branching to minimize the number of point-to-point comparisons. Meanwhile, GPGPU computing offers the potential for parallelizing skyline computation across thousands of cores. However, atte...

We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is lik...

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting tra...

Branching stochastic processes can be considered as models in population dynamics, where the objects have a random lifetime and reproduction following some stochastic laws. Typical examples are nuclear reactions, cell proliferation and biological reproduction, some chemical reactions, economics and financial phenomena. In this survey paper we try to present briefly some of the most important an...

It is known that branching strands of cardiac tissue can form a substrate for very slow conduction. The branches slow conduction by acting as current loads drawing depolarizing current from the main strand ("pull" effect). It has been suggested that, upon depolarization of the branches, they become current sources reinjecting current back into the strand, thus enhancing propagation safety ("pus...

Branching morphogenesis is a critical step in the development of many epithelial organs. The phosphoinositide-3-kinase (PI3K) pathway has been identified as a central component of this process but the precise role has not been fully established. Herein we sought to determine the role of PI3K in murine lung branching using a series of pharmacological inhibitors directed at this pathway. The pan-...

This note discusses the peridynamic horizon (the nonlocal region around a material point), its role, and practical use in modeling. The objective is to eliminate some misunderstandings and misconceptions regarding the peridynamic horizon. An example of crack branching in a nominally brittle material (homalite) is addressed and we show that crack branching takes place without wave interaction. W...