Definition. Let X be a metric space with metric d. (a) A collection {G α } α∈A of open sets is called an open cover of X if every x ∈ X belongs to at least one of the G α , α ∈ A. An open cover is finite if the index set A is finite. (b) X is compact if every open cover of X contains a finite subcover. Definition. Let X be a metric space with metric d and let A ⊂ X. We say that A is a compact s...

A recent study reveals that the propagation of intercellular calcium signals is closely associated with the generation of convergent extension movements during Xenopus gastrulation. Such signals provide a mechanism whereby large populations of cells can communicate to generate orchestrated cell movements.

Epithelial cells frequently display--in addition to the common apical-basolateral polarity--a polarization within the plane of the epithelium. This is commonly referred to as planar cell polarity (PCP) or tissue polarity. Examples of vertebrate PCP include epithelial patterning in the skin and inner ear, and also the morphogenetic movements of mesenchymal cells during convergent extension at ga...

We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre’s condition Sr . In analogy with the sequentially CohenMacaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton...