Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applications of such quantum groupoid and quantum algebroid representations to quasicrystalline structures...

In earlier work the authors introduced dynamic asymptotic dimension, a notion of dimension for topological dynamical systems that applies to many interesting examples. In this paper, we use finiteness of dynamic asymptotic dimension to get information on the K-theory of the associated crossed product C ̊algebras: specifically, we give a new proof of the Baum-Connes conjecture for such actions. T...

We apply the idea of a topological quantum field theory (TQFT) to maps from manifolds into topological spaces. This leads to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed d-dimensional manifolds and (d + 1)-dimensional cobordisms endowed with homotopy classes of maps into a given space. For a group π, we introduce cohomologica...

in this paper, we presented algebroids and crossed modules of algebroids. we also presented pullback crossed module of algebroids and we then define pushout crossed modules of algebroids.

We present a construction which shows that the Tiordered reflection (i ∈ {0, 1, 2}) of a partially ordered topological space (X, τ,≤) exists and is an ordered quotient of (X, τ,≤). We give an explicit construction of the T0-ordered reflection of an ordered topological space (X, τ,≤), and characterize ordered topological spaces whose T0-ordered reflection is T1-ordered. 2000 AMS Classification: ...

The brainstem auditory pathway of the chicken were used to examine the relationship between the maintenance of dendrites following denervation and the successful reinnervation (sprouting) by surviving afferents. In the system the third-order cells in n. laminaris receive spatially segregated binaural innervation from n. magnocellularis. Afferents from the ipsilateral n. magnocellularis innervat...