Dipole statistics of discrete finite images: two visually motivated representation theorems

Dipole statistics of discrete finite images: two visually motivated representation theorems.

A discrete finite image I is a function assigning colors to a finite, rectangular array of discrete pixels. A dipole is a triple, ((dR, dC), alpha, beta), where dR and dC are vertical and horizontal, integer-valued displacements and alpha and beta are colors. For any such dipole, D(I)((dR, dC), alpha, beta) gives the number of pixel pairs ((r1, c1), (r2, c2)) of I such that I[r1, c1] = alpha, I...

Two geometric representation theorems for separoids

Separoids capture the combinatorial structure which arises from the separations by hyperplanes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid S can be represented by a family of convex sets in the (|S| − 1)dimensional Euclidian space. The geometric dimension of the separoid is the minimum dimension where it can be represented a...

Event Structures as Presheaves -Two Representation Theorems

The category of event structures is known to embed fully and faithfully in the category of presheaves over pomsets. Here a characterisation of the presheaves represented by event structures is presented. The proof goes via a characterisation of the presheaves represented by event structures when the morphisms on event structures are “strict” in that they preserve the partial order of causal dep...

Two Embedding Theorems for Finite Lattices

Introduction. One of the important unsolved problems of lattice theory is the problem of embedding every finite lattice in a finite partition lattice. The partition lattice can be considered as the lattice of subspaces of a suitable geometry. By slightly loosening the conditions on this geometry it will be shown that every finite lattice can be embedded in the lattice of subspaces of a finite g...

Convergence Theorems For Two Finite Families of Uniformly L-Lipschitzian Mappings