Edge-coloured cube decompositions

An edge-colored graph is a graph H together with a function f:E(H) → C where C is a set of colors. Given an edge-colored graph H, the graph induced by the edges of color c C is denoted by H(c). Let G, H, and J be graphs and let μ be a positive integer. A (J, H, G, μ) edge-colored graph decomposition is a set S = {H 1,H 2,...,H t} of edge-colored graphs with color set C = {c 1, c 2,..., c k} such that Hi ≅ H for 1 ≤ i ≤ t; Hi (cj) ≅ G for 1 ≤ i ≤ t and ≤ j ≤ k; and for j = 1, 2,..., k, each edge of J occurs in exactly μ of the graphs H 1(c j ), H 2(c j ),..., H t (c j ). Let Q 3 denote the 3-dimensional cube. In this paper, we find necessary and sufficient conditions on n, μ and G for the existence of a (K n ,Q 3,G, μ) edge-colored graph decomposition. © Birkhäuser Verlag, Basel 2007.

Local edge detectors: A substrate for fine spatial vision at low temporal frequencies in rabbit retina

Visual acuity is limited by the size and density of the smallest retinal ganglion cells, which correspond to the midget ganglion cells in primate retina and the beta- ganglion cells in cat retina, both of which have concentric receptive fields that respond at either light- On or light- Off. In contrast, the smallest ganglion cells in the rabbit retina are the local edge detectors ( LEDs), which respond to spot illumination at both light- On and light- Off. However, the LEDs do not predominate in the rabbit retina and the question arises, what role do they play in fine spatial vision? We studied the morphology and physiology of LEDs in the isolated rabbit retina and examined how their response properties are shaped by the excitatory and inhibitory inputs. Although the LEDs comprise only similar to 15% of the ganglion cells, neighboring LEDs are separated by 30 - 40 mu m on the visual streak, which is sufficient to account for the grating acuity of the rabbit. The spatial and temporal receptive- field properties of LEDs are generated by distinct inhibitory mechanisms. The strong inhibitory surround acts presynaptically to suppress both the excitation and the inhibition elicited by center stimulation. The temporal properties, characterized by sluggish onset, sustained firing, and low bandwidth, are mediated by the temporal properties of the bipolar cells and by postsynaptic interactions between the excitatory and inhibitory inputs. We propose that the LEDs signal fine spatial detail during visual fixation, when high temporal frequencies are minimal.

Edge-to-edge matching - The fundamentals

The basis of the present authors' edge-to-edge matching model for understanding the crystallography of partially coherent precipitates is the minimization of the energy of the interface between the two phases. For relatively simple crystal structures, this energy minimization occurs when close-packed, or relatively close-packed, rows of atoms match across the interface. Hence, the fundamental principle behind edge-to-edge matching is that the directions in each phase that correspond to the edges of the planes that meet in the interface should be close-packed, or relatively close-packed, rows of atoms. A few of the recently reported examples of what is termed edge-to-edge matching appear to ignore this fundamental principle. By comparing theoretical predictions with available experimental data, this article will explore the validity of this critical atom-row coincidence condition, in situations where the two phases have simple crystal Structures and in those where the precipitate has a more complex structure.