Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K-n, is a 3-factorization of & in which the components of each factor are cubes. We show that there exists a cube factorization of & if and only if n equivalent to 16 (mod 24), thus providing a new family of uniform 3 -factorizations as well as a partial solution to an open problem posed by Kotzig in 1979. (C) 2004 Wiley Periodicals, Inc.

A PbS quantum-cube: conducting polymer composite for photovoltaic applications

We have developed a new non-polar synthesis for lead sulfide (PbS) quantum-cubes in the conjugated polymer poly-2-methoxy, 5-(2-ethyl-hexyloxy-p-phenylenevinylene) MEH-PPV. The conducting polymer acts to template and control the quantum-cube growth. Transmission electron microscopy of the composites has shown a bimodal distribution of cube sizes between 5 and 15 nm is produced with broad optical absorption from 300 to 650 nm. Photoluminescence suggests electronic coupling between the cubes and the conducting polymer matrix. The synthesis and initial characterization are presented in this paper. (C) 2003 Elsevier B.V. All rights reserved.

Helicoverpa armigera (Hubner): can wheat stubble protect cotton plants against attack?

When investigating strategies for Helicoverpa armigera (Hubner) control, it is important to understand oviposition behaviour. Cotton (Gossypium hirsutum) was sown into standing wheat (Triticum astivum L.) stubble in a closed arena to investigate the effect of stubble on H. armigera moth behaviour and oviposition. Infrared cameras were used to track moths and determine whether stubble acted as a physical barrier or provided camouflage to cotton plants, thereby reducing oviposition. Searching activity was observed to peak shortly before dawn (03:00 and 04:00 h) and remained high until just after dawn (4 h window). Moths spent more time resting on cotton plants than spiralling above them, and the least time flying across the arena. While female moths spent more time searching for cotton plants growing in wheat stubble, the difference in oviposition was not significant. As similar numbers of eggs were laid on cotton plants with stubble (3.5/plant SE +/- 0.87) and without stubble (2.5/plant SE +/- 0.91), wheat stubble does not appear to provide camouflage to cotton plants. There was no significant difference in the location of eggs deposited on cotton plants with and without stubble, although more eggs were laid on the tops of cotton leaves in wheat stubble. As the spatial and temporal distribution of eggs laid on the cotton plant is a crucial component of population stability, eggs laid on the upper side of leaves on cotton plants may be more prone to fatalities caused by environmental factors such as wind and rain. Therefore, although stubble did not influence the number of eggs laid, it did affect their distribution on the plant, which may result in increased mortality of eggs on cotton plants sown into standing wheat stubble.

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.

Optical trapping of a cube

The successful development and optimisation of optically-driven micromachines will be greatly enhanced by the ability to computationally model the optical forces and torques applied to such devices. In principle, this can be done by calculating the light-scattering properties of such devices. However, while fast methods exist for scattering calculations for spheres and axisymmetric particles, optically-driven micromachines will almost always be more geometrically complex. Fortunately, such micromachines will typically possess a high degree of symmetry, typically discrete rotational symmetry. Many current designs for optically-driven micromachines are also mirror-symmetric about a plane. We show how such symmetries can be used to reduce the computational time required by orders of magnitude. Similar improvements are also possible for other highly-symmetric objects such as crystals. We demonstrate the efficacy of such methods by modelling the optical trapping of a cube, and show that even simple shapes can function as optically-driven micromachines.