Barlow's tables of squares, cubes, square roots, cube roots, reciprocals of all integer numbers up to 10,000.

Mode of access: Internet.

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.

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.

The Boundary Descriptors of the n-dimensional Unit Cube Subset Partitioning

* The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be

Distance Distributions and Energy of Designs in Hamming Spaces

We obtain new combinatorial upper and lower bounds for the potential energy of designs in q-ary Hamming space. Combined with results on reducing the number of all feasible distance distributions of such designs this gives reasonable good bounds. We compute and compare our lower bounds to recently obtained universal lower bounds. Some examples in the binary case are considered.

L'échantillonnage équilibré par la méthode du cube et la méthode rejective

Au cours de ces dernières années, les techniques d’échantillonnage équilibré ont connu un regain d’intérêt. En effet, ces techniques permettent de reproduire la structure de la population dans des échantillons afin d’améliorer l’efficacité des estimations. La reproduction de cette structure est effectuée par l’introduction des contraintes aux plans de sondage. Encore récemment, des nouvelles procédures d’échantillonnage équilibré ont été proposées. Il s’agit notamment de la méthode du cube présentée par Deville et Tillé (2004) et de l’algorithme réjectif de Fuller (2009). Alors que la première est une méthode exacte de sélection, la seconde est une approche approximative qui admet une certaine tolérance dans la sélection. Alors, après une brève présentation de ces deux méthodes dans le cadre d’un inventaire de pêcheurs, nous comparons à l’aide de simulations Monte Carlo, les plans de sondage produits par ces deux méthodes. Aussi, cela a été l’occasion pour nous de vérifier si ces méthodes modifient les probabilités de sélection des unités.