Constructing a class of designs with proportional balance

Text serves as a sequel to 'Computational and Constructive Design Theory,' c1996; containing research papers and surveys of recent research work on design construction and computer-aided study of designs. For researchers in theory of computational designs.

Public choice of species for the ark: Phylogenetic similarity and preferred wildlife species for survival

Humans play a role in deciding the fate of species in the current extinction wave. Because of the previous Similarity Principle, physical attractiveness and likeability, it has been argued that public choice favours the survival of species that satisfy these criteria at the expense of other species. This paper empirically tests this argument by considering a hypothetical ‘Ark’ situation. Surveys of 204 members of the Australian public inquired whether they are in favour of the survival of each of 24 native mammal, bird and reptile species (prior to and after information provision about each species). The species were ranked by percentage of ‘yes’ votes received. Species composition by taxon in various fractions of the ranking was determined. If the previous Similarity Principle holds, mammals should rank highly and dominate the top fractions of animals saved in the hierarchical list. We find that although mammals would be over-represented in the ‘Ark’, birds and reptiles are unlikely to be excluded when social choice is based on numbers ‘voting’ for the survival of each species. Support for the previous Similarity Principle is apparent particularly after information provision. Public policy implications of this are noted and recommendations are given.

The mu-way intersection problem for m-cycle systems

An m-cycle system of order upsilon is a partition of the edge-set of a complete graph of order upsilon into m-cycles. The mu -way intersection problem for m-cycle systems involves taking mu systems, based on the same vertex set, and determining the possible number of cycles which can be common to all mu systems. General results for arbitrary m are obtained, and detailed intersection values for (mu, m) = (3, 4), (4, 5),(4, 6), (4, 7), (8, 8), (8, 9). (For the case (mu, m)= (2, m), see Billington (J. Combin. Des. 1 (1993) 435); for the case (Cc,m)=(3,3), see Milici and Quattrochi (Ars Combin. A 24 (1987) 175. (C) 2001 Elsevier Science B.V. All rights reserved.

The metamorphosis of lambda-fold 4-wheel systems into lambda-fold 4-cycle systems

A 4-wheel is a simple graph on 5 vertices with 8 edges, formed by taking a 4-cycle and joining a fifth vertex (the centre of the 4-wheel) to each of the other four vertices. A lambda -fold 4-wheel system of order n is an edge-disjoint decomposition of the complete multigraph lambdaK(n) into 4-wheels. Here, with five isolated possible exceptions when lambda = 2, we give necessary and sufficient conditions for a lambda -fold 4-wheel system of order n to be transformed into a lambda -fold Ccyde system of order n by removing the centre vertex from each 4-wheel, and its four adjacent edges (retaining the 4-cycle wheel rim), and reassembling these edges adjacent to wheel centres into 4-cycles.

Star decompositions of cubes

Let Sk denote the complete bipartite graph K-1k and let e,, denote the ii-cube. We prove that the obvious necessary conditions for the existence of an S-k-decomposition of Q(n) are sufficient.

A new integrable model of strongly correlated electrons with Hubbard-like interaction

A new completely integrable model of strongly correlated electrons is proposed which describes two competitive interactions: one is the correlated one-particle hopping, the other is the Hubbard-like interaction. The integrability follows from the fact that the Hamiltonian is derivable from a one-parameter family of commuting transfer matrices. The Bethe ansatz equations are derived by algebraic Bethe ansatz method.

On single laws for varieties of quasigroups associated with extended cycle systems

It has been previously shown by Lindner and Rodger that quasigroups associated with 2-perfect extended m-cycle systems can be equationally defined if and only if m is an element of {3, 5, 7}. In this paper we present a single identity for each such m which is equivalent to the identities given for these varieties.

Proportionally balanced designs: Some further results and general constructions

Proportionally balanced designs were introduced by Gray and Matters in response to a need for the allocation of markers of the Queensland Core Skills Test to have a certain property. Put simply, markers were allocated to pairs of units in proportions that reflected the relative numbers of markers allocated in total to each unit. In this paper, the first author extends the theoretical results relating to such designs and provides further instances, and two general constructions, in the case that the design comprises blocks of precisely two sizes.

Smallest weak and smallest totally weak critical sets in the latin squares of order at most seven

A critical set in a latin square of order n is a set of entries in a latin square which can be embedded in precisely one latin square of order n. Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one latin square of order n. In this paper we find smallest weak and smallest totally weak critical sets for all the latin squares of orders six and seven. Moreover, we computationally prove that there is no (totally) weak critical set in the back circulant latin square of order five and we find a totally weak critical set of size seven in the other main class of latin squares of order five.