Common multiples of complete graphs and a 4-cycle

A graph G is a common multiple of two graphs H-1 and H-2 if there exists a decomposition of G into edge-disjoint copies of H-1 and also a decomposition of G into edge-disjoint copies of H-2. In this paper, we consider the case where H-1 is the 4-cycle C-4 and H-2 is the complete graph with n vertices K-n. We determine, for all positive integers n, the set of integers q for which there exists a common multiple of C-4 and K-n having precisely q edges. (C) 2003 Elsevier B.V. All rights reserved.

On the forced matching numbers of bipartite graphs

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M, such that S is contained in no other perfect matching of G. This notion has arisen in the study of finding resonance structures of a given molecule in chemistry. Similar concepts have been studied for block designs and graph colorings under the name defining set, and for Latin squares under the name critical set. There is some study of forcing sets of hexagonal systems in the context of chemistry, but only a few other classes of graphs have been considered. For the hypercubes Q(n), it turns out to be a very interesting notion which includes many challenging problems. In this paper we study the computational complexity of finding the forcing number of graphs, and we give some results on the possible values of forcing number for different matchings of the hypercube Q(n). Also we show an application to critical sets in back circulant Latin rectangles. (C) 2003 Elsevier B.V. All rights reserved.

The Fa,b,c conjecture, I

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and one part of the conjecture was proved. The conjecture completely determines all finite members of the family. Here we prove another part of the conjecture and foreshadow a paper which completes the proof of the other three parts.

The Fa,b,c conjecture is true, II

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. (c) 2006 Elsevier Inc. All rights reserved.

Standardising bull breeding soundness evaluations and reporting in Australia

There is substantial variation in bull breeding soundness evaluation procedures and reports in Australia; the situation is compounded by difficulties in interpretation and the validity of many reports. In an effort to overcome this, the scientific literature was reviewed [Fordyce G. In: Fordyce G, editor. Bull fertility: selection and management in Australia. Eight Mile Plains, Australia: Australian Cattle Vets; 2002] and the needs of stakeholders were considered in preparing a manual, Evaluating and Reporting Bull Fertility [Entwistle KW, Fordyce G. Evaluating and reporting bull fertility. Eight Mile Plains, Australia: Australian Cattle Vets; 2003.] that outlined standards for assessing and reporting bull breeding soundness. A new recording and reporting system, called Bull Reporter, is based on standards from this manual and groups bull fertility traits into five summary categories: Scrotum, Physical, Crush-side Semen, Sperm Morphology, and Serving. The client will generally select which categories they wish to have included in the evaluation to suit their specific purposes. While there is adequate room for comments, the veterinarian is not required to make an overall judgment of whether the bull has normal capacity to sire calves under natural mating management, but ensures the standards for each selected category are met. Professional, standardised, easy-to-read reports are produced either electronically [Entwistle KW, Fordyce G. Evaluating and reporting bull fertility. Eight Mile Plains, Australia: Australian Cattle Vets; 2003.] or manually. A bull owner or their agent signs the certificate to affirm that bulls have not undergone procedures to rectify faults which may have otherwise caused them to fail the standards. An accreditation system for assessing sperm morphology was established because of its demonstrated relationship with pregnancy rates and because of the difficulties in achieving consistent and accurate assessments among laboratories. It is considered that Bull Reporter is applicable to beef and dairy bulls across all levels of management, genotypes and environments throughout Australia, with substantial potential for application elsewhere in the world. Crown Copyright (c) 2006 Published by Elsevier Inc. All rights reserved.

Some equitably 3-colourable cycle decompositions

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has n(i) vertices coloured c(i), i = 1, 2,..., k, and (i) - n(j) less than or equal to 1 for any i, j is an element of {1, 2,..., k}, then C is equitably k-coloured. An m-cycle decomposition C of a graph G is equitably k-colourable if the vertices of G can be coloured so that every m-cycle in C is equitably k-coloured. For m = 4,5 and 6, we completely settle the existence problem for equitably 3-colourable m-cycle decompositions of complete graphs and complete graphs with the edges of a 1-factor removed. (C) 2004 Elsevier B.V. All rights reserved.

The impact of market forces on the sustainability of the biotechnology industry

his paper contains a warning for investors, executives, analysts and scientists about the sustainability of the biotechnology industry. The study upon which the paper is based examines the impact of market forces on the biotechnology industry and argues that the short-term focus of market driven policies and practices impacts on the sustainability of firms operating in the industry. The market is represented by the National Association of Securities Dealers, Automated Quotations Market (NASDAQ), considered to be one of the vehicles of the promotion of ''new economy'' companies and principles. Through the application of bibliometric data (using both refereed and non-refereed papers), matched with the long term tracking of the NASDAQ Biotechnology Index, the authors provide a clear indication that the short-term investment thinking is leading an industry that is characterised by long R&D cycles. There is an incompatibility between the shorter-term investment considerations and the long-term scientific developments the biotechnology industry is attempting to achieve. Graphs and illustrations are provided to portray the comparative data.

Single patient (n-of-1) trials with binary treatment preference

The use of a fully parametric Bayesian method for analysing single patient trials based on the notion of treatment 'preference' is described. This Bayesian hierarchical modelling approach allows for full parameter uncertainty, use of prior information and the modelling of individual and patient sub-group structures. It provides updated probabilistic results for individual patients, and groups of patients with the same medical condition, as they are sequentially enrolled into individualized trials using the same medication alternatives. Two clinically interpretable criteria for determining a patient's response are detailed and illustrated using data from a previously published paper under two different prior information scenarios. Copyright (C) 2005 John Wiley & Sons, Ltd.

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.

Evolutionary theory of history

Several attempts have been made recently to apply Darwinian evolutionary theory to the study of culture change and social history. The essential elements in such a theory are that variations occur in population, and that a process of selective retention operates during their replication and transmission. Location of such variable units in the semantic structure of cognition provides the individual psychological basis for an evolutionary theory of history. Selection operates on both the level of cognition and on its phenotypic expression in action in relation to individual preferred sources of psychological satisfaction. Social power comprises the principal selective forces within the unintended consequences of action and through the struggle of individuals and groups in pursuit of opposing interests. The implication for historiography are methodological in that evolutionary theory of history sharpens the focus of explanatory situational analysis, and interpretive in that it provides a paradigmatic metanarrative for the understanding of historical change.