Atomic Latin squares of order eleven

A Latin square is pan-Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i j. A Latin square is atomic if all of its conjugates are pan-Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1-factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan-Hamiltonian Latin square of order n describes a perfect 1-factorization of Kn,n, and vice versa. Perfect 1-factorizations of Kn,n can be constructed from a perfect 1-factorization of Kn+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn-square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self-orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self-orthogonal Latin squares in the same main class as a given Latin square.

Constructing and Deconstructing Latin Trades

This paper discusses existence results for latin trades and provides a Glueing Construction which is subsequently used to construct all latin trades of finite order greater than three.

Sampling phylogenetic tree space with the generalized Gibbs sampler

The generalized Gibbs sampler (GGS) is a recently developed Markov chain Monte Carlo (MCMC) technique that enables Gibbs-like sampling of state spaces that lack a convenient representation in terms of a fixed coordinate system. This paper describes a new sampler, called the tree sampler, which uses the GGS to sample from a state space consisting of phylogenetic trees. The tree sampler is useful for a wide range of phylogenetic applications, including Bayesian, maximum likelihood, and maximum parsimony methods. A fast new algorithm to search for a maximum parsimony phylogeny is presented, using the tree sampler in the context of simulated annealing. The mathematics underlying the algorithm is explained and its time complexity is analyzed. The method is tested on two large data sets consisting of 123 sequences and 500 sequences, respectively. The new algorithm is shown to compare very favorably in terms of speed and accuracy to the program DNAPARS from the PHYLIP package.

Isotopes, wool, and rangeland monitoring: Let the sheep do the sampling

Stable carbon isotope analyses of wool staples provided insight into the vegetation consumed by sheep at a temporal resolution not previously studied. Contemporary Australian and historic South African samples dating back to 1916 were analyzed for their stable carbon isotope ratio, a proxy for the proportion of C-3 and C-4 plant species consumed by animals. Sheep sample vegetation continuously throughout a year, and as their wool grows it integrates and stores information about their diet. In subtropical and tropical rangelands the majority of grass species are C-4. Since sheep prefer to graze, and their wool is an isotopic record of their diet, we now have the potential to develop a high resolution index to the availability of grass from a sheep's perspective. Isotopic analyses of wool suggest a new direction for monitoring grazing and for the reconstruction of past vegetation changes, which will make a significant contribution to traditional rangeland ecology and management. It is recommended that isotopic and other analyses of wool be further developed for use in rangeland monitoring programs to provide valuable feedback for land managers.

Critical sets for families of Latin squares

In this paper I give details of new constructions for critical sets in latin squares. These latin squares, of order n, are such that they can be partitioned into four subsquares each of which is based on the addition table of the integers module n/2, an isotopism of this or a conjugate.

Bacteria in shrimp pond sediments: their role in mineralizing nutrients and some suggested sampling strategies

Strategies for sampling sediment bacteria were examined in intensive shrimp, Penaeus monodon (Fabricius), ponds in tropical Australia. Stratified sampling of bacteria at the end of the production season showed that the pond centre, containing flocculated sludge, had significantly higher bacterial counts (15.5 X 10(9) g(-1) dw) than the pond periphery (8.1 X 10(9) g(-1) dw), where the action of aerators had swept the pond floor. The variation in bacterial counts between these two zones within a pond was higher than that between sites within each zone or between ponds. Therefore, sampling effort should be focused within these zones: for example, sampling two ponds at six locations within each of the two zones resulted in a coefficient of variation of approximate to 5%. Bacterial numbers in the sediment were highly correlated with sediment grain size, probably because eroded soil particles and organic waste both accumulated in the centre of the pond. Despite high inputs of organic matter added to the ponds, principally as pelleted feeds, the mean bacterial numbers and nutrient concentrations (i.e. organic carbon, nitrogen and phosphorus) in the sediment were similar to those found in mangrove sediments. This suggests that bacteria are rapidly remineralizing particulates into soluble compounds. Bacterial numbers were highly correlated with organic carbon and total kjeldahl nitrogen in the sediment, suggesting that these were limiting factors to bacterial growth.

Semiochemicals and social signaling in the wild European rabbit in Australia: II. Variations in chemical composition of chin gland secretion across sampling sites

The volatile components of the chin gland secretion of the wild European rabbit, Oryctolagus cuniculus (L.), were investigated with the use of gas chromatography. Studies of the chemical nature of this secretion by previous workers demonstrated that it was important in the maintenance of social structure in this species. This study identified 34 different volatile components that consist primarily of aromatic and aliphatic hydrocarbons. Especially common are a series of alkyl-substituted benzene derivatives that provide most of the compound diversity in the secretion. Samples of chin gland secretion collected from animals at three different geographical locations, separated by more than 100 km, showed significant differences in composition. This work suggests that variation among populations needs to be considered when undertaking semiochemical research. Alternate nonparametric methods are also used for the analysis of chromatographic data.

Critical sets in direct products of back circulant Latin squares

To date very Few families of critical sets for latin squares are known. The only previously known method for constructing critical sets involves taking a critical set which is known to satisfy certain strong initial conditions and using a doubling construction. This construction can be applied to the known critical sets in back circulant latin squares of even order. However, the doubling construction cannot be applied to critical sets in back circulant latin squares of odd order. In this paper a family of critical sets is identified for latin squares which are the product of the latin square of order 2 with a back circulant latin square of odd order. The proof that each element of the critical set is an essential part of the reconstruction process relies on the proof of the existence of a large number of latin interchanges.

On the distance between distinct group Latin squares

In an article in 1992, Drapal addressed the question of how far apart the multiplication tables of two groups can be? In this article we continue this investigation; in particular, we study the interaction between partial equalities in the multiplication tables of the two groups and their subgroup structure. (C) 1997 John Wiley & Sons, Inc.