Main pavilion, Lake Weyba House, Sunshine Coast

South-West elevation of main pavilion - living, kitchen, dining and study/guest areas.

Bedroom pavilion, Lake Weyba House, Sunshine Coast

As seen from outdoor seating area. Pivoting vent on left and day bed alcove on right. Standard roller doors to clerestory.

Bath house pavilion, Lake Weyba House, Sunshine Coast

South-West elevation. Paved seating area in foreground.

Bath house pavilion, Lake Weyba House, Sunshine Coast

Link to bedroom pavilion on right; roof of main pavilion on left.

North-East elevation, Lake Weyba House, Sunshine Coast

Bedroom pavilion on left, bath house pavilion centre and main pavilion on right.

North-East elevation, Lake Weyba House, Sunshine Coast

As seen from East, looking along pavilions. Bedroom pavilion and day bed alcove in foreground.

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.

A Fibred Tableau Calculus for Modal Logics of Agents

In previous works we showed how to combine propositional multimodal logics using Gabbay's \emph{fibring} methodology. In this paper we extend the above mentioned works by providing a tableau-based proof technique for the combined/fibred logics. To achieve this end we first make a comparison between two types of tableau proof systems, (\emph{graph} $\&$ \emph{path}), with the help of a scenario (The Friend's Puzzle). Having done that we show how to uniformly construct a tableau calculus for the combined logic using Governatori's labelled tableau system \KEM. We conclude with a discussion on \KEM's features.

Trades and Graphs

The trade spectrum of a simple graph G is defined to be the set of all t for which it is possible to assemble together t copies of G into a simple graph H, and then disassemble H into t entirely different copies of G. Trade spectra of graphs have applications to intersection problems, and defining sets, of G-designs. In this investigation, we give several constructions, both for specific families of graphs, and for graphs in general.

Packing complete multipartite graphs with 4-cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given.