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.

John Waldridge, The Leaven of the Ancients: Suhrawardi and the Heritage of the Greeks, SUNY Series in Islam (Albany: State University of New York Press, 1999). Pp. 323

In the Leaven of the Ancients, John Walbridge studies the appropriation of non–Peripatetic philosophical ideas by an anti–Aristotelian Islamic philosopher, Shihab al-Din al-Suhrawardi (d. 1191). He proposes a comprehensive explanation of the origin of Suhrawardi's philosophical system, a revival of the “wisdom of the Ancients” and its philosophical affiliations “grounded” in Greek philosophy (p. xiii). Walbridge attempts to uncover the reasons for Suhrawardi's rejection of the prevailing neo–Aristotelian synthesis in Islamic philosophy, Suhrawardi's knowledge and understanding of non–Aristotelian Greek philosophy, the ancient philosophers Suhrawardi was attempting to follow, the relationship between Suhrawardi's specific philosophical teachings (logic, ontology, physics, and metaphysics), and his understanding of non–Aristotelian ancient philosophy and the relationship between Suhrawardi's system and the major Greek philosophers, schools, and traditions—in particular the Presocratics, Plato, and the Stoics (p. 8). Copyright © 2003 Cambridge University Press

Social-conservatism, Australian politics and cricket: The triumvirate of Prime Minister John Howard, Sir Robert Menzies and Sir Donald Bradman

This Article does not have an abstract.

Cultural frictions: John Romeril's The 'Floating World'

Hailed as an 'unruly masterpiece', John Romeril's The Floating World is one of the few 'new wave' Australian plays representing Australians and their Asian 'others' to be restaged periodically since its premiere in 1974. Paying particular attention to production of the play that have used Japanese theatre forms such as kabuki and bunraku, this article focuses primarily on the ways in which the significations of race have been interpreted by the critical establishment. The fascinating stage history of The Floating World is treated as a barometer of Australian theatre's response to the challenge of representing cultural conflict, during a period marked by public debate about the desirability, and inevitability, of Australia's political, economic and cultural 'enmeshment' with Asia.