CanaDAM 2015
Université de la Saskatchewan, 1 - 4 juin 2015

Combinatorial structures and games

ROBERT CRAIGEN, University of Manitoba
Negacyclic Weighing Matrices  [PDF]

Negacyclic matrices are like circulants except that rows are obtained one negates the first entry after circulating the previous row one position to the right. Negacyclic designs present a remarkably versatile counterpoint to circulants. Delsarte, Goethals \& Siedel showed all Paley-class conference matrices equivalent to negacyclic (though never equivalent to circulant). Negacyclic weighing matrices appear in the work of Butson, Berman, Seberry and others, often serendipitously, and under various names. Their pattern of existence was unmapped until I gave the task to some students in summer 2013. I report our findings, fundamental results about existence, and some open problems.

RYAN HAYWARD, University of Alberta
Some results from the game of Hex  [PDF]

We discuss results --- some mathematical, some algorithmic --- and open problems on Hex and variants: who wins (2k+1)-by-n Cylindrical Hex? what's new in Reverse Hex? how close are computers to solving 11x11 Hex problems? and how easy is it to win 11x11 Hex with a 1-stone handicap?

VILAS KHARAT, SP Pune University, Pune, INDIA
Properties of $\phi$-absorbing primary elements in multiplicative lattice modules  [PDF]

Let $M$ be a lattice module over $C$-lattice $L$, $\phi: M \rightarrow M$. A proper element $P \in M$ is $\phi$-absorbing primary if, $x_{1}x_{2} \cdots x_{n}N \leq P$ and $x_{1}x_{2} \cdots x_{n}N \nleq \phi(P)$ imply $x_{1}x_{2} \cdots x_{n} \leq (P:1_M)$ or $x_{1}x_{2} \cdots x_{i-1} x_{i+1}\cdots x_{n}N \leq \sqrt{P}$. Two characterizations of $\phi$-absorbing primary elements are established and some basic properties are derived. Various generalizations of prime and primary elements are unified as $\phi$-absorbing elements and $\phi$-absorbing primary elements. It is proved that, an element is $\omega$-absorbing primary if and only if it is $n$-almost $n$-absorbing primary for $n \geq 2$.

JOSÉ PLÍNIO SANTOS, State University of Campinas-UNICAMP-Brazil
On a new formalism for the Theory of Partitions  [PDF]

In this talk we have a new formalism in dealing with the Theory of Partitions using ideas from the Multiplicative Theory. We define special additive functions to be used in finding, among others, the number of unrestricted partitions of $n$. Classical results can be expressed in this language and, for same, we have been able to find a proof. This is a joint work with Eduardo Bovo.


Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathmatiques The Fields Institute Pacific Institute for the Mathematical Sciences Socit mathmatique du Canada Université Saskatchewan