MSI Weekly Bulletin - Week starting Monday 13 August, 2007

A $t-(v,k,\lambda_{t})$ design D, for $t\geq 0$, is an ordered pair $(V,B)$, where $V$ is a set of $v$ elements, called points, $B$ is a collection (multiset) of $k$-subsets of $V$, called blocks, such that each $t$-subset of $V$ belongs to exactly $\lambda_{t}$ blocks. A defining set of a $t-(v,k,\lambda_{t}$) design is a set of blocks which is a subset of a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper subsets is a defining set. A smallest defining set is one with smallest cardinality. This talk will summaries the ideas of earlier algorithms and proposes a new and more efficient algorithm that finds all non-isomorphic minimal defining sets of a given $t$-design. The complete list of minimal defining sets of the full $2-(7,3,5)$ design, $2-(15,3,1)$ designs, $2-(25,5,1)$ design and $2-(31,6,1)$ design were found. Some new theoretical technics will also be given for finding the minimal defining sets of $t-(v,k,\lambda_{t})$ designs.

Narrow operators have been called the largest practical class of operators which satisfy the Daugavet equation (||T+Id||=1+||T||). A Banach space is said to have the Daugavet property if every rank one operator satisfies the Daugavet equation. In this seminar I will explore the links between the action of narrow operators and the geometry of spaces with the Daugavet property.

The Wiener-Hopf integral equation is (1) $Z(x)=z(x)+\int_{-\infty}^x Z(x-y)\mu(\dd y),\ \ x\ge 0$. It originates from mathematical physics problems, but if $\mu$ is a probability measure, it has numerous applications to random walks. The classical method of solution involves factorizing $1-\widehat\mu$ as $H_+,H_-$, where $\widehat\mu$ is the Fourier transform of $\mu$ and $H_+,H_-$ are analytic in each their halfplane. It was recognized by Feller that the Wiener-Hopf factors can be given a probabilistic interpretation. I will present an overview of the probabilistic aspects with main emphasis on explicit solutions and a purely probabilistic approach to the integral equation (1). S. Asmussen (1992) Phase-type representations in random walk and queueing problems. {\em Ann. Probab.} {\bf 20}, 772--789. S.Asmussen (1998) A probabilistic view of the Wiener-Hopf equation. {\em SIAM Review} {\bf 40}, 189--201.