Dynamic programming in cricket: optimizing batting order for a sticky wicket

In cricket, a rain-affected pitch can make batting more difficult than normal. Several other conditions such as poor light or an initially lively pitch, may also result in difficulties for the batsmen. In this note, we refer to all of them as ‘sticky wickets’. On sticky wickets, lower order batsmen are often sent into ‘hold the fort’ until conditions improve. In this paper, a stochastic dynamic programming model is used to examine the appropriateness of this policy. The model suggests that the tactic is often optimal when the sticky wicket persists until the end of the day's play, but not often when the sticky wicket is transitory. In some circumstances, it is worthwhile, on a normal wicket near the end of the day, to send in a lower order batsman to hold the fort (a night watchman): when the wicket is sticky, this tactic is even more worthwhile.     

The critical group from a cryptographic perspective

The critical group of a graph is an abelian group that arisesin several contexts, and there are some similarities with thegroups that are used in cryptography. We construct a familyof graphs with critical groups that are cyclic, and discussthe associated computational problems using algorithms basedon the theory of ‘chip-firing’.     

Translation planes of large dimension admitting nonsolvable groups

In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq ³ admittingGL(2,q) as a collineation group.     

Dependence of slow crack growth in polyethylene on butyl branch density: Morphology and theory

A theoretical equation has been developed to described the rate of slow crack growth in an ethylene-hexene copolymer in terms of the basic morphological parameters. These parameters are spacing of the butyl branches, number of tie molecules, and the thickness of the lamellar crystal. Experimentally, the thickness of the lamellae and the long period were determined as functions of the branch density. The calculation of the number of tie molecules is based on the values of the molecular weight and the long period. The model of slow crack growth is based on the rate of disentanglement of the tie molecules. The rate of disentanglement varies inversely with the number of tie molecules and directly with the number of tie molecules that are not pinned by the branches.     

The weak pigeonhole principle for function classes in S 12

It is well known that S ¹₂ cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective (dual) weak pigeonhole principle in S ¹₂ for provably weaker function classes. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transversal-Free Translation Nets

Translation planes of order q ² containing non-Desarguesian Baer subplanes are used to construct transversal-free translation nets with very small deficiencies. Also, a generalization of the ideas of Bruen shows that any non-Desarguesian spread in PG(3, q) produces a transversal-free net of small deficiency.