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Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

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This paper solves the problem of finding exact formulas for the waiting time cdf and queue length distribution of first-in-first-out M/G/1 queues in equilibrium with Pareto service. The formulas derived are new and are obtained by directly inverting the relevant Pollaczek-Khinchin formula and involve single integrals of non-oscillating real valued functions along the positive real line. Tables of waiting time and queue length probabilities are provided for certain parameter values under heavy traffic conditions.

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When claims in the compound Poisson risk model are from a heavy-tailed distribution (such as the Pareto or the lognormal), traditional techniques used to compute the probability of ultimate ruin converge slowly to desired probabilities. Thus, faster and more accurate methods are needed. Product integration can be used in such situations to yield fast and accurate estimates of ruin probabilities because it uses quadrature weights that are suited to the underlying distribution. Tables of ruin probabilities for the Pareto and lognormal distributions are provided.

Andrews and Curtis conjectured in 1965 that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Recent computational work by Miasnikov and Myasnikov on this problem has been based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest) and to consider the length thirteen case for two generators. We prove that, up to equivalence, there is a unique minimum potential counterexample.