Abstract. We design a method of decomposing convex polytopes into simpler polytopes. This decomposition yields a way of calculating exactly the volume of the polytope, or, more generally, multiple integrals over the polytope, which is equivalent to the way suggested in [9], based on FourierMotzkin elimination ([10, pp. 155-157]). Our method is applicable for finding uniform decompositions of certain natural families of polytopes. Moreover, this allows us to find algorithmically an analytic expression for the distribution function of a random variable of the formis a random vector, uniformly distributed in a polytope.1. Introduction. The indefinite integral of a function is in general "smoother" than the function itself. However, it is also usually more difficult to express. Thus, the integral of an elementary function is usually non-elementary. The value of a definite integral may not be a "recognizable" number even if the function is quite simple. The situation is even more difficult for multiple integrals; these can be seldom evaluated exactly. Therefore, there is an abundance of methods for approximating the values of definite integrals.One situation where multiple integrals may be exactly calculated is where the region of integration is a polytope and the function very special. For example, if the function is constant, the problem reduces to the computation of the volume of the polytope. For the problem of calculating more general multiple integrals see, for example, [2] and [8]. While the problem of finding the exact volume of a polytope is ♯P -hard [4], which implies that no efficient algorithm should be expected, there are nevertheless several algorithms for it. These algorithms start with decomposing the given polytope into a union 2000 Mathematics Subject Classification: 65D18, 60-08. Key words and phrases: decomposition of polyhedra, repetitive polyhedron, calculating volume.[243]

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We study the dichotomous choice model under conditions of uncertainty. In this model, a committee of decision makers is required to select one of two alternatives, of which exactly one is correct. A decision rule translates the individual opinions into a group decision. We focus on the direction concerned with the identification of the optimal decision rule under partial information on the decision skills. Namely, we assume the correctness probabilities of the committee members to be independent random variables, selected from some given distribution. In addition, we assume that the ranking of the members in the committee is known. Thus, one can follow rules based on this ranking.One of the commonly used measures of the efficiency of a decision rule is its probability of being optimal. Here we provide a method for an explicit calculation of this probability for any given weighted majority rule for a wide family of distribution functions. Moreover, under the assumption of exponentially distributed decision skills, we provide an improved algorithm for this calculation. We illustrate our results with various examples.

In this paper, we study some periodicity concepts on words. First, we extend the notion of full tilings which was recently introduced by Karhumäki, Lifshits, and Rytter to partial tilings. Second, we investigate the notion of quasiperiods and show in particular that the set of quasiperiodic words is a context-sensitive language that is not context-free, answering a conjecture by Dömösi, Horváth and Ito.

We analyze the statistics of electrostatic energies (and their differences) for a quantum dot system composed of a finite number K of electron islands (metallic grains) with random capacitance-inductance matrix C, for which the total charge is discrete, Q = N e (where e is the charge of an electron and N is an integer). The analysis is based on a generalized charging model, where the electrons are distributed among the grains such that the electrostatic energy E(N ) is minimal. Its second difference (inverse compressibility) χ N = E(N + 1) − 2E(N ) + E(N − 1) represents the spacing between adjacent Coulomb blockade peaks appearing when the conductance of the quantum dot is plotted against gate voltage. The statistics of this quantity has been the focus of experimental and theoretical investigations during the last two decades. We provide an algorithm for calculating the distribution function corresponding to χ N and show that this function is piecewise polynomial.