Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.

It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studing arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in 1 and ζ(4) = π 4 /90 yielding a conditional upper bound for the irrationality measure of ζ(4); (2) a second-order Apéry-like recursion for ζ(4) and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for ζ(2) and ζ(3).

Abstract. We give a detailed survey of the Nelson-Oppen method for combining decision procedures, we show how Shostak's method can be seen as an instance of the Nelson-Oppen method, and we provide a generalization of the Nelson-Oppen method to the case of non-disjoint theories. 1 Introduction Decision procedures are algorithms that can reason about the validity or satisfiability of classes of formulae in a given decidable theory, and always terminate with a positive or negative answer.

Rissanen's Minimum Description Length (MDL) principle is a statistical modeling principle motivated by coding theory. For exponential families we obtain pathwise expansions, to the constant order, of the predictive and mixture code lengths used in MDL. The results are useful for understanding different MDL forms.

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The sum of all the ladder and rainbow diagrams in φ3 theory near 6 dimensions leads to self-consistent higher order differential equations in coordinate space which are not particularly simple for arbitrary dimension D. We have now succeeded in solving these equations, expressing the results in terms of generalized hypergeometric functions; the expansion and representation of these functions can then be used to prove the absence of renormalization factors which are transcendental for this theory and this topology to all orders in perturbation theory. The correct anomalous scaling dimensions of the Green functions are also obtained in the six-dimensional limit.

On the spectra of certain Integro-Differential-Delay problems with applications in neurodynamics by Peter Grindrod and Dimitri PinotsisOn the spectra of certain Integro-Differential-Delay problems with applications in neurodynamics