This paper deals with q{series arising from the study of the transitiveclosure problem in random acyclic digraphs. In particularit presents an identity involvingdivisor generatingfunctions which allows to determine the asymptotic behavior of polynomials de ned by a general class of recursive equations, including the polynomials for the mean and the variance of the size of the transitive closure in random acyclic digraphs.

No abstract

No abstract

In this paper we consider the Markov process defined byP1,1=1, Pn,[lscr ]=(1−λn,[lscr ]) ·Pn−1,[lscr ] +λn,[lscr ]−1 ·Pn−1,[lscr ]−1for transition probabilities λn,[lscr ]=q[lscr ] and λn,[lscr ]=qn−1. We give closed forms for the distributions and the moments of the underlying random variables. Thereby we observe that the distributions can be easily described in terms of q-Stirling numbers of the second kind. Their occurrence in a purely time dependent Markov process allows a natural approximation for these numbers through the normal distribution. We also show that these Markov processes describe some parameters related to the study of random graphs as well as to the analysis of algorithms.

Apesar do problema da quadratura do círculo, isto é, o problema de construir um quadrado tendo a mesma área que a de um círculo dado, permanecer um problema aberto entre os matemáticos do começo do século xvii, e de Descartes ter até mesmo declarado a impossibilidade de sua solução, ele próprio havia fornecido uma solução, datada dos anos de 1625-1628. Neste artigo, examinarei essa solução comparando-a a uma análise feita por Euler um século mais tarde e também a uma solução conhecida desde os antigos e apresentada por Pappus. Interrogar-me-ei, em seguida, sobre as razões que conduziram Descartes a excluir as duas construções por serem inaceitáveis em relação ao ideal de exatidão explicitado em A geometria de 1637.
Although the problem of squaring the circle, that is, the problem of constructing a square having the same area of a given circle, was considered an open problem among early XVIIth century mathematicians, René Descartes affirmed that it could not be solved. On the other hand, he himself provided a solution to the problem, that can be dated in the period, 1625-1628. In this article, I will examine this solution by comparing it to an analysis made a century later by Euler, and to a solution known to the ancients and discussed by Pappus. I will investigate, successively, the reasons that led Descartes to dismiss these two solutions as not acceptable in the light of the ideal of exactness deployed in the Geometry