A Generalized Coupon Collector Problem

This paper presents an analysis of a generalized version of the coupon collector problem, in which the collector receives d coupons each run and chooses the least-collected coupon so far. In the asymptotic case when the number of coupons n goes to infinity, we show that, on average, (nlogn) / d + (n / d)(m − 1)log logn + O(mn) runs are needed to collect m sets of coupons. An exact algorithm is also developed for any finite case to compute the exact mean number of runs. Numerical examples are provided to verify our theoretical predictions.

A Generalized Coupon Collector Problem

This paper presents an analysis of a generalized version of the coupon collector problem, in which the collector receives d coupons each run and chooses the least-collected coupon so far. In the asymptotic case when the number of coupons n goes to infinity, we show that, on average, (nlogn) / d + (n / d)(m − 1)log logn + O(mn) runs are needed to collect m sets of coupons. An exact algorithm is also developed for any finite case to compute the exact mean number of runs. Numerical examples are provided to verify our theoretical predictions.

Lusztig’sa-Function in TypeBnin the Asymptotic Case

In this paper, we study Lusztig’s a-function for a Coxeter group with unequal parameters. We determine that function explicitly in the “asymptotic case” in typeBn, where the left cells have been determined in terms of a generalized Robinson-Schensted correspondence by Bonnafé and the second author. As a consequence, we can show that all of Lusztig’s conjectural properties (P1)–(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the “leading matrix coefficients” introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori-Hecke algebraHnin terms of the Dipper-James-Murphy basis ofHn.

Computer simulations of voting systems

All voting procedures are susceptible to give rise, if not to paradoxes, at least to violations of some democratic principles. In this paper, we evaluate and compare the propensity of various voting rules -belonging to the class of scoring rules- to satisfy two versions of the majority principle. We consider the asymptotic case where the numbers of voters tends to infinity and, for each rule, we study with the help of Monte Carlo methods how this propensity varies as a function of the number of candidates.

Hypergraph Packing and Graph Embedding

We provide sufficient conditions for packing two hypergraphs. The emphasis is on the asymptotic case when one of the hypergraphs has a bounded degree and the other is dense. As an application, we give an alternative proof for the bipartite case of the recently developed Blow-up Lemma [12].

On the motion of particles in non-uniformly vibrating liquid

The motion of a rigid particle in an inviscid non-uniformly vibrating ambient liquid is considered. The vibrations are caused by a dipole changing its strength in time. This model of a vibrator presents an asymptotic case of vibrations caused by a rigid sphere periodically changing its position when the radius and amplitude are small and the velocity is large. It is found that periodic oscillations with zero mean can cause the directed motion of a submerged particle even if its density equals that of the liquid. The direction of the motion is studied. It is shown that particles of density not less that of the ambient liquid are attracted by a vibrator. The direction of motion of lighter particles depends on their initial position.