Raphaël Cerf

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The set of the three dimensional polyominoes of minimal area and of volume n contains a polyomino which is the union of a quasicube j × (j + δ) × (j + θ), δ, θ ∈ {0, 1}, a quasisquare l × (l + ), ∈ {0, 1}, and a bar k. This shape is naturally associated to the unique decomposition of n = j(j +δ)(j +θ)+ l(l+ )+k as the sum of a maximal quasicube, a maximal(More)
We give a short proof of Cramér’s large deviations theorem based on convex duality. This proof does not resort to the law of large numbers or any other limit theorem. The most fundamental result in probability theory is the law of large numbers for a sequence (Xn)n≥1 of independent and identically distributed real-valued random variables. Define the(More)
We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets. Several works have been devoted to deriving limit theorems for random sets. For i.i.d. random compact sets in R, the law of large numbers was initially proved by Artstein and Vitale [1] and(More)
Based on speculations coming from statistical mechanics and the conjectured existence of critical states, I propose a simple heuristic in order to control the mutation probability and the population size of a genetic algorithm. Genetic algorithms are widely used nowadays, as well as their cousins evolutionary algorithms. The most cited initial references on(More)
We consider the classical Wright–Fisher model of population genetics. We prove the existence of an error threshold for the mutation probability, below which a quasispecies is formed. We show a new phenomenon, specific to a finite population model, namely the existence of a population threshold: to ensure the stability of the quasispecies, the population(More)