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(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 study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by ℓ the length of the chromosomes, by m the population size, by pC the crossover probability and by pM the mutation probability. We introduce a parameter σ, called the selection drift, which measures the selection intensity of the fittest(More)
Let ρ and µ be two probability measures on R which are not the Dirac mass at 0. We denote by H(µ|ρ) the relative entropy of µ with respect to ρ. We prove that, if ρ is symmetric and µ has a finite first moment, then H(µ|ρ) ≥ R z dµ(z) 2 2 R z 2 dµ(z) , with equality if and only if µ = ρ. We give an applicaion to the Curie-Weiss model of self-organized(More)