Get A-operator on complete foliated Riemannian manifolds PDF

By Yorozu S.

We provide a generalization of the outcome acquired by means of C. Currais-Bosch. Weconsider the -operator linked to a transverse Killing box v on acomplete foliated Riemannian manifold . less than a undeniable assumption,we end up that, for every belongs to the Lie algebra of the linearholonomy workforce . a different case of our outcome, the model of the foliationby issues, implies the consequences given through B. Kostant (compact case) andC. Currfis-Bosch (non-compact case).

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Example text

The initial population is generated according to an isotropic normal distribution with mean (103, 0, . . , R(0) = 103 and σx1 = σx2 = 1. 42), while the simulation results are averaged over 10 runs of DE experiments. The analytical results well predict the basic trends of simulation progress in spite of a major discrepancy: simulation results show a slower response to the poorly selected initial variances. This might be because the parent population becomes not as close to being normal distributed when the variances increase quickly.

40). As it decreases, the progress rate might become negative for sufficiently large variances. However, this does not conflict with the elitist selection procedure of DE since the elitism merely guarantees the positive progress of the best individual (but not the mean) of the population. 4 (b) – (d). The y-axis represents the relative change, φσ∗ 1 or φσ∗ 2 , of the variance from generation g to g + 1. It is clear that the analytical and experimental results have a good match in most regions except the upper-right corner of the plots.

The ratio σx1 x2 rate. As is shown, the φR∗ curves move upward as the ratio increases. It seems that an ∗ in the progress direction (the direction from the population enhanced variance σx1 mean to the optimum) is able to improve the progress of the algorithm. However, this needs further investigation since the practically achievable progress rate depends not only on the behavior of φR∗ but also on that of φσ∗ 1 and φσ∗ 2 . ∗ and σ ∗ are small and then deThe normalized progress rate climbs fast when σx1 x2 creases slowly after reaching the optimum.

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A-operator on complete foliated Riemannian manifolds by Yorozu S.

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