FIG. 15.2 Lévy process simulation: a) Simulation of the Pareto process Par(α, β, γt; u) as defined in ( 15.30 ), with α = 1.7, β = −0.1, γ = 88.3. The simulation has been performed using ε = 0.01 in the principal value integrals in ( 15.5 ), and this represents the minimum size of step available to the simulation. The simulation was performed for 500 000 jumps, leading to a total time of 500 time units. The figure plots the value of u at times that are multiples of Δt = 0.5 time units. b) The jumps Δu between successive values of u at times that are multiples of Δt = 0.5 obtained from the simulation in a). c) Simulation of the distribution of Par(α, β, γΔt; x) at time Δt = 0.5 obtained from the data of b). More about this image found in Lévy process simulation: a) Simulation of the Pareto process Par(...

FIG. 3.3 a)  Blue curve: Linear plot showing a log-normal distribution p _log‐normal (x), with x₀ = 10, and σ = 1. Pink curve: A comparable Pareto distribution with α = 1.8 and x₀ = 10. b)  Blue curve: Log-log plot of the log-normal distribution function p _log‐normal (x), with x₀ = 10, and σ = 1. Black dots: For comparison, we plot 20 000 simulated points from this distribution, assigned to 200 bins. Pink plot: A Pareto distribution with α = 1.8 and x₀ = 10. It can be seen that this fits the data in the simulated tail as credibly as does the log-normal distribution. More about this image found in a)  Blue curve: Linear plot showing a log-normal distribut...

FIG. 5.9 (a) Comparison between the target colors and the colors obtained with one of the optimal solutions found on the 3D Pareto front. The table below corresponds to the geometrical parameters of the Pareto solution. (b) Reflexion spectra of the optimal structure presented in (a). (c) Distribut... More about this image found in (a) Comparison between the target colors and the colors obtained with one o...

FIG. 5.8 3D Pareto fronts obtained by minimizing (g₁, g₂, g₃), two realizations with different initial random population are reported. Inset: 3D Pareto front shown in the projected a^∗b^∗ plane. More about this image found in 3D Pareto fronts obtained by minimizing (g₁, ...

FIG. 3.2 Log-log plot of population size S versus rank r of the 134 largest U. S. metropolitan areas in 1991 [ 3.6 ]. The pink line represents Zipf’s law, that is, the equation S = 3.9 × 10⁷/r, which closely matches the asymptotic behavior of the data at large r and small S. In contrast, the blue line represents the equation S = 2.2 × 10⁷ r^−η, with η = 0.75, which closely matches the asymptotic behavior of the data at large S and small r. This corresponds to a form of Pareto’s law, that is p(S) = 1.62 × 10⁻¹⁰S^−(1+α), with α = 1/η = 1.33. More about this image found in Log-log plot of population size S versus rank r...

FIG. 3.1 a) Plot of the data on income distribution in the USA in 2007, as reported in [ 3.1 ], together with a corresponding Pareto distribution fit f ( I ) = α I 0 α / I α + 1 , with α = 1.6 and I₀ = $35,126. In b) the same data are plotted on a log-log plot. The value of α chosen fits the slope of the curve for incomes greater than about $60 000, and the value of I₀ ensures that the areas under both curves are the same and equal to one. The data set plotted excludes very high incomes, either positive or negative, such that |I| > $418,000. The full range of incomes is from about −$500 000 to $190 000 000. More about this image found in a) Plot of the data on income distribution in the USA in 2007, as reported...