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Many processes in nature are governed by nonlinear mechanisms that are frequently superposed by pronounced linear components. To characterize such complex dynamics it is desirable to disentangle linear and nonlinear features in empirical data. Quantitative nonlinear measures are also influenced by linear properties of the signals and, for the univariate case, Fourier transform surrogates do not represent properly the null hypothesis of zero nonlinear features. Here we elaborate on the application of a recently published method in Fourier space that does not suffer from comparison with inappropriate surrogate data and that reveals with high sensitivity correlations between Fourier phases and amplitudes. In addition, we propose a simple pre-processing procedure that avoids the mixing of linear and nonlinear features when applying conventional nonlinear measures, which can drastically increase the sensitivity of the statistical evaluation and avoids numerical problems associated with surrogate data. We test our proposal on data derived from numerical models and we analyze electroencephalographic recordings from epilepsy patients as well as heart rate signals from healthy subjects before and during meditation. [ABSTRACT FROM AUTHOR]

Abstract: Pressure transmission and drag force fluctuation in a slowly plunged granular column are measured with various sample preparations, grains shapes, and cyclic penetration tests. First, we slightly modify the nonlinear scaling laws for drag force and pressure transmission, which have been reported in Katsuragi [2012. Nonlinear wall pressure of a plunged granular column. Phys. Rev. E 85, 021301]. Next, we find that the modified scaling relations are almost independent of details of grains and sample preparations, particularly in vertical direction. By ten times cyclic penetration tests, we reveal that the first penetration is quite particular than other successive ones. After the second penetration, the work to penetrate into the granular column becomes independent of the cycle number. Namely, there are two groups, such as the first penetration and others. From the drag force fluctuation analysis, we find that the fluctuation is almost proportional to the square root of the size ratio between grains and the intruder. [Copyright &y& Elsevier]

Abstract: Parabolic reaction–diffusion systems may develop sharp moving reaction fronts which pose a challenge even for adaptive finite element methods. We propose a method to transform the equation into an equivalent form that usually exhibits solutions which are easier to discretize, giving higher accuracy for a given number of degrees of freedom. The transformation is realized as an efficiently computable pointwise nonlinear scaling that is optimized for prototypical planar travelling wave solutions of the underlying reaction–diffusion equation. The gain in either performance or accuracy is demonstrated on different numerical examples. [Copyright &y& Elsevier]

Abstract: The Central Limit Theorem (CLT) and Extreme Value Theory (EVT) study, respectively, the stochastic limit-laws of sums and maxima of sequences of independent and identically distributed (i.i.d.) random variables via an affine scaling scheme. In this research we study the stochastic limit-laws of populations of i.i.d. random variables via nonlinear scaling schemes. The stochastic population-limits obtained are fractal Poisson processes which are statistically self-similar with respect to the scaling scheme applied, and which are characterized by two elemental structures: (i) a universal power-law structure common to all limits, and independent of the scaling scheme applied; (ii) a specific structure contingent on the scaling scheme applied. The sum-projection and the maximum-projection of the population-limits obtained are generalizations of the classic CLT and EVT results — extending them from affine to general nonlinear scaling schemes. [Copyright &y& Elsevier]