Order statistics of 1/f^{\alpha} signals

Order statistics of periodic, Gaussian noise with 1/f^{\alpha} power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d_k= between the k-th and (k+1)-st largest values of the signal. The result d_k ~ 1/k known for independent, identically distributed variables remains valid for 0<\alpha<1. Nontrivial, \alpha-dependent scaling exponents d_k ~ k^{(\alpha -3)/2} emerge for 1<\alpha<5 and, finally, \alpha-independent scaling, d_k ~ k is obtained for \alpha>5. The spectra of average ordered values \epsilon_k= ~ k^{\beta} is also examined. The exponent {\beta} is derived from the gap scaling as well as by relating \epsilon_k to the density of near extreme states. Known results for the density of near extreme states combined with scaling suggest that \beta(\alpha=2)=1/2, \beta(4)=3/2, and beta(infinity)=2 are exact values. We also show that parallels can be drawn between \epsilon_k and the quantum mechanical spectra of a particle in power-law potentials.Comment: 8 pages, 5 figure

Collective motion of groups of self-propelled particles following interacting leaders

In order to keep their cohesiveness during locomotion gregarious animals must make collective decisions. Many species boast complex societies with multiple levels of communities. A common case is when two dominant levels exist, one corresponding to leaders and the other consisting of followers. In this paper we study the collective motion of such two level assemblies of self-propelled particles. We present a model adapted from one originally proposed to describe the movement of cells resulting in a smoothly varying coherent motion. We shall use the terminology corresponding to large groups of some mammals where leaders and followers form a group called a harem. We study the emergence (self-organization) of sub-groups within a herd during locomotion by computer simulations. The resulting processes are compared with our prior observations of a Przewalski horse herd (Hortobagy, Hungary) which we use as results from a published case study. We find that the model reproduces key features of a herd composed of harems moving on open ground, including fights for followers between leaders and bachelor groups (group of leaders without followers). One of our findings, however, does not agree with the observations. While in our model the emerging group size distribution is normal, the group size distribution of the observed herd based on historical data have been found to follow lognormal distribution. We argue that this indicates that the formation (and the size) of the harems must involve a more complex social topology than simple spatial-distance based interactions. (C) 2017 Elsevier B.V. All rights reserved

Distribution of Maximal Luminosity of Galaxies in the Sloan Digital Sky Survey

Extreme value statistics (EVS) is applied to the distribution of galaxy luminosities in the Sloan Digital Sky Survey (SDSS). We analyze the DR8 Main Galaxy Sample (MGS), as well as the Luminous Red Galaxies (LRG). Maximal luminosities are sampled from batches consisting of elongated pencil beams in the radial direction of sight. For the MGS, results suggest a small and positive tail index $\xi$, effectively ruling out the possibility of having a finite maximum cutoff luminosity, and implying that the luminosity distribution function may decay as a power law at the high luminosity end. Assuming, however, $\xi=0$, a non-parametric comparison of the maximal luminosities with the Fisher-Tippett-Gumbel distribution (limit distribution for variables distributed by the Schechter fit) indicates a good agreement provided uncertainties arising both from the finite batch size and from the batch size distribution are accounted for. For a volume limited sample of LRGs, results show that they can be described as being the extremes of a luminosity distribution with an exponentially decaying tail, provided the uncertainties related to batch-size distribution are taken care of

Distribution of Maximal Luminosity of Galaxies in the Sloan Digital Sky Survey

Extreme value statistics (EVS) is applied to the pixelized distribution of galaxy luminosities in the Sloan Digital Sky Survey (SDSS). We analyze the DR8 Main Galaxy Sample (MGS) as well as the Luminous Red Galaxy Sample (LRGS). A non-parametric comparison of the EVS of the luminosities with the Fisher-Tippett-Gumbel distribution (limit distribution for independent variables distributed by the Press-Schechter law) indicates a good agreement provided uncertainties arising both from the finite size of the samples and from the sample size distribution are accounted for. This effectively rules out the possibility of having a finite maximum cutoff luminosit

Distribution of Maximal Luminosity of Galaxies in the Sloan Digital Sky Survey

Extreme value statistics (EVS) is applied to the pixelized distribution of galaxy luminosities in the Sloan Digital Sky Survey (SDSS). We analyze the DR8 Main Galaxy Sample (MGS) as well as the Luminous Red Galaxy Sample (LRGS). A non-parametric comparison of the EVS of the luminosities with the Fisher-Tippett-Gumbel distribution (limit distribution for independent variables distributed by the Press-Schechter law) indicates a good agreement provided uncertainties arising both from the finite size of the samples and from the sample size distribution are accounted for. This effectively rules out the possibility of having a finite maximum cutoff luminosit

Distribution Of Maximal Luminosity Of Galaxies In The Sloan Digital Sky Survey

Extreme value statistics (EVS) is applied to the pixelized distribution of galaxy luminosities in the Sloan Digital Sky Survey (SDSS). We analyze the DR6 Main Galaxy Sample (MGS), divided into red and blue subsamples, as well as the Luminous Red Galaxy Sample (LRGS). A non-parametric comparison of the EVS of the luminosities with the Fisher-Tippett-Gumbel distribution (limit distribution for independent variables distributed by the Press-Schechter law) indicates a good agreement provided uncertainties arising both from the finite size of the samples and from the sample size distribution are accounted for