Power law noise plays an important role in the description of high performance oscillators. Commonly, five types of noise are considered to affect clocks and clock measurements: white phase, flicker phase, white frequency, flicker frequency and random walk frequency. These noise types are distinguished by the slopes of their spectral densities, S,@ =y (on a log-log scale). The noise is inherent...

We investigate the quantum percolation problem in a diluted chain with long-range hopping amplitudes. Each bond is activated with probability p(r) = p1/r , where r is the distance between two sites and α characterizes the range of the interactions. The average participation ratio of all eigenstates is used as a measure of the wave-functions localization length. We found that, above a quantum pe...

Aaron Clauset, 2 Cosma Rohilla Shalizi, and M. E. J. Newman Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA Department of Computer Science, University of New Mexico, Albuquerque, NM 87131, USA Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48...

Power functions (e.g., f(t) = at) describe the relationships among many variables observed in nature. One example of this is the power law of forgetting: The decline in memory performance with time or intervening events is well fit by a power function. This simple functional relationship accounts for a great deal of accumulated data. In this note, we consider a simple yet general memory model i...

Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation ["Time domain wave-equations for lossy media obeying a frequency power-law," J. Ac...

Galaxy kinematics and gravitational lensing are two complementary ways to constrain the distribution of dark matter on galaxy scales. The typical density profiles adopted in dynamical studies cannot easily be studies. Ideally, a mass model should used that has global characteristics realistic distributions, allows for an analytical calculation magnifications deflection angles. A simple with the...

The remarkable discovery of many large-scale real networks is the power-law distribution in degree sequence: the number of vertices with degree i is proportional to i−β for some constant β > 1. A lot of researchers believe that it may be easier to solve some optimization problems in powerlaw graphs. Unfortunately, many problems have been proved NP-hard even in power-law graphs. Intuitively, a t...

How do the lifestyles (free-living unparasitized, free-living parasitized, and parasitic) of animal species affect major ecological power-law relationships? We investigated this question in metazoan communities in lakes of Otago, New Zealand. In 13,752 samples comprising 1,037,058 organisms, we found that species of different lifestyles differed in taxonomic distribution and body mass and were ...

When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distri...