Ana V. Coronado

Size Effects on Correlation Measures

The detection and quantification of long-range correlations in time series is a fundamental tool to characterize the properties of different dynamical systems, and is applied in many different fields, including physics, biology or engineering. Due to the diversity of applications, many techniques for measuring correlations have been designed. Here, we study systematically the influence of the length of a time series on the results obtained from several techniques commonly used to detect and quantify long-range correlations: the autocorrelation analysis, Hurst’s analysis, and detrended fluctuation analysis (DFA). Using the Fourier filtering method, we generate artificial time series with known and controlled long-range correlations and with a broad range of lengths, and apply on them the different correlation measures we have studied. Our results indicate that while the DFA method is practically unaffected by the length of the time series, and almost always provides accurate results, the results from Hurst’s analysis and the autocorrelation analysis strongly depend on the length of the time series.

Comparison of multifractal and thermodynamical properties of fractal and natural spectra

In this paper, we explain the multifractal and thermodynamical properties of deterministic fractal spectra, and compare the most relevant behaviors with those obtained for some natural spectra with multifractal properties. In particular, we compare the heat capacity, C(T), derived from Cantor-set-type spectra with the ones obtained from several atomic spectra. We find that, although many of the properties of deterministic fractals, previously reported, are also reproduced in natural fractals (oscillations of C(T) in log scale), new features appear in the natural case: the mean value of C(T) is not always related to the multifractal properties, and the oscillations are present in cases for which they disappear in deterministic fractals.

On the focal point of a lens: beyond the paraxial approximation

In this paper we study a planar-convex lens where the focal point is calculated numerically and analytically beyond the paraxial approximation within the context of geometrical optics. We consider this problem as an appropriate and useful example to fill the gap found in physics and optics courses between the simplicity of the paraxial approximation and the complexity of the theory of aberrations, and it can be used as an introduction to non-paraxial behaviour even when teaching general physics courses. We show in a simple way how beyond the paraxial approximation the focal distance is not unique, and how it depends on the distance of the incoming ray to the optical axis. We show the importance of the caustic surface, which is calculated analytically, and its effect on the position of the point with the highest concentration of light, which is defined as the optimal focal distance of the lens. Finally, we also present some simulations showing light distributions in screens placed at different distances from the lens, to illustrate our results.

Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations

Fluctuation Analysis (FA) and specially Detrended Fluctuation Analysis (DFA) are techniques commonly used to quantify correlations and scaling properties of complex time series such as the observable outputs of great variety of dynamical systems, from Economics to Physiology. Often, such correlated time series are analyzed using the magnitude and sign decomposition, i.e., by using FA or DFA to study separately the sign and the magnitude series obtained from the original signal. This approach allows for distinguishing between systems with the same linear correlations but different dynamical properties. However, here we present analytical and numerical evidence showing that FA and DFA can lead to spurious results when applied to sign and magnitude series obtained from power-law correlated time series of fractional Gaussian noise (fGn) type. Specifically, we show that: (i) the autocorrelation functions of the sign and magnitude series obtained from fGns are always power-laws; However, (ii) when the sign series presents power-law anticorrelations, FA and DFA wrongly interpret the sign series as purely uncorrelated; Similarly, (iii) when analyzing power-law correlated magnitude (or volatility) series, FA and DFA fail to retrieve the real scaling properties, and identify the magnitude series as purely uncorrelated noise; Finally, (iv) using the relationship between FA and DFA and the autocorrelation function of the time series, we explain analytically the reason for the FA and DFA spurious results, which turns out to be an intrinsic property of both techniques when applied to sign and magnitude series.

First-Passage Time Properties of Correlated Time Series with Scale-Invariant Behavior and with Crossovers in the Scaling

The observable outputs of a great variety of complex dynamical systems form long-range correlated time series with scale invariance behavior. Important properties of such time series are related to the statistical behavior of the first-passage time (FPT), i.e., the time required for an output variable that defines the time series to return to a certain value. Experimental findings in complex systems have attributed the properties of the FPT probability distribution and the FPT mean value to the specifics of the particular system. However, in a previous work we showed (Carretero-Campos, Phys Rev E 85:011139, 2012) that correlations are a unifying factor behind the variety of findings for FPT, and that diverse systems characterized by the same degree of correlations in the output time series exhibit similar FPT properties. Here, we extend our analysis and study the FPT properties of long-range correlated time series with crossovers in the scaling, similar to those observed in many experimental systems. To do so, first we introduce an algorithm able to generate artificial time series of this kind, and study numerically the statistical properties of FPT for these time series. Then, we compare our results to those found in the output time series of real systems and we demonstrate that, independently of the specifics of the system, correlations are the unifying factor underlying key FPT properties of systems with output time series exhibiting crossovers in the scaling.

On the circular orbits of a particle sliding on the inside of a generic surface of revolution

A classical problem in introductory general physics courses is the determination of the properties of the circular orbits of a particle sliding without friction on the inside of an inverted cone. Typically, the aim is to find the speed of the orbit or other physical quantities of interest as a function of the height of the particle. Here, we generalize this problem by studying the properties of the circular orbits of a particle sliding on the inside of generic surfaces of revolution ρ(z) using basic Newtonian mechanics. Given the surface ρ(z), we obtain equations to calculate several physical quantities of interest (namely the speed v, the angular velocity ω, the total mechanical energy E and the angular momentum L) of the circular orbits inside the surface. Conversely, we also study the inverse problem: the determination of surfaces in which any of the physical quantities of interest (v, ω, E or L) associated with the orbits presents a desired functional dependence on the height of the particles orbit. As an example, we find the surfaces in which these quantities are constant in all the circular orbits. Finally, we study such surfaces using analytical mechanics.

Towards a Deeper Understanding of the Complex Behaviour Observed in the Distribution of Words in Written Texts

Here we show that the recently reported presence of long-range correlations in the distribution of words along texts is due to the complex distribution of the keywords, while common words are not correlated. Indeed we prove that the degree of long-range correlations of a word at long scales is a good measure of its relevance to the text. Additionally, we develop a model able to reproduce the spatial distribution of a word in a text, based on the long-range correlations observed for the word. The model not only reproduces the complex behaviour characterized by the presence of correlations at long scales and the degree of relevance of the word, but also the probability distribution of the inter-occurrences distances in the whole range of scales.

COMMENT ON: CONFINEMENT IN A DOUBLE BARRIER STRUCTURE IN THE PRESENCE OF AN ELECTRIC FIELD BY NIMOUR ET AL.

Abstract We comment on some of the results and the related conclusions presented by Nimour et al. [Phys. Lett. A 226 (1997) 393]. The cause given by the authors to explain the field dependence of the resonant energies of a double barrier structure is not correct.

Easy calculation of magnetic flux

In this paper, we give a very simple geometrical method to facilitate the calculation of the magnetic flux in some common cases where the magnetic flux density B produced by an infinitely long, straight wire carrying a steady current is considered. The method is also valid for vector fields with the same symmetry.