Oscar Sotolongo-Costa

Behavior of tumors under nonstationary therapy

We present a model for the interaction dynamics of lymphocytes-tumor cells population. This model reproduces all known states for the tumor. Further, we develop it taking into account periodical immunotherapy treatment with cytokines alone. A detailed analysis for the evolution of tumor cells as a function of frequency and therapy burden applied for the periodical treatment is carried out. Certain threshold values for the frequency and applied doses are derived from this analysis. So it seems possible to control and reduce the growth of the tumor. Also, constant values for cytokines doses seems to be a successful treatment.

Lévy flights and earthquakes

Levy flights representation is proposed to describe earthquake characteristics like the distribution of waiting times and position of hypocenters in a seismic region. Over 7500 microearthquakes and earthquakes from 1985 to 1994 were analyzed to test that its spatial and temporal distributions are such that can be described by a Levy flight with anomalous diffusion (in this case in a subdiffusive regime). Earthquake behavior is well described through Levy flights and Levy distribution functions such as results show.

Tsallis Entropy and the transition to scaling in fragmentation

By using the maximum entropy principle with Tsallis entropy we obtain a fragment size distribution function which undergoes a transition to scaling. This distribution function reduces to those obtained by other authors using Shannon entropy. The treatment is easily generalisable to any process of fractioning with suitable constraints.

Burr, Lévy, Tsallis

The purpose of this short paper dedicated to Prof Constantin Tsallis on his 60th anniversary is to show how the use of mathematical tools and physical concepts introduced by Burr, Levy and Tsallis open a new line of analysis of the old problem of non-Debye decay and universality of relaxation. We also show how a finite characteristic time scale can be expressed in terms of a q-expectation using the concept of q-escort probability. The comparison with the Weron et al. probabilistic theory of relaxation leads to a better understanding of the stochastic properties underlying the Tsallis entropy concept.

Continuous time random walks and south Spain seismic series

Lévy flights were introduced through the mathematical research of thealgebra or random variables with infinite moments. Mandelbrot recognizedthat the Lévy flight prescription had a deep connection toscale-invariant fractal random walk trajectories. The theory of ContinuousTime Random Walks (CTRW) can be described in terms of Lévydistribution functions and it can be used to explain some earthquakecharacteristics like the distribution of waiting times and hypocenter locationsin a seismic region. This paper checks the validity of this assumptionanalyzing three seismic series localized in South Spain. The three seismicseries (Alborán, Antequera and Loja) show qualitatively the samebehavior, although there are quantitative differences between them.

Diffusion regimes in Lévy flights with trapping

The diffusion of a walk in the presence of traps is investigated. Different diffusion regimes are obtained considering the magnitude of the fluctuations in waiting times and jump distances. A constant velocity during the jump motion is assumed to avoid the divergence of the mean squared displacement. Using the limit theorems of the theory of Levy stable distributions we have provided a characterization of the different diffusion regimes.

Universal relaxation in nonextensive systems

We have derived the dipolar relaxation function for a cluster model whose volume distribution was obtained from the generalized maximum Tsallis nonextensive entropy principle. The power law exponents of the relaxation function are simply related to a global fractal parameter α and for large time to the entropy nonextensivity parameter q. For intermediate times the relaxation follows a stretched exponential behavior. The asymptotic power law behaviors both in the time and the frequency domains coincide with those of the Weron generalized dielectric function derived from an extension of the Levy central-limit theorem. They are in full agreement with the Jonscher universality principle. Moreover, our model gives a physical interpretation of the mathematical parameters of the Weron stochastic theory and opens new paths to understand the ubiquity of self-similarity and power laws in the relaxation of large classes of materials in terms of their fractal and nonextensive properties.

Dimensional crossover in fragmentation

Experiments in which thick clay plates and glass rods are fractured have revealed different behavior of fragment mass distribution function in the small and large fragment regions. In this paper we explain this behavior using non-extensive Tsallis statistics and show how the crossover between the two regions is caused by the change in the fragments’ dimensionality during the fracture process. We obtain a physical criterion for the position of this crossover and an expression for the change in the power-law exponent between the small and large fragment regions. These predictions are in good agreement with the experiments on thick clay plates.

Tsallis entropy approach to radiotherapy treatments

The biological effect of one single radiation dose on a living tissue has been described by several radiobiological models. However, the fractionated radiotherapy requires to account for a new magnitude: time. In this paper we explore the biological consequences posed by the mathematical prolongation of a previous model to fractionated treatment. Nonextensive composition rules are introduced to obtain the survival fraction and equivalent physical dose in terms of a time dependent factor describing the tissue trend towards recovering its radioresistance (a kind of repair coefficient). Interesting (known and new) behaviors are described regarding the effectiveness of the treatment which is shown to be fundamentally bound to this factor. The continuous limit, applicable to brachytherapy, is also analyzed in the framework of nonextensive calculus. Here a coefficient that rules the time behavior also arises. All the results are discussed in terms of the clinical evidence and their major implications are highlighted.

Self-similar conductance patterns in graphene Cantor-like structures

Graphene has proven to be an ideal system for exotic transport phenomena. In this work, we report another exotic characteristic of the electron transport in graphene. Namely, we show that the linear-regime conductance can present self-similar patterns with well-defined scaling rules, once the graphene sheet is subjected to Cantor-like nanostructuring. As far as we know the mentioned system is one of the few in which a self-similar structure produces self-similar patterns on a physical property. These patterns are analysed quantitatively, by obtaining the scaling rules that underlie them. It is worth noting that the transport properties are an average of the dispersion channels, which makes the existence of scale factors quite surprising. In addition, that self-similarity be manifested in the conductance opens an excellent opportunity to test this fundamental property experimentally.