Finite Domain Anomalous Spreading Consistent with First and Second Law

After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.Comment: 14 figure

Non-Markovian diffusion equations and processes: analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.Comment: 43 pages, 19 figures, in press on Physica A (2008

Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

In the last decade the subordinated processes have become popular and found many practical applications. Therefore in this paper we examine two processes related to time-changed (subordinated) classical Brownian motion with drift (called arithmetic Brownian motion). The first one, so called normal tempered stable, is related to the tempered stable subordinator, while the second one - to the inverse tempered stable process. We compare the main properties (such as probability density functions, Laplace transforms, ensemble averaged mean squared displacements) of such two subordinated processes and propose the parameters' estimation procedures. Moreover we calibrate the analyzed systems to real data related to indoor air quality

Theory of Earthquake Recurrence Times

The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that, if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan's argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori law of earthquake triggering. We then develop the full theory of the statistics of inter-event times in the framework of the ETAS model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression fits well the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by Omori's law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension $d \approx 1.8$, the average clustering ratio $n \approx 0.9$, and the productivity exponent $\alpha \approx 0.9$ times the $b$-value of the Gutenberg-Richter law.Comment: 30 pages + 13 figure

Continuous time random walk and parametric subordination in fractional diffusion

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a generally non-Markovian diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented L\'evy process, we generate and display sample paths for some special cases.Comment: 28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'. Denton, Texas, August 200

Some aspects of fractional diffusion equations of single and distributed order

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.Comment: 14 pages. International Symposium on "Analytic Function Theory, Fractional Calculus and Their Applications", University of Victoria (British Columbia, Canada), 22-27 August 200

Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel $\sim 1/t^{1+\theta}$ with $0 \leq \theta <1$. In the presence of an epidemic-like process of triggered shocks developing in a cascade of generations at or close to criticality, this "bare" kernel is renormalized into an even slower decaying response function $\sim 1/t^{1-\theta}$. Surprisingly, this means that the shorter the memory of the bare kernel (the larger $1+\theta$), the longer the memory of the response function (the smaller $1-\theta$). Here, we present a detailed investigation of this paradoxical behavior based on a generation-by-generation decomposition of the total response function, the use of Laplace transforms and of "anomalous" scaling arguments. The paradox is explained by the fact that the number of triggered generations grows anomalously with time at $\sim t^\theta$ so that the contributions of active generations up to time $t$ more than compensate the shorter memory associated with a larger exponent $\theta$. This anomalous scaling results fundamentally from the property that the expected waiting time is infinite for $0 \leq \theta \leq 1$. The techniques developed here are also applied to the case $\theta >1$ and we find in this case that the total renormalized response is a {\bf constant} for $t < 1/(1-n)$ followed by a cross-over to $\sim 1/t^{1+\theta}$ for $t \gg 1/(1-n)$.Comment: 27 pages, 4 figure

Deep learning for healthcare applications based on physiological signals: A review

Background and objective: We have cast the net into the ocean of knowledge to retrieve the latest scientific research on deep learning methods for physiological signals. We found 53 research papers on this topic, published from 01.01.2008 to 31.12.2017. Methods: An initial bibliometric analysis shows that the reviewed papers focused on Electromyogram(EMG), Electroencephalogram(EEG), Electrocardiogram(ECG), and Electrooculogram(EOG). These four categories were used to structure the subsequent content review. Results: During the content review, we understood that deep learning performs better for big and varied datasets than classic analysis and machine classification methods. Deep learning algorithms try to develop the model by using all the available input. Conclusions: This review paper depicts the application of various deep learning algorithms used till recently, but in future it will be used for more healthcare areas to improve the quality of diagnosi