The article is devoted to the problem of calculating the probability density of a strictly stable law at x→∞. To solve this problem, it was proposed to use the expansion of the probability density in a power series. A representation of the probability density in the form of a power series and an estimate for the remainder term was obtained. This power series is convergent in the case 0<α<1 and asymptotic at x→∞ in the case 1<α<2. The case α=1 was considered separately. It was shown that in the case α=1 the obtained power series was convergent for any |x|>1 at N→∞. It was also shown that in this case it was convergent to the density of g(x,1,θ). An estimate of the threshold coordinate x_ε^N, was obtained which determines the range of applicability of the resulting expansion of the probability density in a power series. It was shown that in the domain |x|⩾ x_ε^N this power series could be used to calculate the probability density.

The paper considers the problem of calculating the distribution function of a strictly stable law at $x\to\infty$. To solve this problem, an expansion of the distribution function in a power series was obtained, and an estimate of the remainder term was also obtained. It was shown that in the case $\alpha<1$ this series was convergent for any $x$, in the case $\alpha=1$ the series was convergent at $N\to\infty$ in the domain $|x|>1$, and in the case $\alpha>1$ the series was asymptotic at $x\to\infty$. The case $\alpha=1$ was considered separately and it was demonstrated that in that case the series converges to the generalized Cauchy distribution. An estimate for the threshold coordinate $x_\varepsilon^N$ was obtained which determined the area of applicability of the obtained expansion. It was shown that in the domain $|x|\geqslant x_\varepsilon^N$ this power series could be used to calculate the distribution function, which completely solved the problem of calculating the distribution function at large $x$.

The paper studies the issue of the law of the gene expression distribution obtained with the use of the next-generation sequencing technology (NGS). It has been shown that gene expression has the form of a shift-scale mixture of distributions. One of the components of this mixture is fractionally stable distribution with characteristic indicators varying within the range 0.91 <= alpha <= 1.24. 0.17 <= beta <= 0.75. This component describes the distribution of gene expression at values FPKM > 1. The use of Fisher's goodness-of-fit test chi boolean AND 2 does not reject the the hypothesis of fractionally stable distribution. Another component of the mixture appears at expression values FPKM < 1 and may be associated with errors in the sequencing process using the technology of NGS.

The problem of calculating the probability density and distribution function of a strictly stable law is considered at x→0. The expansions of these values into power series were obtained to solve this problem. It was shown that in the case α<1, the obtained series were asymptotic at x→0; in the case α>1, they were convergent; and in the case α=1 in the domain |x|<1, these series converged to an asymmetric Cauchy distribution. It has been shown that at x→0 the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws.

The problem of calculating the Mittag-Leffler function E_ρ,μ (z) is considered in the paper. To solve this problem integral representations for the function E_ρ,μ(z) are transformed in such a way that they could not contain complex variables and parameters. Integral representations written in this form allow one to use standard methods of numerical integration to calculate integrals contained in them. To verify the correctness of the integral representations obtained the function E_ρ,μ(z) was calculated both with the use of obtained formulas and with the use of known representations of the Mittag-Leffler function. The calculation results demonstrate their exact matching. This fact is indicative of the correctness of new integral representations of the function E_ρ,μ(z) that were obtained.

In this paper, we obtain a generalization of the integral representation of the gamma function, which shows that the Hankel contour allows the rotation in the complex plane. The range of allowable values for the rotation angle of the contour is set. Using this integral representation, we obtain a generalization of the integral representation of the Mittag-Leffler function that expresses the value of this function through the contour integral.

This paper considers a method of stochastic solution to the anomalous diffusion equation with a fractional derivative with respect to both time and coordinates. To this end, the process of a random walk of a particle is considered, and a master equation describing the distribution of particles is obtained. It has been shown that in the asymptotics of large times, this process is described by the equation of anomalous diffusion, with a fractional derivative in both time and coordinates. The method has been proposed for local estimation of the solution to the anomalous diffusion equation based on the simulation of random walk trajectories of a particle. The advantage of the proposed method is the opportunity to estimate the solution directly at a given point. This excludes the systematic component of the error from the calculation results and allows constructing the solution as a smooth function of the coordinate.

The process of Levy random walks is considered in view of the constant velocity of a particle. A kinetic equation is obtained that describes the process of walks, and fractional differential equations are obtained that describe the asymptotic behavior of the process. It is shown that, in the case of finite and infinite mathematical expectation of paths, these equations have a completely different form. To solve the obtained equations, the method of local estimation of the Monte Carlo method is described. The solution algorithm is described and the advantages and disadvantages of the considered method are indicated.

The integral representation of the two-parameter Mittag-Leffler function E ρ , μ ( z ) is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function E ρ , μ ( z ) has two forms: the representation “A” and “B”. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed.

Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters ( α , θ ) . A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter λ .

This article studies the properties of the characteristic function of fractional stable distribution expressed through the Mittag-Leffler function. It is shown that the existing integral expression for the Mittag-Leffler function is incorrect, and the corrected integral expression is given. New properties of the characteristic function of the fractional stable law are obtained that make it possible to perform the inverse Fourier transformation. As a result, the integral representations are obtained for the density and distribution function of the fractional stable law. Some properties of these representations are studied, and the results of numerical calculations for the probability density and distribution function are presented.

A generalization of the model of Lévy walks with traps is considered. The main difference between the model under consideration and the already existing models is the introduction of multiplicative particle acceleration at collisions. The introduction of acceleration transfers the consideration of walks to coordinate–momentum phase space, which allows both the spatial distribution of particles and their spectrum to be obtained. The kinetic equations in coordinate–momentum phase space have been derived for the case of walks with two possible states. This system of equations in a special case is shown to be reduced to ordinary Lévy walks. This system of kinetic equations admits of integration over the spatial variable, which transfers the consideration only to momentum space and allows the spectrum to be calculated. An exact solution of the kinetic equations can be obtained in terms of the Laplace–Mellin transform. The inverse transform can be performed only for the asymptotic solutions. The calculated spectra are compared with the results of Monte Carlo simulations, which confirm the validity of the derived asymptotics.

We present an estimation algorithm for the characteristic parameter α of a stable law and obtain an upper bound of the quadratic deviation of the estimator. We apply this algorithm in the description of the fluctuation-induced flux in the edge region of the plasma cord of a controlled fusion experiment.

In our study, we estimate an effect from chromosome aberrations and genome mutations on changes in microRNA expression profiles in cancer cell lines demonstrating different radiosensitivity. Here, cell viability and microRNA spectrum have been estimated 1, 4, and 24 h after irradiation. MiSeq high-throughput sequencing system (Illumina, San Diego, CA, USA) is employed to perform microRNA spectrum estimation. In the K562 cell line, the number of expressed microRNAs in chromosomes demonstrates a more pronounced variation. An analysis of microRNA effects on signaling pathway activity demonstrates differences in post-transcriptional regulation of the expression of genes included into 40 signaling pathways. In the K562 cell line, microRNA dynamics analyzed for their dependence on chromosome localization show a wider scattering of microRNA expression values for a pair of chromosomes compared to the HL-60 cell line. An analysis of microRNAs expression in the K562 and HL-60 cell lines after irradiation has shown that chromosome abnormalities can affect microRNA expression changes. A study of radiation-induced changes of microRNA expression profiles in the K562 and HL-60 cell lines has revealed a dependence of microRNA expression changes on the number of chromosome aberrations and genome mutations.

High throughput technologies opened a new era in biomedicine by enabling massive analysis of gene expression at both RNA and protein levels. Unfortunately, expression data obtained in different experiments are often poorly compatible, even for the same biologic samples. Here, using experimental and bioinformatic investigation of major experimental platforms, we show that aggregation of gene expression data at the level of molecular pathways helps to diminish cross- and intra-platform bias otherwise clearly seen at the level of individual genes. We created a mathematical model of cumulative suppression of data variation that predicts the ideal parameters and the optimal size of a molecular pathway. We compared the abilities to aggregate experimental molecular data for the 5 alternative methods, also evaluated by their capacity to retain meaningful features of biologic samples. The bioinformatic method OncoFinder showed optimal performance in both tests and should be very useful for future cross-platform data analyses.

The aim of the investigation was to study microRNA expression and its influence on signaling pathways activity in radioresistant and radiosensitive cell lines exposed to radiation. Materials and Methods. Radioresistant K562 cell line and radiosensitive HL-60 and Raji cell lines were used in the study. Cell survival was estimated after exposure to 4 Gy gamma radiation. MicroRNA composition was studied 1, 4 and 24 h after radiation exposure using massively parallel sequencing method. Bioinformatics analysis was performed using GenXpro and PANTHER database. Results. After single 4 Gy radiation exposure, the number of cells with the signs of necrosis increased several times in radiosensitive cell lines as compared to control samples. MicroRNA hsa-miR-590-3p was found in each studied cell line at every stage of the experiment. The most significant differences between radioresistant and radiosensitive cell lines were observed in dynamics of microRNA influence on Integrin signaling pathway and General transcription by RNA polymerase I pathway. Conclusion. MicroRNA hsa-miR-590-3p dynamics and expression level were found to correlate with cancer cell radiosensitivity. Its influence on radioresistant and radiosensitive cell lines was different: in radioresistant K562 cell line, the inhibitory effect of microRNA on Integrin signaling pathway was reduced and this effect on General transcription by RNA polymerase I pathway was increased.

Algorithm for statistical estimation of the parameters of fractional-stable distributions is described in this article. This algorithm is constructed on the basis of the method of distance minimization between the empirical and theoretical distributions. As the distance between the two distributions, the χ distance is considered. The main difficulty in dealing with fractional-stable distributions is the absence of explicit expressions for the probability density function. That is why the theoretical density is estimated by the histogram method. The results of the test calculations and the results of the estimation of the quadratic deviation are presented. The results obtained by this estimator are compared with the results obtained by the method of moments. An example of the use of the estimator for the approximation of the experimental data obtained in the investigation of gene expression by RNA sequencing technology is given as well. It is shown that the probability density function of the gene expression in a wide-enough domain can be described by fractional-stable distributions.