Volume 39, Issue 2

The representability of the Burr distribution as a mixture of Weibull distribution is studied in order to justify its utility for modelling the statistical regularities in extreme values registered in non-stationary flows of informative events. A result of [24] is improved by extending the domain of admissible values of the parameters which provide the representability of the (generalized) Burr distribution as a scale mixture of the Weibull distribution. This result gives an argument in favour of application of the Burr distribution as a model of statistical regularities of extreme values registered within moderate regular time intervals, say, daily ( short-term ) extremes. In turn, if we are interested in the statistical regularities of the behaviour of the absolute extreme observation over a long period, say, a decade (the long-term extreme), then it can be noted that the daily extreme values form a sample of the Burr-distributed random variables. As is known, the Burr distribution belongs to the domain of max-attraction of the Fréchet distribution. The problem of improving the accuracy of the approximation of the distribution of the absolute extreme by the Fréchet distribution by the construction of an asymptotic expansion for the distribution of the extreme order statistics in the sample of independent identically Burr-distributed random variables is also considered. These results are illustrated by an example of fitting the Burr distribution to the data representing the extreme values of characteristics of the interplanetary magnetic field.

We study periodic trajectories of nonlinear dynamical systems considered as models of the simplest molecular repressilator. In the phase portraits of these systems, we find hidden attractors and nonlocal oscillations. The cases of nonuniqueness of cycles in these portraits are described as well.

Optimal disturbances of a number of typical stationary solutions of the hepatitis B virus infection dynamics model have been found. Specifically optimal disturbances have been found for stationary solutions corresponding to various forms of the chronic course of the disease, including those corresponding to the regime of low-level virus persistence. The influence of small optimal disturbances of individual groups of variables on the stationary solution is studied. The possibility of transition from stable stationary solutions corresponding to chronic forms of hepatitis B to stable stationary solutions corresponding to the state of functional recovery or a healthy organism using optimal disturbances is studied. Optimal disturbances in this study were constructed on the basis of generalized therapeutic drugs characterized by one-compartment and two-compartment pharmacokinetics.

We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems.

The projected gradient method for matrix completion is generalized towards the higher-dimensional case of low-rank Tucker tensors. It is shown that an operation order rearrangement in the common projected gradient approach provides a complexity improvement. An even better algorithm complexity can be obtained by replacing the completion operator by a general operator that satisfies restricted isometry property; however, such a replacement transforms the completion algorithm into an approximation algorithm.