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We consider various iterative algorithms for solving the linear equation $ax=b$ using a quantum computer operating on the principle of quantum annealing. Assuming that the computer's output is described by the Boltzmann distribution, it is shown under which conditions the equation-solving algorithms converge, and an estimate of their convergence rate is provided. The application of this approach to algorithms using both an infinite number of qubits and a small number of qubits is discussed., Comment: In Russian, 18 pages, 2 figures

The paper presents a stochastic analysis of the growth rate of viscous fingers in miscible displacement in a heterogeneous porous medium. The statistical parameters characterizing the permeability distribution of a reservoir vary over a wide range. The formation of fingers is provided by the mixing of different-viscosity fluids -- water and polymer solution. The distribution functions of the growth rate of viscous fingers are numerically determined and visualized. Careful data processing reveals the non-monotonic nature of the dependence of the front end of the mixing zone on the correlation length of the permeability (describing the medium graininess) of the reservoir formation. It is demonstrated that an increase in graininess up to a certain value causes an expansion of the distribution shape and a shift of the distribution maximum to the region of higher velocities. In addition, an increase in the standard deviation of permeability leads to a slight change in the shape and characteristics of the density distribution of the growth rates of viscous fingers. The theoretical predictions within the framework of the transverse flow equilibrium approximation and the Koval model are contrasted with the numerically computed velocity distributions., Comment: 18 pages, 10 figures

Random walks are powerful tools to analyze spatial-temporal patterns produced by living organisms ranging from cells to humans. At the same time, it is evident that these patterns are not completely random but are results of a convolution of organisms' sensor-based information processing and motility. The complexity of the first component is reflected in the statistical characteristics of trajectories produced by an organism -- when it is, e.g., foraging or searching for a mate (or a pathogen) -- and therefore some knowledge about the component can be obtained by analyzing the trajectories with the standard toolbox of methods used for random walks. Here we consider trajectories which appear as the results of a game played by two players on a finite square lattice. One player wants to survive, i. e., to stay within the interior of the square, as long as possible while another one wants to reach the adsorbing boundary. A game starts from the center of the square and every next movement of the point is determined by independent strategy choices made by the players. The value of the game is the survival time that is the number of steps before the adsorption happens. We present the results of a series of experiments involving both human players and an autonomous agent (bot) and concentrate on the probability distribution of the survival time. This distribution indicate that the process we are dealing with is more complex than the standard random walks.

The graded viscosity banks technology (tapering) for polymer flooding is studied for several different models of mixing zones behavior. Depending on the viscosity function the limiting polymer injection profile is rigorously derived for the transverse flow equilibrium model, for the Koval model and for the Todd-Longstaff model. The potential gain in the polymer quantity compared to the profile with a finite number of slugs is numerically estimated.

The paper investigates the linear growth of the mixing zone during polymer slug injection into a water reservoir. The velocities of the slug front and of the boundaries of the mixing zone are analyzed as key parameters. Using two different numerical methods (finite volumes and finite elements), the impact of the slug size, reservoir dimensions, Peclet number, and viscosity curve shape on the corresponding velocities is examined. Notwithstanding the realization of the solution by two computational schemes, the simulation results coincide with sufficient accuracy. The numerically obtained velocities are compared with theoretical estimates within the transverse flow equilibrium approximation and Koval model. Based on the comparison pattern, recommendations are presented on the use of specific analytical methods for estimating the growth rate of the mixing zone depending on the characteristics of the polymer.

We prove that for any closed manifold of dimension 3 or greater that there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a hyperbolic set is an intrinsic property for premaximality. Lastly,we review some results due to Anosov that concern premaximality., Comment: 25 pages, 4 figures