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A mathematical model for reducing the pressure of natural gas in a direct acting pressure regulator is developed. It is shown that, if an output-input pressure ratio typical for practice is lower than a critical value, the pressure reduction process comes down to two stages: partial expansion of gas with throttling in the orifice plate of the pressure regulator and subsequent expansion in its casing. The values of the gas temperature in the characteristic sections of the pressure regulator are determined. It is established that the temperature drops at the first stage of pressure reduction and rises as the flow in the pressure regulator casing expands and decelerates. A method for minimizing the possibility of formation of gas hydrates in the pressure regulator is proposed.

In this paper, we consider conditions under which the gas enclosed in the main crack located at the edge of a coal or rock formation can produce fracture of the formation. Kinetic theory is developed for two competing physical processes: formation unloading due to rock pressure and gas filtration from the crack cavity into the surrounding massif. The first process promotes fracture, and the second leads to a decrease in the gas pressure causing he fracture. The evolution of the crack is determined by the ratio of the rates of these processes. It is found that a modified Griffiths criterion is a necessary but not sufficient condition for fracture. For formation fracture, it is also necessary that the unloading rate to the filtration rate exceed a certain threshold value.

A size-dependent cracked Timoshenko beam model is established based on the nonlocal strain gradient theory and flexibility crack model. Expressions of the higher-order bending moment and shear force are derived. Analytical expressions of the deflection and rotation angle of the cross section of a simply supported microbeam with an arbitrary number of cracks subjected to uniform loading are obtained. The effects of the nonlocal parameter, the material length scale parameter, the presence of the crack, and the slenderness ratio on the bending behaviors of the cracked microbeam are examined. It is found that the material length scale parameter plays an important role in the cracked microbeam bending behavior, while the nonlocal parameter is not decisive. Furthermore, the cracked microbeam also exhibits a stiffening or softening effect depending on the values of the two scale parameters; if the two parameters are equal, the bending deformation of the nonlocal cracked microbeam may not be reduced to that of the classical elastic cracked Timoshenko beam. Additionally, the influence of the size effect on beam stiffening and softening becomes more significant as the slenderness ratio decreases.

A model of disk-like cracks based on amplitude-frequency spectra of acoustic emission, detected in the fracture of a concrete sample, is used to restore their size distribution function, as well as corresponding distributions of porosity and specific area of internal surface of the material. Changes in these characteristics of a solid in a time interval between the instances of detection of the spectra are studied.

A prototype of a pulsed X-ray apparatus with and operating voltage of 600-800 kV based on a combined helical generator has been developed and tested. Compared to the classical helical generator, the total length of the helical winding is increased by adding a single-bus line, which allows matching of the wave propagation time along the helical generator line to the oscillation period in the generator with a Tesla transformer. It is shown that the proposed transformer has high efficiency. A theoretical model describing the operation of the combined generator is proposed.

We consider two related discrete optimization problems of searching for a subset in a finite set of points in the Euclidean space. Both problems are induced by the versions of the fundamental problem in data analysis, namely, by selecting a subset of similar elements in a set of objects. In each problem, an input set and a positive real number are given, and it is required to find a cluster (i.e., a subset) of the largest size under constraints on the value of a quadratic clusterization function. The points in the input set which are outside the sought for subset are treated as the second (complementary) cluster. In the first problem, the function under the constraint is the sum over both clusters of the intracluster sums of the squared distances between the elements of the clusters and their centers. The center of the first (i.e., the sought) cluster is unknown and determined as the centroid, while the center of the second one is fixed at a given point in the Euclidean space (without loss of generality in the origin). In the second problem, the function under the constraint is the sum over both clusters of the weighted intracluster sums of the squared distances between the elements of the clusters and their centers. As in the first problem, the center of the first cluster is unknown and determined as the centroid, while the center of the second one is fixed in the origin. In this paper, we show that both problems are strongly NP-hard. Also, we present the exact algorithms for the cases of these problems in which the input points have integer components. If the space dimension is bounded by some constant, the algorithms are pseudopolynomial.

An automated system for identifying the conditions of homogeneous and heterogeneous reactions includes mathematical modeling of a chemical reaction, determination of optimization criteria conditions for variable parameters, setting and multipurpose optimization problem solution and an optimal control problem, development of efficient algorithms for a computing experiment. Modeling and optimization of a homogeneous and heterogeneous catalytic reactions are carried out. Optimal conditions for carrying out reactions to achieve the specified criteria are determined.

Using additional boundary conditions and additional unknown functions in the integral method of heat balance, we consider the method of obtaining analytical solutions to the thermal conductivity problem associated with the separation process of thermal conductivity of two phases with respect to time, which allows reducing the solution of partial differential equations to the integration of two ordinary differential equations for some additional desired functions. The first stage is characterized by a rapid convergence of the analytical solution to an exact one. For the second stage, the exact analytical solution has been obtained. Additional boundary conditions for both phases are in such a form that their execution by a desired solution be equivalent to realization of the original equation at boundary points and at a front of the temperature perturbations. It is shown that the implementation of the equations at the boundary points leads to its execution also inside the domain.

In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the space of square integrable functions instead of the classical H (div;Ω) space. The velocity and the pressure are approximated by a P ₀²- P ₁ pair which satisfies an inf-sup condition. Firstly, we solve the original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton's iteration on the fine grid twice. It is shown that the algorithm can achieve an asymptotically optimal approximation as long as the mesh sizes satisfy h = O ( H ⁶ |ln H |²). As a result, solving such a large class of nonlinear equations will not be much more difficult than solving one linearized equation. Finally, a numerical experiment is provided to verify the theoretical results of the two-grid method.

Randomized algorithms of Monte Carlo method are constructed by the combined realization of the base probabilistic model and its random parameters for investigation of the parametric distribution of linear functionals. The optimization of algorithms with the use of the statistical kernel estimator for the probability density is presented. The randomized projection algorithm for estimating a nonlinear functional distribution as applied to the investigation of criticality fluctuations for the particles multiplication process in a random medium is formulated.