Kalashnikov N.P., Matyatina A.N., Olchak A.S.
Numerous experiments demonstrated that the thermal power released in a collision of metal objects (fast projectiles and meteorites) with a solid obstacle can significantly exceed the kinetic energy of the projectile whose velocity exceeds a certain threshold. The effect is so noticeable that some researchers attributes the anomalous heat release to the quantum and even nuclear nature. A less extravagant theoretical explanation of this effect associated with the explosive oxidation of a metal evaporating upon impact has been proposed in this work. The oxidation rate and the chemical explosion development time are estimated. The critical threshold velocity of the metal object is calculated above which the converted kinetic energy is sufficient for the immediate evaporation of a significant part of the metal, accompanied by additional energy release due to the explosive exothermic chemical oxidative reaction of hot atomic metal vapor in the oxygen-containing atmosphere. The results are in complete agreement with observed facts and experiments. DOI: 10.1134/S2304487X1805005X
Belaya E.A., Gryaznova M.S., Kolmogortsev A.M.
Ferrites are compounds of iron (III) oxide with oxides of other metals. Due to the unique structure of their crystal lattice, ferrites have magnetic properties and are widely used in various fields such as radio engineering, electronics, and medicine. A scheme of ion-exchange synthesis of nickel ferrite involving a pre-synthesized cation-exchange material, which serves as an auxiliary phase, has been presented in this work. It has been established that ionite acts as an additional phase in the process of synthesis. The samples obtained have been studied by the X-ray diffraction analysis, differential thermic analysis, and scanning electron microscopy. The thermal analysis has shown that the smooth heating of the cation exchange material processed by metallic salts results in successive processes of its decomposition, burning of the carbon residue, and decomposition of iron and nickel salts with the formation of a nickel ferrite phase at 815°C. The X-ray phase analysis has revealed the formation of a spinel structure with the spatial group Fd-3m. It has been established that the proposed scheme of synthesis allows obtaining round particles with a size up to 100 nm. The scanning electron microscopy studies have also shown that the typical particle sizes in nickel ferrite samples calcined at temperatures from 500 to 1000°C are 20–100 nm. DOI: 10.1134/S2304487X18050024
Two boundary value problems for a nonlinear evolutionary partial differential equation have been considered. The possibility of the well-known Landau–Hopf scenario of passage to turbulence has been shown for these boundary value problems. А plan as the cascade of the Andronov–Hopf bifurcation to realization the scenario proposed later by F. Takens has been implemented. The studied boundary value problems can be interpreted as generalized forms of the well-known mathematical “multiplier–accelerator” model of macroeconomics or a form of the generalized Van der Pol equation with distributed parameters. The justification of the results is based on the application of methods of qualitative theory of partial differential equations such as the method of integral manifolds, the Poincaré–Dulac theory of normal forms, asymptotic methods of analysis of dynamic systems with infinite-dimensional phase space, and perturbation theory for linear differential operators. In particular, asymptotic formulas for the solutions forming invariant tori are obtained by application of the generalized Krylov–Bogoliubov method. DOI: 10.1134/S2304487X18050085
Polyanin A.D., Sorokin V.G.
Problems of the linear stability and instability of stationary (and some nonstationary) solutions of initial-boundary value problems for various classes of nonlinear partial differential equations are investigated. Reaction–diffusion equations, Klein–Gordon equations, nonlinear telegraph equations, and Cattaneo–Vernotte differential-difference equations have been considered. All equations contain a kinetic function of the general form f(u, w), which arbitrarily depends on the unknown function u = u(x, t) and the function w = u(x, t – τ), where τ is the delay time. Dispersion equations for the spectral parameter are derived. The stability and instability criteria for solutions are given in the form of inequalities for the derivatives of the kinetic function. For reaction–diffusion equations with delay, the representation of the spectral parameter in terms of the Lambert function and the characteristic parameters of the problem are obtained. It is proved that stationary solutions of Cattaneo–Vernotte differential-difference equations for an arbitrary kinetic function are always unstable in the linear approximation. Two classes of nonlinear partial differential equations with delay are also described, any solutions of which are unstable for an arbitrary kinetic function (these results are exact and obtained without linearizing the equations). It is shown that problems with initial data and some initial-boundary value problems for these classes of equations are ill-posed in the sense of Hadamard. DOI: 10.1134/S2304487X18050103
The well-known Solow mathematical model of macroeconomics is considered. It has been proposed to take into account the delay factor, which is typical of macroeconomic processes. In this case, it has been shown that the introduction of delay changes the dynamics of solutions in this mathematical model. It has been shown that a stable periodic solution describing macroeconomic cycles appears if stability is lost by a positive equilibrium state. Asymptotic formulas for the solutions forming this cycle have been obtained. In particular, it has been found that cycles with a long period can appear, which can be interpreted as N.D. Kondratiev long waves. The analysis of the mathematical model is based on the use of methods of the qualitative theory of differential equations such as the method of integral manifolds, the Poincaré–Dulac theory of normal forms, and the asymptotic methods of analysis of dynamic systems with infinite-dimensional phase space. DOI: 10.1134/S2304487X18050097
Savyolova T.I., Sirenchenko I.I.
Most of the natural and artificial crystalline substances are polycrystals, i.e., aggregates of single crystals. The physical and mechanical properties of polycrystals are largely determined by the crystallographic orientation and mutual arrangement of grains. When crystallites are oriented, they indicate the presence of a texture, which is mathematically described by the orientation distribution function. The texture is often axial and appears if certain crystallographic directions in all grains are parallel to any external direction. These circumstances determine the urgency of research and development of methods for modeling the orientation distribution function and pole figures for Gaussian axial textures. The existing Neumann method is inefficient in time because a certain fraction of random numbers is not used in modeling and is generated in vain. The problem of modeling the orientations of a Gaussian axial texture of type has been considered in this work. A new Monte Carlo method has been developed to mathematically model the Gaussian axial component. Numerical examples of the use of the proposed algorithm for various sharpness parameters ε have been given. The results of the computations have been compared by the χ 2 criterion. The above algorithm combined with a specialized Monte Carlo method for “peak” normal distributions is a way of modeling any orientation distribution function by the Monte Carlo method. DOI: 10.1134/S2304487X18050115
—The yield of products is a key indicator of its economic efficiency. In the microelectronic industry, the yield is determined not only by technological factors, but also by the quality of product design. Technology in general is not ideal, so the technological reasons for reducing the yield of integral circuits are taken into account during their design, which is most relevant for integral circuits with increased reliability requirements. The main catastrophic and parametric reasons for loss of the yield of nanometer IP blocks and VLSI, which can be random or systematic, have been analyzed in this work. The optical correction of proximity effects is an efficient method but not enough. For this reason, process design kits for the chip designers sometimes include special rules or recommendations taking into account various technological features for minimizing defects. These rules increase the chip area and reduce the performance because of longer distances between structures. The suggested yield improvement design methods are applied on the levels of standard cells and IP-blocks and provide the trade-off between the electrical and geometrical parameters. The methods are schematic and constructive-topological and are based on taking into account variations in process parameters, defectiveness of contact windows and interconnects, and placement and routing of elements. It has been shown that taking into account the reasons of yield loss and applying proposed methods during the project design make it possible to increase the yield by 15–20%. DOI: 10.1134/S2304487X18050061
Gur'eva V.M., Kotov Yu. B., Semenova T.A., Yablokova M.E.
A mathematical prognostic rule for distinguishing pregnant patients with pre-eclampsia by the 24-hour monitoring has been developed. The statistical retrospective analysis for 38 curves of the blood pressure and pulse rate for pregnant women with diabetes mellitus has been carried out. Median smoothing of curves excludes sharp jumps of measurements due to external noise or recording failures and keeps slow processes occurring in a patient organism. The patients have been a priori classified by an experienced doctor among four classes: (i) almost healthy, (ii) with a small hypertension, (iii) with a more serious hypertension, and (iv) with severe hypertension. Processing of curves by nonparametric statistic methods selects the informative cases of 24-hours monitoring for diagnostic purposes. An algorithm for revealing patients with risk of pre-eclampsia has been developed. The mathematical equivalent of the empirical doctor’s assumption has been given. A decision rule for pre-eclampsia diagnostics using 24-hour blood pressure curves has been obtained. The results can be interesting for monitor developers, obstetricians curing the pregnant women with diabetes, and general practitioners. DOI: 10.1134/S2304487X18050048
A method for counting the number of arithmetic operations at program runtime is proposed. To use it, the source code of the program or function in the C or C++ programming language is needed. The analyzed area of the code is surrounded especially developed functions, but its modification is not required. To implement the dynamic counting, C ++ programming language features such as the overloading of arithmetic and logical operations, static members of the class, replacement of identifiers using a preprocessor are used. When the modified program is executed, the number of operations of a given type is counted for specific inputs. The proposed method allows determining the exact number of operations performed, depending on the input data taking into account actually called functions, executed branches of conditional operators. It also allows estimating the overhead costs when implementing the algorithm and getting the requirements for a computer system in which the tested algorithms are supposed to be used. In contrast to the asymptotic estimate, the proposed method allows estimating not only the growth rate, but also a constant. Estimates of the proposed technique and asymptotic estimates for well-known algorithms are compared. Along with counting operations on real numbers, the proposed method makes it possible to obtain statistics of the application of operations potentially for any type of data used in the program and for any type of operations not necessarily arithmetic. Thus, the scope of the method is much wider than the analysis of numerical algorithms. DOI: 10.1134/S2304487X18050073
Bykhovets E., Kryanev A.V., Tatarinova A.N., Orozbekov A.M.
A scheme and an algorithm for detecting anomalous readings of direct-charge sensors measuring the energy release in cores of nuclear reactors and correcting the calculated energy release taking into account the readings of sensors have been discussed. The scheme for detecting anomalous readings of sensors is based on the robust method for detecting anomalous realizations of observed values in the presence of a noise component. Thus, the reconstruction of the energy release in the core of a nuclear reactor is based on the energy release distribution values obtained by the numerical solution of the neutron-physical boundary value problems and on the readings of in-reactor control sensors after filtering the detected anomalous readings of the sensors. The task of the reconstruction consists in combining the calculated values and the sensor readings, providing the minimum average error of the reconstructed values. The results of the restoration of the energy release in the core of a VVER-1000 reactor. The scheme and algorithm for detecting anomalous readings of sensors measuring the energy release in the cores of nuclear reactors and correcting the calculated energy release values taking into account the readings of the sensors provide the operative reconstruction of the energy release distribution. DOI: 10.1134/S2304487X18050036