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We present a synthesis of numerical modeling data for the evolution of mantle-crust systems in oceanic and continental spreading zones from decompression melting with the associated generation of mafic magmas and fluid release in their crystallization to mineral deposition in the crust. Model parameters were chosen to match those appropriate for natural magmatic-fluid systems in slow-spreading mid-ocean ridges (MOR) and the Siberian trap province. The evolution of a melting region was modeled for two cases: (i) a hot spot beneath a mid-ocean ridge, with 7-10 km thick oceanic crust underlain by metasomatized lithosphere, and (ii) a melting region beneath anomalously thick crust. Magmatic systems beneath thick crust were found out to be more compact and symmetrical and undergo a longer evolution.
Numerical modeling for continental melting zones with regard to the lithospheric structure and the size of the juxtaposed cratons and plates allowed the following inferences: (1) the extent of the predicted lithospheric melting region slightly exceeds the length of the respective lava field, (2) the melting zone has a layered structure (therefore, melts derived from a relatively homogeneous substrate should be homogeneous and of the same type), (3) magma chambers are relatively independent, which provides a qualitative explanation for the known cyclicity of lava compositions and the spatial distribution of major-element compositions of rocks in igneous provinces.
The behavior of the compositions of fluids outgassing at the solidus boundary from the crystallizing basaltic melt were computed using the Selektor software in a flow reactor and a step source modifications. Modeling shows that a quasi-steady temperature profile of a fluid-magmatic system related to a 30-40 km deep magma source sets up for 0.5 to 1 Myr. We infer that uncondensed reduced fluids vent on the seafloor and produce graphite and Fe, Ti, and Mn ferrite deposits found in the crest of the Mid-Atlantic Ridge. The numerical results were supported by physical modeling of carbon precipitation during the interaction of synthetic gas with mafic and ultramafic minerals. Carbon-related mineralization associated with gas condensation is controlled by the relative contents of C, H, Cl, F, and S in magmatic fluids. The composition of outgassing fluids changes notably within the liquidus-solidus range in crystallizing magma. During retrograde boiling, fluid separates into a low-density fraction and a brine.
Retrograde boiling in magma chambers and auto-metasomatism of igneous rocks are similar in mid-ocean ridges and in the Siberian trap province in the case of very low wall-rock assimilation and contamination of mafic melts. Assimilation of crustal material, especially carbonates and salt-bearing rocks, coal beds, hydrocarbons, or oil water, by mafic melts produces anomalous magmatic fluids with up to 60 rel.% total hydrocarbons, including 45-50% CH4, and total H₂O, H₂S, N₂, and with H₂ two orders of magnitude higher than CO₂ and CO. These very fluids promote the formation of mineral deposits hosted by igneous rocks.

We report results of computer modeling of physicochemical ore formation processes at mercury deposits accumulated during the development of secondary-hydrothermal and mixed-fluid ore-forming systems. Exogenous chloride brines, oil waters of artesian basins, and petroleum pools are shown to serve as secondary mercury reservoirs and geochemical barriers.
Modeling of possible mechanisms of mercury transfer and deposition in the form of cinnabar (α - HgS) was performed for ore-forming solutions of different compositions. Four main thermodynamic models have been constructed using the Chiller program: (1) simple cooling (cooling only), for recent thermal springs, (2) mixing of high-chloride hydrothermal solutions with cold hydrosulfuric waters (mixing model), for telethermal deposits, (3) isoenthalpic boiling (P = f (T)), and (4) solution-rock interaction (rock titration model).

As inferred from melt inclusions in minerals of volcanogenic rocks, the ore-magmatic pyrite systems of Rudny Altai and Tuva had quite a specific history. Volcanism manifestations in these regions have both similarities and differences.
The chemical composition of melt inclusions in quartz of acid volcanic rocks from the pyrite deposits of Rudny Altai is close to the chemical composition of the rocks, and in REE content they correspond to island-arc magmas. At early stages of development of a magmatic system, acid melts formed, which had high temperature (1230-1250

In order to investigate the force-free fields it is proposed to use the computerized tomography methods. For the inversion of the ray transformation, the method of multipole fields expansion has been developed. This method is based on the expansion of a vector field and the ray transformation over the special basis of vector-functions. Analytical expressions for the ray transform of the basis vector-functions and the results of computer simulation are given.

In this paper, experience in the conducted investigation of the stochastic cellular automata models of forming stable oscillations and autowaves in active media is generalized. As a result, the concept of stochastic cellular automaton (CA), corresponding to the asynchronous CA with probabilistic transition rules, is formulated. The formal notions of a stochastic CA and a stochastic CA model are given. Properties of the CA models and methods of their synthesis, using a specified set of elementary physical and chemical transformations, are described. The possibility of the autowave and oscillatory processes simulation is shown on an example of the carbon monoxide oxidation reaction on the platinum catalyst with reconstructing its surface structure. The CA-simulation enabled to reveal the range of reaction parameters values, at which stable oscillations of the reagents concentration occur, and to observe autowaves over the platinum surface. Considerable attention has been given to a high efficiency of the stochastic CA parallel implementation, which demands preliminary transformation of the asynchronous mode to the block-synchronous one with validation of its equivalence to the asynchronous mode. The latter is done for the investigated reaction CA model by means of the comparative statistical analysis of the simulation results.

We consider the problem of estimating parameters of linear mathematical models. It is proved that due to the choice of weights in the least squares method it is possible to obtain solutions by minimizing any penalty functions of a wide class, including those of the Holder norms. A limitation on a set of solutions resulting from the variation of the weights in the least squares method has been determined. The possibility of the practical use of the established theoretical facts is illustrated by the ecology-mathematical models.

The interpolation problem of a one-variable function, which can be considered as a solution of a boundary value problem for an equation with a small parameter ε with a higher derivative is investigated. The application of the Lagrange interpolation for such a function on a uniform grid can result in serious errors. In the case of the Shishkin mesh, ε-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The ε-uniform error estimates of the Newton-Cotes formulas on such meshes are obtained. Numerical experiments have been carried out. The results obtained confirm the theoretical estimates.

This paper presents a family of hybrid linear multistep methods (LMM) with a second derivative term for the numerical solution of stiff initial value problems (IVPs) for ordinary differential equations (ODEs). The methods are stiffly stable for the step number k≤ 7.

The method for solving the inverse eigenvalue problem for the product of matrices of second and third orders is proposed. The necessary and sufficient conditions for the existence of the problem solution have been obtained.

We consider an inverse boundary value problem of heat conduction. To solve it, we propose a new approach based on the Laplace transform. This approach allows us to confine the original problem to solving the Volterra equations of the first kind. We have developed algorithms of the numerical solution to the resulting integral equations. The algorithms developed are based on the application of the product integration method and the quadrature of middle rectangles. A series of test calculations were performed to test the efficiency of the numerical methods.