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Geological, geochronological, and geochemical isotope studies are carried out for metamorphosed volcanic rocks and dolerites of the Maltsevka sequence of the Elash Group in the Biryusa block of the Siberian craton. It is found that mafic igneous rocks (dolerites and basaltic andesites) are close in composition to intraplate basalts. Flat or slightly fractionated REE patterns ((La/Yb) _n = 1.3-2.3) and positive ε_Nd( T ) values of +3.7 and +4.1 are observed. It is assumed that the depleted asthenospheric mantle and, possibly, plume mantle were the sources of these rocks, while the lithospheric mantle had no significant effect. Meta-andesites of the Maltsevka sequence belong to the tholeiitic series and have high La, Th, and U contents. Pronounced negative Nb and Ti anomalies are observed in the multielement patterns of these rocks, along with negative ε_Nd( T ) values of -4.6. It is assumed that meta-andesites resulted from the late Archean crustal melting with the participation of the mantle material. Metarhyolites prevalent in the Maltsevka sequence are divided into two groups similar in REE composition to А-type and I-type granites. A-type metarhyolites show high contents of Zr, Y, Nb, Th, and REE (except for Eu) and positive ε_Nd( T ) values of +2.2 and might have resulted from the melting of a source with geochemical isotope parameters close to those of mafic igneous rocks of the Maltsevka sequence. I-type metarhyolites have low contents of Y, Yb, Zr, and Nb but high contents of Th and show negative ε_Nd( T ) values of -3.7. They might have resulted from the melting of lower crustal diorite-tonalite rocks with addition of juvenile mantle material to the magma generation area. U-Pb zircon dating of metarhyolites of the Maltsevka sequence corresponding to A- and I-type granites showed that they are close in age, 1872 ± 10 and 1874 ± 10 Ma, respectively, which agrees with the age estimated earlier for granitoids of the Sayan complex of the Biryusa block. The similar ages and structural positions, along with the localization within the same structure, made it possible to unite volcanic rocks of the Elash Group and granitoids of the Sayan complex of the Biryusa block into a Paleoproterozoic volcanoplutonic association. The rocks of the association form the Sayan-Biryusa volcanoplutonic belt stretching for about 300 km along the zone of junction of the Biryusa block of the Angara fold belt and the Archean Tunguska superterrane of the Siberian craton. The belt is part of the large Paleoproterozoic South Siberian postcollisional magmatic belt formed at the final formation stage of the Siberian craton, when it was possibly part of the Paleoproterozoic Columbia supercontinent.

Composite dikes (mingling dikes) of Western Sangilen (southeastern Tuva) are believed to indicate a shift from tectonic contraction, related to the mountain fold system formation, to postcollisional extension that ended with the disintegration of the collisional orogen. However, the position of composite dikes in the general sequence of magmatic events was based only on their geologic location and has not been confirmed by U/Pb isotope-geochronological data. The proposed study is concerned with the geologic structure, composition, and isotopic age (485-490 Ma) of West Sangilen composite dikes. The isotopic age of the host granites from the Matut pluton is 509 Ma; thermal events of Riphean (679-934 Ma) and early Paleozoic (484 Ma) ages are reflected in zircons from gneiss-granites of the Erzin metamorphic complex. The obtained data are consistent with the earlier determined sequences of West Sangilen tectonometamorphic events. Basites from the examined composite dikes are strongly different in the contents of indicator elements (Rb, Nb, Ta, Zr, and Hf) and the Nb/Y, Zr/Y, Th/Ta, Zr/Nb, and Nb/Th ratios, which is indicative of different (probably, different-depth) mantle sources. The protolith of the salic component of the combined dikes might have been rocks similar in composition to the autochthonous/parautochthonous gneiss-granites of the Erzin metamorphic complex.

In this paper, we obtain the existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing an explicit Green's function and fixed point index theorem combining with an operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish a theorem on the existence of positive solutions for fourth-order boundary value problem with variable parameter. Finally, we give an example to illustrate our results.

The existence and uniqueness of the generalized solution of the boundary value problem for the system of magnetoporosity equations in the dissipative approximation have been proved. The results of the numerical solution obtained by the finite element method of the test boundary value problem of magnetoporosity in the frequency domain are presented.

The Kalman filter algorithm is currently one of the most popular approaches to solving the data assimilation problem. The major line of the application of the Kalman filter to the data assimilation is the ensemble approach. In this paper, we propose a version of the Kalman stochastic ensemble filter. In the algorithm presented the ensemble perturbations analysis is attained by means of transforming an ensemble of forecast perturbations. The analysis step is made only for a mean value. Thus, the ensemble π-algorithm is based on the advantages of the stochastic filter and the efficiency and locality of the square root filters. The numeral method of the ensemble π-algorithm realization is proposed, the applicability of this method has been proved. This algorithm is implemented for the problem in the three-dimensional domain. The results of the numeral experiments with the model data for estimating the efficiency of the offered numeral algorithm are presented. The comparative analysis of the root-mean-square error behavior of the ensemble π-algorithm and the Kalman ensemble filter by means of the numeral experiments with a one-dimensional Lorentz model is made.

This paper realizes a symbiosis of two approaches to the numerical solution of second order ODEs with a small parameter having singularities such as interior and boundary layers, namely, the application of both compact schemes of high orders and layer-resolving grids. The generation of layer-resolving grids, based on estimates of solution derivatives and formulations of coordinate transformations eliminating solution singularities, is a generalization of the methodology early developed for the first order scheme. This paper presents the formulas of the coordinate transformations and numerical experiments for the schemes of the first, second, and third orders on uniform and layer-resolving grids for the equations with boundary, interior, exponential and power layers of the first and second scales. The experiments conducted confirm the uniform convergence of the numerical solutions of equations with the help of compact schemes of high orders on the layer-resolving grids. By using the transfinite interpolation methodology or numerical solutions to the Beltrami and diffusion equations in a control metric, built by the coordinate transformations eliminating the solution singularities, the developed technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers.

The algorithms for solving the inverse source problem for the production-destruction type systems of nonlinear ordinary differential equations with measurement data in the form of time series are presented. The sensitivity operator and its discrete analogue on the basis of adjoint equations are constructed. This operator binds the perturbations in the unknown parameters of the model to those of the measured values. The operator allows one to construct a family of quasi-linear operator equations linking the required unknown parameters and the data of the inverse problem. The Newton-Kantorovich type method with right-hand side r- pseudoinverse matrices is used to solve the equations. The algorithm is applied to solving the inverse source problem for the atmospheric impurities transformation model.

This paper is concerned with investigating and solving the mixed initial boundary value problem for the heat conduction equation. The statement of the problem includes the three intervals: the first one (from 0 → T₁) describes heating the combustion chamber, the second (from T₁ → T₂) - cooling the chamber and a slower cooling of its wall. Finally, the third interval describes natural cooling of the chamber wall when the chamber has the temperature coinciding with that of environment. The validity of the application of the Fourier transform with respect to this problem has been proved. This made possible to transform the governing equation to the ordinary differential equation. By using the resulting equation, the inverse boundary value problem for the heat conduction equation by applying the nonlinear method of projection regularization was solved and the error of approximate solution was obtained.

One of widespread approaches to solving the Cauchy problem for the Laplace equation is to reduce it to the inverse problem. As a rule, an iterative procedure to solve the latter is used. In this study, an efficient direct method for the numerical solution of the inverse problem in the rectangular form is described. The main idea is based on the expansion of the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of the Laplace operator.

The main objective of this experimental work is to determine the physico-mechanical properties of tuffs used as building stone and to investigate the relationships between basic mechanical properties (compressive strength, flexural tensile strength, loss of volume by abrasion and impact strength) as well as physical and index properties (apparent porosity, dry unit weight, water absorption, P-wave velocity, Brinell hardness and point load index) of tuffs which are relatively easy to implement and low cost. The rock type investigated in this study was tuffs. Statistical analyses were performed to correlate the different properties. The results show that there are good and satisfactory relationships between the mechanical and physical-index properties of tuffs.