The article focuses on differential geometry of ρ-dimensional complexes C^ρ of m-dimensional planes in the projective space P ⁿ containing a finite number of developable surfaces.

This article relates to researches on projective differential geometry based on the moving frame method by E. Cartan and method of exterior differential forms. These methods make it possible to study from a single viewpoint differential geometry of submanifolds of different dimensions of a Grassmann manifold and also generalize the results to wider classes of manifolds of multidimensional planes.

In order to study such submanifolds we apply the Grassmann mapping of the manifold G(m, n) onto the (m + 1)(n − m)-dimensional algebraic manifold Ω (m, n) of the space P ^N, where N =N =  (ⁿ⁺¹_m+1)−1.  

Primary task of differential geometry of submanifolds of Grassmann manifolds is to carry out uniform classifications of various classes of such submanifolds, to determine their structure and–a related task–to define the degree of freedom of their existence, and also to research the properties of submanifolds of various classes.

The intersection of an algebraic manifold Ω (m, n) with its tangent space T_l Ω (m, n) represents the Segre cone C_l(m + 1, n − m). This cone is of dimension n and carries plane generatrices with dimensions m + 1 and n − m intersecting in straight lines. The projectivization P B_l(2) of this cone is the Segre manifold S_l(m, n − m − 1).

The Segre manifold S_l(m, n−m−1)s is invariant under projective transformations of the space P ^{(m+1)(n−m)−1} = P T_l Ω (m, n), which is the projectivization with the center at point l of the tangent space T_l Ω (m, n) to the algebraic manifold Ω (m, n). The Segre manifold S_l(m, n − m − 1) is used for classification of the considered submanifolds of the Grassmann manifold G(m, n), and also for interpretation of their properties in projective algebraic manifold terms. Classification of submanifolds of the Grassmann manifold G(m, n) is based on various configurations of plane P T_l Ω (m, n) and on the Segre manifold S_l(m, n − m − 1). The purpose of this article is to prove geometrically a theorem for determining the order of the Segre manifold S_l(m, n − m − 1).

The article is devoted to the study of behavior of the solution to a second-order parabolic equation with Tricomi degeneration on the lateral boundary of a cylindrical domain Q^T , where Q is a stellar region whose boundary ∂Q is an (n−1)-dimensional closed surface without boundary of class C^1+λ, 0 < λ < 1. We study the question of unique solvability of the first mixed problem for the equation with the boundary and initial functions belonging to spaces of type L_p, p > 1. This topic goes back to the classical works of Littlewood–Paley and F. Riesz devoted to the boundary values of analytic functions. All directions of taking boundary values for uniformly elliptic equations turn out to be equal, and the solution has a property similar to the continuity with respect to a set of variables. In the case of degeneracy of the equation on the boundary of the domain when the directions are not equal, the situation becomes more complicated. In this case, the statement of the first boundary value problem is determined by the type of degeneracy.

We consider autonomous differential equations of the second order the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuous coefficients. In addition, it is assumed that the leading coefficient and the free term do not vanish. Such equation define on the cylindrical phase space a dynamical system without singular points and closed trajectories homotopic to zero. Structurally stable are equations for which the topological structure of the phase portrait of the corresponding dynamical system does not change under small perturbations in the class of such equations. An equation is structurally stable if and only if all of its closed trajectories are hyperbolic. Structurally stable equations form an open and everywhere dense set in the space of the equations under consideration. The paper investigates equations of the first degree of structural instability – structurally unstable equations for which the topological structure of the phase portrait does not change when passing to a sufficiently close structurally unstable equation. The set of equations of the first degree of structural instability is an embedded smooth submanifold of codimension one in the space of all equations under consideration; it is open and everywhere dense in the set of structurally unstable equations and consists of equations that have a single nonhyperbolic closed trajectory – a double cycle.

The present work covers mathematical modeling of the process of dissociation (decomposition) of natural gas hydrate of the Sredneviluysky gas condensate field in a laboratory sample of natural sandstone. Initially, the porous medium, filled with natural gas, water, and hydrate, is in thermobaric conditions meeting the stable state of the gas hydrate. Then, the pressure is released from one side of the cylindrical hydrate sample, which causes its decomposition. The mathematical model of the decomposition process takes into account the two-phase filtration of gas and water, the throttling effect, convective heat exchange, heat absorption during hydrate dissociation, and the kinetics of this process. The developed model and its implementation algorithm are tested against the results of a known experimental work. As a result of the computational experiment, distributions of gas pressure and temperature, hydrate and water saturation are obtained. Furthermore, the duration of the hydrate dissociation process is estimated with varying some initial parameters.

A linear two-dimensional problem in the form of dynamic equations of porous media for the components of velocities, stresses and pressure is considered. The dynamic equations are based on conservation laws and consistent with the thermodynamics conditions. The medium is considered to be ideal (there is no energy loss in the system) isotropic and two-dimensional inhomogeneous with respect to space. For the numerical solution of the problem posed, the method of integrating the integral Laguerre transform with respect to time with finite-difference approximation in spatial coordinates is used. The solution algorithm employed makes it possible to efficiently carry out simulations in a complex porous medium and to study the wave effects arising in such media.

With the use of convolutional neural networks, we solve inverse problems of exploration seismology to determine the spatial position and physical characteristics of geological fractures, such as the proportion of excess surface and the nature of saturation. The training and validation sets were formed using numerical modeling by the grid-characteristic method on unstructured meshes in the two-dimensional case. The continuum mechanics equations were used, while the fractures were specified discretely in the integration domain; this approach made it possible to obtain the most detailed patterns of wave responses.