Vol 10, No 1 (2016)

We study a partially invariant solution of rank 2 and defect 3 of the equations of a viscid heat-conducting liquid. It is interpreted as a two-dimensional motion of three immiscible liquids in a flat channel bounded by fixed solid walls, the temperature distribution on which is known. From a mathematical point of view, the resulting initial-boundary value problem is a nonlinear inverse problem. Under some assumptions (often valid in practical applications), the problem can be replaced by a linear problem. For the latter we obtain some a priori estimates, find an exact steady solution, and prove that the solution approaches the steady regime as time increases, provided that the temperature on the walls stabilizes.

For the general solution of two-dimensional equations of dynamics of a transverse isotropic medium with the Carrier–Gassmann condition, we give a representation in terms of two resolvent functions satisfying two separate wave equations. The problem of reflection of plane waves from a rigid wall and a free surface is solved. The coefficients of reflection and transformation of the plane waves are found. These formulas yield a solution for isotropic media too. Some special cases are consideredwhere the shapes (amplitudes) of the reflectedwaves are not uniquely determined, but linearly related with the shape of the incident wave.

We propose a method for finding sources of discrete dynamical systems of the circulant type with a q-valued arbitrary function at vertices. We find all sources, all fixed points, and some cycles, as well as lengths of some maximal chains outside cycles for the systems with Boolean threshold functions of at most three variables at the vertices.

Some nonstationary mathematical model is constructed to describe the vortex motion of an incompressible polymeric liquid. In the stationary case, several partial solutions for this model are found. The derivation of a variant of this model is given for the case when the pressure along the axis is independent of time.

Some results are presented of the numerical study of periodic solutions of a nonlinear equation with a delayed argument in connection with themathematical models having real biological prototypes. The problem is formulated as a boundary value problem for a delay equation with the conditions of periodicity and transversality. A spline-collocation finite-difference scheme of the boundary value problem using a Hermitian interpolation cubic spline of the class C¹ with fourth order error is proposed. For the numerical study of the system of nonlinear equations of the finitedifference scheme, the parameter continuation method is used, which allows us to identify possible nonuniqueness of the solution of the boundary value problem and, hence, the nonuniqueness of periodic solutions regardless of their stability. By examples it is shown that the periodic oscillations occur for the parameter values specific to the real molecular-genetic systems of higher species, for which the principle of delay is quite easy to implement.

Using the finite elementmethod we designed and implemented a nonlinear geomechanical model of the rock mass in the vicinity of the protective dam of tailing facilities in the permafrost of the Kumtor mine (the Kyrgyz Republic). The model takes into account the information on the structure of the object, the data on the strain-strength, thermophysical and filtration properties of frozen and thawed soil, as well as on the seasonal variations in air temperature. Numerical experiments showed that, under constant external conditions, the invariant properties of the rock mass, and the position of the neutral layer, the zero isotherm separating frozen and thawed rocks reaches a stationary position in 12–15 years after filling the storage; minor damage to the antiseepage screen can lead to a vast destruction area in the body of the dam. Using synthetic data, we demonstrated the solvability of the inverse boundary-value problem of determining the timing and location of damage in the anti-seepage screen from the data of piezometric measurements at several observation wells.

We study the symmetric properties of APN functions as well as the structure and properties of the range of an arbitrary APN function. We prove that there is no permutation of variables that preserves the values of an APN function. Upper bounds for the number of symmetric coordinate Boolean functions in an APN function and its coordinate functions invariant under a cyclic shift are obtained. For n ≤ 6, some upper bounds for the maximal number of identical values of an APN function are given and a lower bound is found for different values of an arbitrary APN function of n variables.

Under study are the polytopes of (0, 1)-normalized convex and 1-convex (dual simplex) n-person TU-games and monotonic big boss games.We solve the characterization problems of the extreme points of the polytopes of 1-convex games, symmetric convex games, and big boss games symmetric with respect to the coalition of powerless agents. For the remaining polytopes, some subsets of extreme points are described.