Vol 52, No 1 (2019)

We consider quasilinear parabolic systems of equations with nondiagonal principal matrices. The oblique derivative of a solution is defined on the plane part of the lateral surface of a parabolic cylinder. We do not assume smoothness of the principal matrix and the boundary functions in the time variable and prove partial Hölder continuity of a weak solution near the plane part of the lateral surface of the cylinder. Hölder continuity of weak solutions to the correspondent linear problem is stated. A modification of the A(t)-caloric approximation method is applied to study the regularity of weak solutions.

We considered a self-diffeomorphism of the plane with a fixed hyperbolic point at the origin and a non-transverse point homoclinic to it. Periodic points located in a sufficiently small neighborhood of the homoclinic point are divided into single-pass and multi-pass points depending on the location of the orbit of the periodic point with respect to the orbit of the homoclinic point. It follows from the works of W. Newhouse, L.P. Shil’nikov, B.F. Ivanov and other authors that for a certain method of tangency of the stable and unstable manifolds there can be an infinite set of stable periodic points in a neighborhood of a non-transverse homoclinic point, but at least one of the characteristic exponents of these points tends to zero with increasing period. Previous works of the author imply that for a different method of tangency of the stable and unstable manifolds there can be an infinite set of stable single-pass periodic points, the characteristic exponents of which are bounded away from zero in the neighborhood of a non-transverse homoclinic point. It is shown in this paper that under certain conditions imposed primarily on the method of tangency of the stable and unstable manifolds there can be a countable set of two-pass stable periodic points, the characteristic exponents of which are bounded away from zero in any neighborhood of a non-transverse homoclinic point.

The method of consecutive projections (MCP) is popular due to the simplicity of its implementation and efficient use of memory. The main idea of the method is that a convex set is represented as a finite or infinite intersection of a set of simple convex (elementary) sets. Then it is projected on the sets external to the current point. The projection on these elementary sets is very simple because they are usually semispaces. It is proved that the iterative process of consecutive projections converges, and its modifications that ensure final convergence are developed. Three subproblems are solved within the MCP at each iteration: find an elementary set for projection, determine the direction, and calculate the step length in this direction. In this paper, we make several simple propositions that make it possible to combine these three problems and accelerate the convergence of the method for solving a special class of problems called deconvolution problems.

This study is devoted to the issue of constructing robust T-optimal discriminating designs for two competing trigonometric regression models, which differ by three trigonometric functions at most. To solve this problem, we propose using a Bayesian and standardized maximin approaches. The robust T-optimal discriminating designs were found explicitly in a number of particular cases. In the general case, on account of the complexity of the optimization problem, the corresponding optimal designs are not easy to find and have to be determined numerically. The results are illustrated by means of several examples.

The effect of the vibrational level of a molecule on the relaxation time of its rotational energy is studied within the state-to-state kinetic theory approach. The rotational levels of molecules are described by the non-rigid rotator model, while the interaction between molecules is described by the variable soft sphere model. This model is used to calculate the N₂-N, O₂-O, and NO-O collision cross sections for different vibrational and rotational levels of molecules. The rotational energy relaxation time is introduced for each vibrational level using the methods of the kinetic theory of nonequilibrium processes. The relaxation times are numerically calculated within a broad temperature range and compared with the relaxation time determined by the well-known Parker formula. The effect of various multi-quantum rotational transitions on the accuracy of the rotational relaxation time calculation is analyzed, and the convergence of the solution with an increase in the maximally possible number of quanta transmitted in the course of transition is demonstrated. It has been established that the vibrational state of a molecule has an appreciable effect on the rotational energy relaxation time in the state-to-state approach, and using the Parker formula may lead to a noticeable error in the calculation of state-to-state transport coefficients. The Parker formula provides a satisfactory agreement with the results obtained via the averaging of state-resolved relaxation times with a Boltzmann vibrational energy distribution in the one-temperature approximation at moderate temperatures.

The influence of the relative thickness of a “stiff” viscoelastic polymer layer and the orientation of reinforcing layers of the bearing layer on the values of their natural frequencies and loss factors of damped vibrations of an unsupported two-layered composite plate has been investigated. It has been established that each natural vibration mode of a two-layered plate corresponds to a certain effective relative thickness of an isotropic viscoelastic polymer layer. A further increase in the relative thickness of plates with an orthotropic bearing layer depending on ambient temperature may result in both an increase and decrease in natural frequencies without any considerable change in their energy dissipation. For plates with a monoclinic bearing layer, the increase in relative thickness of the isotropic viscoelastic polymer layer is accompanied by both a decrease and increase of loss factor for different vibration modes. It is shown that a change in the stacking sequence of reinforcing layers in the bearing layer brings about mutual transformation domains of coupled vibration modes. It has been established that the mutual transformation of the natural shapes of coupled modes of an unsupported quasi-uniform rectangular plate vibrations appears if one of the natural shapes has an even number of quarter-wavelengths in at least one of the plate directions, whereas another natural shape has an odd number of those. It has been demonstrated that the temperature-frequency relationship of elastic-dissipative characteristics of the “stiff” viscoelastic polymer has a considerable effect upon natural frequencies and loss factors of all the investigated vibration modes for an unsupported rectangular two-layered plate.

The theory of figures of celestial bodies, which are in the state of hydrostatic equilibrium under the action of pressure, gravitational, and centrifugal forces, took the form of a rigorous mathematical theory in the second part of the 20th century. Fundamental physical laws served as its basis. The Huygens–Roche figure (its total mass is concentrated in the center, while the rotating atmosphere takes the equilibrium form) plays an important role in the theory. Properties of the figure are carefully examined. In particular, it is known that each isobar (surface of equal pressure) itself is one of the three-parameter family of Huygens—Roche surfaces. However, as far as we know, convexity (or its absence) has not been discussed in the literature. Meanwhile, there are non-convex figures between equilibrium ones. In the present paper, we find the curvature of the meridional section of an arbitrary Huygens—Roche figure both in closed form and in the form of a series in powers of the Clairaut parameter, which is basic in the theory of equilibrium figures. We succeeded in proving that the curvature is positive and is bounded away from zero. Hence, every surface of the family of Huygens—Roche figures is convex and has no points of flattening. Moreover, none of the curves on its surface has points of straightening.