Том 94, № 1 (2016)

New results concerning improved local integrability and continuity of the densities of solutions to stationary Kolmogorov equations with diffusion matrices of low regularity and with locally unbounded drift coefficients are obtained.

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.

Conditions are presented under which two-part trigonometric systems arising in mixed type equations form a Riesz basis in the space of Lebesgue square integrable functions. For such systems, biorthogonal systems can be obtained in explicit form. As a result, an integral representation of the solution to the Frankl problem in a special domain can be found. The results are extended to two-part systems of broader functions.

A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero.

Expansion–contraction boundary value problems on manifolds with boundary are studied. The trajectory symbols of such problems are calculated, an analogue of the Shapiro–Lopatinskii condition is obtained, and the corresponding finiteness theorem is given.

A spectral problem for the one-dimensional Dirac system is considered. The question of the number of zeros for the components of the eigenvector functions of this problem is studied.

A numerically implementable mathematical model of a positive isotropic “broken” exponentially correlated field is constructed. A one-dimensional field distribution is determined by the first two moments for a given probability of zero, which defines the “degree of the breakage.” The corresponding distribution density of nonzero values is studied numerically.

Sufficient conditions for the asymptotic stability on a half-line of the solutions of a linear homogeneous Volterra-type integro-differential equation of order 3 in the case where the solutions of the corresponding linear homogeneous differential equation are asymptotically unstable are determined. A new method is proposed, and an illustrative example is constructed.

M.A. Sychev has recently shown that conditions necessary and sufficient for the lower semicontinuity of integral functionals with p-coercive integrands are W^1,p-quasi-convexity and the so-called matching condition (M). Condition (M) is so general that there is the conjecture that is always holds in the case of continuous integrands. The paper develops relaxation theory (construction of lower semicontinuous envelopes) under the assumption that condition (M) holds. It turns out that, in this case, the theory has very good structure. Applications of general relaxation theory to particular cases, including the theory of strong materials, are also discussed.

The goal of this study is to prove an existence and uniqueness theorem for the solution of a stochastic differential equation with Lévy noise in the case where the drift coefficient can be discontinuous. Additionally, the differentiability of the solution with respect to the initial condition is proved.

The Jacobian of a graph is defined as the maximal Abelian group generated by flows obeying two Kirchhoff’s laws. This notion, also known as the Picard group, sandpile group, or critical group, has been extensively studied by many authors in the past decade. This is an important algebraic invariant of a finite graph. At the same time, the structure of the Jacobian is known only in particular cases. The paper is devoted to the study of the structure of the Jacobian group for circulant graphs. For the simplest graphs in this family, the Jacobian group is explicitly described, and in the general case, and effective algorithm for calculating it is proposed.

The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ₁,ξ₂,...) of degree d in independent standard Gaussian random variables ξ₁ (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B^1/d (R¹) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on R^k to possess a density in the Nikol’skii–Besov class B^α(R)^k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R^k via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.

Exact solutions of a nonlinear integro-differential equation with quadratically cubic nonlinear term are found. The equation governs, in particular, stationary shock wave propagation in relaxing media. For the exponential kernel the shapes of both compression and rarefaction shocks having a finite width of the front are calculated. For media with limited “memorizing time” the difference relation permitting the construction of wave profile by the mapping method is derived. The initial equation is rather general. It governs the evolution of nonlinear waves in real distributed systems, for example, in biological tissues, structurally inhomogeneous media and in some meta-materials.

The integrodifferential Kolmogorov–Feller equation describing the stochastic dynamics of a system subjected to a regular “force” and a random external disturbance in the form of short pulses with random “amplitudes” and occurrence times is considered. The equation is written in differential form. A method for finding the regular force from a given stationary probability distribution is described. The method is illustrated by examples.