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Vol 72, No 1 (2017)
- Year: 2017
- Articles: 8
- URL: https://journal-vniispk.ru/0027-1322/issue/view/10014
Article
The variational approach to estimation of the dielectric permittivity of a unidirectional fibrous composite
Abstract
Based on the dual variational formulation of an electrostatic problem in an inhomogeneous anisotropic dielectric, we construct two-side bounds for possible values of second rank tensor components for the dielectric permittivity of a unidirectional fibrous composite taking into account mutual positions of anisotropic reinforcing fibers in the plane perpendicular to their axes. We consider variants of arrangement of cross sections of fibers such that the composite is transversal-isotropic or orthotropic relative to the property of dielectric permittivity. Some estimates of the greatest error appearing in the choice of true values of each component of the dielectric permittivity tensor of the composite in the form of a half-sum of its boundary values are obtained. A successive improvement of two-side bounds for domains where the components of this tensor should lie is carried out. A quantitative analysis of calculation dependencies is presented.
1-9
Probabilities of high extremes for a Gaussian stationary process in a random environment
Abstract
Let ξ(t) be a zero-mean stationary Gaussian process with the covariance function r(t) of Pickands type, i.e., r(t) = 1 − |t|α + o(|t|α), t → 0, 0 < α ≤ 2, and η(t), ζ(t) be periodic random processes. The exact asymptotic behavior of the probabilities P(maxt∈[0,T]η(t)ξ(t) > u), P(maxt∈[0,T] (ξ(t) + η(t)) > u) and P(maxt∈[0,T] (η(t)ξ(t) + ζ(t)) > u) is obtained for u → ∞ for any T > 0 and independent ξ(t), η(t), ζ(t).
10-14
Level lines of harmonic functions related to some Abelian integrals
Abstract
The geometry of level lines of harmonic functions being real parts of some Abelian integrals is studied. Such harmonic functions appear in the study of asymptotics of solutions to second-order differential equations, and the corresponding level lines are related both to the distribution of eigenvalues of a non-self-adjoint Sturm–Liouville problem and to location of trajectories of the corresponding quadratic differentials.
15-23
Integration of Banach-valued functions and Haar series with Banach-valued coefficients
Abstract
It is proved that for any Banach space each everywhere convergent Haar series with coefficients from this space is the Fourier–Haar series in the sense of Henstock type integral with respect to a dyadic differential basis. At the same time, the almost everywhere convergence of a Fourier–Henstock–Haar series of a Banach-space-valued function essentially depends on properties of the space.
24-30
Generalization of Lundberg’s inequality for the case of stock insurance company
Abstract
The ruin probability of an insurance company paying dividends according to a barrier strategy with a step barrier function is considered. Upper bounds for the probability of ruin are obtained within the framework of Sparre Andersen and Cramer–Lundberg risk models.
31-34
Brief Communications
Countability of the set of closed overclasses of some minimal classes in the partly ordered set ℒ23 of all closed classes of three-valued logic that can be mapped homomorphically onto two-valued logic
Abstract
The following theorem is proved: the set of closed classes containing some minimal classes in the partly ordered set ℒ23 of closed classes in the three-valued logic that may be mapped homomorphically onto the two-valued logic is countable.
35-36
The example of a differential system with continual spectrum of wandering exponent
Abstract
Some linear homogeneous two-dimensional differential system with piecewise continuous bounded coefficients is constructed so that the set of wandering exponents of its different solutions contains a segment of the real axis.
37-40
Multidimensional system of Diophantine equations
Abstract
An asymptotics for the number of solutions to a system of three Diophantine equations of additive type in six variables is found. Each additive summand of these equations is a simplest form whose degree in each variable does not exceed 1.
41-43