Vol. 46 No. 3 (229) (2012)

 Let D be a bounded convex domain in the Euclidean plane and we choose uniformly and independently two points in D. How large is the mean distance m(D) between these two points? Up to now, there were known explicit expressions for m(D) only in three cases, when D is a disc, an equilateral triangle and a rectangle. In the present paper a formula for calculation of mean distance m(D) by means of the chord length density function of D is obtained. This formula allows to find m(D) for those domains D, for which the chord length distribution is known. In particular, using this formula, we derive explicit forms of m(D) for a disc, a regular triangle, a rectangle, a regular hexagon and a rhombus.

The famous Theorem of Yu. Lubich allows us in the language of the almost periodicity to get the criterion of completeness of the eigenvectors of a Hermitian compact operator in a weakly complete Banach space. In this paper this result is strengthened for the representation generated by the operation of the generalized shift.

It is well known that free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free distributive lattice on n free generators is isomorphic to the lattice of monotone Boolean functions of n variables. In this paper we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. Keywords: Antichain, monotone Boolean function, De Morgan function, free De Morgan algebra.

We continue the study started in [1] regarding the Taylor–Maclaurin representation of functions. In the present paper we obtain a more convenient representation for the function R_n(x), the residual term of the generalized Taylor–Maclaurin formula.

A new statistic called “generalized sample range” is introduced and estimation of its expectation is provided. Maximum value of expectation of new range is determined. Characterization of corresponding distribution, which affords that maximum is obtained. The asymptotic behavior and limiting relations of the distribution and its characterization are considered.

We consider the degenerate nonself-adjoint differential equation of fourth order $Lu\equiv (t^{\alpha} u^{\prime\prime})^{\prime\prime} + au^{\prime\prime\prime} − pu^{\prime\prime} + qu = f$ , where $t \in(0, b), 0\leq\alpha\leq 2, \alpha ≠ 1,~a,~p,~q$ are the constant numbers and $a ≠ 0, p > 0, f \in L_2(0, b)$. We prove that the statement of the Dirichlet problem for the above equation depends on the sign of the number a (Keldysh Teorem).

The contact problem has been considered for elastic composite (piecewisehomogeneous) infinite plate consisting of two semi-infinite plates interlinked along the common straight border. Parallel to this line of heterogeneity of these semi-infinite plates with different elastic properties, an elastic piecewise-homogeneous infinite stringer is continuously glued over its full length and width on the upper semi-infinite plate, the layer of glue during the deformation being in the state of pure shear. The contacting triple (plate–glue–stringer) is simultaneously deformed by codirectional concentrated forces applied to the stringer and uniformly distributed horizontal tensile stresses of piecewise–constant intensity acting at infinity on the plate. According to the generalized Fourier integral transform, under certain conditions the solution of contact problem under consideration reduces to a solution of functional equation in the Fourier transforms of an unknown function on the real axis. A closed form solution of the contact problem in question is given in an integral form. As a result of investigations it was shown that due to the presence of the layer of glue the tangential contact forces have no singularities in the points of application of forces and in sections of semi-infinite stringer attachment.

In the present paper the completeness and the minimality of the set of the following built-in constants of Backus FP system are proved: Identity, Head, Tail, Append left, Equals, Composition, Constuction, Condition, Constant. 

Huygens–Fresnel principle for dynamical diffraction of X-rays in a crystal with weak deformation field of its space lattice is stated with a view to construction of the wave functions of diffracted beams in the crystal lattice at an arbitrary distance r of the point source of primary radiation from the crystal. In particular, the well-known approximations of an incident spherical $(r = 0)$ and plane $(r\rightarrow\infty)$ waves result as the limiting cases of the problem under consideration. Some features of the interference absorption (the Borrmann effect) of X-ray wave packages in a crystal with weak field of lattice displacements were discussed.

The possibility of localized surface plasmons formation in spherical structure based on metamaterials, with both the positive and negative dielectric constants, are considered. In contrast to ordinary localized surface plasmons, where the field decreases according to the power law with increasing distance from metallic nanoparticles, in case of strongly localized surface plasmons the field decreases according to the exponential law. Owing to that good preconditions are created for formation of structures with controllable parameters, where the wave energy is concentrated within nanometric ranges as is the case in atoms.

The interplay of the Rashba and Dresselhaus spin-orbit as well as the Fröhlich type electron-phonon interactions on the energy dispersion relation of the spin subbands in a two-dimensional electron gas in semiconductor polar heterostructures is studied theoretically. The Rayleigh-Schrodinger perturbation theory has been used to obtain in closed form the basic parameters of the polaron state (self-energy and effective mass) as a function of the Rashba and Dresselhaus coupling strengths.