Vol. 46 No. 1 (227) (2012)

In the paper considers continuality in a parameter of generalized eigenfunc-tions of one class of differential operators. Properties of one class of functions constructed by means of the generalized eigenfunctions and used in the theory of integral equations are studied.

In this paper the convergence in $L_1[0, 1]$ of some irregular linear means of Fourier–Walsh series of integrable functions after correcting these functions on sets of small measure is studied.

The problem of the existence of non trivial constant weight perfect codes in the $B^n$-space defined over $GF(2)$ remains unsolved up to now. It has been proved in the present paper that the problem of the existence of constant weight perfect codes is equivalent to the problem of the existence of $D$-representable codes in the fixed layer.

In the present paper the results of [1] are generalized, more general systems of operators generated by Riemann–Liouville integral and derivative and general systems of functions generated by Mittag–Leffler type functions are introduced as well.

Contact problem has been considered for an infinite composite elastic (piecewise homogeneous) plate consisting of two semi-infinite plates having different elastic characteristics, that are linked to each other along the common straight border. It was assumed that an infinite elastic stringer (IES) is glued over its full length and width to the upper surface of infinite composite elastic plate parallel to the line of heterogeneity of the mentioned two semi-infinite plates and have different elastic properties.The layer of glue being in the state of pure shear. The contacting triple (plate–glue–stringer) is simultaneously deformed by a concentrated force applied to IES and by uniformly distributed horizontal tension stresses of constant intensity acting at infinity on the infinite composite elastic plate. The solution of contact problem under consideration is reduced under certain condition to the solution of functional equation according to Fourier transformant of required function. The closed-form solution of the contact problem at issue is constructed in the integral form. The tangential contact and normal stresses arising in IES have been determined. Asymptotic formulae describing the behavior of stresses both near and far from the origin of force have been obtained.

The diffraction of plane shear electro-elastic wave on a thin semi-infinite $\Phi_{\infty}(x, y),~w_{\infty}(x, y)$ metallic layer piezoelectric-vacuum space is considered in the presence of a thin infinite (ground shield) metallic layer in vacuum. The problem is reduced to the solution of a functional equation in the theory of analytical functions. The problem admits closed solution describing the wave fields in vacuum and in the piezoelectric medium. The presence of the semi-infinite metallic layer leads to a diffraction of a waves, a result of which surface electroelastic waves occur.

In this paper independence numbers of the powers of $C_5$ graph is investigated. Independence number of the 3^rd degree of $C_5$ is calculated and a method is given that can help calculate independence numbers of higher degrees of $C_5$. Indepen-dence number of the 3^rd degree of $C_5$ is also calculated by the given method.

In the present article logic programs (both with and without negation) that do not use functional symbols are studied. Three algorithmic problems for functional symbol-free programs are investigated: the existence of a solvable interpreter, the problem of Δ-equivalence and the problem of logical equivalence. The first two problems are known to be decidable for functional symbol-free definite programs. We show that the third one is also decidable for such programs. In contrast, all three problems are shown to be undedicable for functional symbol-free general programs.

A public-key encryption scheme is proposed, which uses a special case of Sims’ algorithm for construction of “strong” generator sets for permutation groups.

Modelling of a RF magnetic field’s configuration near the flat coil face of the tunnel diode based single-layer flat-coil-oscillator (SFCO) has been done – that serves as a sensor in one of most of effective methods for study of superconducti-vity (the SFCO technique). Particularly, the distribution of the measuring magnetic field of flat coil is investigated. An analytical formula has been derived, which allows to determine the parallel and perpendicular to coil face components of the measuring RF magnetic field at any point near the coil. The results of calculations are aimed at an achievement of the optimum sensitivity of the SFCO technique during the experimental study of fine peculiarities of superconductivity that are important for understanding the phenomenon.

In the present paper it has been shown that nodes of any finite set $X\subset\mathbb{R}^d$ can be made independent by arbitrarily small perturbation, in other words, the set $X$ can be approximated by sets of independent nodes. In the case of #$X =\dim\Pi_n^d$ the set $X$ can be approximated by sets of poised nodes.

A round rod of variable cross-section is considered, to which a centralized force is applied. An optimization problem is stated for obtaining such a regularity of transverse section variability, for which the value of maximum deflection of the rod would be the least. The case of pin-jointing of the rod was considered and the problem was solved by means of variational method. An analytical solution to this problem was obtained, according to which the corresponding plots of unknown functions were constructed.