Vol. 47 No. 1 (230) (2013)

In the paper inverse $\pi$-extensions of the semigroup $Z_+$ are studied. It is shown that $\pi$-extension of the semigroup $Z_+$ is inverse, if and only if its $\pi$-extension coincides with $p(Z_+)$. The existence of a non-inverse $\pi$-extension for any abelian semigroup is proved.

Denote the space of all bivariate polynomials of total degree $\leq n$ by $\Pi_n$. We study the $n$-independence of points sets on quartics, i.e. on algebraic curves of degree 4. The $n$-independent sets $\mathcal X$ are characterized by the fact that the dimension of the space ${\mathcal P}_{\mathcal X}:=\{p\in \Pi_n : p(x)=0, \forall x\in \mathcal X\}$ equals $\dim \Pi_n-\#\mathcal X.$ Next, polynomial interpolation of degree $n$ is solvable only with these sets. Also the $n$-independent sets are exactly the subsets of $\Pi_n$-poised sets. In this paper we characterize all $n$-independent sets on quartics. We also characterize the set of points that are $n$-complete in quartics, i.e. the subsets ${\mathcal X}$ of quartic $\delta,$  having the property $p\in\Pi_n,~p(x)=0~\forall  x\in {\mathcal X} \Rightarrow p=\delta q,~q\in \Pi_{n-4}.$

In the paper the continuity of one conformal invariant extremal length is considered. A counterexample is constructed disproving the result of P.M. Tamrazov on the continuity of the extremal distance between two sets. Then some sufficient conditions for the continuity are given.

The paper studies the linear operators depending on normal pair of weight functions $\{\varphi, \psi\}$ in the Banach spaces $L^p(B_n)$. Here $B$ is the unit ball in the complex space $\mathbb C^n$. In particular, we study the question: for which values of $p$ these operators are bounded projectors.

Mean field games theory is a recent area of study introduced by Lions and Lasry in a series of seminal papers in 2006. Mean field games model situations of competition between large number of rational agents that play noncooperative dynamic games under certain symmetry assumptions. In this paper we consider quasivariational mean field games system with additional dependence on a velocity field of the players. We obtain certain estimates for the solutions to this system. In a forthcoming paper we intend to obtain an existence result using this kind of estimates and continuation method.

The contact problem for an elastic half-plane strengthened at its boundary by heterogeneous (piecewise-homogeneous) elastic overlay consisting of two semiinfinite pieces and one finite piece having different elastic characteristics has been considered. It was supposed that the contact interaction between the deformable foundation and an overlay was realized through a shear interlayer having different physico-mechanical properties and geometric configuration. The contact triple (overlay, interlayer, half-plane) was assumed to be deformed under the action of horizontal forces applied to the overlay. Using generalized Fourier transform the determinational problem of unknown contact stresses are reduced to the systems of Fredholm’s second kind integral eqations with two unknown functions within the different intervals, which in the region in the large change values of ratio of the problem characteristic parameters in the $B$ Banach space may be solved by the method of successive approximations. Possible particular cases have been observed and the character of change contact stresses is illustrated in different contact parts.

The problem of beam bending and proper frequency under load has been considered. The displacement, stresses and the frequency of the beam have been calculated using finite elements such as the Beam, Shell, Plane and Solid. The calculations have been carried out by applying the ANSYS software package and comparisons show that the results of modeling using the Plane elements prove more accurate.

The method for content adaptation of the interactive multimedia publications to various mobile devices is proposed. A comprehensive set of utilities is built for generation of the optimized output view in any of the broadly used forms, including HTML, Flash, etc. The implementation of the method is performed in a way, which allows an easy adoption of the new source applications and/or output views.

The algorithm for finding the set of ”nearest neighbors” in a set using compact blocks and hash functions is known (Elias algorithm). In this paper hash coding schemas associated to coverings by spheres of the same radius are considered. In general, such coverings can be obtained via perfect codes, and some other generalizations of perfect codes such as uniformly packed or quasi perfect codes. We consider the mentioned algorithm for Golay code and for two-error-correcting primitive BCH codes of lenght $2^m-1$ for odd $m$. A formula of time complexity of the algorithm is obtained in these cases.

The method for content adaptation of the interactive multimedia publications to various mobile devices is proposed. A comprehensive set of utilities is built for generation of the optimized output view in any of the broadly used forms, including HTML, Flash, etc. The implementation of the method is performed in a way, which allows an easy adoption of the new source applications and/or output views.

In this paper resolution strategies in fuzzy logic are investigated. Some new deduction strategies are introduced in fuzzy logic, analogous to those of classic logic, aimed to narrow deduction search space. In particular, semantic resolution is redefined for fuzzy logic, and some special cases are considered. The completeness of these strategies is proved.

For different types of DNA sequences it is demonstrated that a calorimetric melting curve calculated as a temperature dependence of a relative heat absorbance caused by DNA helix-coil transition is very close to the “real” melting curve that is the temperature dependence of the fraction of melted base pairs. There is the same closeness for the calorimetric differential melting curves (DMC) and the values of melting temperatures and temperature melting ranges. A simple procedure of recalculation of a calorimetric DMC into a real DMC is proposed.