Vol. 49 No. 2 (237) (2015)

In this paper we consider two independent and identically distributed lines, which intersect a planar convex domain D. We evaluate the probability P_D, for the lines to intersect inside D. Translation invariant measures generating random lines is obtained, under which P_D achieves its maximum for a disc and a rectangle. It is also shown that for every p from the interval [0,1/2] and for every square there are measures generating random lines such that P_D = p.

It is proved, that if the order of a splitting automorphism of n-periodic product of cyclic groups of order r is a power of some prime, then this automorphism is inner, where $n \geq 1003$ is odd and r divides n. This is a generalization of the analogue result for free periodic groups.

In the present paper we deduce closed form expressions for the entries of the Moore–Penrose inverse of a special type upper bidiagonal matrices. On the base of the formulae obtained, a finite algorithm with optimal order of computational complexity is constructed.

The conjugate space of $\beta$-uniformly closed, symmetric algebras of functions on topological spaces are investigated. An application of such spaces in algebras of almost periodic functions is given.

We prove that for any perfect set P of positive measure, for which 0 is a density point, one can construct a function f (x) continuous on [0, 1) such that each measurable and bounded function, which coincides with f (x) on the set P has diverging Fourier–Walsh series at 0.

The problems of contact interaction are observed for an elastic half-plane and the infinite plate, which are strengthened, along the line (in the plane) by two finite overlays (stringers) with different elastic characteristics and constant small thickness. The contact interaction between deformable foundations and overlays is realized through a shear layers (in form of glue layers) having different physical–mechanical properties and geometric configuration. The determination problem of unknown contact stresses are reduced to the systems of Fredholm’s integral equations of the second kind within the different finite intervals, which in the certain region of the change of characteristic parameter typical to the problems, may be solved by the method of successive approximations. Possible particular cases are observed and the character and behavior of contact stresses are illustrated.

A total coloring of a graph G is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges get the same color. An interval total t-coloring of a graph G is a total coloring of G with colors 1,2, . . . ,t such that all colors are used and the edges incident to each vertex v together with v are colored by d_G(v)+1 consecutive colors, where d_g(v) is the degree of a vertex v in G. A block graph is a graph, in which every 2-connected component is a clique. In this paper we prove that all block graphs are interval total colorable. We also obtain some bounds for the smallest and greatest possible number of colors in interval total colorings of such graphs.

Typed $\lambda$-terms that use variables of any order and don't use constants of order $>1$ are studied in the paper. An algorithm of translation of typed $\lambda$-terms to untyped $\lambda$-terms is presented. According to that algorithm, each typed term $t$ is mapped to an untyped term $t'$. We study in which case typed terms $t_1$, $t_2$ such that $t_1 \to \to_{\beta\delta} t_2$ correspond to untyped terms ${t_1}',{t_2}'$ such that ${t_1}' \to\to_\beta {t_2}'.$

The study of spectral compression of randomly modulated optical signals is reported. The study of the spectral compression process is carried out for the additive noise model for initial pulses with different coherence time. The studies show that the ratio of spectral compression increases with decreasing of the coherence time of the initial signal.

The mass defect of the strange stars has been studied in the present work. The strange quark matter with minimal energy consisted of u, d, s quarks and electrons was observed in the framework of the bag model. For certain values of the bag parameters the integral parameters for three models of the strange stars with maximal masses twice exceeding sun mass were obtained depending on central density. The dependencies of inner energy E_in and bond energy Δ₂M on total mass M for data of the strange star models were presented. It was shown that the packing coefficient of strange stars consisted of strange quark matter is significantly higher than the analogous one of neutron stars.

The general solution of quadratic equations in free group using an automat, specified by the certain oriented graph is presented.