Vol. 51 No. 2 (243) (2017)

In the present paper we consider the Toeplitz-$T_{\bar{h}}^{ \alpha}$ and differentiation-$D^\delta $ operators on the Besov spaces $B_p(\beta)$ for all $0< p<\infty.$ We show that $T_{\bar{h}}^{ \alpha}: B_p(\beta)\rightarrow B_p(\beta)$ for $\bar h\in H^\infty(B^n)$ and $D^\delta :B_p(\beta)\rightarrow B_p(\widetilde\beta)$, where  $\widetilde\beta=\beta +p\delta .$

In this article the “bonus hunger” behavior for the Alternative bonus–malus system (BMS) is discussed. The Alternative BMS is a model, where the next premium is the combination of the previous premium and the aggregate claim amount. The key characteristics for the comparison are the discounted premium reduction for some time horizon and the entire claim amount. Existence of the steady state for the BMS discussed in this paper was proved and the probability of claiming for the general model and for its steady state was found out.

In an earlier work of author a theorem on recovery of Franklin series from its sum under some conditions was obtained. In present article it is shown that to some extent these conditions are necessary.

In the present paper a special class of infinite nonlinear system of algebraic equations with Teoplitz–Hankel matrices is studied. Above mentioned class of equations has direct applications in radiative transfer theory. Existence componentwise positive solutions for the system in space l1 are proved and some examples for mentioned equations, representing separate interest are given.

A model of elastically clamped support for an inner edge of an axisymmetric annular round bending plate is proposed. Values of parameters of the support as well as relationship between them is determined. With the collocation method the problem of bending of a cylindrically orthotropic annual plate of a variable thickness under a uniformly distributed load is solved taking into account the transverse shear. It is assumed that the inner edge of the plate is elastically fastened and the outer one is hingedly supported. Based on the analysis of calculated values of immeasurable quantities qualitative conclusions are drawn.

In this paper typed and untyped functional programs are considered. Typed functional programs use variables of any order and constants of order $\le$1, where constants of order 1 are strong computable, $\lambda$-definable functions with indeterminate values of arguments. The basic semantics of a typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. The basic semantics of an untyped functional program is an untyped $\lambda$-term, which is defined by means of a fixed point combinator. An algorithm that translates typed functional program $P$ into untyped functional program $P'$ is suggested. It is proved that the basic semantics of the program $P'$ $\lambda$-defines the basic semantics of the program $P$.

Central moments of function of the differential melting curve (DMC) of DNA have been calculated through expansion of block DNA DMC to Gaussian components. Calculations showed that the first moment of DNA DMC characterizes the melting temperature, the second is DNA melting interval. Central moments of the third and fourth orders characterize distribution of nucleotides along DNA chain. It was shown that the distribution of nucleotides along DNA chain differs from quasi-arbitrary distribution.

In this paper the Jacobson radical of an algebra$F\langle X\rangle / H$ is studied, where FhXi is a free associative algebra of countable rank over infinite field $F$ and $H$ is a homogeneous ideal of the algebr$F\langle X\rangle$. The following theorem is proved: the Jacobson radical of an algebra $F\langle X\rangle / H$ is a nil ideal.

We prove that if an automorphism j of the relatively free group of the group variety, defined by the identity relation $[x^n; y] = 1$, acts identically on its center, then j has either infinite or odd order, where $n\geq 665$ is an arbitrary odd number.

Considered the effects of external optical signals on the entangled states of the Fock of Kerr qubits in a nonlinear resonator. The necessary time dependence of the Hamiltonian for the realization of negation, Hadamard and controlled negation operators for the algebra of quantum logic have been found.