Vol. 47 No. 3 (232) (2013)

We derive new asymptotic formulae for the norming constants of Sturm– Liouville problem, which generalize and make more precise previously known formulae, by taking into account the smooth dependence of norming constants on boundary conditions.

The work is devoted to the investigation of one class of Volterra type nonlinear integral equations on positive half-line. The specified class of equations except for self-interest in mathematics has also important applications in physical kinetics. The combination of special factorization methods with methods of construction of invariant cone segments allows us to construct a non-negative solution of initial equation, and investigate integral asymptotics of that solution at infinity.

The present paper is devoted to the $\beta$-uniform algebra of bounded generalized analytic functions on the “generalized disk”. The issues related to the well-known corona problem for this topological algebra are investigated.

In the paper we prove that for some type of general Haar systems (particularly for classical Haar system) and for any $\varepsilon>0$ there exists a set {$E\subset (0,1)^2,|E|>1-\varepsilon$}, such that for every $f\in L^1(0,1)^2$ one can find a function $g\in L^1(0,1)^2$, which coincides with $f$ on $E$ and Fourier--Haar coefficients $\{c_{(i,k)}(g)\}_{i,k=1}^\infty$ are monotonic over all rays.

We introduce the Banach spaces $h_{\infty}(\varphi)$, $h_{0}(\varphi)$ and $h^{1}(\eta)$ of functions harmonic in the unit ball in $\mathbb{R}^n$, depending on weight function $\varphi$ and weighting measure $\eta$. The paper studies the following question: for which $\varphi$ and $\eta$ we have $h^{1}(\eta)^*\sim h_{\infty}(\eta)$ and $h_{0}(\varphi)^*\sim h^{1}(\eta)$. We prove that the necessary and sufficient condition for this is that certain linear operator, which projects $L^\infty(d\eta\, d\sigma)$ onto the subspace $\varphi h_{\infty}(\varphi)$, is bounded.

In the paper we consider systems of operators generated by Weil integral and derivative, and functions generated by exponential type functions. For a certain class of functions a generalization of Taylor–Maclaurin type formula is obtained in a neighborhood of $+\infty$.

A structure consisting of a layer and a semi-space, made of elastic piezoelectric materials, is considered. Unlike the known approaches to the problem, it is assumed that the layer and the semi-space can freely slide relative to each other. The problem of Gulyayev–Bleustein type surface wave propagation is investigated for four different variants of boundary conditions at the external surface of the layer. It is established, that in one case there exist two Gulyayev– Bleustein type waves, in two other cases there exists one such wave for each case, and in the last case there is no surface wave of mentioned type.

The present article is devoted to the termination of logic programs, which do not use functional symbols (FSF programs). A program $P$ is terminating with respect to a goal $G$, if the SLD-tree of $P$ and $G$ is finite. In general, FSF programs are not terminating. A transformation is introduced, by which any FSF program is transformed into another, not FSF program, which is shown to be terminating with respect to the permitted goals of the original program. The program obtained via transformation and the original program are $\Delta$-equivalent.

This study was carried out because of a widely spread mistaken viewpoint that melting temperature $(T_m)$ cannot be used as a reasonable parameter in the case of multi-peak differential melting curves (DMC) of DNA. However, there were no theoretical or experimental evidences or disclaimers of this viewpoint. In this work such study has been carried out. We have tested various definitions of $T_m$. Indeed, some of them give unreasonable dependences of Tm on relative concentration of modifications $(r_b)$. At same time, the average temperature of the helix-coil transition calculated as the integral over the temperature of the product of DMC and temperature gives reasonable smooth dependences $T_m(r_b)$. The same definition is the best for the dependence of $T_m$ on the average GC-content.

In the present paper it is shown that for every number $k\not\equiv 1\pmod{60},$ the equation $\dfrac{5}{k}=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}$ has at least one solution $(x, y, z)\in N.$