Vol. 50 No. 1 (239) (2016)

We construct a sequence of polynomials with respect to the Haar system and show that they form democratic bases in $L^1(0;1).$

In this paper a new differentiation and divided difference formula for rational functions is proved. The main result is a connection between divided differences of two rational functions with the same numerator, where the knots of one divided difference coincide with the zeros of the denominator of another rational function.

The present paper is a direct continuation of the papers [1, 2]. We obtain intermediate results, which will be used in the next final fourth part of this study, where a definitive solution to the Moore–Penrose inversion problem for singular upper bidiagonal matrices is given.

Let the sequence $\{a_{k}\}_{k=1}^{\infty},$ $a_{k}\searrow0$ with $\{a_{k}\}_{k=1}^{\infty}\notin l_{2},$ and Walsh system $\{W_{k}(x)\}_{k=0}^{\infty}$ be given. Then for any $\epsilon>0$ there exists a measurable set $E\subset\lbrack0,1]$ with measure $|E|>1-\epsilon$ and numbers\ $\delta_{k}=\pm1, 0$ such that for any $p\in[1, 2]$ and each function $f(x)\in L^{p}(E)$\ there exists a rearrangement $k\to\sigma(k)$ such that the series $\displaystyle\sum _{k=1}^{\infty}\delta_{\sigma(k)}a_{\sigma(k)}W_{\sigma(k)}(x)$ converges to $f(x)$ in the norm of $L^{p}(E)$.

Denote the space of all bivariate polynomials of total degree n$\leq n$ by $\Pi_n$. We are interested in n-poised sets of nodes with the property that the fundamental polynomial of each node is a product of linear factors. In 1981 M. Gasca and J. I.Maeztu conjectured that every such set contains necessarily $n+1$ collinear nodes. Up to now this had been confirmed for degrees $n \leq 5$. Here we bring a simple and short proof of the conjecture for $n = 4$.

In this paper a special class of systems of Boolean equations is investigated. For a “typical” case of such systems an asymptotic estimate for the number of solutions is determined.

Secure computation platforms are becoming one of the most demanded cryptographic tools utilized in diverse applications, where the performance is critical. This point makes important the optimization of every component of secure computation systems. Oblivious Transfer (OT) is a fundamental cryptographic primitive heavily used in such protocols. Most of the OT protocols used today are based on public-key cryptography, hence their efficiency suffers heavily from the number of modular exponentiation operations done. OT extensions were introduced to reduce the number of basic OT protocol execution rounds requiring public-key cryptography operations. Recently a white-box cryptography based OT protocol (WBOT) was introduced that avoids using expensive public-key operations. In this article extension protocols for WBOT are presented, that further improve the novel approach by dramatically decreasing the protocol invocation count required.

In the paper we consider the problem of the transmission of limited sound beams. The transmission of such beams is described by a nonlinear partial differential equation. In the paper we solve this equation by the latticecharacteristics method. Some numerical results are obtained for a special case.

Helix–coil transition of complexes DNA with two ligands of different binding mechanisms to native and melted regions of biopolymer is considered using the most probable distribution method. It was shown that obtained biphasic behavior of helix–coil transition curves depends on both binding affinity and concentration of ligands in solution. Thermodynamic behavior of the ligands having higher and lower affinity to the native DNA is compared.

In the paper influence of Sodium dodecyl sulfate (SDS) on planar bilayer lipid membranes (BLM) was investigated. It was shown that the presence of SDS and it’s concentration increasing leads to the loss of stability of the BLM, which is associated with the decrease in the value of the linear tension of pore edge in BLM, because of positive spontaneous curvature of the SDS molecule. It is also shown that the number of lipid pores on the BLM increases as a result of increase in probability of the pore formation, with reduction of the value of linear tension.

In the paper is presented existence of an increasing sequence of natural numbers $M_{\nu}, \nu=0,1,... ,$ such that for any $\varepsilon>0$ there exists a measurable set $E$ with a measure $\mu E>1-\varepsilon,$ such that for any function $f\in L^{1}[0,1]$ one can find a function $g\in L^{1}[0,1],$ which coincides with the function $f$ on $E$, and for any $\alpha \neq-1,-2,...$ the Cesaro means $\sigma_{M_{\nu}}^{\alpha}(x,\tilde{f}),~\nu=0,1,...,$ converges to $g(x)$ almost everywhere on [0,1].