Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean Spaces

Ukrainian Mathematical Journal - We establish new criteria of compatibility for a linear system of equations (equivalent to the Kronecker–Capelli theorem) or inequalities (equivalent to the...     

Integral Newton-Type Polynomials with Continual Nodes

We construct an integral Newton-type interpolation polynomial with a continual set of nodes. This interpolant is unique and preserves an operator polynomial of the corresponding degree.     

Structure and hydration of polycytidylic acid from the data of infrared spectroscopy, EHF dielectrometry and computer modeling

The aim of this paper is to summarize the original results concerned with the elucidation of the role of water environment in the formation of different structures of polyribocytidylic acid (poly(rC)), depending on the pH, temperature and ion content. To solve this problem, we studied the hydration of poly(rC)-K⁺ in films differing in water content, by methods of infrared spectroscopy and piezogravimetry, and in solution, using the dielectric measurements at a wavelength of 7.6 mm (extremely high frequencies — EHF). The experimental results were confirmed by a Monte Carlo simulation of the interaction between water molecules and single-strand and double-strand poly(rC) fragments in clusters of 800 water molecules. A model of hydration of double-strand complex of poly(rC)⁺-poly(rC) has been proposed based on our results and the known X-ray parameters of the complex. The obtained results and proposed structure of poly(rC)⁺-poly(rC) suggest that, the stabilization of this complex occurs due to intra- and inter-chain water bridges, together with the hydrogen bonds between neutral and protonated cytosines in pairs.     

On Weak Convergence of an Iteration Process with Minimal Norm in a Hilbert Space

We show that in a Hilbert space with a measure and a minimal norm on a chosen sequence of nodes, the interpolation process converges to the interpolated polynomial operator.     

Analysis of the Accuracy of Interpolation of Entire Operators in a Hilbert Space in the Case of Perturbed Nodal Values

In a Hilbert space with Gaussian measure, we obtain an estimate for the accuracy of interpolation of an entire operator in the case where its values are perturbed at nodes and determine the value of the degree of an interpolation polynomial the exceeding of which does not improve the estimate of the accuracy of interpolation.     

On continual interpolation nodes for operators in linear topological spaces

We establish conditions for the existence of continual nodes for interpolation polynomials of the integral type. This result is generalized to the case of multivariable operators. Some examples of these interpolants are analyzed.     

On the problem of Hermite interpolation of operators in a Hilbert space

The problem of Hermite operator interpolation with interpolational conditions containing Gateaux higher-order differentials in arbitrary directions is investigated. A necessary and sufficient condition for solvability of this problem in a Hilbert space is established, and the set of all Hermite operator polynomials and its subset of interpolants preserving operator polynomials of the same degree are described. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 38–48.     

A matrix modification of formulas for the Hermite interpolation of operators in a Hilbert space

Necessary and sufficient conditions for the existence of an Hermite operator polynomial in a Hilbert space are obtained, and Hermite interpolating operator formulas containing vectors and matrices of lower (in comparison with E. G. Kashpur, V. L. Makarov, and V. V. Khlobystov, “On the problem of Hermite interpolation of operators in a Hilbert space”, Vychisl. Prikl. Mat., No. 78, 38–48 (1994)) dimension are derived. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 28–37.     

Hydration of DNA-binding biological active compounds: EHF dielectrometry and molecular modeling results

We report hydration properties of several DNA-binding ligands (pharmaceutical drug caffeine; mutagens proflavine, ethidium bromide, propidium iodide; polyamines putrescine, spermine), which are able to interact with DNA in different modes via external binding, intercalation or minor groove binding. We show that the detection of the bound water molecules and the estimation of their amount in aqueous solutions of ligands can be efficiently carried out using the measurements of complex dielectric permittivity of the solutions in the millimeter range of radio waves. Our dielectrometric data are combined with the results of the molecular modeling including quantum chemical calculations and Monte Carlo simulations. We show that number of water molecules able to form hydrogen bonds with donor–acceptor groups of ligands correlates with hydration numbers taken from the literature or obtained in EHF dielectrometry experiment. The latter indicates that the EHF dielectrometry method is sensitive for the tightly bound water molecules.     

Interpolation of polynomial operators in a Hilbert space

An interpolant with orthonormal nodes is constructed for a polynomial operator of a given degree defined on an abstract Hilbert space. An estimate for the interpolation accuracy is derived in the metric of the space of operator values, while in the case where nodes are elements of a basis, the pointwise convergence of the interpolational operator process in this metric is proved as the number of nodes increases. Bibliography: 7 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 3–9