On the mean-value theorem for analytic functions

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Volyn University, Lutsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1143–1147, August, 1997.     

On the interpolation of analytic mappings

Let (E ₀,E ₁) and (H ₀,H ₁) be two pairs of complex Banach spaces densely and continuously embedded into each other, E ₁ ? E ₀ and H ₁ ? H ₀ and also let $\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } $ . By E _θ = [E ₀, E ₁]_θ and H _θ = [H ₀, H ₁]_θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by B _θ(0,R) we denote an open ball of radius R in the space E _θ. Let Φ: B ₀(0,R) → H ₀ be an analytic mapping taking B ₁(0,R) into H ₁, and let the estimates $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball B _θ(0,r) of radius r ∈ (0,R) in the space E _θ into H _θ and $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $ .     

Approximation of functions of Dirichlet class by Fejér means

For the Dirichlet classes D _p of holomorphic functions in the disk, we obtain the exact orders of best polynomial approximations and of upper bounds for deviations of Fejér means of Taylor series in the Hardy spaces H _p.     

The role of water H-bond imbalances in B-DNA substate transitions and peptide recognition revealed by time-resolved FTIR spectroscopy

The conformational substates B(I) and B(II) of the phosphodiester backbone in B-DNA are thought to contribute to DNA flexibility and protein recognition. We have studied by rapid scan FTIR spectroscopy the isothermal B(I)-B(II) transition on its intrinsic time scale. Correlation analysis of IR absorption changes occurring within seconds after a reversible incremental growth of the DNA hydration shell identifies water populations w(1) (PO(2)(-)-bound) and w(2) (non-PO(2)(-)-bound) exhibiting weaker and stronger H-bonds, respectively, than those dominating in bulk water. The B(II) substate is stabilized by w(2). The water H-bond imbalance of 3-4 kJ mol(-1) is equalized at little enthalpic cost upon formation of a contiguous water network (at 12-14 H(2)O molecules per DNA phosphate) of reduced ν(OH) bandwidth. In this state, hydration water cooperatively stabilizes the B(I) conformer via the entropically favored replacement of w(2)-DNA interactions by additional w(2)-water contacts, rather than binding to B(I)-specific hydration sites. Such water rearrangements contribute to the recognition of DNA by indolicidin, an antimicrobial 13-mer peptide from bovine neutrophils which, despite little intrinsic structure, preferentially binds to the B(I) conformer in a water-mediated induced fit. The FTIR spectra resolve sequential steps leading from PO(2)(-)-solvation to substate transition and eventually to base stacking changes in the complex. In combination with CD-spectral titrations, the data indicate that, in the absence of a bulk aqueous phase, as in molecular crowded environments, water relocation within the DNA hydration shell allows for entropic contributions similar to those assigned to water upon DNA ligand recognition in solution.     

Peculiarities of the Binding of Some Small Ligands to DNA

The comparative study of the interaction of ethidium bromide, acridine orange, and isoquinoline alkaloids such as sanguinarine and berberine with DNA in aqueous solutions using the optical spectroscopy methods is presented. The dependences of the spectral characteristics of ligands on the concentration ratio P/D between the DNA base pairs and ligands molecules in solutions are considered. The character of binding is found to depend on P/D. The parameters of the binding with DNA are determined for the external binding and the intercalation, by using the modified Scatchard and McGhee–von Hippel equations. The influence of a ligand form on binding ways is shown.     

Dirac system with potential lying in Besov spaces

We study the spectral properties of the Dirac operator L_P,U generated in the space (L₂[0, π])² by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y₁, y₂)^t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ_n}_n∈Z of the operator L_P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B_1,∞^θ, θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L_P,U, and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections.     

Groups generated by 3-state automata over a 2-letter alphabet. II

This is the second in a series of papers presenting results on the classification of groups generated by 3-state automata over a 2-letter alphabet. Among the examples treated here, one can find automata generating the free product of 3 cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups, and other interesting self-similar groups. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Characterization of ultra‐thin polymer films by polarization modulation FTIR spectroscopy

Self assembly monolayers of octadecyltrichlorosilane Cl₃‐Si‐(CH₂)₁₇‐CH₃ and 17‐cyanopentadecyltrichlorosilane Cl₃‐Si‐(CH₂)₁₇‐CN on silicon wafers have been prepared by adsorption from solution. The molecular orientation within the monolayers was investigated by using Polarisation Modulation FTIR spectroscopy. Quantitative analysis reveals that both types of silanes – monofunctionalised and bifunctionalised – form highly ordered monolayers. A high degree of ordering as well as a small tilt angel of the molecular backbones with respect to the surface normal are indicated by the strength of the Si‐O‐Si stretching modes and the weakness of the CH₂ stretching modes. The decomposition of the terminal nitrile group of the substituted silane into a carboxyl group could be identified. The decomposition is caused by a high local HCl concentration, which develops upon binding of 17‐cyanopentadecyltrichlorosilane to the OH groups of the silicon surface.