Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].     

Oxidation of α-methylstyrene on an MCM-22 encapsulated (<Emphasis Type="Italic">R,R</Emphasis>)-(−)-<Emphasis Type="Italic">N,N′</Emphasis>-bis(3,5-di-<Emphasis Type="Italic">tert</Emphasis>-butylsalicylidene)-1,2-cyclohexanediaminocobalt(II) catalyst

(R, R)-(−)-N, N′-bis(3,5-di-tert-butylsalicylidene)-1,2-cyclohexanediaminocobalt(II) was encapsulated into MCM-22 using the zeolite synthesis method. The encapsulated catalyst proved to be active in the oxidation of α-methylstyrene with NaOCl with higher specific activity than the homogeneous catalyst. At the same time, this encapsulated catalyst was completely inactive in the hydrolytic kinetic resolution of racemic styrene oxide. This observation is in a good correlation with the assumption of the cooperative bimetallic mechanism proposed by Annis and Jacobsen.     

Conventional and unconventional infrared spectrometry and their quantitative interpretation

In condensed matter, optical properties can be described by a dielectric function (DF), and the structures observed in spectra are then related to the poles and zeros of the DF. As an example, model functions are calculated by a fit to measured spectroscopic data for polystyrene and silica. The first material shows weak, narrow bands and the latter strong, broad bands and a negative real part of the DF.Based on these model DFs, spectra are simulated which are expected to be obtained by “conventional” methods such as transmittance or reflectance measurements, or by “unconventional” methods such as reflectance at oblique incidence, diffuse reflectance, photoacoustic spectroscopy and attenuated total reflectance. A variety of simulated, typical spectra are plotted as a small “atlas”. Conditions are discussed that allow a straightforward procedure for interpreting the spectra quantitatively, i.e., the evaluation of the resonance frequency and the concentration of the oscillators under consideration.It is shown that for systems characterized by weak, narrow oscillator lines, mostly an intuitive interpretation is possible, looking only at the position and strength of “lines” in the spectra. Materials showing strong polar vibrations, however, require more sophisticated procedures for interpreting the spectra.     

Characteristics of excited-state intermediates of TiO2–Y-Zeolite and MCM41 encapsulating photosensitive molecules: design of new photocatalysts

Some photosensitive molecules, such as p-N,N′-dimethylaminobenzoic acid (DMABA), Nile Red, heteropolytungstic acid (H₃PW₁₂O₄₀, HPA) and metalloporphyrins, have been entrapped onto nano-scale pores or channels of TiO₂-modified Y-Zeolite (TiO₂-Y-Zeolite) and MCM41 (TiO₂-MCM41) and their excited-state intermediates have been characterized in terms of the excited-state dynamics by using laser spectroscopic techniques. Through these studies, it has been found that the photo-induced electrons are generated from the intramolecular charge transfer (ICT) state of DMABA, Nile Red or metalloporphyrin (MnTPP(Cl)), followed by transferring to the TiO₂-Y-Zeolite or TiO₂-MCM41 more efficiently as compared to the bulk TiO₂, NaY-Zeolite or MCM41. The efficient photoinduced interfacial electron transfer causes the rapid formation of radicals of those photosensitive molecules (a few tens ps). It has been also found that these photophysical properties can be applied to develop the new photocatalyst as observed by the efficient photocatalytic activities of the DMABA or Nile Red-entrapped TiO₂-Y-Zeolites for the photoreduction of an azo-dye such as Methyl Orange in water. On the other hand, in case of HPA-entrapped TiO₂-Y-zeolite, the electron generated from the excited-state TiO₂ is transferred to HPA, followed by formation of the reduction product, heteropoly blue (HPB) which is also generated by UV irradiation of HPA. This electron transfer is analogous to the Z-scheme mechanism of plant photosynthetic systems showing two photon reactions. Because of this photoelectron transfer mechanism, the HPA-entrapped TiO₂-Y-zeolite has demonstrated the synergistic enhancement of the photocatalytic decomposition of Methyl Orange and hydrogen generation from photolysis of water.     

MCM-41分子筛负载钼钴系催化剂的噻吩HDS研究

通过氧吸附量、噻吩吸附热及反应速率常数的测定，研究了MoO3/MCM-41、MoO3-CoO(NiO)/MCM-41系列催化剂，发现，对于MoO3/MCM-41催化剂，当MoO3的质量分数（以MCM－41为底数，即MCM－41＝1g时，MoO3含量为0．15g，下同）从15％增加到20％时，其噻吩的加氢硫（HDS）活性增大，至25％时活性下降，所对应的氧吸附量（mL/g催化剂）也是先增大后减少，并且两者有很好的线性对应关系，而且噻吩吸附热则基本保持不变，采用不同的MoO3-CoO(NiO)浸渍顺序制备的MoO3-CoO(NiO)/MCM-41催化剂中，先浸渍CoO(NiO）再浸渍MoO3的催化剂，其噻吩HDS活性明显优于对其它浸渍顺序制备的催化剂，同时催化剂的氧吸附量和噻吩吸附热也最大。     

A gas-phase electron diffraction study of the molecular structure of tetravinylsilane

The molecular structure of tetravinylsilane has been studied by gas-phase electron diffraction. The radial distribution curve suggests the absence of conformers having vinyl double bonds staggered with respect to the SiC₄ skeleton. Of the eclipsed or approximately-eclipsed conformers, the one with S₄ symmetry gives the best fit with experiment, although a small admixture of a C₁ conformation cannot be ruled out. Least-squares refinement gave the following values for the independent structural parameters (lengths, r_a basis; angles, r_α basis): C-H = 1.118 ± 0.003 Å, CC = 1.355 ± 0.002 Å, Si-C = 1.855 ±0.002 Å, ∠SiCC = 124.0 ± 0.3°, ∠SiCH = 118.4 ± 1.0°, torsion angles CSiCC are 17.5 ± 0.6° from the eclipsed conformation. During the refinement the vibrational amplitudes u and perpendicular amplitude corrections K were held constant at calculated values. The CC bond length provides evidence of interaction between the vinyl π-bonds and the vacant d-orbitals of silicon.     

Pore Sizes of Solids

A new, fast, and simple method based on size exclusion chromatography permits determination of the volumes of open and closed pores of rigid solids such as silica or porous glasses, together with the specific surface and the pore-size distribution for pore diameters ranging from 10 to 4000Å. The results are in good agreement with those obtained by classical methods (capillary condensation, mercury porosimetry).     

Polymer-silica nanocomposites prepared by sol-gel technique: Nanoindentation and tapping mode AFM studies

Polymer-silica nanocomposites based on poly(2-hydroxyethyl acrylate) (PHEA) have been prepared by the simultaneous polymerization of the organic and the silica phases in a sol-gel process with the silica precursor tetraethyl orthosilicate (TEOS). The structure of this system is investigated using atomic force microscopy (AFM) in the tapping mode and in nanoindentation experiments. The structure of the PHEA/silica hybrids strongly depends on the ratio of both components in the system. For silica weight fractions lower than 0.15, the system consists of aggregated silica particles dispersed in the organic matrix; above that concentration of silica the structure is co-continuous with that of the organic matrix, similarly to two interpenetrated networks.