Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order $k\geqslant 2$ spanned by $\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}$ on each subinterval $[x_i,x_{i+1}\rangle\subset [0,1]$ , i?=?0,1, ...l. Most of the paper deals with non-polynomial case m _i ,n _i ?∈?[4,?∞?), and polynomial splines known as VDP–splines are the special case when m _i , n _i are integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator $\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}$ . Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.     

Electron slip influence on quasimolecular X-rays in heavy ion collisions

The rotation of the quasimolecular axis in the heavy ion collision is treated dynamically by taking into account the 2pπ-2pσ electron slip. This is done by the correct quantum mechanical method in the perturbation approximation for low ion velocities. The various radiative processes resulting from initial alignments are discussed. The spectra and anisotropies are calculated. Their dependence on the electron slip is investigated.     

alpha-phenylpropionic acid derivatives. Synthesis and dethiation of 5-benzoylbenzo[b]thiophene-3-carboxylic acid

The total synthesis of the title compound 8 started with p-thiocresol which was acylated with oxalyl chloride to give compound 1 . This product underwent a condensation reaction with chloroacetic acid under basic conditions yielding compound 2 . Two different synthetic pathways were used to convert compound 2 into the title compound 8 . The first consisted in decarboxylation of 2 to 3 , which was then converted to the ester 4 , which was brominated and the product 5 was subjected to a Friedel-Craft's reaction with benzene. The resulting benzyl derivative 6 was oxidized to the benzoyl stage i.e. compound 7 , which was finally hydrolyzed to 8 . The second pathway was similar to the first one so that the steps of esterification, bromination, Friedel-Craft's alkylation and oxidation started with the dicarboxylic acid 2 . Thus compounds 12–16 were obtained, and the last product was decarboxylated to 8 . The yields in both procedures were similar. Finally, the dethiation of compound 8 with Raney nickel afforded compounds 18, 19 and 20 .     

THE EFFECTS OF LOW DENSITY LIPOPROTEINS ON UPTAKE OF PHOTOFRIN II

The cellular uptake of Photofrin II (PII) was studied using fluorescence imaging and chemical extraction. The influence of serum and low density lipoproteins (LDL) was examined under a variety of experimental conditions employing cultured human cells of different origins as well as a subcutaneously SMT-F tumor implanted in mice. Results showed that serum inhibited PII uptake. In general, LDL also inhibits PII uptake with the exception of an initial increase in the first 10-30 min when the cellular concentration of PII was measured by fluorescence imaging instead of chemical extraction. Our results suggest a possible de-aggregation process occurring upon internalization or binding of PII to LDL.     

Manipulating adenoviral vector ion‐exchange chromatography: Hexon versus fiber

The serotype specificity of adenovirus ion‐exchange chromatography has previously been studied using standard particle‐based columns, and the hexon protein has been reported to determine retention time. In this study, we have submitted Adenovirus type 5 recombinants to anion‐exchange chromatography using methacrylate monolithic supports. Our experiments with hexon‐modified adenoviral vectors show more precisely that the retention time is affected by the substitution of amino acids in hypervariable region 5, which lies within the hexon DE1 loop. By exploring the recombinants modified in the fiber protein, we have proven the previously predicted chromatographic potential of this surface constituent. Modifications that preserve the net charge of the hexon protein, or those that cause only a small charge difference in the fiber protein, in addition to shortening the fiber shaft, did not change the chromatographic behavior of the adenovirus particles. However, modifications that include the deletion of just two negatively charged amino acids in the hexon protein, or the introduction of a heterologous fiber protein, derived from another serotype, revealed recognizable changes in anion‐exchange chromatography. This could be useful in facilitating chromatography‐approach purification by creating targeted capsid modifications, thereby shifting adenovirus particles away from particular interfering substances present in the crude lysate.     

N‐Benzyloxy‐1H‐benzotriazole‐1‐carboxamide: a hydrogen‐bonded tetramer based upon a rare R44(20) structural motif

The title compound, C₁₄H₁₂N₄O₂, is the first example of a heterocyclic substituted hydroxamic derivative. The asymmetric unit consists of two molecules. The molecules are linked into centrosymmetric R₄⁴(20) tetramers by four strong hydrogen bonds of the N—H...O and N—H...N types. These tetramers are connected through C—H...O interactions into a three‐dimensional network.     

The associated variety of an induced representation

This paper studies the behavior of the associated variety under induction from real parabolic subgroups. We derive a formula for the associated variety of an induced module which is analogous to the formula for the wave front set of a derived functor module obtained by Barbasch and Vogan.