Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.     

Multipole expansion of diatomic overlap

A systematic extension of Ruedenberg's expansion formula is applied to evaluate two-electron integrals occurring in calculations on molecular structure. Minimum STO basis sets are used for all SCF-calculations within the framework of the MEDO-method (Multipole Expansion of Diatomic Overlap). The errors due to this approximation scheme are almost negligible compared to those introduced by the truncated basis set: LiH, Li₂ and N₂ are chosen as examples.     

Synthesis of angiotensins,bradykinins and substance P octapeptides in which the residues Phe and Tyr have been replaced with car and of [Car1, Leu5]-enkephalin

Thirteen peptides, analogues of bradykinin (BK), enkephalin, Substance P (SP) and [Sar¹]-angiotensin II ([Sar¹]-AT_II) have been synthesized by the solid-phase method. In all these peptides the residue Phe and Tyr were substituted with the boron-containing amino-acid L -o-carboranylalanine (Car). The purity and identity of the peptides were established by TLC., cellulose-electrophoresis and amino acid analysis.     

Fractals and analytical chemistry — an old and a new look

Summary The impact of the concept of fractals on the foundations of analytical chemistry is discussed. Examples from such diverse subfields like geochemistry, reactions in turbulent flow systems or particle technology illustrate the new look, quantifying what was so far left to intuition of the expert — complexity. These developments led to philosophical problems resolvable in the unity of arts and sciences.     

PHOTOCHEMISTRY OF REDUCED LUMIFLAVIN IN ITS ACIDIC FORM

Abstract— The photooxidation of reduced lumiflavin in its acidic form, LfH 3 ⁺, takes place in two consecutive steps. Upon illumination of LfH 3 ⁺ in its absorption band at 313 nm the semiquinone, LfH 3 ⁺, is formed. Two LfH 2 ⁺ ions are consumed for every LfH 2 ⁺ formed. Illumination of the semiquinone in its absorption band at 495 nm causes further oxidation so that the oxidized LfH⁺ ion is formed. In this reaction one LfH 3 ⁺ ion is photolyzed for every LfH⁺ formed. In addition, a hydrogen atom is released in the photooxidation of LfH 2 ⁺. Mechanisms for the two photoreactions are proposed.     

Ein fünffach koordinierter ionischer zirkonium(IV)-komplex mit der (π-C5H5)2ZrIV-Baueinheit: [(π-C5H5)2Zr(H2O)3]2+(CF3SO3−)2·THF

The ionic complex [(π-C₅H₅)₂Zr(H₂O)₃]²⁺(CF₃SO₃^?)₂·THF, which corresponds to the 18-electron rule, is formed in the reaction of (π-C₅H₅)₂Zr(CF₃SO₃)₂(THF) with H₂O in tetrahydrofuran. It crystallizes in the hexagonal space group P6₃ with Z = 6 and unit cell dimensions at ? 100°C of a 21.945(5) and c 8.711(3) Å. The geometry of the (π-C₅H₅)₂Zr moiety (length of the vectors between Zr and the C₅ ring centroids: 2.210 and 2.193 Å; angle between these vectors: 129.0°; angle between the C₅ ring normals: 128.3°) agrees with that of neutral, four-coordinate (π-C₅H₅)₂ZrX₂ compounds. The three H₂O ligands lie in the plane that bisects the angle between the C₅ ring planes. The ZrO distances are 2.239(7), 2.195(7), and 2.261(7) Å. The CF₃SO₃^? anions and the THF molecule of crystallization are packed around the complex cation in such a way that their oxygen atoms point towards the H₂O ligands. The CF₃ sides of the anion, on the other hand, are clustered together so as to produce hydrophobic domains in the crystal structure.     

Photocyclisierung von 1-[3-(Chloracetylamino)propyl]-3-methylindol. 12. Mitteilung über synthetische Indol-Verbindungen

Photochemical cyclization of 1-[3-(chloroacetylamino)propyl]-3-methyl-indole . Photochemical cyclization of 1-[3-(chloroacetylamino)propyl]-3-methyl-indole involving loss of HCl leads to the three isomers 2, 3 and 4 , the structures of which have been determined by physical and chemical methods.     

Optical sectioning in wide-field microscopy obtained by dynamic structured light illumination and detection based on a smart pixel detector array

Optical sectioning in wide-field microscopy is achieved by illumination of the object with a continuously moving single-spatial-frequency pattern and detecting the image with a smart pixel detector array. This detector performs an on-chip electronic signal processing that extracts the optically sectioned image. The optically sectioned image is directly observed in real time without any additional postprocessing.