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.     

Topological Simplicity of a Certain LF-Algebra

We show that the LF-algebra considered by Akkar and Nacir in [1] is topologically simple.     

Deposition of boron nitride on graphite at 1300 K

A method for the deposition of BN onto graphite and other substrates is described. Boron trichloride (BCl₃) and ammonia (NH₃) diluted with Ar were used as reacting gases. The deposition process was carried out at 1300 K as well as lower temperatures in an open system at pressures of 1 atm. The consequences of the introduction of hydrogen to the system were considered. It was demonstrated that the replacement of argon with hydrogen increases the efficiency of the process as well as the theoretical rate of BN deposition. However, the acceleration of the deposition seems to be unprofitable, because the resulting supersaturation leads to the formation of an amorphous phase. The modification of the experimental conditions were proposed.Dedicated to Professor Dr. rer. nat. Dr. h.c. Hubertus Nickel on the occasion of his 65th birthday     

On Numerical Solution of Shape Inverse Problems

The new method is proposed for the numerical solution of a class of shape inverse problems. The size and the location of a small opening in the domain of integration of an elliptic equation is identified on the basis of an observation. The observation includes the finite number of shape functionals. The approximation of the shape functionals by using the so-called topological derivatives is used to perform the learning process of an artificial neural network. The results of computations for 2D examples show, that the method allows to determine an approximation of the global solution to the inverse problem, sufficiently closed to the exact solution. The proposed method can be extended to the problems with an opening of general shape and to the identification problems of small inclusions. However, the mathematical theory of the proposed approach still requires futher research. In particular, the proof of global convergence of the method is an open problem.     

Determination of Emodin and Phenolic Acids in the Petioles of Rheum undulatum and Rheum rhaponticum

JPC – Journal of Planar Chromatography – Modern TLC - Emodin and twelve phenolic acids (ellagic, gallic, protocatechuic, homoprotocatechuic, caffeic, p-hydroxybenzoic, p-coumaric,...     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Sensitised decomposition of microcystin-LR using UV radiation

UV radiation was applied to degrade cyanobacterial hepatotoxin, microcystin-LR in the presence of phycocyanin as a model natural sensitiser. The concentrations of both the toxin and the pigment used in the experiments were higher by several orders of magnitude than found in the environment. The photoreaction parameters were optimised. The process was found to be of limited use for water treatment due to its low efficacy. Additionally, pronounced UV-induced bleaching of the pigment significantly reducing the photoreaction rates of the toxin was observed for the highest UV radiation intensities applied.     

Study on the electroreduction of In(III) ions at gold from acidic bromide solutions by the rotating ring-disc electrode technique

The mechanism of the indium electrodeposition on gold from acidic KBr solutions has been investigated by means of the rotating ring-disc electrode technique. The measurements showed that the overall electrode reaction In(III) + 3 e In consists of two successive charge transfer steps, i.e. In(III) + 2 e In(I) and In(I) + e In, with soluble monovalent indium as an intermediate. The current efficiency for In(I) generation under steady-state conditions is 100% in the case of low disc current densities. The disproportionation of In(I) in the bulk of solution occurs in slightly acidified solutions,pH=2, in addition to the In(I) electrochemical reduction. Based on the ring current transients, it was established that the two-electron transfer step is preceded by the formation of a thin passive film of indium metal.

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Untersuchung der elektrochemischen Reduktion von In(III)-Ionen an Gold in sauren Bromidlösungen mit der rotierenden Ring-Scheiben-Elektrode

Zusammenfassung Mit Hilfe der rotierenden Ring-Scheiben-Elektrode wurde der Mechanismus der kathodischen Abscheidung von Indium auf Gold in sauren KBr-Lösungen untersucht. Aus den Messungen folgt, daß die Bruttoreaktion In(III) + 3e In über zwei aufeinanderfolgende Durchtrittsreaktionen, In(III) + 2 e In(I) und In(I) + e In, mit löslichem einwertigem Indium als Zwischenprodukt abläuft. Bei kleinen Stromdichten werden die In(I)-Ionen mit 100%iger Stromausbeute erzeugt. In schwächer sauren Lösungen (pH=2) konnte außer der elektrochemischen Reduktion von In(I) zusätzlich dessen Disproportionierung innerhalb der Lösung ermittelt werden. Auf Grund der Ringstrom-Zeit-Kurven wurde auch festgestellt, daß die Bildung von einer dünnen In-Metallschicht, die ein passives Verhalten aufweist, dem Zweielektronenübergang vorangegangen ist.

     

Electrochemical studies of Tl(I) crown-ether complexes in nonaqueous media

Summary Complexation equilibria of the Tl(I) ion with 18-crown-6 and dibenzo-18-crown-6 were studied polarographically in 10 nonaqueous solvents. The stability of the complexes is strongly influenced by the nature of solvents and varies with their Lewis basicities. It has been found that the logK _s value (K _s is the stability constant of the complex) can be well described by empirical relation logK _s=a DN+b, whereDN stands for the Gutmann donor number anda andb mark the regression coefficient. Addition of the second explanatory parameter, the acceptor number, is not statistically significant. This result is in agreement with the predominant role of Tl(I) ion solvation.

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Elektrochemische Untersuchungen von Tl(I)-Kronenetherkomplexen in nichtwäßrigen Medien

Zusammenfassung Es wurden die Komplexierungsgleichgewichte des Tl(I)-Ions mit 18-Krone-6 und Dibenzo-18-krone-6 polarographisch in 10 nichtwäßrigen Lösungsmitteln untersucht. Die Stabilität der Komplexe wird sehr stark vom Solvens beeinflußt, wobei eine starke Abhängigkeit von der Lewis-Basizität beobachtet wird. Es wurde festgestellt, daß die logK _s-Werte (K _s ist die Komplexstabilitätskonstante) gut mit der empirischen Beziehung logK _s=a DN+b beschrieben werden können, wobeiDN die Gutmann'sche Donorzahl unda undb die Regressionskonstanten bedeuten. Hinzunahme der Akzeptorzahl als zweiten Parameter bleibt statistisch insignifikant. Dieses Ergebnis stimmt mit dem dominierenden Einfluß der Tl(I)-Ionensolvatation überein.

     

Kinetics and mechanism of the first aquation stage for the [Cr(pic)3]0 and [Cr(pic)2(OH)]2 0 complexes in HClO4 solutions

Aquation of [Cr(pic)₃]⁰ and [Cr(pic)₂(OH)]₂ ⁰ in aqueous HClO₄ solutions leads to formation of the common product – [Cr(pic)₂(H₂O)₂]⁺. The first, reversible stage, the ring opening via Cr—N bond breaking in [Cr(pic)₃]⁰ is followed by the second, rate-determining step – one-end bonded pic ligand liberation. In the case of the [Cr(pic)₂(OH)]₂ ⁰ complex, the first faster stage produces the singly bridged dimer, which undergoes cleavage into the parent monomers in the second, much slower step. The subsequent aquation of [Cr(pic)₂(H₂O)₂]⁺ is extremely slow and leads to [Cr(pic)(H₂O)₄]²⁺ formation, which practically does not undergo further ligand substitution under the conditions applied. Kinetics of the first aquation stage for [Cr(pic)₃]⁰ and of the second step for [Cr(pic)₂(OH)]₂ ⁰ were studied spectrophotometrically in the 0.1–1.0 M HClO₄ range at I = 1.0 M. The observed pseudo-first order rate constant for [Cr(pic)₃]⁰ decreases with [H⁺] increase according to the rate law: k _obs = k ₁ + k _–1 Q ₁/[H⁺], where k ₁ and k _–1 are the rate constants of the forward and the reverse processes in the unprotonated substrate and Q ₁ is the protonation constant of the pyridine nitrogen atom. In the case of the [Cr(pic)₂(OH)]₂ ⁰ complex, the rate for the singly bridged dimer cleavage does not depend on [H⁺]. The activation parameters for the chelate-ring opening in [Cr(pic)₃]⁰ and for the singly bridged dimer cleavage have been determined and discussed. Some kinetic data of the slow, second aquation stage for the [Cr(pic)₃]⁰ complex and of the fast, first aquation stage for the doubly bridged dimer have been studied; for both reactions the rate increases linearly with the increase in [H⁺].