A systematic formula for the asymptotic expansion of singular integrals

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are improper, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

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Résumé Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont impropres c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Nomenclature D domain of integration - f(x) weight function - FP Finite Part - h(x) weight function - I ,I _o bounded intervals - j, J integers - K(x, ) singular kernel - L, L integers - M integer defining the approximation order - P _k (x) Legendre polynomial - R set of real numbers - R(ß) equals 1 if is an integer and 0 if not - R _{f, J} ,R _{K, L} remainders of Taylor developments - S () equals either 1 or the sign function:sgn() - t, u, v, x variable of integration - , real numbers - homogeneity order of the kernel - F () Euler's integral (gamma function) - small parameter - [.] integer part of     

Investigation of the formation process of MCs+-molecular ions during sputtering

In secondary ion mass spectrometry, the detection of MCs+ clusters (with M an element of the specimen) under a Cs bombardment is frequently used for the quantification of major elements. Despite some very good results obtained by this method, some problems still remain. In order to gain some more insight into these problems, the formation mechanism of the MCs+ clusters is investigated using a Monte Carlo model. It is shown that the majority of the constituent particles of the formed clusters are initially first or second neighbor atoms at the surface and that the velocity distribution of the MCs+ clusters becomes broader and peaked at higher velocities with increasing surface binding energy of the M atom. In addition, it is demonstrated that the interaction potential between the M and Cs+ particle has no influence on the velocity distribution of the MCs+ clusters. On the other hand, the cluster formation probability, defined as the probability that a sputtered M and Cs+ particle will form a MCs+ cluster, is extremely sensitive to this interaction potential. It is also shown that the cluster formation probability decreases with increasing surface binding energy. Finally, a good correspondence is obtained between the calculated and experimental velocity distributions of MCs+ clusters sputtered from different monoatomic materials. As a consequence, the Monte Carlo model and the discussed results can be validated.     

Third-order nonlinear optical figure of merits for conjugated TTF-quinone molecules

We have revealed a substantial enhancement of third-order optical figure of merits by the synthesis of a compact molecule possessing the tetrathiafulvalene (TTF) group with two backside C=O groups. Addition of the saturated methylene chain substantially suppresses the third-order optical figure of merits and even local optical hyperpolarizabilities at lambda = 532 nm. Another TTF-derivative molecule possessing ethylenic and acetylenic chains demonstrates large hyperpolarizabilities; however, generally, the figure of merit factor decreases due to the increasing optical losses as a consequence of enhanced linear absorption. At the same time, both of the chromophores have a large nonlinear optical response. General approaches for search and design of the third-order optical materials with improved properties are given.     

Synthesis of functionalized rab GTPases by a combination of solution- or solid-phase lipopeptide synthesis with expressed protein ligation

Prenylated proteins with non-native functionalities are generally very difficult to obtain by recombinant or enzymatic means. The semisynthesis of preparative amounts of prenylated Rab guanosine triphosphatases (GTPases) from recombinant proteins and synthetic prenylated peptides depends largely on the availability of functionalised prenylated peptides corresponding to the proteins' native structure or modifications thereof. Here, we describe and compare solution-phase and solid-phase strategies for the generation of peptides corresponding to the prenylated C terminus of Rab7 GTPase. The solid-phase with utilisation of a hydrazide linker emerges as the more favourable approach. It allows a fast and practical synthesis of pure peptides and gives a high degree of flexibility in their modification. To facilitate the analysis of semisynthetic proteins, the synthesised peptides were equipped with a fluorescent group. Using the described approach, we introduced fluorophores at several different positions of the Rab7 C terminus. The position of the incorporated fluorescent groups in the peptides did not influence the protein-ligation reaction, as the generated peptides could be ligated onto thioester-tagged Rab7. However, it was found that the positioning of the fluorescent group had an influence on the functionality of the Rab7 proteins; analysis of the interaction of the semisynthetic Rab7 proteins with REP (Rab escort protein) and GDI (guanosine diphosphate dissociation inhibitor) molecules revealed that modification of the peptide side chains or of the C-terminal isoprenoid did not significantly interfere with complex formation. However, functionalisation of the C terminus was found to have an adverse effect on complex formation and stability, possibly reflecting low structural flexibility of the Rab GDI/REP molecules in the vicinity of the lipid-binding site.     

Influence of the coordination number of nitrogen on the structure of 6aλ4-thia-1,3,4,6-tetraazapentalenic systems. Crystal structure analyses of 3-(2-pyridylimino)-3H-[1,2,4]thiadiazolo[4,3-a]pyridine and its 1-methylated salt

Structural data were obtained by X-ray crystallography for the title compounds which show that they are essentially planar and exhibit an approximately linear N2-S1-N8 arrangement. In compound 3 the separation between the sulfur atom and the pyridine nitrogen atom (2.61 Å) is larger than the Huggins constant energy distance (2.58 Å), suggesting that there is little or no bonding between them. The methylated salt 4 , on the contrary, has a closer S…N(pyridine) distance (2.19 Å) with an estimated bond dissociation energy of 6 kcal/mole.     

Method Development for the Speciation of Chromium in River and Industrial Wastewater Using GFAAS

A simple, sensitive and inexpensive method for the speciation of Cr(VI) and Cr(III) in river and industrial wastewater was developed, optimized and validated. Samples were collected, filtered and then chromium species were determined in the filtrate. In the filtrate, total Cr was determined directly by Graphite Furnace Atomic Absorption Spectrometry (GFAAS). A portion of the filtered sample was buffered to pH 6.4, extracted with Amberlite LA-2/MIBK and finally Cr(VI) was determined in the organic extract and Cr(III) in the aqueous phase using GFAAS. The method was evaluated by analysis of the certified reference material, CRM 544, and good accuracy was obtained. The limit of detection for Cr(VI) and Cr(tot)/Cr(III) was found to be 0.30 and 0.08 µg L⁻¹, respectively, and a precision expressed as relative standard deviation of better than 11% was achieved for nine replicates. A number of water samples from the Tinishu Akaki River, Ethiopia, and wastewater samples from a chemical plant in Germany, were analysed. In addition to the GFAAS method, Cr(tot) was also determined using ICP-MS (in CRM 544, lyophilized water, and Tinishu Akaki River water samples) and Cr(tot) and Cr(VI) using UV-VIS spectrophotometry (in CRM 544 and industrial wastewater samples). Good agreement between the different methods was found.     

Comparison of mono- and polyatomic primary ions for the characterization of organic dye overlayers with static secondary ion mass spectrometry

Organic carbocyanine dye coatings have been analyzed by time-of-flight static secondary ion mass spectrometry (TOF-S-SIMS) using three types of primary ions: Ga(+) operating at 25 keV, and Xe(+) and SF(5) (+) both operating at 9 keV. Secondary ion yields obtained with these three primary ions have been compared for coatings with different layer thickness, varying from (sub)-monolayer to multilayers, on different substrates (Si, Ag and AgBr cubic microcrystals). For (sub)-monolayers deposited on Ag, Xe(+) and SF(5) (+) primary ions generate similar precursor ion intensities, but with Ga(+) slightly lower precursor ion intensities were obtained. Thick coatings on Ag as well as mono- and multilayers on Si produce the highest precursor and fragment ion intensities with the polyatomic primary ion. The yield difference between SF(5) (+) and Xe(+) can reach a factor of 6. In comparison with Ga(+), yield enhancements by up to a factor of 180 are observed with SF(5) (+). For the mass spectrometric analysis of dye layers on AgBr microcrystals, SF(5) (+) again proves to be the primary ion of choice.     

Influence of peak measurement parameters on the quality of chiral electrophoretic separations

The effect of nine peak measurement/analysis parameters on chiral capillary electrophoresis (CE), followed by UV detection, was studied. The parameters pertained UV-detection (detection wavelength, reference wavelength, and wavelength bandwidths), signal processing (data acquisition rate, type and amount of filtering) and peak detection (detection threshold and peak width). The influence of these factors on the chiral separation of dimethindene enantiomers was studied at two different concentrations (i.e., at high and low signal-to-noise (S/N) ratio) by the means of experimental designs. The electropherogram characteristics considered were the resolution between the two enantiomers, the peak areas, and the S/N ratio. A D-optimal design was first used as screening design to identify the most critical parameters. Afterwards, a modelling of the different responses as a function of these critical parameters was performed based on the results of a face-centered central composite design. The results showed that the signal-processing parameters should be carefully selected when developing a CE separation since very important variations in the separation, the S/N ratio and the peak area of the substances can occur by setting these parameters at different levels. The detection wavelength should also be carefully chosen for optimal peak area measurement. The role of these parameters becomes more important with decreasing concentration of the analytes (i.e., low S/N ratio). This study showed that the peak measurement/analysis parameters should be optimized as the chemical and physical parameters of a method. They also should always be well specified in order to allow a good transfer of a method from one instrument to another.     

Evaluation of an accurate calibration and spectral standardization procedure for Raman spectroscopy

Due to modern developments Raman spectroscopy has evolved into a fast vibrational technique. Detailed fingerprints in combination with non-destructivity and minimal sample preparation has allowed the construction of reference libraries in a variety of research fields. Long-term stability and comparability are important characteristics when developing reference libraries. In addition, small shifts in highly similar spectra of different samples may limit the full potential of Raman spectroscopy. Since libraries often contain a large number of different and/or highly similar spectra, it is important that each data point in all the spectra corresponds to the exact Raman wavenumber. This is often not the case, due to shifts in optical pathway and/or shifts in laser wavelength. This paper describes a complete calibration protocol (wavelength and intensity) and evaluates the procedure for both short and long term stability, by means of 60 randomly selected measurement sessions spread over a period of nine months. A two-step standardization procedure is proposed to deal with spectral shifts.