Polylogarithmic functional equations: A new category of results developed with the help of computer algebra (MACSYMA)

The interrelation of polylogarithmic functional equations and certain numerical results, known as ladders, is discussed, and leads to a consideration of three new, single-variable functional equations at the second order. Two of these families each contain six leading terms whose interrelationship constitutes a constraint on the integration process, but the third has only a single leading term with no such constraints. It is shown how this functional equation can be integrated to the third order, and the process reduced to an algorithm — actually a sequence of instructions — for incorporation into a computer program for symbolic manipulation. The procedure utilizes results from Kummer's equations to cancel out, in sequence, terms which do not vanish, or do vanish, with the variablez. Arguments are all of the form ±z ^p (1–z) ^q (1+z) ^r , and the process is algebraicized by using a (p,q,r,s) notation (withs=±1) to represent such terms. Application of the procedure leads to an integration to the fourth and fifth orders, the latter exhibiting 55 transcendental terms. The first step for the transition to the sixth order can also be achieved but the subsequent steps are frustrated by the restricted forms that the Kummer equations take at the fifth order — it is not possible to create the needed equations in a form which vanishes withz; this corresponding to the elimination of the (5) constant in the extension of the numerically determined ladders to the sixth and higher orders. The existence of the higher-order ladders strongly suggests functional equations af these orders, but the present process has not yet been successful in finding them. The new equations have, however, produced ladders that were inaccessible from Kummer's equations, and had heretofore been only obtainable numerically, up to the fifth order. The method which was developed should be capable of generalization to other systems of equations characterized by the appearance of arguments with recurrent factors. Some new feature, however, will need to be determined before the barrier to the sixth order can be breached.     

Long-time and unitary properties of semiclassical initial value representations

We numerically compare the semiclassical "frozen Gaussian" Herman-Kluk propagator [Chem. Phys. 91, 27 (1984)] and the "thawed Gaussian" propagator put forward recently by Baranger et al. [J. Phys. A 34, 7227 (2001)] by studying the quantum dynamics in some nonlinear one-dimensional potentials. The reasons for the lack of long-time accuracy and norm conservation in the latter method are uncovered. We amend the thawed Gaussian propagator with a global harmonic approximation for the stability of the trajectories and demonstrate that this revised propagator is a true alternative to the Herman-Kluk propagator with similar accuracy.     

Zur spektralanalytischen Bestimmung von SiO2 in quarzhaltigen Flu?spatkonzentraten

Zusammenfassung Die quantitative spektralanalytische Bestimmung von SiO₂ in quarz — haltigen Flußspatkonzentraten wurde bei Anregung im Gleichstrom- und Wechselstromkohlebogen untersucht, wobei der letzte eine größere Genauigkeit lieferte. Als Bezugselement für Si konnte Ca selbst gewählt werden. Das Verfahren bedient sich einer Eichkurve, die für SiO₂-Gehalte zwischen 0,01 und 10% aufgestellt wurde. Der mittlere Fehler der Einzelmessung beträgt ±8,0%.Das beschriebene Wechselstrom-Kohlebogenverfahren dürfte auch für andere quantitative Bestimmungen von Verunreinigungen und Spurenelementen geeignet sein, wie sie bei laufender Betriebskontrolle und bei geochemischen Untersuchungen erforderlich sind.Wärend der Drucklegung erschien eine Arbeit von G. A. Monnot (Spectrochimica Acta 6, 153 [1954]) über eine spektralanalytische SiO₂-Bestimmungsmethode in Flußspat, deren Genauigkeit zwischen 0,05 und 5 % SiO₂-Gehalt ± 10% beträgt. Die Anregung erfolgt mit Hochspannung aus dem multisource-Gerät. Die Methode arbeitet mit einer Art elektrostatischen Sprühbogens, mit dem in 20 sec 20 mg Substanz zur Verdampfung und Anregung gelangen.     

Methylrhenium oxides

Reactions of the rhenium(VII) precursors Re₂O₇, acetyl perrhenate, trifluoroacetyl perrhenate, chlorotrioxorhenium and trimethylsilyl perrhenate are performed with various common tin free methylating agents. The yields of MTO and the products of partial reduction, bis[dimethyl(μ-oxo)oxo-rhenium(VI)] and (μ-oxo)bis[trimethyloxorhenium(VI)], are quantified by NMR spectroscopy. With aluminium- and copper-containing methylating agents none of the above mentioned compounds are formed, solely perrhenate and rhenium(VI) oxide are detected. The best result is achieved with trifluoroacetyl perrhenate and dimethylzinc, yielding >60% MTO.     

Neue Laboratoriumsapparate

Ohne Zusammenfassung     

Einen Gasentwicklungsapparat

Ohne Zusammenfassung     

Conserved dynamics and interface roughening in spontaneous imbibition: A critical overview

Imbibition phenomena have been widely used experimentally and theoretically to study the kinetic roughening of interfaces. We critically discuss the existing experiments and some associated theoretical approaches on the scaling properties of the imbibition front, with particular attention to the conservation law associated to the fluid, to problems arising from the actual structure of the embedding medium, and to external influences such as evaporation and gravity. Our main conclusion is that the scaling of moving interfaces includes many crossover phenomena, with competition between the average capillary pressure gradient and its fluctuations setting the maximal lengthscale for roughening. We discuss the physics of both pinned and moving interfaces and the ability of the existing models to account for their properties. Received 17 February 1999 and Received in final form 24 November 1999     

Differential non-destructive image current detection in a fourier transform quadrupole ion trap

Dual-detector differential non-destructive Fourier transform detection in a quadrupole ion trap is shown to improve signal intensity and reduce noise compared with spectra recorded using a single detector. A larger area detector in each end-cap electrode is machined to fit its hyperbolic shape and so minimize field imperfections on the z-axis. Argon, acetophenone and bromobenzene spectra were recorded to allow a comparison between single- and dual-detector (differential) modes of detection and to demonstrate the improvement achieved with differential detection. Copyright 1999 John Wiley & Sons, Ltd.     

Pseudo-steady-state solutions for solidification in a wedge

Polubarinova-Kochina's analytical differential equation methodis used to determine the pseudo-steady-state solution to problemsinvolving the freezing (solidification) of wedges of liquidwhich are initially at their fusion temperature. In particular,we consider four distinct problems for wedges which are: freezingwith the same constant boundary temperature, freezing with thesame constant boundary heat fluxes, freezing with distinct constantboundary temperatures and freezing with distinct constant fluxesat the boundaries. For the last two problems, a Heun's differentialequation with an unknown singularity is derived, which in bothcases admits a particularly elegant simple solution for thespecial case when the wedge angle is . The moving boundariesobtained are shown pictorially.