On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.     

Untersuchung der Autoprotolyse von Carbonsäuren mittelsMO-SCF-Berechnungen

Zusammenfassung Am Beispiel der dimeren Ameisensäure wird untersucht, inwieweit semiempirischeMO-SCF-Methoden (CNDO/2) ein Bild des Autoprotolysevorganges von Carbonsäuren wiedergeben können. Die aus den Berechnungen abzuleitende Reaktionsenthalpie liegt beträchtlich über den mit Hilfe elektrostatischer Modelle berechneten Werten. Für nicht zu große Abstände liefert die verwendete Methode plausible Ergebnisse für die Beschreibung des beginnenden Dissoziationsschrittes. An Hand der Ladungsverteilung zeigt sich, daß innerhalb dieses Teilbereiches bereits gewisse Erkenntnisse über den Reaktionsablauf der Autoprotolyse gewonnen werden können.

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Investigation of the autoprotolysis-reactions of carboxylic acids by means of MO-SCF-calculations

The applicability of semiempiricalMO-SCF-methods (CNDO/2) to describe autoprotolysis reactions of carboxylic acids is investigated by calculations on dimeric formic acid. The calculated energy values are compared with those obtained by simple electrostatic model calculations, proving such a model to be unsatisfactory. The method used here leads to reasonable results and enables a discussion of the phenomena accompanying dissociation within not too big distances. The calculated charge distributions give some informations about the mechanism of autoprotolysis within this range.

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Mit 1 Abbildung     

Bor-tris-(trifluoromethansulfonat), B(OSO2CF3)3 in Trifluoromethansulfonsäure – ein neues „supersaures System”︁

Boron-tris(trifluoromethane-sulfonate), B(OSO₂CF₃)₃, in Trifluoromethane Sulfonic Acid — a New ?Super Acid System”? The system of boron-tris(trifluoromethane-sulfonate) in excess trifluoromethane sulfonic acid, was investigated as a proton donor. The Hammett acidity functions, H₀, were determined in relation to the concentration of B(OSO₂CF₃)₃ using various indicators. The system is comparable to SbF₅ · 3 SO₃ in HSO₃F, the most acidic system known until now.     

Versuche Zur Elektrochemischen Fluorierung anorganischer Verbindungen

Zusammenfassung Über die Ergebnisse der Elektrofluorierung von Kalium- und Ammoniumsulfat und-peroxodisulfat, Sulfamid, Amidosulfonsäure, Ammoniumperchlorat, Schwefelstickstoff, Quecksilber(II)-cyanid und Suspensionen von elementarem Schwefel in Fluorwasserstoff bei 0°C bzw. in Kalium- oder Ammoniumhydrogenfluoridschmelzen bei 80°C wird berichtet.Die sich daraus im Hinblick auf die hauptsächlichen Elektrolysenprodukte und auf explosive Bestandteile der Anodengase ergebenden allgemeinen Gesichtspunkte sowie die verschiedenen möglichen Primärreaktionen an der Anode werden diskutiert.Mit 11 AbbildungenHerrn Prof. Dr.E. Hayek zum 60. Geburtstag in aufrichtiger Verehrung und Dankbarkeit gewidmet.     

Über eine einfache Darstellungsweise von Arsentrifluorid und eine Molekülverbindung mit Schwefeltrioxyd

Zusammenfassung AsF₃ und SO₃ treten zu einer unzersetzt destillierbaren Molekülverbindung 2 AsF₃·3 SO₃ zusammen. Diese entsteht nicht bei der Umsetzung von HSO₃F mit As₂O₃, die Reaktion läßt sich aber vorteilhaft zur Herstellung von AsF₃ in einer Glasapparatur verwenden.     

Two-dimensional non-linear evolution equations: The derivation and the transient wave solution

The general theory of two-dimensional evolution equations describing transient wave propagation in non-linear continuous media is presented. The ray method is used and the two-dimensional evolution equations for plane and cylindrical wave-beams are obtained. The transient wave solutions are discussed briefly. A transformation of variables is proposed that permits the transformation of the two-dimensional evolution equation into a one-dimensional evolution equation with coordinate-dependent coefficients. A breakdown time analysis is carried out for this case. The dispersion relations for plane and cylindrical wave-beams are presented. The non-linear dispersion relation is obtained by making use of a series representation.     

Deformation waves in thermoelastic media and the concept of internal variables

Summary Nonlinear thermoelastic continua with Fourier's type of heat conduction illustrate complex systems composed of internal and reaction-diffusion subsystems. An evolution equation is derived for the observable variable (deformation) when temperature effects are viewed as internal processes. This approach, which pertains to the so-called instantaneous wave analysis, shows that thermal losses accompanying the deformation wave correspond to the low-frequency approximation.     

Error bounds for quadrature methods involving lower order derivatives

Quadrature methods for approximating the definite integral of a function f(t) over an interval [a,b] are in common use. Examples of such methods are the Newton–Cotes formulas (midpoint, trapezoidal and Simpson methods etc.) and the Gauss–Legendre quadrature rules, to name two types of quadrature. Error bounds for these approximations involve higher order derivatives. For the Simpson method, in particular, the error bound involves a fourth-order derivative. Discounting the fact that calculating a fourth-order derivative requires a lot of differentiation, the main concern is that an error bound for the Simpson method, for example, is only relevant for a function that is four times differentiable, a rather stringent condition. This paper caters for functions for which derivatives exist only of order lower than normally required. A number of quadrature methods are considered and error bounds derived involving only lower order derivatives that can be used depending on the smoothness of the function.