Search results for "General equation"

Period-multiplying bifurcations and multifurcations in conservative mappings

1983

The authors have investigated numerically and analytically the period-doubling bifurcations and multifurcations of the periodic orbits of the conservative sine-Gordon mappings. They have derived a general equation for the appearance of multifurcations in conservative mappings. In agreement with many recent studies, they also find evidence that such mappings possess universality properties. They also discuss the role of multifurcations in conservative mappings exhibiting chaotic behaviour.

1973

In the present paper the connection between an arbitrary polydisperse property and the molecular weight distribution is investigated using exact mathematical methods. On the basis of the superposition principle a general equation is derived which can be applied e.g. to the polydispersity problems connected with the Gel Permeation Chromatography, to the light scattering from dilute solutions of high polymers, or the ultracentrifuge. From this point of view, each of these phenomena is described by help of a linear operator with a special kernel, the analytical or numerical inversion of the operator yielding the molecular weight distribution of the system by help of an uniform algorithm. In de…

Analyse zweier charakteristischer Verteilungen von Polystyrol durch Kolonnenfraktionierung

1962

Zwei Polystyrole werden radikalisch derart polymerisiert, das die Molekulgrose einmal durch Kombination (Polystyrol A), das andere Mal durch Ubertragung (Polystyrol B) bestimmt ist. Die hierbei zu erwartenden Molekulargewichtsverteilungen konnen durch die allgemeine Koppelungsgleichung (1) mit k = 2 bzw. k = 1 beschrieben werden, wobei kleine Abweichungen, die aus der Kinetik folgen, rechnerisch berucksichtigt werden. Die Polystyrole werden in einer Kolonne nach BAKER-WILLIAMS fraktioniert und das Ergebnis nach 3 Methoden ausgewertet. Aus der Fraktionentabelle last sich direkt Pw und Pn ausrechnen, wenn die 1. Fraktion eine entsprechende Korrektur erfahrt. Verschiedene graphische Auswertung…

On the integration of kinetic models using a high-order taylor series method

1992

A general equation to derive kinetic models up to any order is given. This equation greatly facilitates the application of the Taylor series method to the integration of kinetic models up to very high orders. When dealing with non-stiff models, computing time is always reduced by increasing the integration order, at least up to the 20th order. When the model is stiff, the integration order should be optimized; however, a twelfth order is recommended to integrate weakly stiff models. The use of an algorithm which permits the immediate calculation of the integration step size required to maintain a given accuracy leads to further reductions in computing time. When implemented as recommended h…

Extension of the linear solvent strength retention model including a parameter that describes the elution strength changes in liquid chromatography.

2020

Modelling the retention behaviour of solutes in liquid chromatography, based on the composition of the mobile phase is a common task in the chromatographic practice. Along the development of liquid chromatography (LC), several models have been proposed to help in understanding the retention mechanisms, and especially, allow the prediction of retention times with optimisation purposes. Particular models are used for different LC modes, such as normal phase (NPLC), reversed phase (RPLC), hydrophilic interaction (HILIC), and micellar (MLC). In this work, a general equation is proposed that includes a parameter (the elution degree, g), which characterises the way the elution strength varies wit…

INFLUENCE OF SEED TREATMENT ON GERMINATION PATTERN OF CHAMAEROPS HUMILIS

2015

Dwarf fan palm (Chamaerops humilis L.) is an important Mediterranean species well known for many uses such as gardening, environmental restoration, and technological uses. Dwarf fan palm is usually propagated from seed, which takes up to 100 days to germinate. The germination pattern of dwarf palm seeds was studied during a 100-days length period in relation to different pre-sowing treatments (hot water, sulphuric acid, or mechanical scarification) in comparison to untreated control. A covariance analysis was performed to test the effect of time within seed-treatment treatment. All germination patterns were significantly fitted by a S-shaped (sigmoidal) distribution function with the genera…