Compressibility effects in packed and open tubular gas and supercritical fluid chromatography

The influence of the pressure drop on the efficiency and speed of anal. in packed and open tubular supercrit. fluid chromatog. (SFC) is described: methods previously developed to describe the effects of mobile phase compressibility on the performance of open tubular columns in SFC have been extended to packed columns. The Horvath and Lin equation has been used to elucidate the influence of variations in velocity, diffusivity, and capacity factor along the column on the overall efficiency of packed column SFC. In packed columns, in contrast with the situation in open tubular columns, because the diffusion coeffs., the increase in both linear velocity and capacity factor which result from a significant pressure drop cause the plate height to increase along the column. The effect of liq. decompression along the length of the column on the speed of anal. in SFC has been studied and numerical expressions derived which enable calcn. of compressibility correction factors for the plate height. Both the f1 and f2 correction factors remain very close to unity for acceptable pressure drops, which means that the pressure drop has virtually no effect on the no. of plates generated per unit time for an unretained component. For retained species, the decompression of the mobile phase across the column causes the capacity factor to increase and hence leads to increased anal. time

A fundamental study of the effects of modifiers in supercritical fluid chromatography

The effects of organic modifiers on retention and peak shape in packed-column supercritical fluid chromatography were studied. The adsorption behavior of different modifiers was investigated on stationary phases consisting of hydrocarbon monolayers chemically bonded on silica. Adsorption isotherms for several modifiers were recorded using breakthrough measurements. The results were compared with those obtained by a simple method based on the injection of different sample sizes. Modifiers were selected to reflect various types of interactions with the silica support. Isotherms were found to be approximately Langmuirean. Experiments were performed to elucidate the influence of adsorbed molecules on the retention of selected test solutes. The introduction of a solvent modifier can lead to a substantial change in the mobile phase density, the effect of which is comparable with that of an increase in the density of pure carbon dioxide. Our results confirm that the effects of low concentrations of modifiers (between 0 and 2 percent) in packed-column SFC are largely due to deactivation of residual silanol groups on the silica support. The accessibility of the active sites was found to depend strongly on the size and structure of the modifier molecules. The decrease of retention due to the addition of low concentrations of modifiers could be described accurately by a model derived from Langmuir adsorption behavior. Some semi-quantitative rules for the selection of modifiers and the required concentrations for optimum deactivation of the support are discussed

Robust option replication for a Black-Scholes model extended with nondeterministic trends

Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and Scholes model. This memory or nondeterministic trend behavior is often seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension of the Black and Scholes model by adding a term to the original Wiener term involving a smoother process which accounts for these effects. The problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing replicating portfolio for a European option without any further knowledge of the extension and that, as a consequence, the classical concept of volatility needs to be re-interpreted.Electrical Engineering, Mathematics and Computer Scienc

Effects of modifiers in packed and open-tubular supercritical fluid chromatography

The applicability of packed and open columns for supercritical fluid chromatography using pure carbon dioxide for the elution of a number of selected test components was investigated. It is showns that the number of solutes that can be eluted as symmetrical peaks is much larger in open-tubular capillary superficial fluid chromatography. For strong hydrogen donors and acceptors, however, poor peak shapes were observed on both types of columns. In both cases, modifiers are needed. Modifiers may effect retention by increasing the mobile phase polarity and density, and by deactivating active sites. The relative influence of these processes was studied in packed and open columns. The effects of various modifiers in packed and open columns were compared under identical operating conditions. The influence of temperature on retention and peak shape was studied for pure carbon dioxid and carbon dioxide—modifier mixture. The influence of modifiers on retention was found to be larger at lower operating temperatures. For components with high affinities for silanol groups the influence of polar modifiers on the capacity factor and the peak shape was found to be of comparable magnitude in packed and open columns. Columns packed with alkyl-modified poly(styrene—divinylbenzene) and with porous carbon coated with poly(ethylene glycol) were used to study the influence of modifiers without interference from silanol groups. Polar and non-polar modifiers were used to study the relative influence of the polarity and density effects of the modifiers

Mobile and stationary phases for SFC: effects of using modifiers

The effects of modifiers on the mobile-phase and the stationary-phase properties in packed-column supercritical-fluid chromatography were studied. Modifiers, may affect retention and improve peak shapes and efficiencies, because of (i) increased mobile-phase polarity, (ii) increased mobile-phase density, (iii) stationary-phase deactivation, and (iv) increased solvation or swelling of the stationary phase. In this paper these four parameters are evaluated and evidence to establish their relative importance is considered. The introduction of a modifier can lead to a substantial increase in the density of the mobile phase. The reliability of several methods for calculating the critical properties and densities of binary fluids is investigated. The Chueh and Prausnitz method gives the most accurate results for the calculation of the critical properties. The Lee and Kesler equation of state yields accurate density estimates. Adsorption isotherms have been measured on several stationary phases for packed-column SFC. These data suggest that stationary-phase deactivation is the most important effect of adding modifiers. Adsorption data on different stationary phases can also be used to judge their applicability. Relevant indicators are the maximum amount of modifier that can be adsorbed on the surface (saturation level) and the initial steepness of the isotherm (surface activity). The latter parameter can be related more closely to the peak shape observed in the absence of modifiers. All silica-based materials studied so far give rise to broad and highly asymmetrical peaks for certain classes of polar solutes. Different silica-based materials appear to differ in degree of activity, but the kind of interactions are the same. On a polystyrene-divinylbenzene stationary phase the mobile-phase modication effect could be studied without the interference of active sites

Effects of modifiers in packed and open-tubular supercritical fluid chromatography

The applicability of packed and open columns for supercritical fluid chromatography using pure carbon dioxide for the elution of a number of selected test components was investigated. It is showns that the number of solutes that can be eluted as symmetrical peaks is much larger in open-tubular capillary superficial fluid chromatography. For strong hydrogen donors and acceptors, however, poor peak shapes were observed on both types of columns. In both cases, modifiers are needed. Modifiers may effect retention by increasing the mobile phase polarity and density, and by deactivating active sites. The relative influence of these processes was studied in packed and open columns. The effects of various modifiers in packed and open columns were compared under identical operating conditions. The influence of temperature on retention and peak shape was studied for pure carbon dioxid and carbon dioxide—modifier mixture. The influence of modifiers on retention was found to be larger at lower operating temperatures. For components with high affinities for silanol groups the influence of polar modifiers on the capacity factor and the peak shape was found to be of comparable magnitude in packed and open columns. Columns packed with alkyl-modified poly(styrene—divinylbenzene) and with porous carbon coated with poly(ethylene glycol) were used to study the influence of modifiers without interference from silanol groups. Polar and non-polar modifiers were used to study the relative influence of the polarity and density effects of the modifiers

Forward and reverse representations for Markov chains

In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny [G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Transition density estimation for stochastic differential equations via forward-reverse representations, Bernoulli 10 (2) (2004) 281-312] for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.Transition density estimation Forward and reverse Markov chains Monte Carlo simulation Estimation of risk

Compressibility effects in packed and open tubular gas and supercritical fluid chromatography

The influence of the pressure drop on the efficiency and speed of anal. in packed and open tubular supercrit. fluid chromatog. (SFC) is described: methods previously developed to describe the effects of mobile phase compressibility on the performance of open tubular columns in SFC have been extended to packed columns. The Horvath and Lin equation has been used to elucidate the influence of variations in velocity, diffusivity, and capacity factor along the column on the overall efficiency of packed column SFC. In packed columns, in contrast with the situation in open tubular columns, because the diffusion coeffs., the increase in both linear velocity and capacity factor which result from a significant pressure drop cause the plate height to increase along the column. The effect of liq. decompression along the length of the column on the speed of anal. in SFC has been studied and numerical expressions derived which enable calcn. of compressibility correction factors for the plate height. Both the f1 and f2 correction factors remain very close to unity for acceptable pressure drops, which means that the pressure drop has virtually no effect on the no. of plates generated per unit time for an unretained component. For retained species, the decompression of the mobile phase across the column causes the capacity factor to increase and hence leads to increased anal. time