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Minimization Procedure (minimization + procedure)
Selected AbstractsSelf-similar solution of a plane-strain fracture driven by a power-law fluidINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2002J. I. Adachi Abstract This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power-law rheology and behaviour index n,0. The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self-similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self-similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed. Copyright © 2002 John Wiley & Sons, Ltd. [source] Collisionally induced dephasing and rotational energy transfer in the CO2 Fermi dyad ,blue' Q-branch 1388 cm,1JOURNAL OF RAMAN SPECTROSCOPY, Issue 8 2007V. G. Arakcheev Abstract In this paper we present a comparative analysis of two alternative approaches to the description of the rotational energy transfer (RET) in the ,blue' (1388 cm,1) Q-branch of the CO2,1/2,2 Fermi dyad based on the simplified spectral exchange (SE) and rotational relaxation (RR) models. The analysis was carried out using a special approximation error minimization procedure for fitting the theoretical model to experimental impulse responses obtained by time-domain CARS at low and moderate densities. At densities ,, 1 amagat, the collisional dephasing (CD) contribution to the linewidth substantially prevails over the width attributed to the rotational structure. The impulse response beating analysis and the fitting results clearly proved that in contrast to the ,red' (1285 cm,1) band, the SE model is more appropriate in comparison with the RR model for a description of the RET in the ,blue' Q-branch. Copyright © 2007 John Wiley & Sons, Ltd. [source] Thermodynamic Reassessment of ZrO2,CaO SystemJOURNAL OF THE AMERICAN CERAMIC SOCIETY, Issue 5 2009Kun Wang In this work, several thermodynamic assessments adopted widely for ZrO2,CaO system are reviewed and examined, the existing discrepancies are summarized, and a new assessment is carried out based on the formation enthalpy of two compounds (CaZr4O9 and Ca6Zr19O44) and experimental activity data concerning cubic solid solution. The thermodynamic parameters of all phases have been optimized by the least squares minimization procedure, a self consistent set of the optimized Gibbs energy parameters has been derived, which can be safely used to extrapolate into the multicomponent system. Compared with experimental data and the results in this work as well as the results reported previously, it is demonstrated that the present thermodynamic assessment is in better agreement with most of the experiments. [source] A note on estimation by least squares for harmonic component modelsJOURNAL OF TIME SERIES ANALYSIS, Issue 5 2003A. M. Walker Abstract. Let observations (X1,,,Xn) be generated by a harmonic model such that Xt=A0 cos ,0t+B0 sin ,0t+,t, where A0,B0,,0 are constants and (,t) is a stationary process with zero mean and finite variance. The estimation of A0,B0,,0 by the method of least squares is considered. It is shown that, without any restriction on , in the minimization procedure, the estimate is an n -consistent estimate of ,0, and hence () has the usual asymptotic distribution. The extension to a harmonic model with k>1 components is discussed. The case k=2 is considered in detail, but it was only found possible to establish the result under the restriction that both angular frequencies lie in the interval [source] A circular statistical method for extracting rotation measuresMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 2 2001S. Sarala We propose a new method for the extraction of rotation measures from spectral polarization data. The method is based on maximum likelihood analysis and takes into account the circular nature of the polarization data. The method is unbiased and statistically more efficient than the standard ,2 procedure. We also find that the method is computationally much more convenient than the standard ,2 procedure if the number of data points is very large. We find that for most sources the method gives results very close to the standard ,2 minimization procedure. We give results for all the cases for which the method gives significantly different results from ,2 minimization. We also make a ,3 fit to the data in order to extract non-Faraday rotation behaviour for those sources for which a large number of data points are available. [source] Prediction of flammability speciation for the lower alkanes, carboxylic acids, and estersPROCESS SAFETY PROGRESS, Issue 1 2007M. Palucis Abstract A Gibbs energy minimization procedure is used to predict the flammability envelopes of alkanes, carboxylic acids, and acetates. In addition to providing the calculated adiabatic flame temperature (CAFT), the product profiles reveal regions of incomplete combustion products and the onset of methane formation above 0.0001 mole fraction. Temperatures at the predicted onset of methane production appear to be closely related to the temperature at the upper flammability limit (UFL). Although a fixed CAFT value could be related to the lower flammability limit (LFL), it was found that this was not the case with the UFL and only for acetic acid could a single CAFT value of 1200K be used to conservatively predict the flammable region. Rather, in general, a single CAFT value could not conservatively predict the upper flammable region. The predictions also reveal local maxima and minima in the concentrations of reaction products. These maps of incomplete combustion products for the flammability region predict that incomplete combustion products are produced at fuel/oxygen ratios significantly below the fuel/oxygen stoichiometric ratio. This region decreases as the limiting oxygen concentration is approached. © 2006 American Institute of Chemical Engineers Process Saf Prog, 2006 [source] | ||||||||||