Milne method

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

Example 5.1.2 Solving the microbial growth model by Milne method... 

In recent years the Milne method has been successfully applied to the complex energy resonances by Korsch et al. (Korsch and Laurent, 1981 Korsch et al., 1981, 1982). 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

Milnes, T. A., Connolly, J. S., Inman, R. E., Reed, T. B., and Seibert, M. Research Overview of Biological and Chemical Methods and Identification of Key Research Areas. SERI/TR-33-067, Solar Energy and Research Institute, Golden, CO. 1978. 61 p. 

The third column of Table 1 is calculated by applying the recursion relation to the values shown in the second column, etc. It corresponds to the method of Milne It is apparent that the convergence becomes much more rapid with each successive column. For this particular example foe same Limiting values is obtained as either n or m becomes very large. 

Milnes and Mostaghaci [5.5] compared the consequences of different drying methods on the density, the sinter rate and micro structures of sublimated TiO-, suspensions. Evaporation of water in a micro-oven and by radiation heating leds to strongly bound agglomerates, while freeze drying resulted in softly bound secondary clusters. The freeze dried powder reached in 2 h of sintering 98 % of the theoretical density, while differently dried powders needed twice as much time and had a less fine microstructure. 

Witchel, H.J., Milnes, J.T., Mitcheson, J.S., and Hancox, J.C., Troubleshooting problems with in vitro screening of drugs for QT interval prolongation using HERG K+ channels expressed in mammalian cell lines and Xenopus oocytes, J. Pharmacol. Toxicol. Methods, 48, 65-80, 2002. 

M71 Solution of ordinary differential equations predictor-corrector method of Milne 7100 7188... 

The module included here is based on the fourth order method of Milne (ref. 4), where the predictor... 

An important question is the relative numerical efficiency of the two methods or, more generally, the two families of methods. At a fixed step size the predictor - corrector methods clearly require fewer function evaluations. This does not necessarily means, however, that the predictor - corrector methods are superior in every application. In fact, in our present example increasing the step size leaves the FTunge - Kutta solution almost unchanged, whereas the Milne solution is deteriorating as shown in Table 5.1. 

Milne, R. G. and Luisoni, E. (1977) Rapid immune electron microscopy of virus preparations, in Methods in Virology (Maramorosch, K. and Koprowski, H. eds.), Academic Press, New York, Vol. 6, pp. 265-281. 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

RP-HPLC correlation, McDuffie 1981) (22°C, shake flask-GC, Bowman Sans 1983b) (shake flask-GC, Schimmel et al. 1983) (HPLC-RT correlation, De Kock Lord 1987) (HPLC-RT correlation, Liu Qian 1988) (shake flask/slow-stirring method, De Bruijn et al. 1989) (Worthing Hance 1991 Tomlin 1994 Milne 1995) (Montgomery 1993) (RP-HPLC, Saito et al. 1993) (RP-HPLC-RT correlation, Sicbaldi Finizio 1993) (recommended, Sangster 1993) (shake flask-HPLC, Ankley et al. 1994) (recommended, Hansch et al. 1995) (RP-HPLC-RT correlation using short ODP column, Donovan Pescatore 2002) (literature-derived value LDV, Muir et al. 2004)... 

MODIFICATIONS OF THE METHOD This method has been widely used for studying carbohydrate and lipid intermediary metabolism (Herling et al. 1998) as well as drug metabolism (Milne et al. 1997 and Milne et al. 2000, Vuppugalla 2004). Many variations have been reported predominantly with respect to the animal species used. Chaib et al. (2004) compared isolated perfused livers of rats with those of guinea pigs. Den Butter (1994) used livers from rabbits. Further modifications are related to the direction of perfusion via hepatic artery or portal vein or both simultaneously or in connection with the isolated jointly perfused small intestine (Stumpel et al. 1997 and Stumpel et al. 2000) as well as the continuous perfusion in a recirculated (see above) or open (non-recirculated) manner (Lopez et al. 1998). 

Milne, T. A., Brennan, A. H., and Glenn, B. H., Sourcebook of Methods of Analysis for Biomass and Biomass Conversion Processes. Elsevier Science New York, 1990 p. 341. 

The matrix elements known, the set (6.2.7) can now be integrated with any of the standard methods for the numerical solution of coupled differential equations (see, e.g., Milne (1970)). In order to be able to interpret the results obtained by the Sturmian method in the physical space of SSE states, we need the overlaps between the Sturmian states (6.2.3) and the SSE states (6.1.24). The result is... 

Milne S, Ivanova P, Forrester J, Brown HA. Lipidomics an analysis of cellular lipids by ESl-MS. Methods 2006 39 92-103. 

Milne et al. recently described a new assay for the measurement of TP activity in cell lines and tissues, which is sensitive, automated, and allows the processing of multiple assays simultaneously (15). This method exploits the ability of TP to convert 5-FU to 5-FDUR in the presence of deoxyribose-1-phosphate (dR-l-P). 

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