Muller’s method

Methods such as Graeffe s root-squaring method, Muller s method, Laguerre s method, and others exist for finding all roots of polynomials with real coefficients. [12 and others]... 

In recent years, several calculational procedures have been proposed for solving distillation problems in which a chemical reaction occurs on each plate. Suzuki, et al.24 used a procedure based on Muller s method whereas, Jelinek and Hlavacek14 used a relaxation method. Komatsu and Holland17 and Nelson19 proposed different formulations of the Newton-Raphson method. 

Where K-values are nonlinear in pressure and temperature and are composition dependent, algorithms such as those in Figs. 7.6 and 7.7 can be employed. For solving (7-19) and (7-21), the Newton-Raphson method is convenient if X-values can be expressed analytically in terms of temperature or pressure. Otherwise, the method of false position can be used. Unfortunately, neither method is guaranteed to converge to the correct solution. A more reliable but tedious numerical method, especially for bubble-point temperature calculations involving strongly nonideal liquid solutions, is Muller s method. ... 

Unlike Muller s method, this algorithm has a single solution ... 

Muller s method has to select the new point between the two solutions of the interpolating parabola. This difficulty is enhanced by interpolating the function with a higher degree polynomial. 

We will mainly follow here Von Smoluchowski s and Tuorila s treatment which, though less exact than Muller s methods, shows the principal facts very clearly and leads to results that are not much less accurate than MOller s. 

E. Muller, ed., Elouben-Weyl s Methods in Organic Chemisty, 4th ed., Vol. 13, Part 7, Georg Thieme, Stuttgart, Germany, 1975. 

Briand, J.P., Muller, S., and Van Regenmortel, M.H.V. (1985) Synthetic peptides as antigens pitfalls of conjugation methods./. Immunol. Meth. 78, 59-69. 

D. Muller S. Abramovich-Bar, R. Shelef D. Sonenfeld A. Levy S. Kimchi and T. Tamiri Improved methods in the post-explosion analysis of TATP. In Proc. 7th Int. Symp. on Analysis and Detection ofExplosives, Qinetiq, Sevenoaks, Edinburgh, U.K., 2001, p. 321. 

Electrolytic Methods.—According to Muller s experiments,3 tellurium is not deposited by electrolysis from solutions in which it is present in the sexavalent condition, so that telluric acid must first be reduced to tellurous acid. A weight of tellurous acid not exceeding 0-25 gram is dissolved in 175 c.c. of 2N H2S04 and electrolysed for two and a half hours between platinum electrodes. The grey deposit of tellurium is washed with water and alcohol and dried in a desiccator over sulphuric acid. In this method the average error is estimated to be 0T per cent. By this process tellurous acid may be estimated in the presence of telluric acid. After the tellurium from the tellurous acid has been removed, the solution is boiled with hydrochloric acid to reduce the telluric acid to tellurous acid and the estimation of the tellurium carried out as before.4... 

Nielsen K V, Muller S, Agerholm I, Poulsen T S, Matthiesen S H, Witton C J, Schonau A. PNA suppression method combined with fluorescence in situ hybridization (FISH) technique. In PRINS and PNA Technologies in Chromosomal Investigation. Frank Pellestor, eds. NOVA Publishers,... 

By the late 1930s, the editors of Analytical Edition of Industrial and Engineering Chemistry—Analytical Chemistry sought to bring the contents of the publication more in line with changes in the field—that is, they sought more articles on instrumental methods. The entire October issues for 1939, 1940, and 1941 focused on instrumentation Muller wrote the entire 1940 and 1941 October issues. In January 1946, Muller s "Instrumentation in Analysis" became the first monthly column in the journal, a column he continued to write until 1968. 

Muller s "Instrumentation in Analysis" essays are of interest for many reasons we can find in them an explicit articulated definition of modern objective instrumental methods they clearly show how these developments were tied to industrial needs and, because of this connection, we can see how technological values of blackboxing, standardization, and cost efficiency get tied to this emerging notion of instrumental objectivity. Muller serves as an excellent witness to changes in this notion of objectivity and the tensions that produced them. 

What then do we learn from Muller s editorials We learn how, within analytical chemistry, objectivity came to be understood instrumentally and that the best instrumental objective methods were automatic methods. There was a strong economic need in industry for more efficient methods of analysis. Despite Muller s pleas, a science of instrumentation with its own academic departments did not develop in the university setting. In the commercial setting, economic values could not be separated from other desirable values in developing new analytical instruments. In short, instrumental objectivity came to incorporate values from the marketplace. 

The stage temperatures are determined from the calculated stage compositions by bubble point calculations. The calculations are carried out iteratively using Muller s algorithm (Wang et al., 1966). The authors have determined that convergence by this method is more reliable than the Newton-Raphson technique. 

P. Muller, S. Kopke and G. M. Sheldrick, Is the bond-valence method able to identify metal atoms in protein structures Acta Crystallogr., D59, 32-37 (2003). 

Between the direct methods, the X-ray diffrac-tometric sin v /s-method is the most important one (Macherauch and Muller 1961). With this method, according to Bragg s diffraction condition, net plane distances dhki are measured, i.e., existing strains are measured directly. As strain of a specimen is direction dependent, it is necessary to incline the specimen around the angle (Fig. 6). 

An novel method for the DKR of a 2-methyl-substituted unsymmetrical 1,3-diketone via enzymatic reduction was described by Muller s group in 2001. As shown in Scheme 3.6, the enantiopure anti product was produced in excellent diastereoselectivity and moderate yield. 

Boeckler C, Frisch B, Muller S et al (1995) Immunogenicity of new heterobifunctional cross-linking reagents used in the conjugation of synthetic peptides to liposomes. J Immunol Methods 191 1—10... 

Bachinski and Muller (l9Tl)- High-temperature, metastable portions of the two-phase region in the system microcline - low albite were determined at 6OO to 900°C and 1 atm by means of alkali ion-exchange and exsolution - dissolution experiments. Compositional data are fitted to two-coefficient Margules expressions using the method of Thompson (1967) and the "r-s" method of Thompson and Waldbaum (1969b). 

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