Browning s method

Initially, 4-methylthiazole (7), R, = Me, Rj = H, was obtained in low yield (<40%) from chloroacetone (102,127). Kurkjy and Brown (364) using bromoacetone with dioxane as solvent increased the yield to 73%, whereas it only yields 39% in benzene (426). Kurkjy and Brown s method was extended later to 4-alkylthiazoles by Metzger and Carrega (455) with 75 to 90% yields. 4-Vinyl- and 4-isopropenylthiazoles (629), 4-adamanthylthiazole and derivatives (705), 4-carboranylmethylthiazole (706), 4,5-dialkylthiazoles (156, 220, 426, 703, 810), 4,5-... 

Unlike the other alternative methods, analytical expressions of partial derivatives are required and the Jacobian must be evaluated in the Newton-Raphson method. These requirements sometimes prove to be the undoing when the method is applied to complicated equations. Brown (B12) has developed a modification to the Newton-Raphson method, which requires only some of the partial derivatives to be calculated. We have tested Brown s method on our sample problems but have found that it actually required more computing time than the unmodified Newton-Raphson method. 

HS, S, HCCU, CO3, RR NH, RR NCOO", H+, OH- and H2O. Hence there are twenty-three unknowns (m and Yj for all species except water plus x ). To solve for trie unknowns there are twenty-three independent equations Seven chemical equilibria, three mass balances, electroneutrality, the use of Equation (6) for the eleven activity coefficients and the phase equilibrium for xw. The problem is one of solving a system of nonlinear algebraic equations. Brown s method (21, 22) was used for this purpose. It is an efficient procedure, based on a partial pivoting technique, and is analogous to Gaussian elimination in linear systems of equations. 

A special instance of Brown s method is the dehydration of ce-(4-ethoxynaphthyl)benzoin (163) in the presence of SnCl2 + HC1 + AcOH instead of the normal reduction of the carbonyl group, cyclodehydration occurs, giving l,2-diphenyl-5-ethoxynaphtho[2,l-6]furan (164).416 The reaction also occurs under the influence of H2S04 in AcOH or of A1C13 in anisole. 

It has been found that monosubstituted acetylenes can be obtained by Brown s method (equation 164) only when lithium acetylide is replaced by the lithium acetylide-ethylenediamine complex. High yields of terminal alkyl and cycloalkyl-acetylenes are obtained. It has also been demonstrated that the migration of an alkyl group proceeds with retention of configuration, as shown in equation (168). ... 

SOSNLE Brown s method Numerical Math. Div., Sandia Natl. Lab., Albuquerque, NM 87185... 

Brent s and Brown s methods are variations of Newton s method that improve convergence. The calculation of the elements in in Eq. (L.ll) and the solving of the linear equations are intermingled. Each row of Jt is obtained as needed using the latest information available. Then one more step in the solution of the linear equations is executed. Brown s method is an extension of Gaussian elimination Brent s method is an extension of QR factorization. Computer codes are generally implemented by using numerical approximations for the partial derivatives in J. ... 

Selim s method Brown s method Present method... 

Brown s ketone synthesis (2, 148 149). Brown s method for conversion of dienes into cyclic ketones via carbonylation of trialkylboranes has been used in a synthesis of ( )-3-methoxy-l,3,5(10)-estratriene (4). The key step is hydro-boralion of the diene 1 with thexylborane to form 2, which is immediately treated with carbon monoxide and then oxidized to give the ketone 3. This is cyelized to the A - -estratetraene and then reduced to give 4. 

The analysis of the significance of differences between pairs of scaled mean scores presents serious statistical problems which have been overcome using Brown s method (.60). For a full discussion, see Macdonald, Hough, Blizard and Perry, in preparation. The results allow the following preference rank to be concluded... 

Before we discuss this approach further, it is worth contrasting it to Brown s method. Brown related acidity to the average expected valence of OH... O bonds to the base as a whole, and that expected valence was determined by assuming the valence sum rule holds. Hiemstra and coworkers assumed bond strengths for OH... 0 bonds to a single O atom on the base, and related acidity to the deviation of the valence sum on that O from the ideal 2 v.u. Thus, a fundamental feature of the MUSIC method is that it assumes the valence sum rule is not obeyed. 

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