Brute force methods

The brute-force method does not rely on mathematical functions/relationships, algorithmic iterations, or gradient profiling. It is the original and classic method 

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Start with 1 mol of and let it age for 10 billion years or so. Refer to Table 2.1. What is the maximum number of atoms of " Po that will ever exist Warning This problem is monstrously difficult to solve by brute force methods. A long but straightforward analytical solution is possible. See also Section 2.5.3 for a shortcut method. 

The brute force method depends on a systematic variation of all involved coefficients over a reasonable parameter space. The combination yielding the lowest goodness-of-fit measure is picked as the center for a further round with a finer raster of coefficient variation. This sequence of events is repeated until further refinement will only infinitesimally improve the goodness-of-fit measure. This approach can be very time-consuming and produce reams of paper, but if carefully implemented, the global minimum will not be missed, cf. Figures 3.4 and 4.4. 

The algebraic/iterative and the brute force methods are numerical respectively computational techniques that operate on the chosen mathematical model. Raw residuals r are weighted to reflect the relative reliabilities of the measurements. 

The next step, the grid search, is essentially a brute-force method in which the experimental range is divided into a grid of specific size and methodically searched. The method is called an exhaustive grid search. From an input of the desired criteria, the program prints out all points (formulations) that satisfy the constraints. 

In order to recover both amines in ophcaUy achve form the amide is hydrolyzed chemically by reachon with NaOH in aqueous ethylene glycol at 150 °C. This brute force method would certainly lead to problems with amines containing other functional groups and is in stark contrast to the elegant enzymatic procedure used for the first step. Hence, an overall greener process can be obtained by employing an enzymatic deacylation step in what we have called an easy-on/easy-off process... 

Clearly, computational resources are not yet to the point that brute force methods will suffice for high-precision calculations. Physical and chemical intuition play an important role in constructing appropriate trial wave functions and, despite the complexity and size of the basis sets used in these calculations, can improve the convergence of the basis significantly. It is important, therefore, to understand the nature of an atomic or molecular system in terms of its physical as well as its more formal mathematical properties. 

The method of successive approximations has been conveniently described by Wigley ( ) where either a "brute force" method or a "continued fraction" method can be used. 

The brute force method is the classical approach where mass action expressions are substituted directly into the mass balance conditions and solved for total concentrations which are then compared to the analytical values. In the continued fraction method, the non-linear equations are rearranged to solve for free ion concentrations which are initially assumed to be equal to the total concentrations, as detailed by Wigley (42). These two methods are best illustrated by a simple example. Assume a solution which contains free Ca2+ ions, free CQ ions, and only one ion pair CaCO. The mass balance s conditions are given by... 

Spectrometer does not fall far behind the theoretical limitations, if it does at all. The quality of the pulses (uniformity and constancy of the flip angles and the rf phases) is sufficiently high that pulse errors hardly play a role as a resolution-limiting factor. The tightest theoretical limitation is the necessarily finite width of the rf pulses, which is particularly acute for the BR-24 sequence. The next significant step to enhance the resolution in solid state proton m.p. spectroscopy may well require either 90° pulses shorter than, say, 500 ns (this would be the brute force method) or another clever idea. 

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