The Brute Force Method

In general, the local sensitivity matrix can only be determined numerically. If the original system of kinetic differential equations can be solved numerically, then the local sensitivity matrix can also be calculated using finite-difference approximations (see Eq. (5.2)). To calculate the sensitivity matrix in this way, we have to know the original solution and the m solutions obtained by perturbing each parameter one by one. All in all, the kinetic system of ODEs has to be solved m + 1) 

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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 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. 

The brute-force method of control based on a harmonic modulation of the normal load L t) has been studied within various approaches that include the generalized Tomlinson model [245], one-dimensional rate-state models... 

Finally, the use of 5 gives us the ability to solve the differential equations arising from an almost arbitrary stimulus, in an elegant way, as opposed to the brute-force method in the time domain. The technique used to do this is the Laplace transform. ... 

The procedure outlined above in which each parameter is incremented one-at-a-time and the system is solved to determine X (t) by (4.A.5) is usually called the brute-force method. For ni parameters, the set of n governing equations (4.A.3) must be solved m times. 

An alternative to the brute-force method is to differentiate (4.A.3) with respect to each Ay,... 

Quaternary systems, such as surfactant-alcohol-oil-water (SAOW), are analyzed in a quaternary diagram made in a regular tetrahedron as indicated in Fig. 16. The brute force method that consists of selecting hundred.s of composition points in the diagram (located at some grid pattern) and in analyzing the phase behavior in all the.se points is of course too tedious to be carried out. 

The main difficulty when working with thin conducting polymer membranes is the lack of quantitative theory of ion diffusion within the membrane. Various theoretical schemes and approximations have been suggested, but the most difficult problem seems to be in the analytical solution or even approximation for the boundary problem of the combined Nernst-Planck and Poisson equations. The latter equation comes from the fact that electroneutrality cannot be assumed to prevail inside the thin membrane. Doblhofer et al." have made an attempt to solve the problem numerically, but even then certain initial approximations were made. Also the brute force method of finite differences does not allow to see clearly the influence of different parameters. 

A related topic is dispensing tests. These are especially important for the in-tank toilet bowl cleaners. Usually the brute force method is used whereby the product is placed in an actual toilet tank, and the number of flushes to exhaust the product is measured. Alternatively, the product can be placed in a container of water the time required for the solid tablet or puck to dissolve completely is measured. (This is not relevant for liquid automatic cleaners.) The difference between the two methods is that the first uses repeated aliquots of fresh water to dissolve the item (as in real use), whereas the second uses the same volume of water for the whole experiment. 

Delocalized electronic systems present special problems in molecular mechanics. From what has been described previously, one clearly needs to know properties on a bond basis in order to carry out molecular mechanics calculations. That is, for each bond one must know the stretching force constant, the value for 1q, and other items. If we talk about an x-y bond, for example, these numbers are usually constants and present no particular problem, as long as their values are known. However, in delocalized system, there are a whole spectrum of bonds which vary in length, and in force constant and other properties, between X and Y. Consider naphthalene as an example. The bonds are somewhat benzenoid, but not exactly. Some are longer, some are shorter. How do we describe this with molecular mechanics There is the brute force method, where one simply gives the different atoms in naphthalene, for example, different atom type numbers, and then gives different bond properties to the different bonds. This would be OK for this molecule, but if one wants to consider a whole series of molecules, where bond lengths vary incrementally... 

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

In essence, the brute-force method starts with an intentionally too strong mobile phase. .. ttiis is key. This ensures that all analytes will elute in a reasonably short time. In fact, under these conditions, many or all analytes may co-elute. The first mobile phase composition tested may be 90/10 acetonitrile/water. The next step is to sequentially weaken the mobile phase until an acceptable separation is achieved. For example, the next mobile phase may be 80/20 acetonitrile/water. Assessment of the optimal conditions (including separation, elution time, and peak shapes) dictates the subsequent tests. For example, severe tailing indicates that a buffer system (e.g., acetate) or mobile phase modifier (e.g., THF) be added in the next iteration. This flexibility is not an inherent aspect of the triangulation optimization method. The aspect of experience comes into pl with the choice and level of buffers and mobile phase modifiers. Sequential modifications to the mobile phase ate made until the optimal separation is achieved. 

Figures 2, 3, 4, 5, and 6 are diagrams representing the best lines through a considerable amount of data. Some of the data were found to scatter considerably, particularly in the regions near the pseudocritical point and near the triple point. Several times, apparently valid liquid samples were taken in a region which should have contained only vapor and solid, indicating a strong tendency toward a metastable liquid state. Consequently, the brute force method of obtaining a large number of points was used to define the boundaries of the liquid-vapor region.

The simulated spectrum is computed by the use of eq. (25), wherein the integrals are converted into discrete sums. It is clear from (25) that, in particular, one needs to know the resonant field values for the various transitions, as well as their transition probabilities for numerous orientations of the external magnetic field over the unit sphere over the unit sphere. A considerable saving of computer time can be accomplished if one uses numerical techniques to minimize the number of required diagonalizations of the SH matrix in the brute-force method. That is, when one uses the known resonant-field value at angle (0,(p) to calculate the one at an infinitesimally close orientation, (0 -i- 80, (p + 8(p), known as the method of homo-... 

Full Update. There are two approaches to fully updating the internal cost tables. Both can be quite slow. The most obvious approach uses brute force, simply updating each cost in the table. A more elegant approach searches for only those costs that need to be updated. But once a good portion of the data path has been filled in, this search may take longer than the brute force method. Currently only the brute force method is used by EMUCS. 

Of course, the brute force method of reacting silica with ammonia or N2/H2 gases at temperatures in excess of 1,200°C will also yield crystalline sihcon nitride ceramics. Another route that does not involve chlorinated precursors consists of sintering a polymeric precursor such as poly[(methylvinyl)silazane]— [(CH3SiHNH)os(CH3SiCH = CH2NH)o.2] . 

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