Original parameters

The important aspect of this problem is that while the penultimate model involves a four dimensional parameter space, the model discrimination problem can be reduced to a two dimensional space by dealing with functions of the original parameters. This approach requires that probabilities for array locations in the four dimensional r, r/, rj, rj ) space be mapped to array locations in the ((ri-r/), (rj-rj )) space. 

Using the above transformation, we are able to perform an unconstrained search over k,. For any value of k the original parameter k, remains within its limits. When k, approaches very large values (tends to infinity), k, approaches its lower limit, kmm j whereas when k, approaches very large negative values (tends to minus infinity), k, approaches its upper limit, kmax>1. Obviously, the above transformation increases the complexity of the mathematical model however, there are no constraints on the parameters. 

In the example above, the sums-of-squares surface is transformed to that shown by Fig. 10. The best point estimates of the parameters of kQ and E are thus more readily obtained than k0 and E the initial-parameter estimates are less critical, and the estimation routine converges more rapidly to the minimum. Although the correlation between the parameter estimates has been reduced by this reparameterization, the size of the confidence region of the original parameters k0 and E will not change. 

Clearly, these are very close to the original parameters in (H.l). 

Since the truncation of Eq. (4.14) is an approximation, the shifts may be under- or overestimated. When the shifts are added to the original parameter values, that is, X2 = Xj + <5X, the elements of the design matrix A, which depend on X, change. Many iterations may be needed before convergence can be reached, if it is indeed reachable. Strategies for coping with difficult refinements have been discussed by Watkin (1994). 

We decided to use the same procedure employed in the earlier infrared studies, retaining the original parameters, that is, sample rate, number of passes, type of smoothing, and so on. For the procedure to work properly in this form, it was critical that the data be scaled to lie between 0 and 1 and that sample spacing relative to sx(x) be approximately the same as that used in the original infrared studies. Here we used 52 points per electron volt. 

Here rref is a suitable reference temperature (e.g. the mean temperature in the experimental field) and Aj and Bj the modified adaptive parameters obviously related to the original parameters kQj and Ej. Replacing Eq. (36) with Eq. (37) can... 

To obtain the explicit fonn, note, first, that all tenns a J are of the form Gj/(ai i(j22 - o2 2) But, the denominator in these ratios will be cancelled as it appears in both the inverse matrix and in the vector. Therefore, in terms of the original parameters, (after cancelling n), we obtain... 

After the steady-state analysis, we now use the differential equations for the system (4.75), (4.76), (4.77) to draw 3D and 2D phase plots of the underlying dynamical system behavior for the original parameters given on p. 183. 

This code is typically called by a MATLAB command such as f luidbed(4000,. 4,. 9,1,1, 45/99,10 s8,10 sll,.4,.6,18,27,1,0,10), which uses a large time limit of 4000 dimensionless time units so that the individual solution curves of the IVPs have time to run to the global steady state at (0.17619, 0.72225, 0.86446). We do this with the original parameter data of p. 183, a controller gain of K = 10, and several different initial values. Since the value of K is greater than 4, the middle steady state with maximal x i> yield is both unique and (statically) stable. 

Finally, the parameter describing the dephosphorylation of the internalized receptor is mapped according to a standard lumping reduction [6]. This means that the reduced parameter p D in Adi is equal to the corresponding original parameter pd times the fraction of the total amount of phosphorylated receptors that is made up oflRi- P-ins... 

Here, (k + Ak) represents the original parameter vector k to whose ith element, M is added. A separate calculation must be performed for each element of k. In other words, the derivatives with respect to the elements in k must be calculated one at a time. It is probably most instructive to study the MATLAB code in MATLAB Box 7.5b, where this procedure is defined precisely. 

Once least squares values of the /3 s were obtained, it was desirable to extract from them as much information as possible about the original parameters. To do so, we make one further statement concerning the relations between the rate constants for mutual termination of polymeric radicals of different size. It has been shown (2) that termination rates in free radical polymerizations are determined by diffusion rates rather than chemical factors. The relative displacement of two radicals undergoing Brownian motion with diffusion coefficients D and D" also follows the laws of Brownian diffusion with diffusivity D = D -J- D" (11). It... 

Then the following functions of the original parameters which are of interest were expressed in terms of the estimated / s ... 

The estimated values of the p s and of the original parameters are shown in Table V. Despite the wide range of initiator and monomer concentrations used, it is not possible to obtain precise estimates of this many parameters from the data. In particular, ps is very poorly defined for this system. Notice that the geometric mean approximation is equivalent to fis = 1 (see Equation 15). For a diffusion-controlled bimolecular reaction the arithmetic mean is appropriate as shown above, and this is reflected in the fact that p3 is significantly less than 1. 

The ion guide produces an ion beam independently of the original parameters of the beam delivered by the source and with highly favourable properties for the oa-TOF mass spectrometer. As a result of the decoupling of the ion source and mass analyser, the performance of the instrument is independent of source conditions and this leads to improved resolution and sensitivity. 

The parameters so obtained differed little from the original parameters. Accordingly, the latter parameters were employed in all subsequent calculations, except for the outer two Co s-functions and the outer p-function, whose GTO exponents were scaled by 2.0 in the Co(NH3)5 calculations, based on minimization of the SCF energy for the Aj g state. 

The ry is an interatomic distance, the parameters, z, a, b, c, are related to the atomic species and A B, r hoh, r and gr are related to 0-H pairs. The values of these parameters are given elsewhere [7]. The distances between the H atoms of adjacent water molecules are very different and could lead to considerable orientational variation of the force constants, much as one would expect to find in a polarisable model. However, the original parameters [7] failed to produce a stable structure for ice Ih [74]. A low value for was used in order to stabilise the structure and frequencies distribution fimction similar to earlier work were obtained [72]. 

To rewrite this expression through the original parameters, we should take into account that E/Eth = e/eth and 4/72 = (x/7 - lAl hf + 1-... 

The idea of the following closed representation of B , which can be found in [Gold87] with a close look, is quite easy In each iteration, the original parameter y is squared, and sometimes a 4 is multiplied to it. All the jjrevious 4 s are also squared. Thus, if the length of b is Z, the result must be 4 ° for some p. 

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