Response unimodal

Figure 14-5 A response surface that is not strongly unimodal... 

The theory necessary for understanding two-station tracer measuring techniques is outlined in Appendix 1. An arbitrary, but unimodal, impulse of tracer is created in a system inlet and the outlet response recorded, see Fig. 21 (Appendix 1). Then, the mean, Mj, of that which resides between the points at which inlet and outlet pulses are observed and recorded is equal to the difference in means of these two signals. Similarly, the variance, T2, and the skewness, T3 are equal to the differences in these respective moments between inlet and outlet. This enables the system transfer function to be defined in terms of a few low-order moments via eqns. (A.5) or (A.9) of Appendix 1, this in turn defining the system RTD. Recall that system moments and moments of the system RTD are one and the same. 

Many patients in clinical practice have already failed or had a partial response to one or more initial treatments, especially when treatment was unimodal. Though robust responses may still occur, on probabilistic grounds, such patients may be expected to have a poorer response to another unimodal treatment in the... 

This minimum is responsible for the diamond and graphite lattices with = 109° and 120° respectively having the smallest and second smallest values of the normalized fourth moment, and hence the shape parameter, s, in Fig. 8.7. This is reflected in the bimodal behaviour of their densities of states in Fig. 8.4 with a gap opening up for the case of the diamond cubic or hexagonal lattices. Hence, the diamond structure will be the most stable structure for half-full bands because it displays the most bimodal behaviour, whereas the dimer will be the most stable structure for nearly-full bands because it has the largest s value and hence the most unimodal behaviour of all the sp-valent lattices in Fig, 8.7, We expected to stabilize the graphitic structure as we move outwards from the half-full occupancy because this... 

Figure 5. Unimodal (Gaussian) response curve with its three ecologically important parameters maximum (c), optimum (u), and tolerance ft). Vertical axis species abundance. Horizontal axis environmental variable. The range of occurrence of the species is seen to be about 4t. (Reproduced with permission from reference 45. Copyright 1987 PUDOC-Centre for Agricultural Publishing...

Here it is advantageous that the number of experiments can be determined before the work starts. The method is adequate for unimodal response functions. 

The unimodality constraint allows the presence of only one maximum per profile (see Figure 11.7) [42, 55, 60], This condition is fulfilled by many peak-shaped concentration profiles, like chromatograms or some types of reaction profiles, and by some instrumental signals, like certain voltammetric responses. It is important to note that this constraint does not only apply to peaks, but to profiles that have a constant maximum (plateau) or a decreasing tendency. This is the case for many monotonic reaction profiles that show only the decay or the emergence of a compound [47, 48, 51, 61], such as the most protonated and deprotonated species in an acid-base titration, respectively. 

For the optimization situation in which two or more independent variables are involved, response surfaces can often be prepared to show the relationship among the variables. Figure 11-12 is an example of a unimodal response surface with a single minimum point. Many methods have been proposed for exploring such response surfaces to determine optimum conditions. 

In addition to the method of steepest ascent and descent, many other strategies for exploring response surfaces which represent objective functions have been proposed. Many of these are based on making group experiments or calculations in such a way that the results allow a planned search of the surface to approach quickly a unimodal optimum point. 

Roth et al. [80] proposed a method to determine the state of membrane wear by analyzing sodium chloride stimulus-response experiments. The shape of the distribution of sodium chloride in the permeate flow of the membrane revealed the solute permeation mechanisms for used membranes. For new membranes the distribution of sodium chloride collected in the permeate side as well in the rejection side was unimodal. For fouled membranes they noted the presence of several modes. The existence of a salt leakage peak, as well as an earlier detection of salt for all the fouled membranes, gave evidence of membrane stmcture modification. The intensive use of the membranes might have created an enlargement of the pore sizes. Salt and solvent permeabilities increased as well. While this is a difficult paper to follow, it may be of use to those who want to develop new methods for measuring membrane degradation. 

To deduce a particle size distribution, the detector response must be deconvoluted by means of a simulation calculation. The scattering particles are assumed to be spherical in shape, and the data are subjected to one of three different computational methods. One system uses the unimodal model-dependent method, which begins with the assumption of a model (such as log normal) for the size distribution. The detector response expected for this distribution is simulated, and then the model parameters are optimized by minimizing the sum of squared deviations from the measured and the simulated detector responses. The model parameters are finally used to modify the originally chosen size distribution, and it is this modified distribution that is presented to the analyst as the final result. 

A second approach uses the unimodal model-independent method, which begins with the assumption that the size distribution consists of a finite number of fixed size classes. The detector response expected for this distribution is simulated, and then the weight fractions in each size class are optimized through a minimization of the sum of squared deviations from the measured and simulated detector responses. The third system uses the multimodal model-independent method. For this, diffraction patterns for known size distributions are simulated, random noise is superimposed on the patterns, and then the expected element responses for the detector configuration are calculated. The patterns are inverted by the same minimization algorithm, and these inverted patterns are compared with known distributions to check for qualitative correctness. 

Fuertes and Nevskaia [139] developed a vapor deposition polymerization (VDP) method to prepare OMCs. Carbon precursor FA was infiltrated into the pores via vapor-phase adsorption at room temperature. When ordered SBA-15 silica was used as a template, the resultant carbon possessed a unimodal pore structure similar to that of CMK-3. However, when a disordered mesoporous silica was used as a template, mesoporous carbon with a well-defined bimodal pore system (mesopores centered at 3 and 12 nm) was obtained, as can be seen from Figure 2.19. A mechanism responsible for the formation of such carbons was subsequently proposed [140] based on the degree of carbon infiltration, which can be controlled with the VDP method. Kruk et al. [141] described a polymerization method for carbon infiltration, which was believed to ensure uniform filling [142] and avoid the formation of nontemplated carbon. 

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