Parameter-dependent initial value sensitivity

Sensitivities of Parameter-Dependent Initial Value Problems... 

To overcome this difficulty in [Bock87] a correction is made at the switching point which takes into consideration the dependence of the switching point on the initial values and parameters. All perturbed trajectories are forced to switch at the same point and the sensitivity matrices are corrected by a term depending on the different switching points. 

Due to the nrmlinear dependence of the a priori, unknown residual sequence E on the parameter vector 0, the last equation leads to a nonlinear-weighted least squares problem, which has to be tackled by nrmlinear optimization methods. However, nonUnear least squares techniques are sensitive to the initial parameter values and if no acciuate estimates are available, the nuniniization procedure is very likely to converge to a local minimum. In order to avoid potential inaccurate convergence problems associated with arbitrary initial estimates, initial values for the coefficients of projection may be obtained by identifying conventional ARMA models for each of the K data... 

ODEs depending on parameters. The question addressed there is about the sensitivity of the model with respect to parameter changes. Consider the ordinary initial value problem (IVP) ... 

Criteria for sensitivity, B and b, are also criteria for validity of the early R-A approximation (ERA), which says that R-A occurs virtually when m = 1 = I. While B for most free-radical polymerizations lies within a narrow range, which exceeds the critical value, b varies widely from subcritical to critical values, depending strongly uponcholceof Initiator and feed parameters [lio and Tq. Decreasing values of b generally depress the critical value of a slightly. Computed R-A... 

The model captures the features seen in the experiment very well. The initial rate of polymerization and the conversion at the onset of autoacceleration are nearly identical with the experimentally generated values as is the rate of autodeceleration. The conversion and value of maximum rate are within 5%. The difference in maximum rate can be ascribed to the high sensitivity of autoacceleration on the At parameter a small change in can greatly influence the rate of polymerization during autoacceleration. The conversion at which the maximum rate occurs is dependent upon fcp, i.e., the free volume (and conversion) at which autodeceleration sets in. The simulated rate also shows a tail around 80% conversion, which can be ascribed to the DSC not capturing polymerization at high conversion and low rate. 

The chapter is divided into three sections the first part is concerned with the derivation of 3D-LogP descriptor and the selection of suitable parameters for the computation of the MLP values. This study was performed on a set of rigid molecules in order, at least initially, to avoid the issue of conformation-dependence. In the second part, both the information content and conformational sensitivity of the 3D-LogP description was established using a set of flexible acetylated amino acids and dipeptides. This initial work was carried out using log P as the property to be estimated/predicted. However, it should be made clear that, while the 3D-LogP descriptor can be used for the prediction of log P, this was not the primary intention behind its the development. Rather, as previously indicated, the rationale for this work was the development of a conformationally sensitive but alignment-free lipophilicity descriptor for use in QSAR model development. The use of log P as the property to be estimated/predicted enables one to establish the extent of information loss, if any, in the process used to transform the results of MLP calculations into a descriptor suitable for use in QSAR analyses. 

The PCS technique has demonstrated an ultra-sensitive ability to probe the conductive properties of the layered superconductor SnNbsSe We have shown that PCS is a unique tool for the detection of nanoclusters which in our case were likely formed by a small fraction of unreacted initial substances (A15 and dichalcogenide) used to synthesize the ternary compound. Finally, we have determined the value and the temperature dependence of the energy gap parameter in the SnNb5Se9 phase. 

Thus, at high I, the pair population is a considerably smaller fraction of the total OH population than the initial fraction given by a Boltzmann distribution at the flame temperature. For example, for the nominal values of 14 and 0.4 A for Oq and Oy, the infinite-intensity fraction is < 1% of the total while the zero-intensity value is 4%. This result is generally valid for the entire range of parameters inserted into the model, which represent physically realistic energy transfer rates. However, the precise numerical values depend sensitively on the actual parameters inserted. These facts form the central conclusions of this study (4). A steady state model with no dummy level and a different set of rate constants and level structure (5) shows some similar features. 

The mass-transfer coefficient is sensitive to several factors, including Henry s constant of the contaminant, the packing factor, and the temperature of the ambient air and water to be treated. An HTU value, calculated at 20°C from Eq. (7), would require a fivefold increase if ambient water and air temperatures of 5°C and -12°C, respectively, were encountered (9). Therefore, the equations presented are recommended for initial design work and evaluation of pilot studies or field data. Data from pilot studies are required to provide dependable values for the mass-transfer coefficient and the effects on removal efficiencies produced by varying system parameters. An analytical program... 

We have seen that the logistic map can exhibit aperiodic orbits for certain parameter values, but how do we know that this is really chaos To be called chaotic, a system should also show sensitive dependence on initial conditions, in the sense that neighboring orbits separate exponentially fast, on average. In Section 9.3 we quantified sensitive dependence by defining the Liapunov exponent for a chaotic differental equation. Now we extend the definition to one-dimensional maps. 

Nonradiative Deactivation Involving a Second Excited State. A somewhat different situation is presented by the pressure effects reported for the MLCT emissions from the ruthenium(Il) complex Ru(bpy)f+. At ambient temperature, in a fluid solution this species shows little unimolecular photochemistry and relatively small emission quantum yields (ff>r < 0.1) [32]. Initial pressure studies on the luminescence of this ion in 18°C aqueous solution detected little sensitivity to pressure [60], as might be expected for a weakly coupled nonradiative mechanism owing to the low compressibility of water. However, detailed studies by Fetterolf and Offen [32,61] painted a more complex picture. These workers probed the temperature dependence of AF and confirmed the small negative value at low temperature but also demonstrated a remarkable temperature dependence for this parameter. 

State is highly sensitive to the medium. There is a change of polarity of the reaction states from initial to transition state with a volume decrease known as electro-strictive shrinkage. Electrostriction can be critical to rate enhancement under pressure. In terms of kinetic parameters it is related to the pressure dependence of the dielectric constant (dins/dp). Some representative values of this quantity are listed in Table 10.15 [50]. 

It is thus possible to calculate the time needed to attain a given exudation value. In Figs. 2.30a and b one can see the lines of equal time values required for isolation of 2 wt% of the inhibiting liquid. The isolation rate at the initial stage of exudation is seen to have a strong dependence on the film-filling degree of the Cl. It is also a parameter that is sensitive to variations in the PhCI ratio, which is specifically evident at their low thermodynamic compatibility (Fig. 2.30b). 

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