Parametric representation

Parametrized representations of individual damping dispersion functions were first obtained [127] by fitting ab initio damping functions [74] for Ft-Fl interactions. The one-parameter dampmg fiinctions of Douketis et al are [127] ... 

EXP AC analyzes an interconnected network of building rooms and ventilation systems. A lumped-parameter formulation is used that includes the effects of inertial and choking flow in rapid gas transienl.s. The latest version is specifically suited to calculation of the detailed effects of explosions in the far field using a parametric representation of the explosive event. A material transport capability models the effects of convection, depletion, entrainment, and filtration of... 

Despite the obvious correspondence between scaled elasticities and saturation parameters, significant differences arise in the interpretation of these quantities. Within MCA, the elasticities are derived from specific rate functions and measure the local sensitivity with respect to substrate concentrations [96], Within the approach considered here, the saturation parameters, hence the scaled elasticities, are bona fide parameters of the system without recourse to any specific functional form of the rate equations. Likewise, SKM makes no distinction between scaled elasticities and the kinetic exponents within the power-law formalism. In fact, the power-law formalism can be regarded as the simplest possible way to specify a set of explicit nonlinear functions that is consistent with a given Jacobian. Nonetheless, SKM seeks to provide an evaluation of parametric representation directly, without going the loop way via auxiliary ad hoc functions. 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

Langenbucher F. Parametric representation of dissolution rate curves by the RRSBW distribution. Pharm Ind 1976 38 472-477. 

Equation 4.51 is an integral equation that can be used to determine D(c ) by a graphical construction or numerical solution. The derivative required in Eq. 4.51 is provided by the measured concentration profile at time t and the integration is performed on the inverse of c x) [6]. However, this historically important method is only moderately accurate, and it would be preferable to obtain diffusion profiles for various assumed diffusivities as a function of concentration by computation. D(c) could be deduced by fitting calculated results for a parametric representation of D(c) to an experimentally determined diffusion profile. 

We also introduce the parametric representation, which, although not necessary in the case of a single component, is the key to the case of many... 

Rearrange the problem. Do not get fixed ideas on which variables are dependent and which independent. The use of parametric representations (see The Use of Parametric Representations in Chapter 3) and the hodograph transformation come under this rubric. Even more radically, the shift to a different type of model (e.g., the wave model of Westerterp see also General Observations and Forming the Model in Chapter 1) is a possibility. 

C. M. Roth and A. M. Lenhoff, "Improved parametric representation of water dielectric data for Lifshitz theory calculations,"). Colloid Interface Sci., 179, 637-9 (1996), present another set of parameters for water. 

Fig. 18.2. ADAS408 generates recombination, ionization and radiated power data from parametric representations of the various coefficients. Electron temperature and density ranges are specified on the processing screen along with atomic masses. This code can accept special filter files which describe the soft X-ray pass band of arbitrary window/detector combinations and modify the calculated radiated power. ADAS408 delivers output data in the ADAS data format adfl 1 - a principal data class accessed by plasma modeling codes...

Figure 3.27 Parametric representation of the acoustoelectric response. (Reprinted with permission. See Ref. [48J. 1989 ieee.)...

Parametric representation of the low-energy portion of the water-carbon dioxide G3(MP2)//B3LYP electronic energy surface. Variations on the theme developed by Professor B. Widom in his PhD dissertation ... 

The value of the parameter a, which lies between 0 and 1, must be determined for each system by comparison with more accurate methods, but once this has been done the Whitten-Rabinovitch method gives an accurate parametric representation of the required density of states. 

The parametric representation of unimolecular rate coefficients, both at the low pressure limit and in the fall-off region has been the result of a monumental work by Troe and his coworkers [67-71]. This work is tremendously useful for applications of unimolecular reactions to complex systems, for example to combustion or to atmospheric chemistry, as it permits accurate representation of the fall-off without having to to a time-consuming and difficult Master Equation calculation. 

The above determination of the values of the characteristic roots requires a knowledge of the reaction time. As we have seen from the parametric representation of the reaction path in terms of bj, we need only a knowledge of the compositions along reaction paths to determine the ratios X,/Xy. According to Eq. (57), a graph of In bi vs In bj is a straight line with a slope of Xt/X, consequently, any curved reaction path that contains sufficient 6,- and bj for accurate plotting can be used to determine X[Pg.231]

Vbias is the bias voltage and p (0 < p< 1) a parametric representation of the bias voltage distribution in the tuimeling gap. Equation (6-10) gives a maximum when t = = j, i.e. close to the equilibrium... 

These equations form a parametric representation for the integral surface z = z(x ... ), provided that the parametric variables in the initial conditions can be inverted to obtain (i ... as functions of This is the case when the Jacobian... 

The condition in Eqnation (2.290) gives parametric representations by the eqnations... 

Thus, laser spectroscopy, and in particular ionisation spectroscopy involving several photons, allow one to study the mechanism of autoionisation itself, as opposed to testing the quality of atomic wavefunctions used to obtain pre-diagonalised states. Thereby, interest in the subject of interacting resonances and their parametric representation is enhanced (see in particular section 8.20). 

For example, let us consider simple molecules in solution. Using the grand canonical formalism, we can find a parametric representation of the dependence of the osmotic pressure with respect to the concentration... 

The first transformation and the simplest one establishes a correspondence between the O(n) model and a solid-on-solid model. In particular, this transformation shows us why in the parametric representation (12.4.3), the number n appears in the form of a cosine. For this reason, the end of this section will be devoted to a description of this transformation. 

Let us consider monodisperse polymers in solution and let us study this system in the framework of the standard continuous model. As was shown in Chapter 10 [see (10.5.2)], the osmotic pressure is given in terms of the number C of polymers per unit volume by the parametric representation... 

As was previously shown, H(q) is given in terms of C by the following parametric representation... 

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