Presentation of numerical results

Textual descriptions of the exposure assessment results might be useful if statements about the mean, the central tendency estimate (median) or a selected quantile of the exposure distribution are given without a description of uncertainty. However, each of the point estimates mentioned will have a different level of uncertainty with respect to model assumptions, database and calculation method. A typical wording to describe results might be, for example  

Taking into account the restrictions/the uncertainty/the lack of sufficient data/. we assume/expect/calculate/... that the exposure of the population/of the most exposed group/the highest exposed individual/... 

A textual presentation of results should in no way lead to a mixture of verbal descriptions of numerical results with an evaluation of results. In 2000, the European Commission gave this clear advice  

Although the advice is given mainly for the description of risk assessment results, it holds completely for exposure assessment, since the quantitative input for risk assessment is exposure assessment and uncertainty analysis. Since any description of the resulting exposure distribution(s) in terms such as very low , low , fair , moderate , high or extreme includes an evaluation, it must be defined and justified (EnHealth Council, 2004). Those communicating the results of an exposure assessment frequently use comparative reporting schemes, such as the 50%/majority/. .. /95% of data/measurements/individuals show exposure values/estimates/measurements lower than the tolerable daily intake 

The presentation of results in graphical formats is state of the art (USEPA, 2000). Although we do not have uniformly accepted recommendations for the presentation formats, the targets of the visualization approaches seem to be in accordance with the following criteria to describe the expected variability in the target population  

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The next three subsections describe the background and principles of random error treatment, and they introduce two important quantities standard deviation a- and 95 percent confidence limits. The four subsections following these— Uncertainty in Mean Valne, Small Samples, Estimation of Limits of Error, and Presentation of Numerical Results—are essential for the kind of random error analysis most frequently required in the experiments given in this book. The Student t distribution is particularly important and useful. 

We present here numerical results illustrating that the solutions of the radial equations (eq.(5) for the hydrogen-like case and eq.(14) for polyelectronic atoms) are weakly dependent of e in a finite volume. 

The general form of g P-) is quite complicated. We shall present some numerical results of gj, (2) as a function of parameters h, K, and q. Here, it is instructive to present the formal form of the pair correlations (in the limits m —> o<> and A, 0) for the case AT= 1. The first few pair correlations are... 

In reality however, situations also exist where a more complex form of the rate expression has to be applied. Among the numerous possible types of kinetic expressions two important cases will be discussed here in more detail, namely rate laws for reversible reactions and rate laws of the Langmuir-Hinshelwood type. Basically, the purpose of this is to point out additional effects concerning the dependence of the effectiveness factor upon the operating conditions which result from a more complex form of the rate expression. Moreover, without going too much into the details, it is intended at least to demonstrate to what extent the mathematical effort required for an analytical solution of the governing mass and enthalpy conservation equations is increased, and how much a clear presentation of the results is hindered whenever complex kinetic expressions are necessary. 

In this section we present some numerical results to examine the efficiency of the new introduced methods. Consider the numerical integration of the Schrodinger equation ... 

Turning to electric fields and classical Maxwell-Boltzmann statistics, soluble analytical models now exist which allow calculations of non-degenerate electron densities as a function of thermodynamic state in intense electric fields (low density high temperature). Semiclassical methods are available for switching on atomic potentials to models studied presently, though numerical results are not yet available here. 

Figure 7 presents the numerical results, obtained for k dependent thermal conductivity k) by combining the MD simulations and GCM approach for KrAr and LiF. The difference between these two cases is seen on the qualitative level. For a mixture of neutral particles (right) the expected behavior, described by the Lorentzian-like function, is observed. Otherwise, we found the increase in A(k) when k becomes larger for molten LiF, and a well pronounced peak is seen at wavenumber kp where the first peak of static structure factor Sxx(k) is located (see Fig. 1). 

It is important to ensure that the graphics used in the presentation of the results of the pharmacometric analysis of clinical trial data do not lie or mislead. The way to do this is to avoid visual distortion. That is, the pharmacometrician has to ensure that the visual representation of the data is consistent with the numerical representation. This is in terms of volume, area, and so on. 

In the present paper we investigate the convergence properties of the RS, SRS, and HS perturbation series for He2 and HeH2 molecules, i.e. for the interaction of two ground-state two-electron systems. These perturbation formalisms correspond to none, weak, and strong symmetry forcing, respectively. In Sec. II the perturbation equations of the RS, SRS, and HS theories are briefly summarized. In Sec. Ill the computational details of the calculations are presented. The numerical results are presented and discussed in Sec. IV. 

We present now numerical tests of the performance of the various approximants to the SS-MRCC formalisms discussed in this article for some prototypical systems. We classify the applications in different groups, delineated in separate subsections, each discussing results of a given approximant. We have organized the sequence of the presentation of our results in such a way that more sophisticated approximants appear later. This will facilitate our discussions regarding the relative efficacies of the corresponding approximations, as more refined hierarchies are undertaken. 

First the equations for the modified Murphree plate efficiency are developed and then the corresponding equations for the Murphree plate efficiency are developed. Expressions for the vaporization efficiencies for each of these efficiencies are then presented. The section is concluded by the presentation of numerical examples for binary mixtures which demonstrate that essentially the same compositions of the vapor leaving a plate are predicted by vaporization efficiencies corresponding to the modified Murphree efficiency model as are predicted by the Murphree plate efficiencies. This result suggests that the correlations for the film coefficients developed by others4 for use with Murphree efficiencies may be used in the prediction of vaporization efficiencies for multi-component systems. 

One of the challenges faced by the experiment team is the comparison of numerical results with measurements and observations. Each team will be required to provide output at specific locations where point measurements will be made. All results will be presented and compared to actual measurements in TecPlot. 

The EEM formalism represents a comprehensive and internally consistent framework for the quantitative as well as qualitative understanding and computation of atom-in-a-molecule sensitivities. The method is direct, due to an adequate separation of the variables, allowed by a spherical-atom approximation. The potential for studying molecules, (ionic) solids and molecule-surface interactions has been fully demonstrated. There are several parameterizations possible, all of them relying on quantum-mechanical calculations for estimating atomic electronegativities and hardnesses. At present, the numerical results are conform with a Mulliken population analysis on STO-3G wavefunctions, but there is no reason why other more sophistieated approaches could not be used. Its simplicity forms a powerful tool for the experimental chemist, who is advised to include the environment into the models, avoiding isolated-atom approaches whenever possible. 

More technical presentations of the results just discussed can be found in [2, App. E] (least-squares adjustment) and [3] (CODATA 2002 adjustment of the fundamental constants). Important online resources include values of the adjusted constants and data from the calculations (list of adjusted variables Z, numerical input values Q, etc.). 

The paper is organized as follows. The problem formulation - parameter identification of the non-linear dynamic model of an E. coli cultivation process - is given in Sect. 2. In Sect. 3 the hybrid schemes between GA and FA are presented. The numerical results and discussion are presented in Sect. 4. Conclusion remarks are done in Sect. 5. 

In Section 2 we set forth the model for excited states of two-electron atoms that is provided by the large-dimension limit and then very briefly describe the method that we used to calculate the expansion coefficients. Further deteuls can be found in Re s. [6] emd [7] and in a forthcoming publication [12]. In Section 3 we present our numerical results and discuss implications of this work. 

We desire to acknowledge the assistance we received, in a considerable part of the numerical calculations, from Mr K, v a n N c s, chem. docts., now once again of the laboratory of the Bataafsche Petroleum Maatschappij, Amsterdam, to whom we arc also indebted for valuable contributions in the final presentation of the results of the theory of Part II. 

We use the experimental results of the solvation Gibbs energies of argon, methane, and ethane to estimate the structural change in the solvent induced by placing such solutes at a fixed position in the liquid. Table 3.7 presents such numerical results. The values shown in this table are all positive. This is consistent with similar conclusions reached by many other authors, i.e. these solutes increase the structure of the solvent and therefore may justifiably be referred to as structure markers or structure promoters. ... 

In Section 5.1 we describe the independent electron pair approximation (lEPA). We use an approach that leads quickly to the computational formalism but which may give the misleading impression that lEPA is an approximation to DCI. After showing what is involved in performing pair calculations, we will return to the physical basis of the formalism and show that in fact both lEPA and DCI are different approximations to full Cl. In Subsection 5.1.1 we describe a deficiency of the lEPA, not shared by DCI or the perturbation theory of Chapter 6 namely, that the lEPA is not invariant under unitary transformations of degenerate molecular orbitals. In Subsection 5.1.2 we present some numerical results which show that while the lEPA is reasonably accurate for small atoms, it has serious deficiencies when applied to larger molecules. 

Summary. It is shown, that in complex chemical reaction systems a very high redundancy in the parameter space of kinetic rate constants occurs which renders the determination of kinetic data difficult or often impossible. Two methods which overcome this parameter redundancy are presented. In the first procedure effective parameters are locally defined and adapted during a standard optimization procedure. The second method approximates the kinetic behaviour of measured concentrations with a neural network. Both methods are analysed on the basis of an example reaction for the neural modelling we present also numerical results. 

The degree of protection (P), or percent protection (P = PxlOO), has been used in the past extensively to assess inhibitor effectiveness and to compare the performance of different products. For this purpose, P is customarily plotted against inhibitor concentration. Because 100% protection is approached asymptotically as the concentration increases, differentiation between products is difficult. Alternatively, differentiation is accomplished numerically by comparing percent protection at a given concentration. This comparison neglects the different performance characteristics of inhibitors because, as will be shown below, one product may surpass another at one concentration while lagging behind at another concentration. For these reasons, and essentially totally practical purposes, another presentation of the results was proposed with some interesting conclusions. 

Finally, we present two examples in which parameter estimation and optimal transition between different operating conditions are solved. Finally, a comparison of numerical results, solution time, and number of variables for the resulting NLPs are provided. 

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