Dealing with Dependencies

Another important qnestion when planning a probabilistic assessment is how to deal with dependencies. This also is one of the key issues that were identified for the Pellston workshop. Fnrther work is needed to evaluate these options. Some additional points are made here. 

Even if a correlation is below the conventional level of significance, consideration should be given to whether it might alter the risk estimate, and it may be prudent to include it. When measnred or estimated correlations are nsed to specify dependencies in Monte Carlo models, it is important to check that the matrix of correlations satisfies mathematical constraints (Table 2.3). 

If there are significant spatial or temporal dependencies, it may be necessary to use a spatially or temporally explicit model in order to avoid misleading results. 

Correlation coefficients used to define dependencies in risk models must be positive-definite e.g., for the variables A, B, and C, if the correlations r and s are big and positive, t must be positive too 

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How should we deal with dependencies, including nonlinear dependencies and dependencies about which only partial information is available ... 

The remainder of the article is organized as follows In Section 2, the factor model, which is a typical CCF factor model, is briefly introduced. Uncertainties brought by P factors are discussed and serve as an inspiration of using modified models to deal with dependencies in PR A. The D-S evidence theory and evidential networks are briefly introduced in Section 3. The EN based approach is then discussed in detail in Section 4. In Section 5, a part of a practical probabilistic risk assessment is analyzed using the proposed approach. The conclusion is expressed in the end. 

The Champ-Sons model has been developed to quantitatively predict the field radiated by water- or solid wedge- eoupled transdueers into solids. It is required to deal with interfaces of complex geometry, arbitrary transducers and arbitrary excitation pulses. It aims at computing the time-dependent waveform of various acoustical quantities (displacement, velocity, traction, velocity potential) radiated at a (possibly large) number of field-points inside a solid medium. 

The chemical potential pi, has been generalized to the electrochemical potential Hj since we will be dealing with phases whose charge may be varied. The problem that now arises is that one desires to deal with individual ionic species and that these are not independently variable. In the present treatment, the difficulty is handled by regarding the electrons of the metallic phase as the dependent component whose amount varies with the addition or removal of charged components in such a way that electroneutrality is preserved. One then writes, for the ith charged species. 

These concluding chapters deal with various aspects of a very important type of situation, namely, that in which some adsorbate species is distributed between a solid phase and a gaseous one. From the phenomenological point of view, one observes, on mechanically separating the solid and gas phases, that there is a certain distribution of the adsorbate between them. This may be expressed, for example, as ria, the moles adsorbed per gram of solid versus the pressure P. The distribution, in general, is temperature dependent, so the complete empirical description would be in terms of an adsorption function ria = f(P, T). 

The operation of the STM depends on the conduction of electrons between tip and sample. This means, of course, that insulating samples are, in general, not accessible to STM investigations. Nevertheless, a large body of work [32] dealing with STM characterization of thin organic films on conducting substrates is now in... 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

Load balancing can then be achieved in NAMD 2 by moving compute objects and patches between nodes. But what if a compute object and a patch it depends on are on different nodes Compute objects individually communicating with off-node patches would generate a huge amount of redundant communication. Therefore, patches are represented on other nodes by proxy patches, which implement the same interface as home patches for dealing with compute objects and handling dependencies but receive coordinates from and... 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

M.o. theory has had limited success in dealing with electrophilic substitution in the azoles. The performances of 7r-electron densities as indices of reactivity depends very markedly on the assumptions made in calculating them. - Localisation energies have been calculated for pyrazole and pyrazolium, and also an attempt has been made to take into account the electrostatic energy involved in bringing the electrophile up to the point of attack the model predicts correctly the orientation of nitration in pyrazolium. ... 

Thermal Process. In the manufacture of phosphoric acid from elemental phosphoms, white (yellow) phosphoms is burned in excess air, the resulting phosphoms pentoxide is hydrated, heats of combustion and hydration are removed, and the phosphoric acid mist collected. Within limits, the concentration of the product acid is controlled by the quantity of water added and the cooling capabiUties. Various process schemes deal with the problems of high combustion-zone temperatures, the reactivity of hot phosphoms pentoxide, the corrosive nature of hot phosphoric acid, and the difficulty of collecting fine phosphoric acid mist. The principal process types (Fig. 3) include the wetted-waH, water-cooled, or air-cooled combustion chamber, depending on the method used to protect the combustion chamber wall. 

The main combustion pollutants are nitrogen oxides, sulfur oxides, carbon monoxide, unbumed hydrocarbons, and soot. Combustion pollutants can be reduced by three main methods depending on the location of thek appHcation before, after, or during the combustion. Techniques employed before and after combustion deal with the fuel or the burned gases. A thkd alternative is to modify the combustion process in order to minimise the emissions. 

In the section dealing with electrophilic attack at carbon some results on indazole homocyclic reactivity were presented nitration at position 5 (Section 4.04.2.1.4(ii)), sulfon-ation at position 7 (Section 4.04.2.1.4(iii)) and bromination at positions 5 and 7 (Section 4.04.2.1.4(v)). The orientation depends on the nature (cationic, neutral or anionic) of the indazole. Protonation, for instance, deactivates the heterocycle and directs the attack towards the fused benzene ring. A careful study of the nitration of indazoles at positions 2, 3, 5 or 7 has been published by Habraken (7UOC3084) who described the synthesis of several dinitroindazoles (5,7 5,6 3,5 3,6 3,4 3,7). The kinetics of the nitration of indazole to form the 5-nitro derivative have been determined (72JCS(P2)632). The rate profile at acidities below 90% sulfuric acid shows that the reaction involves the conjugate acid of indazole. 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

The power from the motor shaft to the driven shaft can be transmitted in many ways, depending upon the power and load application. In this chapter we deal with the most common types of driving systems, their influence on the starting characteristics of the motor and their effect on bearings. 

The lone remaining aspect of this topic that requires additional discussion is the fact that the mechanical threshold stress evolution is path-dependent. The fact that (df/dy)o in (7.41) is a function of y means that computations of material behavior must follow the actual high-rate deformational path to obtain the material strength f. This becomes a practical problem only in dealing with shock-wave compression. 

Although the difference in final strength f, integrated through both the actual shock wave and the computational shock wave, will be mitigated by dynamic recovery (saturation) processes, this is still a substantial effect, and one that should not be left to chance. These are very important practical considerations in dealing with path-dependent, micromechanical constitutive models of all kinds. 

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