Number of realizations

Assuming an infinite number of realizations of the process, equilibrium properties can be obtained from the non-equilibrium dynamics by the Jarzynski equality,... 

A stochastic program is a mathematical program (optimization model) in which some of the problem data is uncertain. More precisely, it is assumed that the uncertain data can be described by a random variable (probability distribution) with sufficient accuracy. Here, it is further assumed that the random variable has a countable number of realizations that is modeled by a discrete set of scenarios co = 1,..., 2. 

Pervaporation has become one of the standard membrane technologies with a large number of realized industrial applications. Pervaporation is used for the dehydration of organic compounds, the separation of organic compounds from aqueous solu-... 

Hornberger and Spear s original application of generalized sensitivity analysis (GSA) used a binary acceptance-rejection procedure, i.e., they discarded a Monte Carlo realization if they thought that the prediction was inconsistent with the site-specific data (a nonbehavior ) or kept it if they thought it was consistent (a behavior ). The prior probability on each Monte Carlo realization was the reciprocal of the total number of realizations. After the acceptance-rejection procedure was applied, the updated (posterior) probability on each realization that was classified as a behavior was the reciprocal of the number of behaviors, and the posterior probability on nonbehaviors was zero. 

Another way to treat the influence of the surrounding lattice on a cluster is to use the embedded or immersed cluster technique. A number of realizations of this scheme have been proposed. We shall merely mention the directions of the evolution of these schemes rather than considering the question in detail. One way is to match the cluster and band solutions. This method, generally based on the early works of Grimley, was realized in Pisani (22) within the scope of CNDO/2. The whole system being considered (support and adsorbate) was subdivided into the cluster (adsorbate and a part of adsorbent) and the environment. Then the self-consistent calculation was carried out at a fixed environment. This approach was used to compute the adsorption of H (22) and CO (23) on graphite. 

Having placed all N° r-mers on the lattice, the remaining N° sites are filled with monomers. We further need a factor N° N°l in the denominator, to account for the indistinguishability of molecules 1 and molecules 2. The factor N° already occurs In the equation because the way in which the r-mers have been added already accounts for the indistinguishability of the sites. However, the indistinguishability of the r-mers has not yet been accounted for. Considering all of this, we arrive at the following expression for the number of realizations of the mixture ... 

This cell is very convenient for many electroorganic processes and is probably the most successful system in respect of the numbers of realized industrial processes. 

Temporal evolutions of M(t) are shown in Fig. 2. In order to suppress fluctuations, we have calculated averages over realizations. Throughout this chapter, unless no comments appear, the numbers of realizations are n = 1000,100, and 8 for AT = 100,1000, and 10,000, respectively. We divide... 

We shall see in Chapter 7 that stochastic algorithms can be used to simulate the evolution of a population of particles represented by the NDF. However, the NDF is not a random quantity rather it is the ensemble average of an infinite number of realizations of the stochastic algorithm (assuming that the latter is unbiased). 

The number of ways how the distributions can be established, if the particles at a certain energy level are not distinguishable, can be easily obtained from Table 12.9. For example, for case 1 (2 particles at 1 eu and 2 particles at 3 eu), we have to put a separator after the second particle, like ab cd, to indicate the border of the energy levels. The particles before have the energy 1 and the particles behind have the energy 2. Within a certain energy level, we sort the particles, e.g., ba dc into ab cd and sort out equal patterns. So the numbers of realizations of distributions... 

This exactly corresponds to the heating of an ideal gas on adiabatic compression, during which there is an entropy loss, because space previously accessible to the gas is now no longer available. This reduces the number of realization possibilities and hence the probability of the system. 

In the third method, in order to check the CPA result, we performed a numerical simulation [17] where for a given random distribution the electronic eigenvalues have been calculated numerically and the results been averaged over a large number of realizations. It turned out that the CPA results could be reproduced quite satisfactorily thus establishing the different role of both types of impurities. 

Since this processed data is mathematically similar to the collected experimental data, it can be processed in an identical manner. As an example, we can fit values of Ea and A for the realization and then use these to make predictions at our tanpera-tures of interest. The predictions are saved for later use and the values of and A can optionally be saved for later analysis. This process is repeated to generate an appropriately large number of realizations so that the confidence bounds are consistent. In practice one can select a large number and then repeat the overall process several times to insure that this number is adequately large. If computation time is an issue, then more formal methods for deciding the appropriate number of realizations can be used. 

Ensemble, consisting of a specific number of realizations p Event space domain... 

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