Operators characteristic-value problem

The characteristic-value problem - more often referred to as the eigenvalue problem - is of extreme importance in many areas of physics. Not only is it the very basis of quantum mechanics, but it is employed in many other applications. Given a Hermitian operator a, if their exists a function (or functtofts) g such that... 

Generalization of the characteristic value problem. The characteristic value problem can be formulated as the quest for the irreducible linear manifolds which are invariant under an operator. The principal result of the spectral theory of normal operators can be formulated, from this point of view, as the statement that all irreducible linear manifolds of normal operators are one-dimensional. Similarly, one can ask for irreducible closed linear manifolds which are invariant under a set of operators. Since a closed linear manifold which is invariant under a set of operators is also invariant under the group or algebra generated by these operators, one is naturally led in this way to a linear manifold which belongs to an irreducible representation of a group or an algebra. 

Among all the non-numerical approximation methods, the effectiveness of the variational methods is perhaps most surprising [11]. This method serves to determine characteristic values of linear operators. Since the time variable can be eliminated from almost every reactor equation by transforming it into a characteristic value problem, the variational method should have wide applications in reactor theory. Its use has been limited, so far, because Boltzmann s operator is not self adjoint or normal. Whether this limitation is a necessary one, remains to be seen. The reason for the great accuracy of the variational principle in simple problems of quantum mechanics is that any function which is positive everywhere and has a single maximum can be so well approximated by any other similar function. Thus... 

The recognition accuracy estimation described above faces one very important problem what is the best choice for the threshold value 0 To solve this problem, statistical decision theory is used. ° The basis for this is an analysis of the so-called the Received Operating Characteristic (ROC) curve. By tradition, ROC is plotted as a function of true positive rate TPj TP + FN) (or sensitivity) versus false positive rate FPj TN+FP) (or 1-Specificity) for all possible threshold values 0. Figure 6.5 presents an example of such a ROC curve for the results obtained with our computer program PASS in predicting antineoplastic activity. 

This example illustrates the use of the t statistic for testing a hypothesis related to a population mean value (/x) when the variance (cr ) is unknown. As before, both two-tailed and one-tailed tests are possible, whichever the problem situation demands. Rejection hmits are set based on a chosen type I (a) error, and operating characteristic curves (OC curves) can be constructed to reflect the associated type II ) error risk. Finally, one should note that the t test was necessitated due to the fact that cr was being estimated by 5 from a sample of size n. If n is large enough, then one would expect 5 to closely approximate cr and the Z test can be used anyway. For n a 30 this is generally an acceptable procedure. 

Due to their simplicity of construction and use and the relatively sharp cut-off characteristics, cascade impactors have been widely used for the size classification and size-classified chemical analysis of aerosols. Table 6.1 lists the most important integrating sampling methods with their main characteristics. Table 6.2 gives the most important differential, size-resolving methods used to sample and measure atmospherie aerosol particles. The section of the particle size distribution and the modes that dominate the sensitivity of the methods are indicated. The upper and lower size limits are nominal values for the most commonly used forms of the techniques. Cost, complexity of operational requirements, calibration problems, and the demands of the particular evaluation to be used also affect the choice of methods. For example, chemical analysis usually requires that a sample be collected, then taken to the evaluation device. 

The term characteristic value is not used in this work. Thus, if L is a completely continuous linear operator, the values of the parameter X for which the problem... 

In addition to the elimination of partial solutions on the basis of their lower-bound values, we can provide two mechanisms that operate directly on pairs of partial solutions. These two mechanisms are based on dominance and equivalence conditions. The utility of these conditions comes from the fact that we need not have found a feasible solution to use them, and that the lower-bound values of the eliminated solutions do not have to be higher than the objective function value of the optimal solution. This is particularly important in scheduling problems where one may have a large number of equivalent schedules due to the use of equipment with identical processing characteristics, and many batches with equivalent demands on the available resources. 

Is one model scale sufficient or should tests be carried out in models of different sizes One model scale is sufficient if the relevant numerical values of the dimensionless numbers necessary to describe the problem (the so-called process point in the pi space describing the operational condition of the technical plant) can be adjusted by choosing the appropriate process parameters or physical properties of the model material system. If this is not possible, the process characteristics must be determined in models of different sizes, or the process point must be extrapolated from experiments in technical plants of different sizes. 

A second area in our combustion work is in direct-combustion. Large industrial combustors are too costly to use for experimentation. However, we have a unique 500 lb/hr. pulverized coal/oil furnace which closely simulates the performance, in other words, the value of the heat per unit volume of commercial unit. Our main thrust in using this is that of resolving applied problems. With this combustor, we ve studied the handling, pulverizing, combustion, and fouling characteristics of SRC-I fuel and operated it on these fuels first during October, 1974. 

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