Invertibility criteria

Three definitions are given about the power line concepts. A power line characterizes the way energy flows between two points in a system. So, talking about inverse models (here implicitly between the inputs and the outputs), the input/output (I/O) power line concept is defined. Finally, the invertibility criteria presented in Section... 

Bicausality is the extension of causality for obtaining the inverse model equations directly from a bond graph representation. The way bicausality is assigned in a bond graph depends on the results of the invertibility criteria and of the structural analysis in terms of I/O power line (see Definition 6.2) sets and I/O causal path (see Definition 6.5) sets. 

In the bond graph representation in preferential integral causality, choose a set of disjoint I/O power lines associated with a set of disjoint I/O causal paths of minimal order. If such sets do not exist then the model is not invertible (criteria I and 2), and the procedure stops. 

If the previous phase has been executed the different invertibility criteria have been verified and the bicausality is already assigned. 

The quahty of incoming raw sugar is paramount for efficient operation. Polarization is a universal quaUty criterion. Color, ash (inorganic), invert sugar, moisture, dextran content, and grain size are other criteria that may be included in raw sugar purchase contracts. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

Now we can understand why we cannot draw a Fischer projection sideways. If we did, we would be inverting the stereocenter. To draw the enantiomer of a Fischer projection, do not turn the drawing sideways. Instead, you should use the second method we saw for drawing enantiomers (place the mirror on the side of the compound and draw the reflection). Recall that this was the method that we used for drawings where wedges and dashes were implied but not shown. Fischer projections are another example of drawings that fit this criterion ... 

The first criterion for the formation of a HIPE is, of course, the presence of two immiscible liquids, one of which is water (or aqueous solution), almost without exception. The nature of the organic, or oil, phase can vary to a considerable extent, although the most stable HIPEs are generally produced with more hydrophobic liquids. However, it is the nature of the surfactant employed to stabilise the HIPE which will ultimately facilitates its formation. Above a certain critical limit of internal phase volume, an emulsion will tend to invert to the opposite type, i.e. an oil-in-water (o/w) emulsion will become the w/o variety, and vice versa. This can be prevented from occurring by careful choice of surfactant, such that it is completely insoluble in the dispersed phase of the emulsion. 

Colour reactions. A good criterion for judging of the genuineness of honey in so far as the addition of commercial invert sugar is concerned is furnished by the colour reactions, when these give distinctly positive results. In this case, since natural honey does not give the colour reactions, it may be concluded that the honey consists partially or wholly of commercial invert sugar. 

The optimal control of a process can be defined as a control sequence in time, which when applied to the process over a specified control interval, will cause it to operate in some optimal manner. The criterion for optimality is defined in terms of an objective function and constraints and the process is characterised by a dynamic model. The optimality criterion in batch distillation may have a number of forms, maximising a profit function, maximising the amount of product, minimising the batch time, etc. subject to any constraints on the system. The most common constraints in batch distillation are on the amount and on the purity of the product at the end of the process or at some intermediate point in time. The most common control variable of the process is the reflux ratio for a conventional column and reboil ratio for an inverted column and both for an MVC column. 

By minimizing the maximum eigenvalue of the information matrix, M, we will be assured that M will be invertible. This would be impossible if the design matrix X and, subsequently, the information matrix M were ill conditioned. A variant of the previously mentioned E-optimality criterion is shown in Equation 8.90,... 

Again, this is a rejection-free algorithm, in the sense that the cluster is always flipped. Just as in the SW algorithm, the cluster-construction process is probabilistic, but the probabilities pij involve energies of individual spin pairs in contrast with an acceptance criterion that involves the total energy change induced by a cluster flip. The implementation can be simplified by a small trick In step (2), each spin j that is added to the cluster can immediately be inverted. This guarantees that a spin is never added twice. Step (5) can then be eliminated. ... 

The third device is to invert the optimal pivoting criterion. Traditionally, the optimal pivot is selected for each column. This ensures the best equation is selected for each variable. 

First, a series of criteria concerns the invertibility checking of a model. An approach based on different I/O causal paths (see Definition 6.8) and the system matrix determinant has been proposed in [40]. Here the approach based on disjoint I/O causal paths (see Definition 6.9) is presented [15, 24, 25]. It uses two structural criteria which, if not verified, enable the inversion process to be stopped early in the procedure. A third criterion is formulated at a behavioral level. This level is called behavioral in the sense that it requires analytical developments based on the constitutive and conservation laws in the bond graph representation. 

Criterion 3 (Junction structure solvabihty) In the bicausal bond graph representation, if for all the sets of disjoint I/O power lines and for all the sets of disjoint I/O causal paths retained for the bicausality assignment, a rum-solvable junction structure appears then the model is not invertible. 

In fact if the specitications in a sizing problem based on the approach of model inversion do not verity this criterion, unit pulses may appear when inverting the equations which is not physically feasible. 

This inclusion of information levels is interesting from a chronological point of view in a design process. In fact, if a property is not verified at a level, it is not at the successive level at all. For instance, concerning non-invertibility of a model, if it is detected at the earliest stage (power lines - Criterion 1, or causal paths - Criterion 2, or junction structure solvability - Criterion 3), the designer will not spend time to go further in his design problem which will be known as ill-posed. 

The acceptance criterion for the resolution of GSI 048 is that the design and operation of the Class IE vital instrument buses, inverters, and uninterruptable power supplies (UPS s) shall provide sufficient assurance that associated batteries, inverters, battery chargers and regulated transformers are available to support safety systems operation, (i.e., there should be a technical specification for the operability of all components which make up the UPS). 

The question now arises as to whether the distribution derived by inverting equation (106) wUl be significantly different from that distribution evaluated assuming the patchwise model, i.e. can the two models be differentiated by analysing isotherm data It is possible that both models will yield equally valid distributions leaving no criterion on which to choose between them. This has indeed been the case found by Rudzinski et when the condensation... 

Numerical tests performed with sets of 25-250 focal mechanisms are presented. The stress tensor is fixed for all datasets. The focal mechanisms are selected to satisfy the Mohr-Coulomb failure criterion (see Fig. 12a, b) and subsequently they are used for the calculation of moment tensors. The moment tensors were contaminated by uniform noise ranging from 0 to 50 % of the norm of the moment tensor (calculated as the maximum of absolute values of the moment tensor eigenvalues). The noisy moment tensors were decomposed back into strikes, dips and rakes of noisy focal mechanisms inverted for stress. The deviation between the true and noisy fault normals and slips attained values from 0° to 25°... 

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