Sufficiency condition numerical

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

In most cases, data that are obtained through theoretical approaches (literature, data bases, software programs) may not be sufficient for final plant design. Experimental work is usually required on various scales depending on the extent of reactivity. Therefore, the application of well designed experimental test methods is of prime importance to define hazardous conditions. Numerous test methods are available using a variety of sample sizes and conditions. 

The last entry in Table 1.1 involves checking the candidate solution to determine that it is indeed optimal. In some problems you can check that the sufficient conditions for an optimum are satisfied. More often, an optimal solution may exist, yet you cannot demonstrate that the sufficient conditions are satisfied. All you can do is show by repetitive numerical calculations that the value of the objective function is superior to all known alternatives. A second consideration is the sensitivity of the optimum to changes in parameters in the problem statement. A sensitivity analysis for the objective function value is important and is illustrated as part of the next example. 

In the derivation of the ES FR, time-reversibility of the equations of motion is required. Thus time reversibility is a sufficient condition for the ES FR. The question arises as to whether time reversibility is a necessary condition. In numerical calculations where irreversibility was introduced by employing an applied field that had no definite parity under time reversal, it was difficult to observe the breakdown of the ES FR. Careful numerical experiments have finally shown that a breakdown indeed occurs and identifies how this comes about. This work confirms that time reversibility and ergodic consistency are necessary and sufficient conditions for ES FR. 

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

The determinant of the - adjacency matrix A. It was observed that this determinant is often equal to zero and this is a necessary and sufficient condition for the presence of non-bonding molecular orbitals in Hiickel theory. The actual numerical value of det A is correlated to the thermodynamic stability of the molecule [Graovac and Gutman, 1978 Trinajstic, 1992]. 

Necessary conditions for the existence of a self-similar solution are that (1) the governing PDE must reduce to an ODE for F as a function ot// alone, and (2) the original boundary and initial conditions must reduce to a number of equivalent conditions for F that are consistent with the order of the ODE. Of course, a proof of sufficient conditions for existence of a selfsimilar solution would require a proof of existence of a solution to the ODE and boundary conditions that are derived for F. In general, however, the problems of interest will be nonlinear, and we shall be content to derive a self-consistent set of equations and boundary conditions and attempt to solve this latter problem numerically rather than seeking a rigorous existence proof. Let us see how the systematic solution scheme based on the general form (3-135) works for the Rayleigh problem. 

In this chapter, we wish to touch on a number of automated AR construction schemes. At the time of writing, research in AR theory has witnessed a shift toward the development of numerical AR constmction algorithms, with less emphasis placed on general AR theory. These developments have arisen primarily out of a practical need to determine candidate regions for complex, higher dimensional problems, which are not easily computed by hand, but which are still important for practical problems of interest. AR construction methods provide a numerical basis wherefrom theoretical predictions may be compared with in the search for a sufficiency condition. Inasmuch as how... 

Thus, the use of a hybrid construction scheme may provide an alternative method for a numerical sufficiency condition. Validation of the AR might then only be applicable to the specific system computed, but this is nevertheless valuable, for at least a theoretically reliable estimate of the AR may be established. 

The creation of spherical droplets has been the focus of numerous studies over the years [11-14]. These show conclusively that for both steady and transient flows the onset and mechanisms of droplet breakup can be correlated with the non-dimensional Weber number. It is the most important dimensionless number characterizing droplet formation and can be applied to determine the threshold of droplet formation. However, the critical Weber number is only a sufficient condition for droplet breakup and not a necessary condition. This means that if the critical Weber number is surpassed in a process certainly droplet breakup will occur. But droplet ejection is also possible at lower Weber numbers. The only necessary condition for droplet formation is that the supplied energy is sufficient to overcome friction losses and the surface energy of an ejected droplet. 

We have shown in our parent UGA-SSMRCC paper [45] that there are two natural ways of choosing the sufficiency conditions, and each leads to a set of UGA-SSMRCC equations for the cluster amplitudes which is inequivalent with the other. However, a study of the numerical performance of both the variants leads us to the conclusion that they produce very close results and no criteria for choosing one over the other can be established [44]. Since the version A involves fewer terms and is simpler in structure, our subsequent applications focused mainly on this alternative. 

Identification by phenotypic analysis of clones isolated in wine often poses two kinds of problems. First, these clones are difficult to multiply in laboratory conditions. Numerous sub-cultures are needed to obtain a sufficient biomass to carry out all of the tests. For the same reasons, the response to biochemical tests in the API tests (Section 4.3.2) can be ambiguous. The change of color of the indicator is not distinct if the strain does not multiply sufficiently in the microtube. Second, the phenotypic character, such as the assimilation of a... 

The implementation of very effective devices on vehicles such as catalytic converters makes extremely low exhaust emissions possible as long as the temperatures are sufficient to initiate and carry out the catalytic reactions however, there are numerous operating conditions such as cold starting and... 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

Biphenyl has been produced commercially in the United States since 1926, mainly by The Dow Chemical Co., Monsanto Co., and Sun Oil Co. Currently, Dow, Monsanto, and Koch Chemical Co. are the principal biphenyl producers, with lesser amounts coming from Sybron Corp. and Chemol, Inc. With the exception of Monsanto, the above suppHers recover biphenyl from high boiler fractions that accompany the hydrodealkylation of toluene [108-88-3] to benzene (6). Hydrodealkylation of alkylbenzenes, usually toluene, C Hg, is an important source of benzene, C H, in the United States. Numerous hydrodealkylation (HDA) processes have been developed. Most have the common feature that toluene or other alkylbenzene plus hydrogen is passed under pressure through a tubular reactor at high temperature (34). Methane and benzene are the principal products formed. Dealkylation conditions are sufficiently severe to cause some dehydrocondensation of benzene and toluene molecules. 

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