Alternative sufficiency condition

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

The two conditions stated above do not assure the occurrence of gelation. The final and sufficient condition may be expressed in several ways not unrelated to one another. First, let structural elements be defined in an appropriate manner. These elements may consist of primary molecules or of chains as defined above or they may consist of the structural units themselves. The necessary and sufficient condition for infinite network formation may then be stated as follows The expected number of elements united to a given element selected at random must exceed two. Stated alternatively in a manner which recalls the method used in deriving the critical conditions expressed by Eqs. (7) and (11), the expected number of additional connections for an element known to be joined to a previously established sequence of elements must exceed unity. However the condition is stated, the issue is decided by the frequency of occurrence and functionality of branching units (i.e., units which are joined to more than two other units) in the system, on the one hand, as against terminal chain units (joined to only one unit), on the other. 

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. 

As it was mentioned earlier, Bastin and Dochain [6] have provided sufficient conditions for the stability of (19) when A t) is proportional to the identity matrix. In such a case, the stability of (19) depends exclusively upon the properties of A t). It was also previously established that when A is not proportional to the identity matrix and it has a deflned structure but it is constant, Chen, [13], provided sufficient conditions for the stability of (19). Then, in this section an alternative issue for the stability of (19) is proposed when A t) is both not proportional to the identity matrix and time-depending. 

From Eqs. (2) and (10) the intrinsic yields qDm are equal to the measured fluorescence yields y (C) at infinite fluor (or quencher) concentrations a sufficient condition that [M] [M] (or [Q] [Q]y2) is established by the complete quenching of molecular fluorescence (Eqs. 1 and 9), as in pure liquids at moderate temperatures or for solutions of fluor in liquid quenchers. Alternatively the intrinsic yields may be computed from the measured yields at the half-quenching concentration [M]i/2 or [Q]y2 (Eqs. 2 and 10), or, following Hirayama and Lipsky,140 from linear plots of y against y which yield qM and qD as intercepts according to the relationship... 

Choice reversal - first choosing A over and then over A - is an even more dramatic form of inconsistency. As mentioned in Section I, some instances of this phenomenon can be explained without any reference to a divided self. This also holds in cases where two corner solutions are deemed equally and optimally good. And, if the choice reversal is itself irreversible, because the preferences of the person simply have changed, there is no paradox at all. Hence, as a necessary but not sufficient condition, we shall have to look for cases in which A and are alternately chosen. 

The geometrical structures of several common conjugated polymers are shown in Fig. 4.1, where the necessary (but not sufficient) condition for conjugation, the alternation of the single and double bonds along the polymer backbone, can be seen. The unique electronic properties of conjugated polymers derive from the presence of -electrons, the wavefunctions of which are delocalized over long portions of (if not the entire) polymer chain, when the molecular structure of the... 

Proof. A sufficient condition for J, evaluated at to have eigenvalues with negative real parts is that if the oflF-diagonal elements are replaced by their absolute values, then the determinants of the principal minors alternate in sign (Theorem A.11). 

From the above discussion it is obvious that the problem of the instability of alternant -electron systems is reduced to the finding of necessary and sufficient conditions for the existence of the number zero in the graph spectrum 120>. This problem has not been solved for the general case, but a set of useful theorems concerning the number zero in spectrum is known. We will denote the number of zeros in the graph spectrum as rj. 

A statement of the alternative means of compliance with the standard from which the applicant is seeking the variance. The statement must contain sufficient detail to support, by a preponderance of the evidence, a conclusion that the employer s proposed alternate methods, conditions, practices, operations, or processes would provide workers with protection that is at least equivalent to the protection afforded to them by the standard from which the employer is seeking the variance. (National defense variances do not require such a statement, and the statement submitted by an employer applying for a temporary variance must demonstrate that the employer is taking all available steps to safeguard workers.)... 

The equality of fugacities is an alternative statement of the necessary and sufficient condition for phase equilibrium and the basis for all such calculations, whether we are dealing with pure substances or with mixtures. By contrast, the equality of the fugacity coefficients is a special result and applies to pure substances only. 

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. 

In alternant hydrocarbons, carbon atoms can be divided into two sets (starred and non-starred) such that an atom of one set is connected only to atoms of the other set. The necessary and sufficient condition for a hydrocarbon to be alternant is to have only even-membered rings. Benzenoid hydrocarbons have only six-membered rings and are therefore alternant or bipartite. 

In order to minimize the risk of microbiological contamination, surfaces must be clean and therefore smooth and accessible. Walls are often tiled or painted while floors are of asphalt, terrazo, or other suitable material. There must be a sufficient fall to the floor to permit efficient drainage. The drains themselves must be well constructed so that they are easy to maintain and trap all odours. Ceilings are so designed that condensation does not fall into the body of the room. Alternatively, air conditioning should ensure that no condensation forms. 

Notice that the sufficient condition for the occurrence of strict Lindemann behaviour, given a strong collision relaxation matrix, is that the d, are constant alternatively, if there is only one grain having a non-zero value of d, the rate is also strict Lindemann, regardless of the form of the assumed relaxation matrix [81.VI]. In the past, it had often been assumed that strict Lindemann behaviour was a strong collision property only, but we now know that near-Lindemann behaviour can often occur in weak collision systems at high temperature, see Chapter 8. 

Appendices follow Chapter 6. In Chapter 2, it has been pointed that local entropy may be expressed in terms of same independent variables as if the system were at equilibrium (local equilibrium). The limitations of Gibbs equation have been discussed in Appendix I. At no moment, molecular distribution function of velocities or of relative positions may deviate strongly from their equilibrium form. This is a sufficient condition for the application of thermodynamics method. Some new developments related to alternative theoretical formalism such as extended irreversible are discussed in Appendices II and III. 

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. 

Table 9.16 only shows the steady-state characteristics of each control structure. However, good steady-state characteristics are not a sufficient condition for good dynamic control system performance. Thus, Aspen Dynamics will be used to evaluate the control system performance for the alternative control structures. Since the unmeasured feed composition changes are the more severe closed-loop test in comparison with the feed rate changes, these load changes will be made in the closed-loop dynamic simulations for comparison. 

The starting point for an alternative formulation of resonance theory is recognition that the necessary and sufficient condition for any one-electron system property (A/) to be representable in the resonance hybrid form (equation 23a) is given by an equation involving localized densities (incoherent... 

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