Equivalence conditions

The carbonyl compound (43) has also been synthesi2ed, and its fluorescence spectmm has been shown to match the bioluminescence spectmm under equivalent conditions (214). The details of the excitation step are unclear and a dioxetanone mechanism (59,215) may apply to the reaction. 

The maximum service temperature is about 60°C lower than that of PTFE for use under equivalent conditions. Continuous service at 200°C is possible for a number of applications. The polymer melts at about 290°C. 

A catalyst supported on y-AFO was prepared from Re2Pt(CO)i2l (Fig. 70) and characterized by IR. X-ray photoelectron spectroscopy (XPS), and TPR. The chemi.sorbed cluster was treated with H2 at about 150 C resulting in fragmentation and formation of rhenium subcarbonyls at 400 C the sample was completely decarbonylated. A catalyst prepared from a mixture of Re3(//-H)3(CO)i2l and PtMe2(// -cod)] and treated under equivalent conditions showed the rhenium to... 

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. 

The final relationship between solution subsets, which we can use to curtail the enumeration of one subset, is an equivalence condition, EQ. Intuitively, equivalence between subsets means that for every solution in one subset, y, we can find a solution in the subset x, which will have the same objective function value, and that plays a similar role in the execution of the algorithm. 

In the case of the flowshop example, we have the following equivalence condition... 

Formally, we define the equivalence condition by specifying a set of necessary and sufficient conditions, which it has to satisfy (Ibaraki, 1978). 

Definition. An equivalence condition, EQ, between two partial solutions, r and y is a binary relationship, x.EQ.y, which has the following properties ... 

Its control information, which is represented by the lower-bound function, g, the dominance conditions, D, and the equivalence conditions, EQ. 

Let A = (F, g, D, EQ, s) be a branch-and-bound algorithm. The algorithm terminates after decomposing exactly L /EQ nodes, provided that L / EQ < 00, where L / EQ denotes the set equivalent classes of solutions induced by the equivalence conditions EQ. 

The example problem was generated by picking the type and size at random, while ensuring that no size and type combination was repeated. This last requirement was imposed to avoid making the equivalence condition appear to perform better due to spurious equivalences. 

If we are successful, then we have verified that for the conditions prevailing in the example, the partial solutions, x and y would indeed be equivalent, as far as Condition-a is concerned. We now move to Condi-tion-b, which ensures that the equivalent node will play an equivalent role in the enumeration as the one that was eliminated. Thus, as we examine the children of x, y, regardless of whether they are members of the feasible set, we would verify that their lower-bound values were equal, and that if we had any existing dominance, or equivalence conditions that the equivalent descendant of x, i.e., xu, participates in the same relationships as does yu. 

The analysis of problem-solving experience that has taken place so far has been based on finding subproblems within an existing branching structure that, when solved, will produce subtrees that satisfy the definitions of dominance and equivalence. As we have noted, this is insufficient for generating new dominance and equivalence conditions because we... 

Thus, the next step in the problem-solving analysis is to use information about the domain of the problem, in this case flowshop scheduling, and information about dominance and equivalence conditions that is pertinent to the overall problem formulation, in this case as a state space, to convert the experience into a form that can be used in the future problem-solving activity. 

We can now identify constraints on the logical analysis which must be satisfied for it to produce the desired dominance and equivalence conditions. 

Simplicity. The complexity of the sufficiency conditions will tend to translate into complex dominance and equivalence conditions. This complexity can take the form of either a large number of predicates to guarantee the conclusion or complex predicates. 

The next section will focus on the representation necessary to express this sufficient theory to the computer, so that it can automatically carry out the reasoning associated with analyzing the examples selected by the syntactic criteria presented in this section. Section V will describe the learning methodology, which, using the representation of Section IV, will generate the new dominance and equivalence conditions. 

The representation introduced in the previous subsection must now be utilized to express the information derived from the problem-solving experience, and required to derive the new control knowledge. Our first step will be to define the types of predicates we require to manipulate the properties of the branching structure and the theory that is needed to turn those properties into useful dominance and equivalence conditions. 

In Section II, we presented the computational model involved in branching from a node, cr, to a node aa,. In this model, it was necessary to interpret the alphabet symbol a, and ascribe it to a set of properties. In the same way, we have to interpret o- as a state of the flowshop, and for convenience, we assigned a set of state variables to tr that facilitated the calculation of the lower-bound value and any existing dominance or equivalence conditions. Thus, we must be able to manipulate the variable values associated with state and alphabet symbols. To do this, we can use the distinguishing feature of first-order predicates, i.e., the ability to parameterize over their arguments. We can use two place predicates, or binary predicates, where the first place introduces a variable to hold the value of the property and the second holds the element of the language, or the string of which we require the value. Thus, if we want to extract the lower bound of a state o-, we can use the predicate Lower-bound Ig [cr]) to bind Ig to the value of the lower bound of cr. This idea extends easily to properties, which are indexed by more than just the state itself, for example, unit-completion-times, v, which are functions of both the state and a unit... 

Notice that the left-hand side of this rule contains two types of clauses. The first type is the variable values of the current state and those necessary to compute the new state, while the second, represented by = computes the value of the variable in the new state. This last clause enables the procedural information about how to compute the state variables to be attached to the reasoning. We must, however, be careful about how much of the computation we hide procedurally, and how much we make explicit in the rules. The level to which computation can be hidden will be a function of the theories we employ to try to obtain new dominance and equivalence conditions. If we do not hide the computation, we will be able to explicitly reason about it, and thus may find simplifications or redundancies in the computation that will lead to more computationally efficient procedures. 

Our last concern is for any change in the control information itself. Such change can be caused by a change in the lower-bound function, or a change in the dominance and/ or equivalence conditions. In the former case, we must again abandon our existing dominances and equivalences, unless the altered lower-bound function, g, satisfies the condition... 

The hermiticity constraint may, then, be transcribed into the following equivalent conditions on the P matrix elements ... 

Note that the equivalent condition 9IV/3c = 0 yields a set of equations which is simply the complex conjugate of Eq. (122). 

Hryant But I don t think that in Drosophila we have the equivalent condition that Andy McMahon is talking about, where in the absence of genes there is no growth. The situations you mention just involve change of the overall fate. 

As anticipated, for an irreversible process the forward peak is located at potentials more negative than if it were reversible (i.e. compared to its standard thermodynamic potential). Moreover, since the above relationship shows that the peak current depends on the square root of the transfer coefficient a, under equivalent conditions the height of the irreversible peak equals 78.6% of the reversible peak given that, as often happens, ot = 0.5... 

Only one group has reported CD of SbiSes. The solution used was potassium an-timonyl tartrate, complexed with triethanolamine and ammonia. Selenosulphate was used as the Se source. No XRD pattern was found, as for the sulphide deposited under equivalent conditions. The bandgap was 1.88 eV, and resistivity O-cm [13,14]. Continued study of this deposition showed the effect of various parameters on deposition rate and film thickness (the latter typically reaching 1 p.m) [15]. This study also described some photoelectrochemical behavior of these films (Chap. 9). 

The enhanced nucleophilicity of weakly solvated fluoride ions, solubilized in non-polar solvents as their alkali metal salts by [18]crown-6, has been studied. The wide range of SN2 reactions possible with this system is illustrated in Table 3. Under equivalent conditions in the absence of crown ether no substitution occurs. Similar effects are seen with many nucleophiles which, even if soluble in the solvent employed, show increased nucleophilic substitution rates in the presence of crown ethers (B-78MI52104). However, the monocyclic crown compound exposes the cation on two sides to approach by the counteranion (see Figures lb, c and d for illustrations of this effect in the crystalline state). The resultant ion pairs that form in non-polar solvents reduce the reactivity of the anion. 

The approach to equilibrium is therefore characterized by the following (equivalent) conditions ... 

Alternatively, for a column vector iq such as (11.141b), the equivalent condition is... 

Boley proceeds to find solutions for short times after melting in powers of the square root of the dimensionless post-melting time and indicates a numerical scheme for continuing the solution. The results are in good agreement with those of Landau for equivalent conditions. 

---