Sub-case

It is important to note that, for important sub-cases of case /), which will be discussed in more detail in Sect. 2.4, there is a low extent of disorder entropy effects, if any, are small and changes of the lattice dimensions are absent or small. These particular disordered forms are not considered as mesomorphic. In such cases, the limiting models which are fully ordered or fully disordered may be designated respectively as ordered or disordered crystalline modifications, if their consideration is useful for the structural description of a polymeric material. Note... 

These modifications constitute important sub-cases of the case of positional disorder for which only some characterizing points of the structure maintain long-range three-dimensional periodicity (indicated as case i in Sect. 2.1). 

An important sub-case of this kind corresponds to the occurrence of long-range positional order of all the atoms in two dimensions within layers of macromolecules (which may be single layers or bilayers, etc.) and disorder in the stacking of such layers, whereas some characterizing points of the layers maintain long-range periodicity and a well defined 3-D lattice. 

Another important sub-case, of disorder in macromolecular crystals, corresponds to the statistical occurrence of two specific orientations only, at well defined positions in a 3-D lattice, of a group of macromolecules or of each single macromolecule. 

Molinari et at. [73] reported a study on various configurations of photocatalytic membrane reactors for the degradation of 4-nitrophenol using Ti02 as catalyst. Two configurations have been studied irradiation of a recirculation tank, with the suspended catalyst confined by means of the membrane, and irradiation of the cell containing the membrane with three sub-cases catalyst deposited on the membrane, catalyst suspended, and catalyst included in a membrane during its preparation. 

A sub case in this category is the benzylation of enolates through the Srn 1 process. In this way, both a-nitro- and a-chloro-4-nitrocumene are alkylated by the enolates of 2-nitropropane, diethylmalonate, or diethyl 2-butylmalonate. A particular case of benzylic C-C fragmentation is the electron transfer photosensitized Cope rearrangement of 2,5-diphenyl-1,5-hexadienes [218,219]. 

Based on this dispersion relation, the following sub-cases can be considered ... 

Prototropy in triazoles is particularly complex when substituents such as OH, SH and NHR are available to donate protons to annular N. A detailed discussion of all possible sub-cases of this type is beyond the scope of this chapter, but the main aspects of this matter are reviewed in a broad critical context (76ahc(s1)i. p. 388, 4i4, 444) tabulating generalizations and their levels of reliability. The section on reactivity (Section 4.12.3) gives examples of the variability of tautomeric preference in some reactions. 

II. The molecule breaks up into atoms, as, for instance, in the case of thermal excitation, or in consequence of absorption of radiation we speak in this case of atomic binding. Here there are several sub-cases to be distinguished ... 

The general adsorption-reaction model given by the set of equations (7-1) and (7-2) can be simphfied and extended to describe two possible sub-cases ... 

Protocooperation differs from mutualism In that In mutualism neither population can survive without the other whereas In proto-cooperation one or both populations can survive without the other. One could define two sub-cases of protocooperation, depending upon whether the presence of the second population Is necessary for... 

This dispersion is for a rectangular lattice (a b), with the longitudinal component as described in Eq. (1). Two sub-cases exist ... 

Assuming the zeolite pellet is initially free of adsorbate, eqs. (10.4-23) and (10.4-24) have been solved by Do (1989). The solutions are summarized below for three sub-cases ... 

B) The two data sets are comparable experimental results that yield indistinguishable values of s, (as determined by the F-test at a specified p-value) this category includes several sub-cases, that differ, e.g., as to whether the two data sets are wholly independent or involve comparisons of data sets that are correlated in some maimer. 

B) This category of t-tests examines the statistical significance of the difference (Xj — X2) between the means of two experimental data sets for which the F-test has shown that the two standard deviations are indistinguishable. However, application of the i-test again varies somewhat with the circumstances, and three sub-cases must be considered (Meier 2000). 

In this sub-case, the ODE system matrix A is not equivalent to a diagonal matrix D, but it is equivalent to a triangular matrix / (the Jordan canonical form). More precisely, there exists a matrix P (it is important to note that in this case not all the columns of P are eigenvectors) such that P A P = and as a consequence, the ODE system X = A X can be easily converted in the following one Y = JY by a simple change of variables X = PY. ... 

As a consequence of this fact, there are no inflexion points in this sub-case. 

This case is the same as the previous one, except for a change in the order of the eigenvalues, so, the main result is the same there are no inflexion points in this sub-case. 

In this new sub-case, the corresponding Jordan canonical form has the following form ... 

Then, in this sub-case, only one inflexion point can be explained. 

A Reactions Package has been created for a specific process i.e. does not belong to any Reactions System. In this case two sub cases can be distinguished ... 

Into how many subleases is each structural case divided How to discriminate between these subleases Many of the logic algorithms listed above fork their non-minimal case into two sub-cases. 

Many logic algorithms are not covered by Schema 8-1 because the non-minimal case is further partitioned into sub-cases, each featuring a different way of combining partial solutions. The enhanced strategy is as follows. 

A schema-variable c represents the number of different sub-cases of the non-minimal case. This new schema supersedes the previous schema if c is bound to 1, and the Discriminatei predicate-variable is bound to a predicate that always holds. 

But this schema is very lengthy, and doesn t sufficiently show the commonalities between the recursive and the non-recursive sub-cases. We thus syntactically merge... 

Instances of Solve may be defined by fairly complex formulas, including divisions into sub-cases and the corresponding discriminating mechanisms, just as in non-minimal cases. But since this is relatively exceptional, we prefer to keep the schema simple, and always treat such a formula as the instance of a single predicate-variable. 

In terms of methods that solve the same (or a similar) problem as the one tackled by our Proofs-as-Programs Method, there first is the method described by [Smith 82] given two formulas F and G, the objective is to find the weakest pre-condition P such that G follows from F a P. Taking F as the initial logic algorithm, G as the property set, and F a P as the final logic algorithm, this problem is a sub-case of ours, because F, G, P are here sets of formulas. Smith s proofs are performed by natural deduction. 

For instance, in our case the MSG Method is sufficient it discovers 2 different ways of computing C from HL and TC, so the schema-variable c denoting the number of sub-cases is instantiated with 2. This is actually shown in full detail in Example 10-2 and Example 10-6, where the following instance of ProcComp is synthesized ... 

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