FREE SCHEMES

Only 2-aminothiazole derivatives are reactive enough toward diazonium salts to undergo the diazo-coupling reaction. The azo group fixes exclusively on the 5-position when it is free (Scheme 62) (351). 

Using our results concerning consistent paths, we can given alternative definitions of free schemes. 

A free scheme always halts if and only if its graph contains no loops since in a free scheme a loop corresponds to an infinite path and hence to an infinite execution sequence. 

We have seen examples of schemes which are not free. There are schemes which are not even strongly equivalent to any free scheme. Such a scheme appears in Example III-2. Intuitively speaking, Example III-2 cannot be "freed" because we must use the two tests as a clock, to see how long it takes to find fn(x) with PCf x)) = 0 and then to run through this cycle again and update x to gnfn(x). To justify this statement formally, we need some additional notation and results. 

THEOREM 3.1H Let P be a monadic free scheme with one output variable z. Then the value language L(P) Is regular. 

This result allows us to conclude that the scheme P in Example III-2 is not strongly equivalent to any free scheme since L(P) = gnfnx n > 1 is not regular. 

We can extend this result a little. If we examine Example III-4, we see that we have altered P(A,B) to form P(A,B) by changing subschema (n+1). Since f and g are assumed to be distinct from the function letters a and b and from each other, if we enter (n+1) with different values for u and v the subscheme will not violate the definition of freeness. But if ws enter (n+1) with val(u) = val(v), then the subscheme behaves like the scheme in Example III-2, which we have seen is not strongly equivalent to any free scheme. Hence, similar arguments will show that now P(A,B) is strongly equivalent to some free scheme if and only if the Correspondence Problem for (A,B) has no solution. 

THEOREM 3.17 It is undecidable whether there is a free scheme which is strongly equivalent to a given scheme. 

We stall stow that every liberal scheme is strongly equivalent to a free scheme. The converse is not true. Example III-5 contains a free scheme P which is not strongly equivalent to any liberal scheme. Under free interpretation I, if 2n+l = Min k > 3 k odd, I(T)(f (x)) - 0, then (P,I,x) halts with output (Ax), f2n+1(x)). To compute f n+1(x) and test it, we must first compute fn(x), and store it until f2n+l(x> is tested the time lag n+1 becomes arbitrarily large so either we need arbitrarily many registers - violating the definition of a program scheme - or else we must recompute fn(x) violating the definition of a liberal scheme. 

We demonstrate the fact that every liberal scheme is strongly equivalent to a free scheme by describing a transformation t on schemes such that ... 

A FREE SCHEME WHICH IS NOT STRONGLY EQUIVALENT TO ANY LIBERAL SCHEME... 

The transformation t we saw at the end of the last section, which changes liberal schemes into free schemes, is such a canonical transformation. The corresponding canonical class of schemes is the class of schemes such that tests are applied initially on the input variables and are applied after assignment statements on the program variables involved, and at no other time. This transformation t is clearly recursive and equivalence preserving. The class of free schemes is not a canonical form class, since, as we saw, there are schemes not strongly equivalent to any free scheme. 

Call a node in a free scheme "live" if there is a path from it to a STOP statement since the scheme is free this path is an execution sequence. Call a node "dead" if it is not live - if there is no path from it to STOP. We have mentioned previously that it is decidable whether there is a path between two nodes in a finite state graph (although it is rot decidable whether there is an execution sequence between two nodes in a scheme that is not free). Hence it is decidable... 

Generally, fourfold and fivefold coordination models are used to describe metalloporphyrins. In the fourfold coordination model, the metal atom is coordinated to the porphyrin nitrogen atoms and the lower coordination site remains free (Scheme lb). The upper coordination site can later be occupied by ligands such as H20, CO, 02, and NO. 

Another alternative to occupancy theory is rate theory. Rate theory was developed by Paton through examination of receptors that bind stimulants.30 Paton proposed that a response is caused by the act of binding, not the state of being bound or free (Scheme 5.8). This seemingly subtle difference shifts the theory away from KD and toward kon and fcoff, the rate constants of association and dissociation. Interestingly, at equilibrium, KD is equal to koa/kon (Equations 5.19-5.21). For this reason, occupancy and rate theory are closely related. 

Model-free scheme the two-loop scheme (5.45)-(5.47) is adopted in conjunction... 

The indicator-free scheme has been characterized using 23-mer oligonucleotides as a model parameters affecting the hybridization assay such as probe immobilization conditions, hybridization time, use of hybridization... 

Thus, the ratio of the force measured by impedance head to the acceleration of outer mount in end-free scheme must remain constant up to the frequency at which wave effect become significant. 

Solvent-free (tetramethylcyclopropyl)methyllithium (40) also forms the hexameric aggregate (39), similar to hexameric cyclohexyllithium. Both hexamers are characterized as a trigonal antiprism with triangular Li faces. The (tetramethylcyclopropyl)methyllithium hexamer (40) was prepared in diethyl ether solution from both the Cl and the Hg compounds (41) as well as from Ae open chain compound (42). In contrast to the cyclohexyllithium hexamer, the hexamer (40) is obtained solvent free (Scheme S). 

In order to eliminate the spurious xc-tail, a self-consistent scheme for the generation of norm-conserving PP has been developed (Engel et al. 2001c). In this parameter-free scheme the screened PP is iterated until its asymptotic structure matches that of... 

Johnson and Dolphini " also reported the preparation of octaethylbenzochlorin 87 via an intramolecular cychzation of the meso-acrylaldehyde-substituted porphyrin 86 under the acidic conditions. The outcome of the ring-forming reaction was dependent on whether the intermediate was metalated or metal free (Scheme 24). 

Several schemes are possible for practical calculations of the main EUE indices. Frequently, all the elements of D are needed, e.g. for computing by Eq. (6.9) and A eff by Eq. (6.15). In a number of cases we can simply exploit the explicit expressions, as in the case of the rather easily performed UHF-like models. When sophisticated multiconfigurational models are used, it is necessary to employ the technique which is elaborated for obtaining D within the restricted active space Cl (RAS-CI) and coupled cluster schemes [39, 42]. However, the direct way is too demanding when large-scale systems need to be addressed. Sometimes, one can employ the RDM-free scheme from [52] that avoids the tedious computations of all matrix elements of D. This scheme (see Eqs. (107) and (111) in [52]) can be applied to the hole-particle quantification scheme described above. The respective technique is based on reverting the obvious relationship which connects expectation values to RDM. Namely, for the given spinless one-particle operator Z we have... 

This section demonstrates how the first three unitary matrices are explicitly constructed and applied to the one-electron operator / (or to some of its parts such as + V uc)- The first transformation has necessarily to be the free-particle Foldy-Wouthuysen transformation Uo, which is followed by the transformation Ui. The third transformation U2 turns out to produce even operators that depend on the parametrization chosen for Uz- Afterwards the infinite-order, coefficient-dependence-free scheme is discussed. 

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