Counter-example

While the Smoliichowski equation is necessary for a Markov process, in general it is not sufficient, but known counter-examples are always non-Gaiissian as well. 

However, this is obviously not always the case, a counter example being the transformation of P-tetralone (23) to (S)—anunotetralin (24), with 2-amino butane serving as the amine donor. The of an (S)-selective aminotransaminase leads to a value of only... 

In principle, the diene can react with dienophiles at either of its faces. Anti 7t-facial selectivity with respect to the substituent at 5-positions was straightforwardly predicted on the basis of the repulsive interaction between the substituent and a dienophile, however, there were some counter examples. The first of them is the syn tr-facial selectivity observed in the reaction between 5-acetoxy-l,3-cyclopentadiene 1 and ethylene reported by Woodward and coworkers in 1955... 

Anti TT-facial selectivity with respect to the sterically demanded substituent in the Diels-Alder reactions of dienes having unsymmetrical tt-plane has been straightforwardly explained and predicted on the basis of the repulsive interaction between the substituent and a dienophile. However, there have been many counter examples, which have prompted many chemists to develop new theories on the origin of 7t-facial selectivity. We have reviewed some theories in this chapter. Most of them successfully explained the stereochemical feature of particular reactions. We believe that the orbital theory will give us a powerful way of understanding and designing of organic reactions. 

Similar accidents have happened when barium perchlorate is added to to C3 alcohols as well as 1 -octanol. The latter alcohol demonstrates the nature of this danger and is a counter-example of the typical observation of stabilisation of unstable species, when the number of carbon atoms increases (whereas it is the case for peroxides and peracids). 

We offer these two counter examples not to castigate the method, for indeed the Epp-Fowler algorithm has worked well for a wide variety of networks ranging up to 307 edges, 170 vertices, and 135 cycles (E2), but as an illustration of the difficulties of constructing a completely reliable algorithm that will work for all cases. 

The net effect of introducing the inertia multiplier is to increase the time step by a factor equal to a. The crucial question is how large a value can a be without substantially distorting the true physical behavior of the system. Yow (Y3) and Wylie et al. (W13) developed correlations for this purpose, but the procedures are apparently somewhat ambiguous and unreliable for pipeline networks. Rachford and Dupont (Rl) gave a counter example for which an apparently reasonable choice of a( = 5) miss the pressure prediction by as much as 100 psi. 

A power law h(co) other functions with that property might be invented. The tan S criterion, therefore, might be not unique. However, no other material has yet been found which also obeys tan 5 = constant in the terminal frequency region, and we suggest the continued use of the tan 8 method for detecting LST until a counter example can be found. 

Formally, this procedure is correct only for spectra that are linear in the frequency, that is, spectra whose line positions are caused by the Zeeman interaction only, and whose linewidths are caused by a distribution in the Zeeman interaction (in g-values) only. Such spectra do exist low-spin heme spectra (e.g., cytochrome c cf. Figure 5.4F) fall in this category. But there are many more spectra that also carry contributions from field-independent interactions such as hyperfine splittings. Our frequency-renormalization procedure will still be applicable, as long as two spectra do not differ too much in frequency. In practice, this means that they should at least be taken at frequencies in the same band. For a counter-example, in Figure 5.6 we plotted the X-band and Q-band spectra of cobalamin (dominated by hyperfine interactions) normalized to a single frequency. To construct difference spectra from these two arrays obviously will generate nonsensical results. 

It turns out that this analysis applies only to systems with a bounded phase space. It is possible that topological restrictions on the accessible phase space - and not only the form of the particular Hamiltonian -play a crucial role in determining when the weak form of the QCT actually applies. For example, this might explain why the open-system quantum delta-kicked rotor is a counter-example to naive expectations regarding the QCT (S. Habib et.al., 2002). 

Incidentally, it may be emphasized that (i) the MYKO-XYL structure constitutes a challenge to group theory in that it has no ternary symmetry and that (ii) the structure of MYKO-CgHg exhibits a tricky counter-example to Gillespie s VSEPR model, the lone pairs of the two exocyclic N atoms on each P being strictly parallel.. . ... 

These upper and lower bounds for the broadened response I(w0 > 0 are the most precise possible, on the basis of the known moments.23 We can show this by counter-example, by constructing distributions having the correct moments and actually attaining the upper or lower bounds to i((t>0,t). Consider, for example, the fraction CM + I(z, Tmax) in Eq. (21), where Tmax is the value of t corresponding to the point on the circle for... 

Hydrate formation is a substantial problem in deepwater production and flowlines. Pipelines that transport condensed hydrocarbon phases such as gas condensate or crude oil have limited possibilities for removing hydrates once the plugs have formed. Earlier work by Scauzillo (1956), indicating that formation may be inhibited by the input of hydrocarbon liquids, cannot be confirmed by thermodynamic calculations, and Skovborg (1993) has found counter-examples. Thus, the construction of large-scale pilot flow loops have been completed by large corporations such as ExxonMobil (Reed et al., 1994), Tulsa University, and Institut Francais du Petrole. Such experiments are discussed briefly in Chapter 6. 

As we have discussed in the outer approximation of v(y), Duran and Grossmann (1986a) made the assumption that y Y instead of y Y n V and introduced the integer cut constraints which they claimed make their assumption valid. Fletcher and Leyffer (1994), however, presented a counter example which can be stated as follows ... 

It has turned out that the computation of metal nuclear shieldings and chemical shifts is much more difficult than the calculation of ligand shifts which were discussed previously. It appears that metal shieldings are more sensitive to the quality of the computed electronic structure and consequently larger influences due to the XC potential are observed. Regarding the 3d metals, it has turned out that hybrid functionals appear to be particularly well suited for NMR computations of the metal shielding constants. This cannot be easily extrapolated to all of the transition-metals, though, since counter examples are known for which nonhybrid functionals perform better. On the other hand, some 4d metals have been treated most successfully with hybrid functionals. 

Phosphines are classical Lewis bases or ligands in transition-metal complexes, but the cationic species shown in Fig. 15.4.l(i) are likely to exhibit Lewis acidity by virtue of the positive charge. Despite their electron-rich nature, an extensive coordination chemistry has been developed for Lewis acidic phosphorus. For example, the compound shown below has a coordi-natively unsaturated Ga(I) ligand bonded to a phosphenium cation it can be considered as a counter-example of the traditional coordinate bond since the metal center (Ga) behaves as a Lewis donor (ligand) and the non-metal center (P) behaves as a Lewis acceptor. 

Remarkably enough, the counter-example eq. (2.49) is well known in the DFT context, and it brought the author [106,113] to the conclusion that the theory employing... 

However, it would be a serious error to confuse electronegativity or ionization energy with hardness . The clearest counter-example is Tl(III) which is a rather soft central atom but which must have a rather high ionization energy. There are other physical properties which accentuate the opposite inequality signs of (1) even more, for instance the electric polarizability. Table 1 contains many values for this quantity a in the unit 10-24 cm3. Gaseous atoms and positive ions have polarizabilities which can be calculated from wavefunctions (9) by evaluating the sum of matrix elements of induced dipole moment ... 

Output sequence encoding methods, expressed by n(CODEs), where n is the number of output units, CODES are Y = Yes (positive) N = No (negative) I = Intron E = Exon O = Other (counter-example) RTL = relative transcription level. 

Granjeon and Tarroux (1995) studied the compositional constraints in introns and exons by using a three-layer network, a binary sequence representation, and three output units to train for intron, exon, and counter-example separately. They found that an efficient learning required a hidden layer, and demonstrated that neural network can detect introns if the counter-examples are preferentially random sequences, and can detect exons if the counter-examples are defined using the probabilities of the second-order Markov chains computed in junk DNA sequences. 

From the fact that T > 0 and that for a cyclic process JdQ/T < 0, can one not also immediately deduce that JdQ < 0 for the same process Discuss this question in detail. (A single counter-example suffices to establish that the proposition is false. Otherwise, a general proof is needed to show the statement to be correct.)... 

Characteristic examples of the use of IR spectra are structural studies on urazoles and triazolinones (66T(S7)213) and triazolinethiones (7UCS(C)ioi6). Tautomeric problems of oxo-hydroxy and thione-thiol systems are simplified by JV-alkylation (73CPBI342). Although earlier views in favour of 0x0 and thio rather than hydroxy and thiol forms have been confirmed, counter-examples have been found 3-hydroxy-2-methyl-5-phenyl-1,2,4-triazole (25) is preferred to the 0x0 tautomer (26) while the isomeric 4-methyl compound (27) follows the general rule (Scheme 9). 

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