R-deficiency measure

Specifically, the internal R-deficiency measure Srji(A), defined as... 

If applied to functional groups, then the internal R-deficiency measure 6r,b(A) refers to electron density contributions actually present within the molecule, whereas the external R-deficiency measure <5r,c(A) may involve fuzzy sets that do not necessarily occur within the given molecule. 

Taking the average of the two measures SRyB(A) and SRyC(A) defines the following R-deficiency measure ... 

Whereas both the external f -deficiency measure SRyc(A) and the R-deficiency measure ARyB(A) are mathematically correct and valid tools for comparisons of the degrees of symmetry deficiencies and their correlations with unusual reactivities of functional groups, the actual physical existence of fuzzy sets involved in the determination of the internal -deficiency measure SRyB(A) nevertheless suggests that this measure is likely to provide more reliable correlations with variations in the chemical reactivity of functional groups. 

The fuzzy Hausdorff-type similarity measures can be used for a direct comparison of a fuzzy set A and its various R sets and / -deficient sets, providing alternative fuzzy symmetry deficiency measures. 

Alternative fuzzy symmetry deficiency measures are defined in terms of a maximal R subset B, maximal mass R subset B, minimal R superset C, and minimal mass R superset C, of fuzzy set A, discussed in Section IV. If fuzzy set D denotes any one of the R subsets B,B or R supersets C,C of fuzzy set A, D G B, B, C,C, then a fuzzy symmetry deficiency measure... 

The dissimilarity of fuzzy sets A and A(R,c) provides a measure of the symmetry aspect R for set A with respect to center c. A large measure of dissimilarity implies a higher degree of symmetry deficiency of fuzzy set A, with respect to symmetry represented by element R. This symmetry deficiency can be described using either one of the fuzzy set dissimilarity metrics. For example, if the fuzzy metric FSNDSM df/A,B) is used, then one obtains the fuzzy symmetry deficiency measure jf JA) ... 

Alternative fuzzy symmetry deficiency measures (. ) of fuzzy set A with respect to reference point c and symmetry represented by element R are defined as... 

Using any one of the versions of the fuzzy Hausdorff-type metrics for the dissimilarity of sets A and An f/fp, for example, the commitment weighted fuzzy Hausdorff-type dissimilarity metric f(A,B), one obtains another generalization of the ZPA continuous symmetry measure of discrete point sets to crisp or fuzzy sets. The corresponding symmetry deficiency measure f(A,A(f ( p p) provides a measure for the symmetry aspect R for crisp or fuzzy set A, with reference to the given positioning P of R with respect to A and to the choice of the associated partitioning of A. 

In discussing the elFect of structure on the stabilization of alkyl cations on the basis of the carbonylation-decarbonylation equilibrium constants, it is assumed that—to a first approximation—the stabilization of the alkyloxocarbonium ions does not depend on the structure of the alkyl group. The stabilization of the positive charge in the alkyloxocarbonium ion is mainly due to the resonance RC = 0 <-> RC = 0+, and the elFect of R on this stabilization is only of minor importance. It has been shown by Brouwer (1968a) that even in the case of (tertiary) alkylcarbonium ions, which would be much more sensitive to variation of R attached to the electron-deficient centre, the stabilization is practically independent of the structure of the alkyl groups. Another argument is found in the fact that the equilibrium concentrations of isomeric alkyloxocarbonium ions differ by at most a factor of 2-3 from each other (Section III). Therefore, the value of K provides a quantitative measure of the stabilization of an alkyl cation. In the case of R = t-adamantyl this equilibrium constant is 30 times larger than when R = t-butyl or t-pentyl, which means that the non-planar t-adamantyl ion is RT In 30= 2-1 kcal... 

Equation (6.87) is the basis for most adsorption measurements associated with liquid solutions. When(d(j/dfi )T is negative, Tg is positive and there is an excess of the solute at the interface. For (dct/dfiB)r > 0, Tg is negative and there is a deficiency of the solute at the interface. In other words, solutes that reduce the surface tension are enriched at the surface. 

The R CrB variables and related hydrogen-deficient carbon stars show strong carbon features (except CH) and very weak Balmer lines. The R CrB stars are surrounded by circumstellar dust shells, and continue to eject puffs of new circumstellar material on time scales of a month or two mass loss rates of order 10- Mo yr- are reported by Walker 1986. The compositions of the R CrB stars and hydrogen-deficient carbon stars are were compiled by Lambert (1986). Hydrogen is extremely deficient in both groups (H/He 10 J-10 °), and C/Fe is enriched by typically an order of magnitude over the solar ratio. C/0 is typically 2 (more than is measured in the AGB stars, and [N/Fe] 1. R CrB itself contains a strong lithium... 

However, when H2 is bound to a highly electrophilic cationic metal center, the acidity of H2 gas can be increased spectacularly, up to 40 orders of magnitude. The pK of H2 can become as low as —6 and thus the acidity of rf -H2 becomes as strong as that in sulfuric or triflic acid. As discussed in reviews by Morris (4,5) and Jia (25) and further work by Morris (26,27), such pK values are usually determined by NMR measurement of the concentrations of M H2 complexes in equilibrium with an external base such as a phosphine or amine. Electron deficient cationic and dicationic H2 complexes with strong short H-H bonds (<0.9 A) and weakly bound H2 such as [Cp Re(H2)(CO)(NO)]+ and [Re(H2)(CO)4(PR3)]+ are among the most acidic complexes (Table I). These acidic complexes typically have relatively high values of Jhd for their r 2-HD isotopomers, although pK values do not correlate... 

Elements are formed by three different mechanisms in stars, as elucidated by Burbidge et al. [82]. In brief, these are the s process, involving the capture of slow neutrons by nuclei the r process, involving the capture of rapid neutrons and the p process, which leads to neutron-deficient nuclides that are generally less abundant than those generated by the s and r processes. The relative abundances of the various isotopes of a given element reveal which processes, or combinations thereof, are involved in their creation in the nuclear reactions that power stars. Measurements of isotopic and elemental abundances in stars thus serve to test theories of stellar processes. 

As previously noted, in a typical process analytical application, the measured data set might consist of spectral data recorded at a number of wavelengths much higher than the number of samples. The rank, R, of the measured matrix of spectra will be equal to or smaller than the number of the samples N. This causes rank deficiency in X, and the direct calculation of a regression or calibration model by use of the matrix inverse using Equation 8.85 and Equation 8.86 is problematic. 

Meinema et al. 363,3665 have measured H NMR benzene solvent shifts for diorganoan-timony(V) /3-diketonates (R = Me, Et, Ph or 1/2 o, o -Qf QH. The observed shifts have been interpreted in terms of a non-specific tengential approach of the benzene molecules to the electron deficient sites of the solute. 

The experiments described here are principally diagnostic in nature where cellular biomass was significantly enhanced in bottles after resource (iron or light) amendment, relative to control (or other) treatments, we infer that algal growth rates in the control (or other) treatments were limited by a deficiency in that resource. The statistical significance of differences between mean values of parameters measured in different treatments were assessed using a two-tailed r-test for comparisons between two treatments, or a one-way analysis of variance (ANOVA) for comparisons between three or more treatments, at a confidence level of 95% (P = 0.05). 

Remondelli P, Mugnoz B, Della Morte R, et al. 1986. Evaluation of the mutagenic potential of thirteen pesticides by measuring both his+ and HGPRT-deficient mutants in Salmonella typhimurium. Med Biol Environ 14 377-386. 

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