Borderline dimensionality

The borderline dimensionality, d. is the hi iest at whidi the d-dependent exponent relations, such as (9.58), (9.59), and (9.61), hold, and the lowest at whidi the exponents have thdr dassical values. Thus, d can always be found as the solution d of the algebraic equation that results from setting the exponents in any of ffie d-dependent exponent relations equal to their dassical values. For example, setting v= and a = 0 in (9.58), or t =0 and 5 = 3 in (9.59), or v= and i. = in (9.61), yidds d = 4. Since t = 0 classically, it follows from (9.59) that the borderline dimensionality d is related to tiie dassical value 5 of the critical-isotherm exponent 5 by... 

In addition to the above chemical reactivity and dimensional stability problems, Minol 11 exhibits borderline initiatability at —65°F with small boosters, but initiates reliably at low temps with standard larger boosters (Ref 43)... 

Fig. 44.22. Three commonly used Kohonen network structures, (a) One-dimensional array (b) two-dimensional rectangular network (each unit, apart from the borderline units has 8 neighbours) and (c) two-dimensional hexagonal network (each unit, apart from the borderline units, has 6 neighbours). (Reprinted with permission from Ref. [70]).

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

Trull, T. J., Widiger, T. A., Guthrie, P. (1990). Categorical versus dimensional status of borderline personality disorder. Journal of Abnormal Psychology, 99, 40-48. 

Compounds of type 13 are species at the borderline between classical aromatics and three-dimensional rx-aromatic clusters (see also Section 3.2.2.4). 

Two-dimensional (2D) More O Ferrall reaction coordinate diagrams (Fig. 2.3) are helpful in rationalizing the differences in the borderline regions for nucleophilic substitution at X-l-Y and X-2-Y that are illustrated by the data in Figure 2.2. These... 

The last of these is perhaps the best criterion for metallic characteristics. Most other types of inorganic substance are effectively insulators, or semiconductors whose conductivity increases with temperature. Semiconductors constitute a borderline area some lie between metals and three-dimensional polymers (e.g. GaAs, isoelectronic and isostructural with elemental germanium), while others have structures which could be described as ionic (e.g. PbS, which has the NaCl structure). 

Another troublesome borderline area is that between ionic solids and three-dimensional polymers. The distinction cannot be made from the structure alone. Electrical conductivity in the molten state does not, as already mentioned, necessarily demonstrate the presence of ions in the solid state and such a test is inapplicable where, as often happens, the substance sublimes or decomposes before melting. There can rarely be any objective means of assigning a compound to one category or the other. We are often persuaded towards one description on aesthetic grounds. For example, the structure of sodium chloride cannot easily be rendered in terms of localised, electron-pair bonds (but this is true also of many unequivocally covalent compounds). Its structure is eminently plausible for an array of cations and anions, however. 

The chemistry of a finely divided solid with high surface area is at the borderline between molecular chemistry and physics of solid surfaces. The surface of a finely divided amorphous silica may be understood as a two-dimensional projection of the three-dimensionally linked silicon dioxide tetrahedra. By this the chemistry of the fumed silica surface is dominated by Si-O-Si units and, in particular, dangling surface Si-0 bonds, which will create silanol groups under the influence of humidity at ambient temperature. 

The mechanistic reinterpretation of the data that had been considered to support an 8 1 mechanism developed from a generalized interpretation of how minimal-energy reaction pathways are determined. Jencks (1985) has demonstrated the importance of mechanisms involving borderline transition states, where the borderline divides concerted and stepwise mechanisms. These occur where necessary intermediates may be too unstable to exist. In terms of the three-dimensional energy diagram, these mechanisms involve reaction paths that approach the corner occupied by the intermediate but do not reach to the corner. Herschlag and Jencks (1986) analyse the kinetic evidence that has been presented in support of a metaphosphate intermediate in a variety of reactions in solution. They conclude that none establishes the existence of metaphosphate ion as an intermediate but that all are expectedly consistent with an unsymmetrically extended transition state in a concerted mechanism (2). 

Fig. 3.12. Discriminant partial-least-squares score plot for the global model of blood-brain barrier (BBB) permeability prediction. The model offers a good discrimination and evaluation of a compound s ability to cross the BBB good permeability, BBB+ (black circles) poor permeability, BBB- (open circles). The model assigned a correct BBB profile to more than 90% of the compounds. A confidence interval is built in the t1-t2 space, where BBB prediction can be borderline and doubtful. (From Crivori P, Crucian G, Carrupt PA, et al. Predicting blood-brain barrier permeation from three-dimensional molecular structure. J Med Chem 2000 43 2204-2216 with permission.)...

At the borderline between the two-dimensional monocyclic polyethers and three-dimensional cryptands is a relatively new type of host ligand, the lariat ether (73,74). 

Fig. 29. Phase diagram a( 7 =0 of a two-dimensional triangular lattice using an effective Hamiltanian mapped into the pseudo spin =1 Blume-Capel Hamiltonian (Ballou et al. 1991). Open circles indicate nonmagnetic sites. The borderlines between the different phases are determined by the following phases I and III, Ji=A-Ji, phases I and II, J =Jphases II and IB, J2=A/9 phases II and IV, y, and are nearest...

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