Pearson distance

Figure 14.12 Notched Izod impact strength data (on crystallized PET) for samples of toughened polymer as a function of the ratio of interparticle distance O, amorphous x, crystalline [28]. Reprinted with permission from Pecorini, T. J. and Calvert, D., in Toughening of Plastics - Advances in Modelling and Experiments, Pearson, R. A., Sue, H.-J. and Yee, A. F. (Eds), ACS Symposium Series, 759, American Chemical Society, Washington, DC, 2000, Ch. 9, pp. 141-158. Copyright (2000) American Chemical Society...

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

WoWj/2 the body-centred cubic structure of W (1 atom in 0, 0, 0 and 1 atom in A, A, /) corresponds to a sequence of type 1 and type 4 square nets at the heights 0 and A, respectively. Note, however, that for a fall description of the structure, either in the hexagonal or the tetragonal case, the inter-layer distance must be taken into account not only in terms of the fractional coordinates (that is, the c/a axial ratio must be considered). For more complex polygonal nets, their symbolic representation and use in the description, for instance, of the Frank-Kasper phases, see Frank and Kasper (1958) and Pearson (1972). 

We underline these results and the implied concepts quoting from a comprehensive review on this subject (Simon 1983). We remember indeed that, ever since it was experimentally possible to determine atomic distances in molecules and crystals, efforts have been made to draw conclusions about the nature of the chemical bonding, and to compare interatomic distances (dimensions) in the compounds with those in the chemical elements. Distances between atoms in an element can be measured with high precision. As such, however, they cannot be simply used in predicting interatomic distances in the compounds. In a rational procedure, reference values (atomic radii) have to be extracted from the individual (interatomic distances) measured values. Various functions have been suggested for this purpose. In the specific case of the metals it has been pointed out that interatomic distances depend primarily on the number of ligands and on the number of valence electrons of the atoms (Pearson 1972). 

Most of the standard clustering algorithms can be directly used for clustering the variables. In this case, the distance between the variables rather than between the objects has to be measured. A popular choice is the Pearson correlation distance, defined for two variables xj and xk as... 

One reason for the failure of the radius ratio rules is that ions do not behave like hard spheres. Even those that are hard in the Pearson (1973) sense can still be compressed. This is clearly seen in the way the bond length varies with the bond valence. If cation anion bonds can be compressed, so can the distance between the 0 ions in the first coordination sphere. The stronger the cation anion bonds, therefore, the closer the anions in the first coordination sphere can be pulled together (Shannon el al. 1975). 

Some years ago in a continuing effort to understand phase diagrams, I had discovered [3] the following empirical rules among more than 300 binary phase diagrams reported in the literature (Hansen, Elliot Shunk) [4,5,6], The metallic radii, Ra, Rb, used are from INTERATOMIC DISTANCES (The Chemical Society, London, 1958) [7] and the structural notation follows that described in Handbook of Lattice Spacing and Structure of Metals (Pearson, 1958) [8]).. 

The Spearman s Rank Correlation is the non-parametric alternative to the Pearson s and Cosine Correlation Distances. No assumption is made concerning the data distribution or the center of the data... 

Thus the partial ionic model of Pearson and Gray (13), and very similar models of greater sophistication (14, 15) show that a very major consideration in causing deviations from the ionic model is the size of the ionization potential of the exposed orbitals of the cations, as compared with the ionic potential of a negative charge placed distance re from the... 

The simplest model consists of two centres, one donor (D) and one acceptor (A), separated by a distance I and contains two electrons. Here we consider this simple system to illustrate some general relations between charge transfer, transition intensities and linear as well as non-linear optical polarizabilities. We will show below that the electro-optic parameters and the molecular polarizabilities may be described in terms of a single parameter, c, that is a measure of the extent of coupling between donor and acceptor. Conceptually, this approach is related to early computations on the behaviour of inorganic intervalence complexes (Robin and Day, 1967 Denning, 1995), Mulliken s model for molecular CT complexes (Mulliken and Pearson, 1969) and a two-form/two-state analysis of push-pull molecules (Blanchard-Desce and Barzoukas, 1998). 

Releases of 1,1-dichloroethane to the environment as a result of industrial activity are expected to the primarily to the atmosphere (see Section 5.2). 1,1-Dichloroethane released to the atmosphere may be transported long distances before being washed out in precipitation. For example, Pearson... 

Fig. 15.15 Complexes containing bent nilrosyl groups (a) (lr(PPh3)2(COi(NO)Cl] anti (b) [Ru(PPh3)2(NO)2CIJ. Phenyl groups have been omitted for clarity distances are in picometers. [Structure (a) from Hodgson, D. J. Payne, N. C. McGinnety, J. A. Pearson. R. G. Ibers, J. A. J, Am. Chem. Soc. 1968, 90. 4 86-4488. Structure (b) from Pierpont. C. G. Eisenberg, R. hiorg. Chem. 1972, II, 1088-1094, Reproduced with permission. ...

In Eq. 2.61 a is a free variable, i.e. the asymmetry parameter, which is refined during profile fitting and z,- is the distance fi om the maximum of the symmetric peak to the corresponding point of the peak profile, i.e. z,-= 20yfc - 20 . This modification is applied separately to every individual Bragg peak, including Kaj and Ka2 components. Since Eq. 2.61 is a simple intensity multiplier, it may be easily incorporated into any of the peak shape functions considered above. Additionally, in the case of the Pearson-VII function, asymmetry may be treated differently. It works nearly identical to Eq. 2.61 and all variables have the same meaning as in this equation but the expression itself is different ... 

The HC algorithm requires choice of a distance measure and linkage method. The distance measure quantifies the similarity or dissimilarity between two gene expression profiles. The Euclidean distance and the Pearson correlation (PC) coefficient have been widely used as distance measures to quantify the similarity between profiles. The centered PC similarity measure, r, between any two series of numbers X = [Xt, X2,..., X and Y = Yi, Y2,..., Y is the familiar PC coefficient used in linear regression. The distance measure is obtained by subtracting the correlation value from unity. The uncentered PC is obtained from the centered PC by setting the means of X and Y to zero. The uncentered PC is defined as... 

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