Correlation distance

More than half of the members of this family have been structurally characterized and the Au-C distances correlated with the donor ability of the /rchemical shifts of 13C resonances in the NMR spectra. According to these results, the /rcarbene ligand appears to be comparable to that of triphenylphosphite P(OPh)3. Owing to the bulkiness of the lBu substituents, the complexes are not aggregated in the crystals. By contrast, the corresponding methyl-substituted compounds are aggregated to form chains via aurophilic contacts and are photoluminescent.18... 

Euclidean distance City block (Manhattan) distance Minkowski distance Correlation coefficient (cos a), similarity... 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

For the p-type copper oxides that superconduct, both carrier concentration and Cu-O distance correlate well with Tc. If one plots either carrier concentration or Cu-O distance vs. Tc, a maximum in Tc is found. The trend is shown schematically in Figure 2. 

On the other hand, WAXS measurements of PE melt clearly indicate a range of intermolecular distance correlations of about 25 A [3]. Together with the relatively high density of polymer melts, the fact that the first interchain halo in WAXS patterns of oriented amorphous polymers tends to lie in the equatorial direction and the relatively high WAXS intensity of the interchain halo support the idea of parallel chain segments on the short range scale. 

The partition of molecular distance correlations into intra- and intermolecular contributions allows us to interpret these correlations in terms of a simple geometrical model. By this means, we are able to elicit structural units as for example segment-clusters that include intermolecular interference phenomena. These clusters are the primary structure units which we call monodomains . These natural units characterize the basic symmetry of the whole structure. If we keep in mind this basic symmetry, we can construct our structure model from a molecular level up to the level of the monodomain treating intra- and intermolecular correlations independently. If we do so, every X-ray pattern can be represented by accounting for the orientation distribution of these monodomains. 

First we discuss and construct monodisperse two-dimensional arrangements of impenetrable cylinders in terms of radial distance correlation functions, the lateral packing fraction and number density. In the second step, these hard cylinders are covered by the mean electronic density functions of the RISA chain segment ensemble. Last of all, the Fourier transformation and final averaging is... 

The radial distance distribution in simple atomic and molecular fluids is determined essentially by the exclusion volume of the particles. Zemike and Prins [12] have used this fact to construct a one-dimensional fluid model and calculated its radial distance correlation function and its scattering function. The only interaction between the particles is given by their exclusion volume (which is, of course, an exclusion length in the one-dimensional case) making the particles impenetrable. The statistical properties of these one-dimensional fluids are completely determined by their free volume fraction which facilitates the configurational fluctuations. 

Any homogeneously distorted two-dimensional coordination scheme should be based upon the distance correlation statistics between next neighbors at least. In the case of a two-dimensional lattice construction, this distance correlation principle has been used by Hosemann and coworkers [13] to generate micro-paracrystals of finite size with the help of a computer. The construction procedure (known as spiral-paracrystal ) terminates if a coordination point cannot be assigned to a lattice point. 

The construction scheme we present in our work does not require a lattice at all. Due to the fact that the distance correlation between three coordination points is considered, the algorithm generates new coordination points in a unique manner. Any correlation of these coordination points to a lattice is disregarded as the generation procedure runs continuously. 

First of all, a distance correlation function gNN(r) of next neighbors must be given. This function has to meet the following conditions ... 

To obtain the radial distance correlation function one has to take the average of many system configurations. The fluctuation parameter g can be derived from these functions easily. 

The intramolecular distances belonging to fixed C-C-distances along the chain (1.54 A and 2.54 A) cannot be resolved due to the smearing effect of the electronic density. The peaks coming from intermolecular distance correlations can be seen clearly, the range of the appropriate short range order is about 25 A which is comparable with results from other authors [3, 7]. 

In order to fit our structure model to the experimental data, we have to vary the model parameters packing density and cylinder length and examine the effects on the RDDF of our model via the structure factors S and F. One can easily see that an increase of packing density causes the intermolecular distance correlation range to increase because mobility and free volume are reduced. In other words one can calculate a realistic value of packing density (Le. cylinders per volume) from the experimental RDDF (Fig. 18). 

Fig. 18. RDDF of the model with variation of the packing density parameter i)L. The range of distance correlation increases with increasing t]L...

In the case of the PE melt, the largest part of the configuration entropy is included in the variety of chain conformations. In our model of the PE melt, these chain conformations cause the diameter of the correlation cylinders in which the chain segments are embedded to fluctuate. This leads necessarily to distance fluctuations of next neighbor segments because of steric hindrance of the local segment exclusion volume. Therefore, the inter- and intramolecular distance correlations cannot be discussed independently. 

Ichikawa M (1978) The O-H vs O- 0 distance correlation, the geometric isotope effect in OHO bonds, and its application to symmetric bonds. Acta Cryst B 34 2074-2080... 

The manner in which sample-to-sample resemblance is defined is a key difference between the various hierarchical clustering techniques. Sample analyses may be similar to one another in a variety of ways and reflect interest in drawing attention to different underlying processes or properties. The selection of an appropriate measure of similarity is dependent, therefore, on the objectives of the research as set forth in the problem definition. Examples of different similarity measures or coefficients that have been used in compositional studies are average Euclidean distance, correlation, and cosine. Many others that could be applied are discussed in the literature dealing with cluster analysis (15, 18, 19, 36, 37). 

Cooper, P. W.,, 4 New Eook at the Run Distance Correlation and Us Relationship to Other Non-Steady-State Phenomena, Proceedings of the Tenth Sjmposium (International) on Detonation, Boston, 1993. 

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