Correlated distances

From the dynamic mechanical spectroscopy, an increase of PTMO molecular weight from 650 to 2000 results in a decrease in both the modulus and the glass transition temperature of the final product. The SAXS results indicate that a correlation distance exists in the samples, and this distance increases as PTMO molecular weight increases. A cluster model is thus suggested to account for the experimental results. 

The General Series Expansion for Stacks. In practice, ID scattering intensities can always be modeled by programming the obvious series expansion of such a structural entity every correlated distance along the stack axis is producing an attenuated oscillation according to Hk (5) that is weighted by the probability of its occurrence under consideration of the zero-sum rule and the related correlation-construction principle (p. 158, Fig. 8.30)... 

In Eq. (13) Mr is the molecular weight of the repeating units from which the polymer chain is composed. Finally, when one combines Eqs. (10) through (13), the correlation distance between two adjacent crosslinks in a swollen hydrogel can be obtained ... 

As a consequence, in a sensor space of dimension N the effective dimension of the sub-space occupied by the data is less than N. This dimension can be precisely evaluated using algorithms developed to describe dynamic systems. An example is the correlation distance that allows evaluating the fractional dimensionality of a data-set [16]. Correlation distance provides an independent way to evaluate the expected reduction of dimension. 

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... 

The specific interfacial surface area, S, defined as the ratio of interfacial surface area. A, to thl volume, V, is calculated from the value of correlation distance obtained by plotting the scattered intensity in the Debye fashion [ 12,38]. 

A) A clear scattering maximum In 24% PBLG/DMF/H2O (trace). The Bragg spacing Is 6ym. B). Photomicrograph of same sample as A between crossed polars. The correlation distance appears to be about 9vtm. 

Objects can be internally correlated in time or space. For mixed objects such as tanks (and the products that have been subjected to storage for longer or shorter times), rivers, lakes, gases and environmental air this correlation can be rendered by auto-correlagrams and modelled by a simple negative exponential function of the correlation distance. 

Reflectance and hydrogen content are the best parameters for correlating distance from an igneous contact within a coal bed. 

Fig. 2. Average correlation distances d between deuterated crosslinks of PS/DVB networks swollen to equilibrium vs gi/3...

As can be expected, the angular scattering intensity distribution has become anisotropic. From the data obtained, two values of the average correlation distance between neighboring nodules can be calculated, parallel and perpendicular to the deformation direction respectively. Although the accuracy of the data is not outstanding, it can be seen that d increases with the macroscopic extension ratio, Ax, whereas dx decreases only slightly (Fig. 10). It is not possible to ascertain experimentally whether dL is proportional to AJ1/2 as it should be if the deformation proceeded at constant volume L e., if the network could be considered incompressible. 

The most important result obtained on the structure of model networks is the existence of a rather well-defined correlation distance between first neighbor crosslinks. The affine character of deformations induced by swelling processes is clearly shown by the fact that, for a given gel, the correlation distance d increases as the cubic root of the volume swelling degree Q up to values of Q of the order of 10. 

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... 

A corresponding inverse triangle inequality can be applied to each triplet to raise values in the lower bound matrix L. Now, a distance matrix D, usually referred to as the trial distance matrix, can be constructed by simply choosing elements dt/ randomly between w/ and lif and used to construct a metric matrix G. A matrix so constructed might be some approximation to the distances in the real molecule, but probably not a very good one. Clearly, every time an element d is selected, it puts limits on subsequent selected distances. This problem of correlated distances is discussed further in the section Systematic Errors and Bias. 

Analysis of these SAXS profiles suggests that this correlation distance (ca. 1/5, an indicator of pseudo-Bragg spacing Figure 6) or average periodicity in electron density fiuctuations is in the order of about 10 nm. This length is not unreasonable when the general chain or coil dimensions of the... 

Figure 7. Proposed model of the local morphological structure in the PTMO-TEOS hybrid systems, in which 1/s is an estimate of the correlation distance, (Reproduced from reference 7. Copyright 1987 American Chemical...

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