Gaussian density distribution

We therefore conclude that the form of 7 -i hypotiiesizedin Eq. (C.18) implies Ekj. (C.24) for Moreover, the case n — 2, proved explicitly above, is nothing but a special case of / as one realizes by comparing Eqs. (C.16) and (C.24) so that Eq. (C.24) completes our proof. 

Finally, we stress that the decomposition of I according to Eq. (C.24) is based on the explicit assumption that the integrand / depends only on distances Zij and not on the individual coordinates Zi and Zj as independent variables. 

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Fig. 77.—Gaussian density distribution of the chain displacement vec tors for chain molecules consisting of 10 freely jointed segments, each of length 1 = 2.5 A. The end-to-end length r is in Angstrom units and W(x, y, z) is expressed in A". ...

Most synthetic polymers in which the monomer units are connected via single bonds have rather flexible chains. The bond torsion energy is relatively small and the units can rotate around their bonds [14,30,31]. Each molecule can adopt a large number of energetically equivalent conformations and the resulting molecular geometry is that of a statistical coil, approximately described by a Gaussian density distribution. This coil conformation is the characteristic secondary structure of macromolecules in solution and in the melt. It is entropically favoured because of its... 

C. Tsoo, C.L. Brooks, Cluster structure determination using Gaussian density distribution global minimization methods, J. Chem. Phys. 101 (1994), 6405... 

Figure 14-10 DoD plot for comparison of two drug assays parametric analysis. A histogram shows the relative frequency of N = 65 differences with the estimated Gaussian density distribution. Parametrically estimated 2.5, and 97.5 percentiles are shown with 90% CIs.

The curves are computed under the assumption that the logarithms of the endurable number of stress reversals demonstrate a spread pattern according to a Gaussian density distribution. The failure rate determined from this shows a tendency to increase along with an increasing number of stress cycles. From a physio-technical point of view, this is understandable, since increasing stress will mean increased wear. 

It has already been established that the results of interior cyclic pressure tests [5-14]-[5-20] demonstrate a spread pattern around the curves of mean endurable dynamic load which are similar to a Gaussian density distribution (Figure 5.17). Furthermore, it has been established that the dispersion of the logarithms of the stress cycles is practically independent of the magnitudes of the amplitudes, i.e., independent of KG. Under these circumstances. Equation (5-27) permits the computation of the failure probability for any combination of stress cycles Nq and N. The result of these computations, together with the stress cycle resistance, has been demonstrated in Figure 5.20,... 

In most electron-diffraction studies both the electron beam and the molecular beam are assumed to have vanishing cross-sections. This approximation is probably satisfactory for the electron beam. The effect of a non-zero cross-section for the molecular beam varies with the density distribution and with the nozzle-to-plate distance. - Rundgren has found that for a Gaussian density distribution with standard deviation, am, the observed values for will be too large by approximately... 

For a Gaussian density distribution with transverse r.m.s. radii ax, ay we have A = AKaxay. If we introduce the beam currents h = niefb (e being the magnitude of the electron charge) we finally have... 

Assuming Gaussian density distributions, we obtain in the limit of a complete overlap, rc,i = Tc,2, an expression identical to the internal excluded volume interaction energy as given by Eq. (2.91), apart from a factor 1/2. Omitting the numerical prefactor of order unity we can write... 

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