Distributional equivalence

Fig. 2.3 (a) Scanning electron micrograph of the morphology of as-received Tego Magnan MgH powder and (b) powder particle size distribution (equivalent circle diameter, ECD)... 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

Flexible molecules, such as those in Fig. 2-7, permit rotational motions of one bond about another, so that a combinatorially huge number of configurations is accessible (Flory 1969). On length scales of tens or hundreds of such monomers, the details of the distribution of allowed bond angles average out, producing in the melt a configuration distribution equivalent to that of a random walk (see Fig. 2-8). Because of the flexibility of these molecules, even in the densely packed melt state, they remain unoriented, or isotropic, at equilibrium. 

Hydrite Properties well-crystallized, axial ratio 10 1, narrow particle size distribution, equivalent spherical particle size 550 nm [2263],... 

All environments that harbor life are stressed by a density of unequilibrated free energy. Trapped free energy is necessary to drive biomass away from the Gibbs equilibrium state [54] and exists only if all the processes active in the system are insufficient to create the equilibrium distribution (equivalently, to ergodically sample the state space in which the equilibrium distribution is the most disordered and thus favored by probability.)... 

Monodisperse PS (Mw = 160,000 Mw/Mn = 1.06) and a narrow distribution PBD (Mw = 78,500 Mw/Mn = 1.23) were compared with their broad distribution equivalents at 10% PS and 6% PBD in S. Phase volumes were the same, which indicates that polymer molecular weight distribution is not a significant variable. 

By contrast, the mean speed follows as the mathematical average of the speed distribution, equivalent to the weighted arithmetic mean of all velocities N number of molecules) ... 

Lagrange Equation of the First Kind With this equation, both geometrical and kinematical constrains can be considered, and it usually leads to a mixed differential and algebraic equation (DAE) as the governing equation of motions. The links (RB) of the mechanism can be considered as a specially distributed, equivalent, system of material particles (of mass w,) that describes the mass inertia of the original system, too, with the so-called natural coordinates (r,). In this way, the definitions of angular velocities and angular... 

This expansion describes an arbitary distribution, P , for n>0 as a perturbation about a T-distribution, equivalent to the Gram-Charlier series resulting from a perturbation about a Gaussian distribution using Hermite polynomials [46]. 

Figure 28.11 showed distributions centered on the specified mean — 1 inch. What if they are not centered Figure 28.12 shows tight distributions, equivalent to acceptable values of Cp, but off the desired mean. The measure Cp corrects for shifts in the mean although the spread is acceptable. Distribution A in this case is centered above the mean distribution B is below, while C is acceptable because it is centered. Both distributions A and B would have Cp ratings in the acceptable range but Cpi, would be unacceptable for both. Distribution C has an acceptable Cpi, because it is centered. 

Every measured value of fraction of one powder Mode of particle size distribution Equivalent Martin diameter (L)... 

A sensible strategy is to define what proportion of the total peak area we wish to measure and set the limits accordingly. Let us say that we would be content to measure 99.7 % of the peak area. If we assume that the peak is Gaussian, taking the data from Chapter 4, Table 4.3 we can see that that would imply a peak region of twice 3 standard deviations width (i.e. both tails of the distribution) equivalent to a total peak width of about 2.5 FWHM. We can then fix the peak integration limits at 1.25 FWHM below and 1.25 FWHM about the centroid. 

The total energy of the electrons should also be dissectable into contributions by the two major terms in equation (2.6). Consider first the monocentric term. The term involving represents the contribution of electrons occupying a distribution equivalent to the AO 0 . This is the n-elec-tron density qi of that position i.e.,... 

The above diseussion assumes fliat die plated electrodes are perfectly conductive. However, the aetual resistance of die plated electrodes is not negligible. When eonsidering the surfaee resistance of die plated electrodes, several authors have proposed the distributed equivalent circuit shown in Fig. 6c (Shahinpoor and Kim 2000 Paquette et al. 2003 Punning et al. 2007 Chen and Tan 2008). In this figure, C is the distributed element of die double-layer capacitance, is that of the electrolyte resistance, is that of die surface resistance of the electrode, and R is... 

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