Location-conditioned statistics

The superscript used in the coefficient matrices in (6.192) is a reminder that the statistics must be evaluated at the notional-particle location. For example, e = e(X r), and the scalar standard-deviation matrix and scalar correlation matrix p are computed from the location-conditioned scalar second moments X )(X, t). 

Study. Their location represents the range of the conditions that should be used in a statistically designed set of experiments for kinetic studies and not actual conditions at the particular location. This way, the same flow is again maintained on all three scales and the troublesome effect of changing flow conditions on scale-up can be eliminated. The proposed idea, for scale-up by scale-down from the view point of the catalyst, was proven to be effective in many industrial projects. 

The relative amount of strand cleavage at each site of AQ-DNA(l) is indicated by the length of the solid vertical arrow shown in Fig. 4. As is often observed, the 5 -G of the GG steps react more often than do the 3 -G. In the case of AQ-DNA(l), the relative reactivity is ca. 1 3, but this ratio depends upon the specific base pair sequence surrounding a GG step, which may be an indication of radical cation delocalization to bases adjacent to the GG sequence. It is worth pointing out again that these reactions are carried out under single-hit conditions where the relative strand cleavage efficiency seen at various locations of AQ-DNA(l) reflect the statistical probability that the radical cation will be trapped by H20 at that site. 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

Consider a highly resolved simulation of a set of microscopic TFM equations for a fluidized suspension of particles in a large periodic domain. The filtering operation does not require a periodic domain however, as each location in a periodic domain is statistically equivalent to any other location, statistical averages can be gathered much faster when simulations are done in periodic domains. After an initial transient period that depends on the initial conditions,... 

A more precise definition would include conditioning on the random initial velocity and compositions /li, , x Uo,. o.Y Vb XIY), V o, y 0- However, only the conditioning on initial location is needed in order to relate the Lagrangian and Eulerian PDFs. Nevertheless, the initial conditions (Uo, o) for a notional particle must have the same one-point statistics as the random variables U(Y, to) and (V. to). 

In Chap. 2 and Chap. 3, Sect. 1.2, the appropriate boundary and initial conditions for reactions between statistically independent pairs of reactants were formulated to model a homogeneous reaction. In these cases, if there is no inter-reactant force, all that is required is one or other reactant to be in vast excess on the other. Since the excited donor or the electron donor has to be produced in situ by photostimulation or high-energy radiation, it is natural to choose [D ] < [A], though there are exceptions. Locating the donor at the origin in a sea of acceptor molecules distributed randomly leads to the initial condition, as before... 

The sterilization process time is determined from the design F value and the product heat transfer data. The sterilization cycle design must be based on the heating characteristics of the load and of containers located in the slowest heating zone of the load. The variation in the rate of heating of the slowest heating zone must be known, so this variation must be determined under fully loaded conditions. The effect of load-to-load variation on the time-temperature profile must also be determined. Then, the statistically worst-case conditions should be used in the final sterilization process design. 

It is beyond the scope of this review to discuss in detail the statistical parameters of subduction zones, and Figure 10 is intended to demonstrate that conditions attained in subducting slabs are highly variable, even for similar convergence parameters, and that interplay between thermomechanical properties and reaction paths are responsible for a complex pattern of fluid release and magma genesis. Furthermore, any correlation of kinematic subduction parameters with volcano location tacitly assumes steady state, which is not necessarily the case. 

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