Single distribution

However, for reasonably wide single distributions lifetime decays can provide a warning that a single discrete model is inappropriate even when the intensity Stern-Volmer plots give no warning of system complexity. 

In Figure 1.13 two recombination steps are clearly distinguishable, especially in the pump-probe data. These have been fitted previously by two separate stretched exponentials. Here, we used a single distribution of relaxation times, which accounted for both recombination steps. 

The solid matrix is assumed to be uniform such that the solid-fluid equilibrium can be formulated with a single distribution ratio ... 

The gas-liquid chromatographic evaluation of n can obviously be performed on a single distribution C E but not on mixtures of distributions. Commercial surfactants always consist of mixed distribution. Only when the procedure is applicable, rt=Lxnrt is calculated from the observed distribution of the molecular fraction (x ) for various values of n. It should be checked that x =l. 

When only variability is quantified, the output is a single distribution representing a best estimate of variation in exposure. This can be used to estimate exposure for different percentiles of the population but provides no confidence intervals and may give a false impression of certainty. 

Depending on the distribution chosen, as few as three fitting parameters may be required to define a distribution of diffusion rates. In some cases, a single distribution was used to describe both fast and slow rates of sorption and desorption, and in other cases fast and slow mass transfer were captured with separate distributions of diffusion rates. For example, Werth et al. [42] used the pore diffusion model with nonlinear sorption to predict fast desorption, and a gamma distribution of diffusion rate constants to describe slow desorption. 

One interesting research issue currently receiving much attention is placement of grid points for the discretization. For a single distributed variable such as temperature, one can see how to place the points, i.e., put more points where the variable is changing more rapidly and fewer where it is not. How is this to be done for many distributed but coupled variables that are changing at quite different rates in different parts of the space This problem is similar to that of slow- and fast-moving units in a dynamic simulation, only here no natural modularity occurs within a flowsheet of interconnected units. 

The (L-N-B)-model represents a highly sophisticated approach to the nonlinear behavior of dynamically excited, filled rubbers. However, it must be criticized that fittings of the storage - and loss - modulus could not be obtained with a single distribution function/la (y). The physical meaning of the density distribution function gla (y) remains unclear, indicating that the consideration of energy dissipation in the (L-N-B)-model is uncompleted. 

In the following the double peak shape will considered as two distinct lifetimes. However, they may simply represent lower and upper bounds of a broad single distribution. The lifetime analysis was performed on the full range of porogen loads. The results are shown in Figures. 20 and 21. 

MIP results do not show the sharp distinction in size between capillary and gel pores postulated by Powers and Brownyard (P20). Mikhail et at. (M69) earlier came to the same conclusion from results obtained by Nj sorption. With increasing age, the volume of pores smaller than about 20 nm increases and that of pores larger than about 200 nm decreases, but there is no sign of two separate distributions (Fig. 8.7). The single distribution... 

The pgf s Fp and F are sufficient for generation of trees and for derivation of statistical averages, and they both are determined by a single distribution Pi. This distribution is obtained from the kinetic differential equations as will be shown on the concrete example of curing of epoxy resins in Section 4. 

As Q increases, the two Gaussians shift toward x = 0 and are merged into a single distribution which peaks at x = 0 when Q reaches the threshold value Qi-... 

Figure 4 compares several of these models with respect to the nature of the constants that each uses. The simplest model (linear sorption or Ai ) is the most empirical model and is widely used in contaminant transport models. values are relatively easy to obtain using the batch methods described above. The Aid model requires a single distribution constant, but the Aid value is conditional with respect to a large number of variables. Thus, even if a batch Aid experiment is carefully carried out to avoid introduction of extraneous effects such as precipitation, the Aid value that is obtained is valid only for the particular conditions of the experiment. As Figure 4 shows, the radionuclide concentration, pH, major and minor element composition, rock mineralogy, particle size and solid-surface-area/solution volume ratio must be specified for each Aid value. 

If we ignore the destruction terms D, the different mutants are fully characterized through their rates for exact replication IV,. This assumption simplifies the following discussion by allowing a single distribution of value parameters f(W) to completely characterize the mutant spectrum, but this is not essential to the argument. Accordingly, we write... 

Data suggest that irritation among office workers does not represent a single distribution, but that susceptible subpopulations exist. Experimental studies, in chambers, show that atopic individuals usually have much lower irritation thresholds for agents commonly found indoors, even when their atopy is not active. In addition, the presence of tear-film instability, from Sicca complex, aging, or other underlying causes, poses an increased risk for irritation. 

Defining a single distribution function, p(x, r, c, t)dxdrdc, as the probable number of particles with internal coordinates in the range dx. about x, located in the spatial range dr about the position r, with a velocity range dc about c. 

The single distribution function /(x, r, t)dxdr thus denotes the probable number of particles within the internal coordinate space in the range dx about x, in the external (spatial) range dr about r at time t. is the mean velocity of all particles of properties x at a location r at time t. The velocity independent birth and death terms are defined by ... 

For flows where compressibility effects in a gas are important the use of the particle mass as internal coordinate may be advantageous because this quantity is conserved under pressure changes [11]. In this approach it is assumed that all the relevant internal variables can be derived from the particle mass, so the particle number distribution is described by the particle mass, position and time. Under these conditions, the dispersed phase flow fields are characterized by a single distribution function /(m, r,t) such that f m,r,t)drdm is the number of particles with mass between m and m+dm, at time t and within dr of position r. Notice that the use of particle diameter and particle mass as inner coordinates give rise to equivalent population balance formulations in the case of describing incompressible fluids. 

Assuming that the particle velocity for a given size is known, v = v(m, r,t), one may express a Boltzmann type of transport equation for the single distribution function ... 

For a one-compartment PK model, the value of Fis constant at all times and hence only a single distribution volume is defined. The volume of distribution for a two-compartment model changes with time, however, and hence several different types of distribution volumes are commonly defined for these models. The first distribution volume term is defined for time... 

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