Propagator probability distribution functions

We note that the approach of section 2 is only appropriate if the geometry is known in advance. For instance, in urban geometries we can in general not assume fiee field propagation. In such a situation it would be more appropriate to use probability distribution functions in terms of characteristic blast sizes and to numerically compute or to estimate the expected overpressures and blast impulses for locations of interest. In this sense piobit functions in terms of the scaled distance incorporate a certain geometry setting and a physical hazard analysis. 

Supplement 1 to the GUM defines the Monte Carlo method as a method for the propagation of distributions by performing random sampling from probability distributions [8]. This method is particularly suitable for models that cannot be linearized or solved with classical methods. How the Monte Carlo method can be used to evaluate uncertainty is explained briefly in [9]. As discussed above, the first step involves defining the dependence of the output quantity as a function of all possible input quantities. This results in the measurement equation Eq. (22.8). After the measurement equation has been formulated, m random samples are generated for each input quantity with the help of a random number generator and probability distribution functions (PDFs) pi,m-... 

Thus, Equation 27 is in this case a possible distribution function. It is of the type of the Schulz-Flory (25) distribution function. The expressions p and alternating polymerization (chain termination). The validity of the Schulz-Flory distribution function in this example of a polymerization with reversible propagation steps is evident. This type of distribution is always present if the distribution of the chain lengths... 

Numerical methods for propagating distributions through models are typically preferred over analytical methods in practice because they are versatile. Such methods can be used with a wide variety of model functional forms, including large black box computerized models, and for a wide variety of probability distributions used to present model inputs. However,... 

There is, however, an important difference between the distributions calculated for equilibrium, bifunctional step-growth polymerization in Chapter 5 and for the free-radical polymerizations with termination by disproportionation and/or chain transfer that are being considered here. Thus p in Eq. (5.81) is the extent of conversion, while P in the above equations is the probability that a propagating radical will continue to propagate instead of terminating. There is a very important second difference. While the distribution functions in the step-growth case apply to the whole reaction mixture, in the free-radical polymerization this distribution applies only to the polymer fraction of the reaction mixture. 

The Full Chain Length Distribution. So far, only the average degree of polymerization has been considered. To calculate the distribution function itself for a steady-state polymerization it is convenient to choose a statistical approach based on kinetic parameters. A probability factor a of propagation is defined as the probability that a radical will propagate rather than terminate. The factor a is the ratio of the rate of propagation over the sum of the rates of all possible reactions the macroradical can undergo. 

Firstly, we assume that termination is solely by disproportionation and that the propagation probability factor is equal for each chain length. The probability for the occurrence of a polymer chain—hence its distribution function—with the length P is given by the probability of P - 1 propagation steps and the probability of one chain-stopping event (termination or transfer). 

The uncertainty of a parameter can be characterised by the upper and lower limits of the parameter or by the expected value and the variance of the parameter. Such descriptions of individual parameter uncertainty can, for example, be obtained from the data evaluation sources introduced in Chap. 3. The joint probability density function (pdf) of parameters gives the most complete information about the uncertainty of a parameter set. Methods of uncertainty analysis provide information about the uncertainty of the results of a model knowing the uncertainty of its input parameters. If such a lack of knowledge of model inputs is propagated through the model system then a model output becomes a distribution rather than a single value. Measures such as output variance can then be used to represent output uncertainty. 

For the larger ring sizes, both propagation and depropagation must be taken into account (Equation 1.3b), and a certain measurable concentration of the unreacted monomer appears when the system comes to thermodynamic equilibrium ([M]e,= 1/Kp = kd/kp)- In this case, the molar mass distribution may be described by the most probable (Flory-Schultz) distribution function [92], for which ... 

This relation is only valid for a crystal with isotropic /-factor. The effect of crystal anisotropy will be treated in Sect. 4.6.2. The function h(6) describes the probability of finding an angle 6 between the direction of the z-axis and the y-ray propagation. In a powder sample, there is a random distribution of the principal axes system of the EFG, and with h 6) = 1, we expect the intensity ratio to be I2J li = I, that is, an asymmetric Mossbauer spectrum. In this case, it is not possible to determine the sign of the quadmpole coupling constant eQV. For a single crystal, where h ) = — 6o) 5 delta-function), the intensity ratio takes the form... 

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