Moments, continuous/discrete

The data obtained from studies 34 36,54,S5) carried out at fixed y and various T (as in Figs. 5 and 61 show that the time dependence of rieff may be approximated by a linear law. The influence of medium deformation on gelatination can not be determined within the limits of experimental uncertainty. This may be seen, for instance, from Fig. 6a where the dependence t eff(T) at 53 °C was obtained for both the discrete (points 1) and continuous deforming modes (points 2). Figure 6 presents only those Tleff which correspond to the moments of discrete measurements. 

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). 

Moments 92. Common Probability Distributions for Continuous Random Variables 94. Probability Distributions for Discrete Random Variables. Univariate Analysis 102. Confidence Intervals 103. Correlation 105. Regression 106. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

The discrete factor solvent number is recognized as a simple bookkeeping designation. We can replace it with the continuous factor dipole moment expressed on a ratio scale and obtain, finally, the response surface shown in Figure 2.13. A special note of caution is in order. Even when data such as that shown in Figure 2.13 is obtained, the suspected property might not be responsible for the observed effect it may well be that a different, correlated property is the true cause (see Section 1.2 on masquerading factors). 

It seems that there is a need to reexamine, some of the basic quantities used in transport processes, like Thiele numbers, attempting to connect them to more chemical quantities. For example, the macroscopic quantity, e the dielectric constant, can be interpreted in terms of dipole moment distribution, and the dipole moment has immediate structural implications. Now to talk of a dielectric constant in the interaction of two atoms would be a rather useless exercise, since the dilectric constant is a continuous matter concept, not a discrete matter concept. In the same... 

The discrete factor solvent number is recognized as a simple bookkeeping designation. We can replace it with the continuous factor dipole moment and obtain, finally, the response surface shown in Figure 2.13. 

There is another characteristic that all these techniques present the current is sampled at a given moment during the application of any individual potential of the sequence (typically at the end of each applied potential), so the response is a discrete collection of pairs of data (potential-current). Conversely, in the case of techniques like Linear Sweep Voltammetry or Cyclic Voltammetry, the current is recorded continuously (see Sect. 5.1).11... 

This doctrine, that quantity must be limited, is intriguing, for it mirrors another metaphysical doctrine of Aristode s, namely that matter cannot exist without form. Aristode does not, to my knowledge, ever explicidy connect quantity as such with matter. But suppose for the moment that they are not accidentally related. Suppose that their connection is guaranteed by the feet that quantity flows from matter, or more precisely, that quantity as such simply is matter of a very special sort, namely indeterminate extension. The following sort of picture would then emerge. Uninformed indeterminate extension would be limidess quantity and hence would be an instance of the actual infinite Aristode denies could exist. Because an actual infinite cannot exist, extension, i.e. quantity, must be limited in some way by form. The metaphysically necessary carving up of matter by form, then, results in discrete and continuous composites of form and matter, which can be measured and counted respectively. It results, in other words, in numerable and countable quantities. 

The MWD may be related mathematically to the so-called moments of a continuous or discrete distribution. If u is a random variable and F u) is its distribution function then the ith-order moment may be defined by the relation ... 

To verify the accuracy of the continuum approximation, we compared the average dipole moments alongthez axis calculated using both the discrete and continuous procedures for integer values of N. 

Mathematically, the infinite set of equations describing the rate of each chain length can be solved by using the z transform method (a discrete method), continuous variable approximation method, or the method of moments [see, e.g., Ray in Lapidus and Amundson (eds.), Chemical Reactor Theory—A Review, Prentice-Hall, 1977]. 

The reconstructed distribution function may be continuous (EQMOM) or discrete (QMOM), but we will assume that it is always realizable (i.e. nonnegative). For the case in which / is a set of weighted delta functions, the computation of the moments and is trivial. With EQMOM the integrals are evaluated using... 

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