Discontinuous distribution

It is useful to distinguish two forms of probability distribution, discontinuous and continuous. As an example of a discontinuous distribution consider the outcome of throwing a die. The chance outcome of a series of throws can be represented as a discontinuous probability distribution. There is only a limited number of possible outcomes and the results can be shown in the form of a block histogram. The second kind of distribution would arise in, e.g., replicate measurements of the fluorescence intensity of a sample. These observations will differ as a result of statistical variation, as discussed in the previous section, but instead of being a single chance event as in the case of the die, many chance factors will contribute to the observed variation in the fluorescence data. The variation observed in this case is an example of a continuous probability distribution. Although it is true in principle... 

From these calculations we conclude that the standard PSD inversion procedures with respect to the isotherm and enthalpy curve data can produce an acceptable PSD even where the underlying structure has a discontinuous distribution of pores and that the enthalpy curve appears to be more sensitive to the presence of the smallest pores. 

The interference function in Eq. 2.14 describes a discontinuous distribution of the scattered intensity in the diffraction space. Assuming an infinite number of points in a one-dimensional periodic structure N -> oo), the distribution of the scattered intensity is a periodic delta-function (see above) and therefore, diffraction peaks occur only in specific points, which establish a one-dimensional lattice in the diffraction space. Hence, diffracted intensity is only significant at certain points, which are determined (also see Eq. 2.13, Figure 2.23, wad Figure 2.25) from... 

Figure 1.6. Illustration of the discontinuous distribution of polymer chain lengths (a) and the apparently continuous distrihution and molar-mass averages (b) from a polymerization ...

Distribution functions can be classified as discontinuous or continuous. Discontinuous distribution functions are subdivided into frequency distributions and cumulative distributions. Continuous distribution functions are further classified as differential and integral distribution functions. 

Discontinuous distribution functions can, of course, be transformed into continuous distribution functions when the difference between two neighboring properties is very small compared with the whole range of property values. Frequency distributions convert to the corresponding differential distributions and cumulative distributions convert to integral distributions,... 

In a discontinuous distribution function, the frequency distribution gives the distribution of the statistical weights (or their fractions) of the components i as a function of the property E. For example, E may be the degree of polymerization. Typical property distributions are... 

A word of caution is in order for those pursuing the lifetime distributions. As mentioned, the Fredholm integral equation (48) is strongly ill-posed, and any result obtained without regularization on the basis of physical grounds should be regarded with suspicion. It seems that more robust solutions are obtained if one confines to discontinuous distributions, i.e., to discrete time constants. One possibility is to solve first the discrete time constants and then to keep these values constant when solving the widths... 

ABSTRACT Type of collapsibility of site is an important content in evaluation of loess collapsibility, the results calculated according to indoor test is often contradict with field test results. In order to solve the contradiction, the paper analyzes the factors that brings about difference in indoor and field test results based on the in-situ immersion test results in loess site completed in recent years, makes a study of the influence of discontinuous distribution of collapsible loess on collapse settlement, and presents the improved empirical formula for collapse settlement calculation under overburden pressure. The results show that indoor collapsibility test succeeds but impairs the correlation between factors such as loess property, stratum configuration, stress history, buried depth and settlement by immersion, during collapse settlement calculation, the correction of these influential factors shall be taken into consideration the standard used for judgment of loess as to whether it is collapsible may be properly heightened, in case of discontinuous distribution of collapsible loess, stress redistribution effect shall be taken into account. 

Based on 10 in-situ immersion tests data and numerical analysis results, this paper gives a adjust method of calculated value of self-weight collapse settlement (computed collapse under overburden pressure) considering the influence of loess deposit age and discontinuous distribution of collapsible loess. 

NUMERICAL CALCULATION OF COLLAPSE SETTLEMENT UNDER THE CONDITION OF DISCONTINUOUS DISTRIBUTION OF COLLAPSIBLE... 

Gradually increase the percentage of collapsible loess in various models to calculate ratio of value of numerical calculation to standard calculation of self-weight collapse settlement (effect of discontinuous distribution of collapsible loess), the result is as shown in Figure 8. 

The improved formula for self-weight collapse settlement enhances the standard for judgment as to whether gj loess is collapsible, considers the influence of discontinuous distribution of collapsible loess, is conductive to solving the problem in difference between calculated value and measured value of self-weight collapse settlement of Q2 loess. 

Numerical analysis result shows that when collapsible loess is in discontinuous distribution, the loess unit is likely to generate redistribution of stress during collapse process, which makes the calculated value of self-weight collapse settlement on the high side. When collapsible loess unit is less in percentage and irregular in distribution, the stress redistribution effect shall be more apparent. 

Campa C, Doulbeau S, Dussert S, Hamon S, Noirot M (2005) DivCTsity in bean caffeine content among wild Coffea species evidence of a discontinuous distribution. Food Chtan 91 633-637... 

The discontinuous nature of propagation is reflected as notches in the spatial extracellular potential field close to the fiber (Figure 4). At radial distances of 10 times the fiber radius or greater, the computed spatial field is a smooth bi-phasic waveshape (Figure 5), and does not reflect the discontinuous distribution of the sources at all. These results imply that while at the tissue level the structural discontinuities affect the extracellular potential field, for global forward prob-... 

Here, the energy levels are very close together and their discontinuous distribution can be replaced by a continuous one and so that that the statistical weight of states corresponding to the layer with a thickness of dz is ... 

Liquid matter, and particularly water, because of high surface-tension forces, tends to occupy that shape and volume consistent with the dissipation of the least surface energy. Gases, however, are relatively unrestricted in their flow by surface-tension forces and therefore tend to occupy all available space. Thus we have two contrasting characterikics of the parent compounds of carbon and hydrogen. Carbon dioxide diffuses throughout the maximum possible space and is therefore of continuous distribution over the earth s surface while water is confined to the least possible space and is of discontinuous distribution. 

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