Mathematical concept probability density function

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

A statistical theory of turbulence which is applicable to continuous movements and which satisfies the equations of motion was introduced by Taylor [159, 160, 161] and [162, 163], and further developed by von Karman [178, 179]. Most of the fundamental ideas and concepts of the statistical turbulence theory were presented in the series of papers published by Taylor in 1935. The two-point correlation function is a central mathematical tool in this theory. Considering the statistics of continuous random functions the complexity of the probability density functions needed in a generalized flow situations was found not tractable in practice. An idealized flow based on the assumption of... 

After the seminal structure building of the QS formalism, several additional studies appeared over time, which developed new theoretical details. Especially noteworthy is the concept of vector semispace (VSS). This mathematical construction will be shown to be the main platform on which several QS ideas are built, related in turn, to probability distributions and hence to quantum mechanical probability density functions. Such quantum mechanical density distributions form a characteristic functional set, which can be easily connected to VSS properties. Construction of the so-called quantum objects (QO) and their collections the QO sets (QOS) (see, for example, Carbo-Dorca ), easily permit the interpretation of the nature of quantum similarity measures for relationships between such quantum mechanically originated elements. Within quantum similarity context, QOS appear as a particular kind of tagged sets, where objects are submicroscopic systems and their density functions become tags. 

The use of probability-density-function analysis, an important topic in statistical analysis, is mentioned with respect to its utility in nondestructive testing for inspectability, damage analysis, and F-map generation. In addition to the mathematical concepts, several sample problems in composite material and adhesive bond inspection are discussed. A feature map (or F map) is introduced as a new procedure that gives us a new way to examine composite materials and bonded structures. Results of several feasibility studies on aluminum-to-alumi-num bond inspection, along with results of color graphics display samples will be presented. 

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