The Beta Function

The Beta function contains two arguments, but these can be connected to Gamma functions, so we define 

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Reinmuth [465] arrived at a solution for the Langmuir isotherm in the form of a series, involving the beta function. Levich et al. [362] had an approximate solution for a general adsorption isotherm. 

Ross (R15) and Ross and Curl (R16) assumed the /3(u, a) term to be given by the beta function, whereas Shiloh et al. (S21) and Delichatsios and Probstein (D5) assumed that a drop breakage results in two equal daughter drops. 

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... 

This beta function is zero in the absence of dipolar interactions. It vanishes for interactions linear in the spin operator and has nonzero values only for bilinear interactions. Furthermore, it is zero for Je approaching zero, so that a nonzero signal indicates residual anisotropic interactions, and it is free of signal attenuation by relaxation. The shape of the beta function has been shown to depend strongly on the strain in rubber samples [Cal3]. 

Beta functions" have been used to represent several sources of error. The beta function is extremely flexible in the sense that it can assume a great variety of shapes depending on values chosen for exponents, as seen in Figure 1, and seems particularly useful in representing negative bias. 

Prom Eq. (45), the emission spectrum (3) follows. Finally, using integral representation for the Beta-function in Eq. (42) we arrive at... 

The first member of (10) is sometimes called the first Eulerian integral, or beta fumtion. It is written B(m, n). The beta function is here expressed in terms of the gamma function. Substitute x = ay/h in the second member of (10), and we get... 

The Beta function can be expressed in alternative form by substituting... 

By definition, P u) tells us how the renormalized u changes with the length scale and is called the beta function in RG. Some more algebra gives... 

The factor of 27t can be absorbed in the definition of u, as have often done. This equation, Eq. 119 is called a renormalization group flow equation with the initial condition u = uq for some L = Lq. It is analytic in e so that various dimensions can be handled with this equation. Initial condition may be taken as u = wp for some L = Lo- In this particular case the beta function is exact to all orders. 

The zeros of the beta function are called fixed points which can be of two types, stable or unstable. For this particular case, the fixed points are u = 0 and u = —e. If we start with a very small u, the flow equation Ldu/dL ss eu shows a growth of u with L if e > 0, i.e., if d < 2. This means u is a relevant variable at the noninteracting point. For the disordered system it translates to relevance of disorder at the pure fixed point. 

Since the beta function is exact, we have obtained the exact correlation length scale exponent for the binding-unbinding critical transition. 

For definition of the beta function and a derivation of Eq. (39), see Ref. [9]. Note also that the notation d) denotes a Pochhammer symbol, defined immediately after Eq. (29). 

For the daughter number distribution, P, the beta function is used... 

Probability density function where B is the beta function. ... 

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