Difference distribution function

Note carefully that the same random variable (function) may have many different distribution functions depending on the distribution function of the underlying function X(t). We will avoid confusion on this point by adopting the convention that, in any one problem, and unless an explicit statement to the contrary is made, all random variables are to be used in conjunction with a time function X(t) whose distribution function is to be the same in all expressions in which it appears. With this convention, the notation F is just as unambiguous as the more cumbersome notation so that we are free to make use of whichever seems more appropriate in a given situation. 

In molecular liquids the situation is slightly more complicated the following points merit discussion. (1) The signal A5 [r, x] is composed of a number of different distribution functions gpv r,t), but this is not a real handicap. If the bond p-v is broken, only the distribution function g v of atoms p, v forming this... 

In order to obtain better agreement with experimental results, the concept of a distribution of correlation times was introduced in nuclear magnetic relaxation. Different distribution functions, G(i c), such as Gaussian, and functions proposed by Yager, Kirkwood and Fuoss, Cole and Cole, and Davidson and Cole (asymmetric distribution) are introduced into the Eq. (13), giving a general expression for... 

Remark 5 Note also that reactors of various distribution functions can be treated with this approach, on the grounds that different distribution functions are usually approximated via cascades of CSTRs. In this case, we can treat the number of CSTRs as a variable or provide a variety of alternative reactors each featuring different numbers of CSTRs. Kokossis and Floudas (1990), present examples for batch, semibatch reactors and different distribution functions. 

Figure 3 Distributions of cadmium intake (ng/kg body weight) from consumption of iceberg lettuce obtained using five different distribution functions of the cadmium concentrations.

Another possible approach solving the equilibrium distribution for an electric double layer is offered by integral equation theories [22]. They are based on approximate relationships between different distribution functions. The two most common theories are Percus-Yevick [23] and Hypernetted Chain approximation (HNQ [24], where the former is a good method for short range interactions and the latter is best for long-range interactions. They were both developed around 1960, but are still used. The correlation between two particles can be divided into two parts, one is the direct influence of particle j on particle i and the other originates from the fact that all other particles correlate with particle j and then influence particle i in precisely... 

The Gaussian distribution function is most valid when the number of data, N, is very large. When there are fewer data, in a smaller sample, a different distribution function is better it was designed by Gosset131 who, in an excess of modesty, would only let it be known as the "Student f" distribution ... 

As was pointed out in the introduction, complex polymers are distributed in more than one direction. Copolymers are characterized by the molar mass distribution and the chemical heterogeneity, whereas functional homopolymers are distributed in molar mass and functionality. Hence, the experimental evaluation of the different distribution functions requires separation in more than one direction. 

For large n this distribution becomes normal (Gaussian), and for r = 0, Q (r) = 0. Depending on the preset value of n, formula (41) will give different distribution functions g(r) with a mean value of r , and the function g(s) will then look like... 

Cao has described the event echo [49] as difference between a joint (correlated) probability to detect photons in time p(ti,T2) and a disjoint (uncorrelated) propability p(ti)p(t2). The difference distribution function is given by... 

We have discussed in Section 5 that both KR and DB results may be expressed in the forms similar to ours. Therefore, their results assume the structure of Eq. (10.1). We can then consider using different distribution functions to evaluate the average length Rq. There are several attempts (38) to obtain good results by adopting various distribution functions, but since our interest has been in more fundamental aspects of the theory we shall omit the discussions. We remark, however, that an elaborate examination of various theoretical formulas has been reported very recently by Burchard (39). 

Different methods of particle size analysis yield different distribution functions as primary information, depending on what parameters are measured in the course of experiment. These parameters may be converted into different ones. It is, however, important for one to realize that such a conversion may yield errors of different magnitudes in different size ranges. 

Since one uses a number of different distribution functions, the average particle size may also be defined in more than one way [40]. In general one may write... 

It is out of the scope of this book to describe the AP mechanics, i. e. microphysics and dynamics (Friedlander 1977, Hinds 1882, Kouimtzis and Samara 1995, Harrison and van Grieken 1998, Meszaros 1999, Spumy 1999, 2000, Baron and Willeke 2001). Here, we only summarize the important topic of atmospheric aerosol size distribution (Jaenicke 1999). Fig. 4.15 shows that the size range covers several orders of magnitude. Therefore, the common logarithm of the radius is useful to describe the different distribution functions dN r)ld gr = f( gr) or dN r)ldr =/(Igr)/2.302 r. N r) cumulative number size distribution (or the integral of radii) having dimension cm , r radius of particle ... 

As was explained in Section 2.9.10, the reduced and oxidized ions of a redox couple interact with the solvent dipoles by ion-dipole interaction. This influences the energy of the electronic states. The fluctuation of the solvent molecules around the ion with only a statistical equilibrium solvation leads to a distribution of the electron energies around a central value of Gaussian form. Two energy distribution functions describe the energy distribution, one for the reduced ions (the occupied states) and the other for the oxidized ions (the unoccupied states). This was shown in Figure 2.33. The development of two different distribution functions is based on stable oxidation states. In each state the ion-dipole interaction can achieve a quasi equilibrium distribution. 

Fiuure S-3. Differential molar distribution of the degree of polymeriiation for three different distribution functions logarithmic normal (LN), Schulz-Flory (SF), and Tung (Tung). Calculations based on = 10,000 and = 2,... 

In any other case it has to be assumed that the unknown sample has a different distribution function of the molar mass and therefore a different distribution function of the intrinsic viscosities of each fraction of the sample. According to Philippoff [72], it could be shown that the measured intrinsic viscosity is a mass average (see Chap. 2) of the intrinsic viscosity distribution ... 

The first application of differentiation to spectroscopy was, of course, the deconvolution of superposed peaks [41-43], which are frequently found in spectral investigations. From then on, many scientists analyzed peak overlapping by summation of synthetic signals, mostly of the Gaussian type [12,14,15, 44-48]. Derivatives, zero crossing, and extrema of different distribution functions such as Gauss, Lorentz, Student, T3, and others, are not difficult to estimate unless there are superpositions of two or more bands. For pure Gaussian functions, see Table 2-3. 

In mixtures of liquid crystals, the molecules of different components may possess different degrees of nematic ordering. In this case the nematic order parameters, S, for different components a are calculated separately using different distribution functions (cos to). 

Distribution functions of the feed copolymer belong only to cloud-point data. On the other hand, each pair of coexistence points is characterized by two new and different distribution functions in each coexisting phase. The critical concentration is the only feed concentration where both parts of the coexistence curve meet each other on the cloud-point curve at the critical point that belongs to the feed... 

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