Distribution functions graphical representation

Mathematical representation of droplet size distribution has been developed to describe entire droplet size distribution based on limited samples of droplet size measurements. This can overcome some drawbacks associated with the graphical representation and make the comparison and correlation of experimental results easier. A number of mathematical functions and empirical equations1423 427 for droplet size distributions have been proposed on the basis of... 

Graphic representation of cumulative frequency distribution of selected effects as a function of concentration is also prepared where ratio of median effective concentration is considered for drug selectivity using different endpoints. 

A graphical representation of the cumulative residence time distribution function is given in Figure 4.97 for a structured well, a laminar flow reactor and an ideal plug flow reactor assuming the same average residence time and mean velocity in each reactor. 

Figure 1 Graphical representation of the radial distribution function D(r) — 4xr2p(r) and of the variable s = (l/(2(37r2)2/3) Vp(f) /p4/3(r) for the beryllium atom.

Figure 3 Graphical representation of the conjoint gradient correction functions - for the beryllium atom - of Pade43 [37], Pearson [36], LLP [28], TF W, B86A [30] and B86B [34] advanced in HKS-DFT. For comparison the exact Hartree-Fock modulating function and radial distribution D(r) = 4nr2p(r) are also shown.

Our proposed method of site energy distribution analysis can, of course, be applied using any function 0(P,T,Q) for the local isotherm (or even graphical representations not amenable to analytical replication). Nor need the function be explicit in 0. As an example of this last situation, it has been suggested (7) that lateral interaction be allowed for within the framework of the Langmuir approach by supposing Q to increase linearly with 0 ... 

Fig. XI.2. Graphical representation of the reaction rate function k E)f the classical energy-distribution function P(E) for a group of s oscillators, and their product k(E)P(E).

Figure 13, which is a graphical representation of the ZND theory, shows the variation of the important physical parameters as a function of spatial distribution. Plane 1 is the shock front, plane 1 is the plane immediately after the shock, and plane 2 is the Chapman-Jouguet plane. In the previous section, the conditions for plane 2 were calculated and u was obtained. From u and the shock relationships or tables, it is possible to determine the conditions at plane 1. Typical results are shown in Table 5 for various hydrogen and propane detonation conditions. Note from this table that (/02/yOi) = 1.8. Therefore, for many situations the approximation that is 1.8 times the sound speed, 02, can be used. 

Figure 5.8a gives the proportions of SO2 in the gas and aqueous phase as a function of pH. For pH < 5, sulfur dioxide occurs mainly in the gas phase for pH > 7, it occurs mainly in the solution phase. The fraction of SO2 in the aqueous phase is given in Figure 5.8b as a function of q (water content) for a few pH values. The double logarithmic graphic representation is particulariy convenient to plot the equilibrium distribution of the aqueous sf>ecies (Figure 5.8c). For a sketch of this diagram it is convenient to recall the following ...

In Chapter 5 we described a number of ways to examine the relative frequency distribution of a random variable (for example, age). An important step in preparation for subsequent discussions is to extend the idea of relative frequency to probability distributions. A probability distribution is a mathematical expression or graphical representation that defines the probability with which all possible values of a random variable will occur. There are many probability distribution functions for both discrete random variables and continuous random variables. Discrete random variables are random variables for which the possible values have "gaps." A random variable that represents a count (for example, number of participants with a particular eye color) is considered discrete because the possible values are 0, 1, 2, 3, etc. A continuous random variable does not have gaps in the possible values. Whether the random variable is discrete or continuous, all probability distribution functions have these characteristics ... 

On a chromatogram the perfect elution peak would have the same form as the graphical representation of the law of Normal distribution of random errors (Gaussian curve 1.1, cf Section 22.3). In keeping with the classic notation, p would correspond to the retention time of the eluting peak while a to the standard deviation of the peak (cr represents the variance), y represents the signal as a function of time X, from the detector located at the outlet of the column (Figure 1.3). 

Fig. 5 shows the usual graphical representation of the spatial distribution of atomic orbitals which is obtained as the square of the angular part of the wave functions provided with the sign of the angular part of the wave functions. 

Fig. 5. Graphical representation of the spatial distribution of atomic s, p and d orbitals (square of real angular functions with indication of the sign of the angular functions)...

The effects of different model parameters on the plasma concentration versus time relationship can be demonstrated by mathematical analysis of the previous equations, or by graphical representation of a change in one or more of the variables. Equation (10.95) indicates that the plasma concentration (Cp is proportional to the dose (Av) inversely proportional to the volume of distribution (V)- Thus an increase in Aw or a decrease in Fboth yield an equivalent increase in Cp as illustrated in Figure 10.20. Note that the general shape of the curve, or more specifically the slope of the line for ln(Cp versus t, is not a function of Av or V. Equations (10.98) and (10.99) show that the... 

The graphical representations of species distributions and/or mineral saturation as a function of E -... 

Where a random variable X is assumed to follow normal distribution it will be described in the fovmXa ) where means is distributed according to . Examples of the graphical representation of the normal distribution are included in Chapter 37. The standard normal distribution is written as N(0,1) with j = 0 and r7 = 0. The cumulative distribution function for the standard normal distribution is given by... 

Figure 5.2 A graphical representation of Eq. (5.5) showing a normal distribution function having a mean of zero and a variance of unity.

Figure 4.36. Graphical representation of the concept of effectiveness factors (a) The effectiveness factor of reaction, rj, as a function of the Damkoehler number of the second kind, Dan, or of the Thiele modulus, (j) (cf. Equ. 4.74). (b) The effective reaction rate, r ff, as a function of the diameter of the particle, d. Part (b) can be used to obtain the value of for a given average diameter, of a population of floes with a distribution d. The range of validity of kinetic control (r js) and of diffusion control (Wfds) is indicated in part a.

Figure 8.11 Graphical representations of the definition and implications of the EPA definition of an MDL. (a). Assumed normal frequency distribution of measured concentrations of MDL test samples spiked at one to five times the expected MDL concentration, showing the standard deviation s. (b) Assumed standard deviation as a function of analyte concentration, with a region of constant standard deviation at low concentrations, (c) The frequency distribution of the low concentration spike measurements is assumed to be the same as that for replicate blank measurements (analyte not present), (d) The MDL is set at a concentration to provide a false positive rate of no more than 1% (t = Student s t value at the 99 % confidence level), (e) Probability of a false negative when a sample contains the analyte at the EPA MDL concentration. Reproduced with permission from New Reporting Procedures Based on Long-Term Method Detection Levels and Some Considerations for Interpretations of Water-Quality Data Provided by the US Geological Survey NationalWater Quality Laboratory (1999), Open-File Report 99-193.

Thus, the graphical representation of the radial distribution function leads to a maximum probability that an electron of the n = 1 layer, that is the Is layer, to be very close to the nucleus. For the n = 2 layer, the graphical representation shows two maxima for the 2s orbital and a curve with a maximum for the 2p orbital. For n = 3, the graphical representation shows 3 maxima for 3s, 2 maxima for 3p, and one maximum for 3d (Figure 2.4) (Pop 2003 Brezeanu et al. 1990 Whitten et al. 1988). 

The first type of graphic representation is that of distribution and logarithmic diagrams, representing species fractions (in linear or logarithmic form) as a function of composition variables of the system [14,15]. The second class is that of the reaction prediction diagrams, and to... 

Finally, by using many rectangles and drawing a smooth curve through their tops, we obtain the particle size distribution curve that is the graphical representation of the fiequency function, or probability density function. Figure 4.4 is an accurate picture of how the particles are distributed among the various sizes it has the same characteristics as Fig. 4.3, but may be amenable to mathematical interpretation. 

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