Probability-logarithmic scale

Since there is a distribution of adherent particles with respect to adhesive force, the detachment velocity will depend on this distribution and on the sizes of the adherent particles. It has been shown experimentally that in the detachment of identical particles, the velocity required for detachment will vary. In Fig. X.3 we show as an example the fractional distribution of particles removed, as characterized by adhesion number, in relation to detaching velocity. A probability-logarithmic scale has been used for these plots. Similar distributions have been obtained for other particle-surface systems [277]. On a probability-logarithmic scale, the distribution of particles removed as a function of detaching velocity is approximated by a straight line. This means that the distribution of detaching velocities, as the distribution of adhesive forces (see Section 3), follows a log-normal law. 

Let us express the results obtained for the detachment of adhering particles as a function of water-flow velocity on the probability—logarithmic scale (Fig. VII.5) for the condition in which the dust-laden surface is parallel to the flow a = 90°). We see from Fig. VII.5 that the distribution of the remaining particles (broken line) when dp is plotted along the x axis deviates from the normal logarithmic law. 

On the basis of Eq. (19), the plot of k0bS/[edta4 ] as a function of the hydrogen ion concentration on a logarithmic scale gave a straight line of slope — 2.0. The slope shows the most probable value for n. Thus, at pH 2 to 4 the substitution reaction proceeds as... 

Figure 8.28. Demonstration of a CDF. Data recorded during non-isothermal oriented crystallization of polyethylene at 117°C. Surface plots show the same CDF (a) Linear scale viewed from the top. (b) Linear scale viewed from the bottom, (c) Viewed from the top, logarithmic scale. Indicated are the determination of the most probable layer thickness, lt, and of the maximum layer extension, le. (d) Viewed from the bottom, logarithmic scale. The IDF in fiber direction is indicated by a light line in (a) and (b) (Source [56])...

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

The calculated values listed in Table 6 are plotted in Figure 13. Note that the probability, P, is plotted on a logarithmic scale. 

Familiar examples are plotting on a logarithmic scale (7=logXr) and the transformation from frequencies to wavelengths (7= l/X). In general the range of 7 differs from that of X. The probability that 7 has a value between y and y + Ay is... 

Figure 4. Cumulative reaction probability for the H2 + OH - H2O + H reaction as a function of total energy for total angular momentum 7 = 0 (Ref. 11) (a) logarithmic scale (b) linear scale.

Figure 1.7 The probability density of the wave packet at time t = 200 on a logarithmic scale. Note that outside the interaction region, we have an exponentially diverging function.

Probably the most widely used type of viscometer in the food industry is the Brookfield rotational viscometer. An example of this instrument s application to a non-Newtonian food product is given in the work of Sarava-cos and Moyer (1967) on fruit purees. Viscometer scale readings were plotted against rotational speed on a logarithmic scale, and the slope of the straight line obtained was taken as the exponent n in the following equation for pseudoplastic materials ... 

Since the particle size is plotted on a logarithmic scale, the presentation of data on a log-probability graph is particularly useful when the range of sizes is large. The geometric standard deviation can be read from the graph, as with the arithmetic distribution, and is given by ... 

Probability graph paper is used in the analysis of cumulative frequency curves for example, graph paper can be used as a rough test of whether the arithmetic or the logarithmic scale best approximates a normal distribution. The scale, arithmetic or... 

Figure 1. Numerically calculated logarithm of survival probability Pit) of the initial state of impurity atoms in a 87Rb BEC plotted versus dimensionless time yt for n = 1014 cm 3 (cs = 0.2 cm/ s) mi = m2, <212 = 3 <222- filled circles Vi = 3 cs, 7 = 1.1 x 103 s 4. Open circles Vi = 7 cs, 7 = 4.4 x 103 s-1. Solid line exponential law exp(— yt). Inset the survival probability of the initial state (on a logarithmic scale) calculated in the HF approximation, plotted versus yapt for K = 10 (straight line, indistinguishable from the exponential decay), K = 1 (long-dashed line), and K = 0.1 (short-dashed line). Note that the inset horizontal axis is scaled by the HF value of the exponential decay rate, 7hf, unlike the horizontal axis of the main plot, which is scaled by the exact value of 7.

Figure 12.17 from Edmister is a plot of (12-58) with a probability scale for 4>a, a logarithmic scale for A and N as a parameter. This plot in linear coordinates was first developed by Kremser. ... 

Figure 15. Reversiblle proton dissociation inside a finite cavity. Left panel [12a] is a representative calculation for the probability of observing a bound pair (i.e., assuming an infinite radiative lifetime, r ). Right panel [4a] compares transient HPTS fluorescence in water (trace A) with its signal when located inside a liposome (trace B). The insert shows the same data on a semi-logarithmic scale.

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