Randomly attributed

The simplest type of solutions which exhibit non-randomness are those in which the non-randomness is attributable solely to geometric factors, i.e. it does not come from non-ideal energetic effects, which are assumed equal to zero. This is the model of an athermal solution, for which... 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

In any particular situation, it is usually possible to give a variety of reasons why the observed quantity behaves in an erratic manner. The observed quantity may be critically dependent on certain parameters and the observed fluctuations attributed to slight variations of these parameters. The implication here is that the observed fluctuations appear erratic only because we have not taken the trouble to make a sufficiently precise analysis of the situation to disclose the pattern the observations are following. It is also possible, in some situations, to adopt the viewpoint that certain aspects of the phenomenon being studied are inherently unknowable and that the best physical laws we can devise to explain the phenomenon will have some form of randomness or unpredictability built into them. Such is the case, for example, with thermal noise voltages, which are believed to be governed by the probabilistic laws of quantum physics. 

Kinetic data for the decompositions of several metal hydrides are summarized in Table 12 to which the following information can be added. The acceleratory period in the decomposition of BeH2 (a < 0.35) is ascribed [673] to the random formation of metal nuclei followed by linear growth. The increase in rate consequent upon exposure to X-irradia-tion is attributed to enhanced nucleation. Grinding similarly increased the... 

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

The effect of a catalyst is important in cationic copolymerizations. Epoxides and /3-lactones form random copolymers only with trialkyl aluminum catalysts. Unusual sequence distributions were observed in the cationic copolymerization of epoxides or lactones using Lewis acids175-177) have been attributed to the di-... 

In the cases we discussed above, students revealed their lack of knowledge of the random distribution of particles, which was consistent with much previous research. This study not only revealed the mis-representation of the diffusion of gases, but also showed the inconsistent mental models that the students held while solving the problems. The result provided some evidence in favor of research that attributes students learning in relation to the context while facing various types of questions. However, this result does not support Vosniadou s framework theory (1994), which implies a consistent mental model used by learners in her study. 

The dependence on film thickness is attributed to the dewetting nucleation, which occurs in the 2.5-4.5 nm thickness range via the formation of randomly distributed droplets rather than the formation of holes. When the initial film thickness exceeds 4.5 nm, dewetting is trigged via nucleation of holes instead of droplets, and for film thickness above 10 nm, dewetting develops slowly via hole nucleation at defects. The different dewetting processes observed for different initial film thicknesses can be explained in terms of the variation of disjoining pressure and the inability of the polymer to spread on its own monolayer. 

Notice that in comparisons such as these sometimes slight inconsistencies in the results can be obtained. In two cases A was considered better than B, and B better than C, yet C was judged superior to A This inconsistency or non-transitivity is known as Simpson s or de Condorcet s paradox. In this particular case it can perfectly well be attributed to random variation. Assessors who are not sure about their conclusion are forced to make a choice, which then can only be a random guess. It is possible, however, to obtain results which are conflicting and statistically significant at the same time A < B and B < C, but C < A. This situation may occur when the attribute to be assessed in the comparisons is open to different interpretations. Actually, this is a case of multicriteria decision making (see Chapter 26) and it may be impossible to rank the three products unambiguously... 

For polymer melts or solutions, Graessley [40-42] has shown that for a random coil molecule with a Gaussian segment distribution and a uniform number of segments per unit volume, a shear rate dependent viscosity arises. This effect is attributed to shear-induced entanglement scission. 

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