Coincidence, probability

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

It is readily apparent that Bell s inequality in the form (16.7) is in disagreement with quantum mechanics. With the coincidence probabilities (16.6), the sense of the inequality is, in fact, reversed since... 

Proton d has not been analyzed, but we will do this by prediction in the next paragraph. Note that three of the coupling constants (aU of which are expected to be small ones) are equivalent, or nearly equivalent. This is either coincidence or could have to do with an inability of the NMR instrument to resolve them more clearly. These coincidences probably result from the fact that the computer did not have enough data points to distinguish the exact apex of each peak. Note that this has clearly happened in Figure 5.24b where one of the differences is 1.11 Hz instead of the expected 1.47 Hz. There may be some second-order interaction (Section 5.10), but it appears that all of the protons are sufficiently well separated form one another that this seems unlikely. More advanced techniques would be required to determine whether the coincidences are real or not. 

An approach to calculate multi-system failure coincidence probabilities... 

The coincidence probability />coin3,2 of v = 3 involved and c = 2 conflicting systems, refer to Fig. 4, yields... 

Moreover, the two systems coincidence probabilities as given in Rakowsky (2007, 2006a, 2005b) correspond with equation 3. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

As discussed in 2-6.3 and 6-4.2, powder ignition via bulking brush discharge requires the coincidence of several factors, each of which has an associated probability. The minimum conditions are... 

Another potential problem is due to rotor instability caused by gas dynamic forces. The frequency of this occurrence is non-synchronous. This has been described as aerodynamic forces set up within an impeller when the rotational axis is not coincident with the geometric axis. The verification of a compressor train requires a test at full pressure and speed. Aerodynamic cross-coupling, the interaction of the rotor mechanically with the gas flow in the compressor, can be predicted. A caution flag should be raised at this point because the full-pressure full-speed tests as normally conducted are not Class IASME performance tests. This means the staging probably is mismatched and can lead to other problems [22], It might also be appropriate to caution the reader this test is expensive. 

Because luciferyl adenylate emitted a red chemiluminescence in the presence of base, coinciding with the red fluorescence of 5,5-dimethyloxylucferin, the keto-form monoanion Cl in its excited state is considered to be the emitter of the red light. Thus, the emitter of the yellow-green light is probably the enol-form dianion C2 in its excited state, provided that the enolization takes place within the life-time of the excited state. Although the evidence had not been conclusive, especially on the chemical structures of the light emitters that emit two different colors, the mechanism shown in Fig. 1.12 was widely believed and cited until about 1990. 

However, in most cases the AW(D) dependencies are distinctly nonlinear (Fig. 9), which gives impulse to further speculations. Clearly, dependencies of this type can result only from mutual suppression of the hydrogel particles because of their nonuniform distribution over the pores as well as from the presence of a distribution with respect to pore size which does not coincide with the size distribution of the SAH swollen particles. A considerable loss in swelling followed from the W(D) dependencies, as shown in Fig. 9, need a serious analysis which most probably would lead to the necessity of correlating the hydrogel particle sizes with those of the soil pores as well as choice of the technique of the SAH mixing with the soil. Attempts to create the appropriate mathematical model have failed, for they do not give adequate results. 

Slow burn-out tends to be associated with high-quality burn-out conditions and to produce a not unduly excessive wall-temperature rise. In fact, there appears to be an extreme condition in which the temperature rise may hardly be noticeable, and it becomes difficult to say whether burn-out has occurred. These circumstances probably coincide with the jump discontinuity in Fig. 3 ceasing to exist for certain values of system parameters. The condition is effectively one in which, at the burn-out point, the heat-transfer coefficient is the same whether the surface is vapor-blanketed or liquid-wetted. 

The assumption that an impurity, probably water, gives rise to bimodal MWD with f-BuX/Et2AlCl system was confirmed directly by experiments. First polymerizations were carried out at —60 °C by using the H20 /Et2AlCl system, i.e., by increasing the moisture content in the enclosure from the usual <30 ppm to 150 ppm. Yields were 30% or less. The GPC trace of these PIB s (Fig. 9) coincides with those... 

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