Distribution, statistical Statistics

Boltzmann distribution statistical distribution of how many systems will be in various energy states when the system is at a given temperature Born-Oppenbeimer approximation assumption that the motion of electrons is independent of the motion of nuclei boson a fundamental particle with an integer spin... 

Figure 6.4 Crystal structure of ar-tetragonal boron. This was originally thought to be B50 (4Bi2 + 2B) but is now known to be either B50C2 or B50N2 in which the 2C (or 2N) occupy the 2(b) positions the remaining 2B are distributed statistically at other vacant sites in the lattice. Note that this reformulation solves three problems which attended the description of the or-tetragonal phase as a crystalline modification of pure B ...

The Central Limit Theorem.—If 4>i,4>a, we identically distributed, statistically independent random variables having finite mean and variance, then... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

Finally, an infinite set of random vectors is defined to be statistically independent if all finite subfamilies are statistically independent. Given an infinite family of identically distributed, statistically independent random vectors having finite means and covariances, we define their normalized sum to be the vector sfn, , sj where... 

At isotopic equilibrium, the tracer will be distributed statistically ... 

Taking a value of 107 for N would mean that in our galaxy (with its perhaps 100 billion stars), there could be several million planets with life forms capable of interstellar communication. However, if these were distributed statistically, the nearest would still be 200 light years away from Earth. One point is important the term probability used in the Drake equation is interpreted in the sense of subjective probability (a term from the nomenclature used by statisticians and probability theorists), as the numerical value of this probability is determined only by the experience of the scientist concerned (Casti, 1989). Casti also provides more information on the Drake factors (apart from the factor fs) in the chapter Where are they then In summary, we can say that the Drake equation is a first attempt to quantify the ETI problem in order to move from the area of science fiction and pure speculation to that of serious scientific debate. 

It should be emphasized that the deviations from the logarithmic correlation obey the normal distribution statistics, allowing one to apply convenient statistical procedures to analytical measurement results. As will be shown below, the logarithmic linear correlation rule was observed for all types of geo-chemical samples, i. e., snow, air, water, and soil. The correlation analysis of the elemental composition of melted-snow fractions showed with confidence level 95% that no significant discrepancy exists between the element composition within the correlation curves (see Table 1) and the corresponding variances are thus homogeneous for any two randomly chosen samples (i. e., points of the territory). 

Liquid crystals are mainly used for decorative purposes in cosmetics. Cholesteric liquid crystals are particularly suitable because of their iridescent color effects, and find applications in nail varnish, eye shadow, and lipsticks. The structure of these thermotropic liquid crystals changes as a result of body temperature, resulting in the desired color effect. In recent times, such thermotropic cholesteric liquid crystals have been included in body care cosmetics, where they are dispersed in a hydrogel. Depending whether this dispersion requires stirring or a special spraying process, the iridescent liquid crystalline particles are distributed statistically in the gel (Estee Lau-... 

Figures 10 and 11. The molecular weight M(v) can then be used to generate molecular weight distribution statistics as summarized in Table I.

Table I indicates good agreement between the molecular weight distribution statistics obtained by coupled GPC/Viscometer method and the nominal values for t BS 706. The discrepancy between the Mark-Houwink parameters obtained here and the reported values for polystyrene standard ( ) in THF at 25°C (i.e., a = 0,706 and k = 1.60 x 10 ) may in part be due to the uncertainty involved in the determination of the dead volume between DRI and viscometer detectors. Our simulation studies over a range of dead volume values (0 to 120u)l) showed that a and k are quite sensitive to the dead volume between the detectors. Larger dead volume results in smaller o and larger k values. This is a direct result of a clockwise rotation of log [q] vs. log M(v) curve (Figure 12) which occurs when the dead volume correction is applied in quantitative analysis. The effect on the molecular weight statistics, however, appeared to...

Detailed structure determinationshave confirmed that in all cases the lattices are built up of eight-membered crown-shaped ring molecules like Sg and Se as was expected due to the morphology of the crystals. All the structures were found to be disordered with sulfur and selenium distributed statistically... 

All choices should be questioned before the final analysis result is adopted. This includes data manipulation methods for dealing with outliers choice of sampling distribution, statistic, or model implementation approaches and number of samples in the simulation. These choices should be well documented in the description of the analysis. 

This hypothesis does not agree with the HRMAS Si-NMR results of Bodart et al. [3], who found that the Al atoms in chabazite are distributed statistically. However, the low resolution of the Si-NMR spectra and the superimposition of different Si(nAl) lines for zeolites of the chabasite group with two independent T sites make these results uncertain. 

Assuming that Ti(IV) is distributed statistically in all tetrahedral positions, it can be easily seen that even for crystallite sizes of 0,2 m the great majority of T1(IV) is located inside the pore structure. Assuming that every Ti(IV) is a catalytic centre with equal activity, diffusion limitations for molecules of different sizes should be observed. This is in fact the case. It has been shown [27] that the rate of oxidation of primary alcohols decreases regularly as the chain length increases, while for iso-butyl alcohol a sudden drop in the rate is observed. Also the reactivity order of olefins on TS-1 is different from the order observed with homogeneous electrophilic catalysts, while as already indicated very bulky molecules are unreactive when TS-1 is used as the catalyst. All these facts can only be interpreted as due to diffusion limitations of the larger molecules, which means that the catalytic sites are located inside the pore structure of the solid. 

This "secondary" molecular weight calibration curve was fit to a polynomial over the retention volume range of the sample. Then the molecular weight distribution statistics are calculated from this "secondary" calibration curve and the DRI trace of the sample under... 

Snedecor GW, Cochran WG (1967) Variance test for homogeneity of binormal distribution. Statistical methods, 6th edn. Iowa State University Press, Ames, pp 240-241... 

A factorial design is a very general scheme to investigate the influence of independent variables on responses. When the response is normally distributed, statistics like the /-test can be used for inference purposes. The ratio SIR as a response can be analyzed with a factorial design but some care must be taken with interpreting the results. Generally, an approximate... 

Area standard deviation 0.06 mm2 Particle distribution statistics... 

At one stage this vibrational photochemistry held the promise of mode selectivity , the possibility of tuning the excitation wavelength to excite one particular vibrational mode in a polyatomic molecule. It would then be possible to select one particular bond for dissociation, simply by changing the wavelength of the irradiation beam according to the vibrational frequency (mode) of that bond. In practice this mode selectivity has not been observed. No matter which specific vibration is excited by multiphoton IR absorption, the energy is very quickly distributed statistically over all the available vibrational modes, so that the weakest bonds dissociate faster. 

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