Geometric Uncertainties

001 A [14]. The coordination number of a H2O molecule varies instantaneously from two [15] to four or even greater [16]. The geometric structure of the weekly ordered H2O liquid was under debate mainly between the monophase of tetrahedrally coordinated configuration with thermal fluctuation [17-20] and the mixed-phase of low- and high-density fragments with thermally modulated fragmental ratios [15, 21, 22]. 

Scientific laws indicate that focusing on the statistic mean of the complete set of all the correlated parameters could be more realistic than on one of them alone at a time in coping with such a strongly correlated and fluctuating system. One may 

Relaxation of the Chemical Bond, Springer Series in Chemical Physics 108, DOI 10.1007/978-981-4585-21-7 38, Springer Science+Business Media Singapore 2014 

38 Molecular Size, Separation, Packing Order, and Density 

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The dc electrical resistivity p was measured versus temperature for a crystal of KjCfio (Fig. 1). Crystals from different preparation batches yielded similar results. Near room temperature the resistivity is about 5 mO-cm, comparable to that obtained for K3C60 films at room temperature (1. 3). However, because of geometrical uncertainties associated with the contact pads, the absolute value of the resistivity should be considered reliable only to within a factor of 2. Below room temperature, p(T) falls in a metal-like fashion with distinct curvature. At T the resistivity drops abruptly to zero, with a transition width < 200 mK. The inset in Fig. 1 shows p(T) near in detail. The temperature has been swept slowly (—50 mK per minute) near the transition temperature showing no difference in Tj between cooling and warming. 

For a reactor diffractometer the resolution width, AQ, arises because the acceptance angle of the colli-mation around the monochromator, the size of the sample and the acceptance angle of the detector system are all greater than zero and also because the monochromator crystal has a mosaic spread. For a treatment of the Q-resolution in terms of independent parameters, these factors may be considered to lead to a geometrical uncertainty in angle. Ad, and an uncertainty in wavelength, AA. Differentiating the... 

For a time-of-flight diffractometer the three major contributions to the Q-resolution are due to the geometrical uncertainty in angle, A6, the uncertainty in flight path, AL, and the uncertainty in time-of-flight, At, leading to the following expression for the total combined resolution width ... 

Chemical analyses should be provided for all anodes used in the offshore and harbor area, together with results for current content in A h kg and current output in amperes [2,3]. The geometric shape and the number of anodes required is determined by these parameters. Expensive calculations for design based on grounding resistances are made only in exceptional cases because in practice there are too many uncertainties and the number and mass of the anodes have to be quoted with a corresponding safety factor. 

In Science, every concept, question, conclusion, experimental result, method, theory or relationship is always open to reexamination. Molecules do exist Nevertheless, there are serious questions about precise definition. Some of these questions lie at the foundations of modem physics, and some involve states of aggregation or extreme conditions such as intense radiation fields or the region of the continuum. There are some molecular properties that are definable only within limits, for example, the geometrical stmcture of non-rigid molecules, properties consistent with the uncertainty principle, or those limited by the negleet of quantum-field, relativistic or other effects. And there are properties which depend specifically on a state of aggregation, such as superconductivity, ferroelectric (and anti), ferromagnetic (and anti), superfluidity, excitons. polarons, etc. Thus, any molecular definition may need to be extended in a more complex situation. 

Other means like the median (see Eq. (4.22)) or the geometrical mean (see Eq. (4.18)) etc. have to be reported in a similar way together with the belonging uncertainty interval, e.g.,... 

Geometrical means have unsymmetrical uncertainty intervals which are characterized by a dispersion factor v (see Sect. 4.1.2, Eq. (4.20)) and a covering factor k (see Sect. 4.2). Corresponding results should be given in the form... 

Each uncertainty factor of 3 is actually the geometric mean of 10, which is 3.16 hence, 3.16x3.16=10. 

Whereas, from all of these informative 1H-PHIP-NMR spectra, the structure of the dihydride intermediate (including geometric details about peculiar bonding therein) can be determined rather exactly and reliably, a degree of uncertainty remains as to whether this intermediate represents the major or the minor diastereomer according to the nomenclature of Halpern [27]. This is the consequence of different kinetic constants associated with the two alternative cycles with different stereochemistry, and which accounts for the major and minor reaction product (Fig. 12.18). In fact, it is the difference in the rate... 

Here, a geometric-based approach capable to regulate the total amount of organic substrate St in spite of uncertainties in either the influent composition or the process kinetics is proposed. In order to emulate real operating conditions, the following assumptions are done. 

The uncertainty in the estimate of Pi is smaller (s, = s x5.04), and the uncertainty in the estimate of Pn is smaller still (Sj, = s x0.04). The geometric interpretation of the parameters Pi and Ph in this model is not straightforward, but Pi essentially moves the apex of the parabola away from Xi = 0, and pn is a measure of the steepness of curvature. The geometric interpretation of the associated uncertainties in the parameter estimates is also not straightforward (for example, P, Pi, and p,i are expressed in different units). We will simply note that such uncertainties do exist, and note also that there is covariance between bg and hi, between and b,i, and between hi and hn. 

At a very basic level, the shapes of the normalized uncertainty and normalized information surfaces for a given model are a result of the location of points in factor space simply because carrying out an experiment provides information - that is, information is greatest in the vicinity of the design. But at a more sophisticated and often more important level, the shapes of the normalized uncertainty and normalized information surfaces are caused by the geometric vibrations of the response surfaces themselves - the more rigidly the model is pinned down by the experiments and the less it can squirm and thrash about, then the less will be the uncertainty and the greater will be the information content. 

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

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