Natural consequences

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

Notice that P f) has a maximum at f — 1/2. The presence of this maximum can be directly traced back to the length-2 loop of the STG shown above, which yields a period-2 component in the spatial configurations. The half-width of the peak (=log(l/g)) is seen to decrease as q decreases, a natural consequence of the period-2 part of the STG being visited more often as q increases. Figure 6.4 shows in empirical estimate of the power spectrum for rule R56 obtained for a size iV = 2 = 2048 system. 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

The DDT mechanism for Case D is identical to that proposed for Case C. However, the model for DDT in Case D will differ from that of Case C because of compaction of the porous bed If from the above discussion the variation of is traced from case to case, it is seen that a minimum in the 8-compaction plane is a natural consequence of the effects of charge permeability and the proposed driving force for DDT The above model for DDT is based on exptl observation with granular and cast HE. However, as already mentioned, DDT in SP and HE should have many similarities. This contention is supported by the observations of Roth (Ref 17) and Wachtel (Ref 29a). Roth s smear camera... 

When using the eighteen electron rule, we need to remember that square-planar complexes of centers are associated with a 16 electron configuration in the valence shell. If each ligand in a square-planar complex of a metal ion is a two-electron donor, the 16 electron configuration is a natural consequence. The interconversion of 16-electron and 18-electron complexes is the basis for the mode of action of many organometallic catalysts. One of the key steps is the reaction of a 16 electron complex (which is coordinatively unsaturated) with a two electron donor substrate to give an 18-electron complex. 

Early attempts to explain the trigger for the tulip flame focused on the pressure wave/flame interactions. This was a natural consequence of the well-documented vibratory behavior of flames seen in the very first streak images recorded [2], and the images of Markstein (like that in... 

The natural consequence of the greenhouse effect is to keep the Earth s temperature over 20 °C higher than it would be if the atmosphere were completely composed of N2 and O2, enabling life to survive. 

For a,co-dicarboxylic acids, the anti-Langmuir concentration profiles are pronounced the most for the compounds with the longest ahphatic chains, as shown in Figure 2.8. From the densitograms obtained, it is apparent that a,(0-dicarboxylic acids are more prone to form anti-Langmuir concentration profiles than monocar-boxylic acids. This suggests that lateral interactions are more effective with dicarboxylic acids than with monocarboxylic ones, which seems a natural consequence... 

An alternative approach to the solution of the system dynamic equations, is by the natural cause and effect mass transfer process as formulated, within the individual phase balance equations. This follows the general approach, favoured by Franks (1967), since the extractor is now no longer constrained to operate at equilibrium conditions, but achieves this eventual state as a natural consequence of the relative effects of solute accumulation, solute flow in, solute flow out and mass transfer dynamics. 

From his early youth, under his father s influence, K. H. Meyer had retained a keen interest in biological problems, as was evident from his study of the phenomena of narcosis, which he pursued during his stay in industry. As a natural consequence, he extended his thoughts to biological problems, and evolved a quantitative theory of muscular contraction (in collaboration with Picken), based on analogies with the elasticity of rubber. With J. F. Sievers, the permeability of synthetic membranes was investigated, and a mathematical treatment of the phenomenon was advanced which was later applied to living membranes. 

Most commercial Fischer-Tropsch refinery designs (Figures 18.1 to 18.9) included the co-production of chemicals with transportation fuels. The chemicals potential of Fischer-Tropsch syncrude has been pointed out repeatedly.38,47,65 66 This is a natural consequence of the properties of Fischer-Tropsch syncrude, that is, richness in linear hydrocarbons, olefins (especially linear a-olefins), and oxygenates. Furthermore, it is sulfur-free and nitrogen-free, which enables access to synthetic routes sensitive to such compounds. 

The formation of carbon chains seems to be a natural consequence of the ion-molecule reactions and would be expected to have low activation energies. We have already mentioned the neutral-neutral radical chemistry that may also lead to carbon chain chemistry. 

The formation of photochemically driven layers in the atmosphere, such as the ozone hole a natural consequence of atmospheric structure... 

In summary, the H-acceptor pairs appear to be very similar to their silicon counterparts, which we have discussed in depth. The H-donor pairs are similar in that the H occupies a silicon-antibonding site however, this is an antibonding site to the defect and not to the host as is found in silicon. It is also interesting to note that the computed hydrogen frequencies appropriate to the latter pairs are better described by theory than the silicon counterparts discussed earlier. It is not clear whether this is a consequence of the electronic-structure method used here, a natural consequence of the differences between the silicon and compound-semiconductor hosts, or simply an accident. 

Thus, free valence persists whenever an atom or a radical undergoes a unimolecular reaction or interacts with valence-saturated molecules (possessing an even number of electrons). This is a natural consequence of conservation of the number of electrons in chemical reactions. Therefore, free valence cannot persist when a radical reacts with a radical. Both reactants have an odd numbers of electrons, and the product formed has an even number of electrons, for example,... 

Luttinger-Tisza method is burdened by independent minimization variables, while analysis of the values of the Fourier components F k) makes it possible to immediately exclude no less than half of the variable set and to obtain a result much more quickly. Degeneracy of the ground state occurs either due to coincidence of minimal values of Vt (k) at two boundary points of the first Brillouin zone k = b]/2 and k = b2/2, or as a result of the equality Fj (k) = F2 (k) at the same point k = h/2. The natural consequence of the ground state degeneracy is the presence of a Goldstone mode in the spectrum of orientational vibrations.53... 

Shulga V., Delayed neuro-endocrine toxicity indused by organophosphorus compounds-natural consequence of poisonous substances application for terrorist purpose , In Medical Aspects of Chemical and Biological Terrorism -Chemical Terrorism and Tramautism, Alexander Monov and Christophor Dishovsky, eds, Publishing House of the Union of Scientists in Bulgaria, 2005 ( in press). 

Another helpful exercise asks the client to walk through a behavior chain to its natural consequences. Many times a client will act without thinking about what the eventual outcome of such an act might be. For example, a client may tell you that he wants to visit a friend at the friend s home. You remember this is one of... 

As a natural consequence of the second law of photochemistry, the sum of the primary quantum yields cannot be greater than unity. 

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