Analytic proof

There is no analytic proof of the Jahn-Teller theorem. It was shown to be valid by considering all possible point groups one by one. The theorem is traditionally treated within perturbation theory The Hamiltonian is divided into three parts... 

Stroboscopic Method.—This method was developed a few years ago by the writer in collaboration with M. Schiffer, and is based on the transformation theory of differential equations.20 We shall give here only the heuristic approach to this method, referring for its analytical proof to other published material.21... 

An analytical proof of the first of these limiting cases follows directly from equations 15.3-3 and -4. As R 0, c Ao - cAo, and 17q, - that for a PFR without recycle. An analytical proof of the second limiting case does not follow directly from these two equations. As R °°, c Ao -> cA1 (from equation 15.3-3), and V/q, -> -(w)(0) (from equation 15.3-4), which is an indeterminant form. The latter can be evaluated with the aid of L Hdpital s Rule, but the proof is left to problem 15-18. 

The limiting cases for equation 15.3-11 are as discussed in Section 25.3.1 for constant density. That is, as R - 0, V is that for a PFR without recycle as R - , V is that for a CSTR (see problem 15-18 relating to an analytical proof of the latter). A graphical construction similar to that in Figure 15.8, but in which l/( -rA) is plotted against /A, can be used to interpret equation 15.3-11 in a manner analogous to that of equation 15.3-4 for constant-density. 

Langner HI, Hillinger HG. 1971. [Taste variation of the egg caused by the deodorant p-dichlorobenzene.] Analytical proof. Berlin Muenchen Tierairztl 84 851. (German)... 

Nothing is known about whether anions can enter the structure of synthetic goethites. Under ambient conditions in an aqueous system. Si interferes with the crystallization of goethite in both acid and alkaline media but, so far, no analytical proof of incorporation in the structure has been obtained (Glasauer, 1995). Furthermore, there is hardly any information about the replacement of Fe in natural goethites by cations other than A1 and by anions. Congruent release of Fe with... 

Results of Ref 5 show that integrating these equations generates coarsening the time dependence of the lateral size, Vc, of the mounds scales with a power of time, Vc f, with exponent n = 1/4. This exponent is associated with the leading A term. Similarly, Stroscio et ah found n = 1/6 numerically when only the A term was present. A detailed analytical proof is given by Golubovic. ... 

Watson and Cherniak (1997) used numerical solutions to investigate the critical cooling rate for the center concentration of zircon core to be affected. Equation 5-142 is similar to their result and may be viewed as the analytical proof of it. The general equation (Equation 5-141) may be applied to diffusion of other species in zircon, as well as other minerals. 

Small spheres can absorb more than the light incident on them. The truth of this assertion follows from simple calculations using (12.11). But analytical proofs have less force, to some minds at least, than geometrical proofs. For this reason, therefore, we consider the interaction of light with a small sphere in a way which, as far as we know, has not been done before. The result is not new knowledge but rather new evidence supporting and firmly implanting in our minds what we already know. 

There is at present no rigorous analytical proof of the asymptotic structure of the KS exchange-correlation potential vxc(r) at a metal surface. For... 

In analytical chemistry, combination of results of several different analytical techniques gives the most reliable results. In a well-equipped off-site laboratory, results from FTIR, MS, and NMR, with other analytical data, can be combined to produce reliable, unambiguous analytical proof to support the chemical disarmament. 

The analytical proof assumes that the radius of curvature of the surface is small compared with the thickness of the transition layer between two phases, otherwise the position arbitrarily chosen for XY is of appreciable importance.3 Gibbs proves that the temperature and chemical potentials of the components are uniform throughout the system when equilibrium exists but we shall take this here as self-evident, as indeed it is from a physical standpoint, considering that temperature and potentials are measures of the escaping tendencies of heat and of each component from the different phases, and therefore equalize themselves automatically. The pressures in the different phases are not, however, equal unless the surfaces are strictly plane, as was pointed out in Chap. I, 15. 

Gibbs s analytical proof of the adsorption formula is, mutatis mutandis, analogous to the analytical deduction of the Gibbs-Duhem relation both depend on the integration of the formula for the increment in energy, followed by differentiation and comparison of the result with the original formula. 

This estimate is independent of a particular location in phase space. This implies that all points of phase space, in particular the period-1 points, are linearly unstable. Thus we have proved that the positively kicked hydrogen atom does indeed not possess any first order eUiptic islands in phase space. It is possible to extend this proof to period-iV points and to show that all period-N points, N integer, are Unearly unstable (Bliimel (1993c)). This implies that the positively kicked hydrogen atom is completely chaotic. We emphasize that, as far as we know, the kicked hydrogen atom is the only model for a physically realizable system where a numerically motivated chaos conjecture was followed up by an analytical proof. In this sense the kicked hydrogen atom is a most remarkable system. 

Computer simulations show that second-order decay occurs even when the column sums of P21 are not all the same in fact, they may differ by orders of magnitude, but no analytic proof of this is available yet. 

In Exercises 2.6.2 and 2.7.7, you were asked to give two analytical proofs that periodic solutions are impossible for vector fields on the line. Review these arguments and explain why they don t carry over to vector fields on the circle. Specifically which parts of the argument fail ... 

There are no analytical proofs for the exact structure of polycyclic naphthenic compounds with number of rings over five. However, based on the results of mass spectral analysis of heavy oil fractions, it can be said that there are polycyclic naphthens with seven or eight rings in their structure. At the moment, it is very difficult to be specific in analytical terms of the exact chemical structure of such molecules. 

Turning to the related problem of the constitution of water, Lavoisier first of all confirmed Cavendishes synthetic experiments in which water was formed from its two con stituent elements, hydrogen and oxygen. Next, he devised an analytical proof by passing steam over iron filings contained in an iron gun-barrel heated to redness as he had surmised, hydrogen issued from the gun-barrel, leaving a solid calx, or oxide as he now called it, of iron. 

Unfortunately, there is no simple analytic proof that this term goes to zero, since conditions at any instant depend on the whole history of the process, as given by two coupled non-linear differential equations in CA and 6. Perhaps the best mathematical proof available is therefore the obvious disappearance of the second term in equation 5.56 as Vv approaches zero. In this limit the TS-CST-SSR behaves as a TS-PF-SSR so that equations 5.55 and 5.56 are the general equations for both cases. 

Similarly, the effect of increasing the flow rate is not immediately clear. As fjj increases, the difference between CA0 and CA will decrease in such a way as to reduce changes in the first term on the right side of equation 5.48. At the same time dCA/dt should decrease since there will be Less variability in CA with clock time. This suggests that the accumulation term should disappear for high flow rates, but again, an analytic proof of this is not simple. 

It is rather straightforward to employ numerical methods and demonstrate that the effectiveness factor approaches unity in the reaction-rate-controlled regime, where A approaches zero. Analytical proof of this claim for first-order irreversible chemical kinetics in spherical catalysts requires algebraic manipulation of equation (20-57) and three applications of rHopital s rule to verify this universal trend for isothermal conditions in catalytic pellets of any shape. 

Compounds formerly named nature identical are to be de ned by law since some years as synthetic products. These molecules have been found in nature by analytical proof and are published in an authorized scienti c journal. The term nature identical is no more valid and allowed in Europe in relation to avor and fragrance substances. Such molecules are identical with those appearing in nature but are produced by a synthetic process. These processes contain undesired by-products. The use of such synthetic compounds is easy to detect, as by-products from manufacturing can easily be detected by GC-MS systems. On the other hand, chiral separation will help to con rm adulteration. 

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