Inequality degeneration

Estimations show that condition (20) usually holds true for reasonable values of the parameters in the absence of degeneration. At the same time, for highly doped semiconductors and also in the formation of an accumulation or inversion layers (i.e., for sufficiently large values of sc ) Lsc may become so small that inequality (20) is not satisfied. 

The convexity of the exponential functions then implies the inequality (Q) > 0, which resembles the second law. The integral fluctuation theorem implies that there are trajectories for which Q is negative with the exception of the degenerate case, p(Q) = d(Q) leading to violation of the second law. One constraint on the probability distribution p(Q). If p(Q) is a Gaussian, the integral flucmation theorem implies the relation ((Q — (O)) ) = 2(Q) between the variance and the mean of Q (Seifert, 2012). 

One can derive the critical phase equation in a much simpler form in many specific cases using Equations 1.1.2--30... 34 as the stability conditions rather than Equation 1.1.2-12. The subsequent line of argument should remain unchanged. The molar Helmholtz potential F is convenient to use for a one-component system with Equations 1.1.2-31 and 1,1.2-32 to express the thermal and mechanical stability conditions, respectively. As the second inequality is stronger, one can confine oneself with it. In this case, the determinant D/r degenerates into one term... 

For the degenerate branching chain reactions the critical phenomena are expressed more distinctly at better fulfillment of the following inequalities... 

The experiments that are most relevant to the present discussion involved the cross metathesis of mixtures of 1-pentene and l-pentene-catalyst system, Calderon and coworkers monitored isotopic scrambling to form the d2 and mass spectrometry). After only short reaction times, the mixture reached 70% of the theoretical scrambling limit, with almost no formation of any octene products. To explain pentene s proclivity toward generating itself rather than octene in the presence of metathesis catalysts, Calderon suggested that either the mono-P-substituted metallacycle was favored over the mono-a-substituted isomer, or the a,a -disubstituted metallacycle was preferred relative to the a,p-substituted one (Scheme 10.4). In other words, degenerate metathesis can be performed by either M=CH2 or M=CHR species. These pathways were indistinguishable, but the results proved that one of the two inequalities must be true [5]. 

When one of the inequalities (8.7) becomes an equality, that is, when the largest of the three tensions equals the sum of the two smaller, the Neumann triangle degenerates to a line the vertex that was previously opposite the longest side comes to lie on that side, as the altitude of the triangle, measured from that vertex to that side, vamshes. Suppose is the largest tension. Then in the limit we are now contemplating... 

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