Strong local minimum

The feasible region is depicted in Figure 6.4 and is nonconvex due to the bilinear inequality constraint. This problem exhibits a strong local minimum at (x, y) — (4,1) with objective equal to -5, and a global minimum at (x, y) = (0.5,8) with objective equal to -8.5. 

When the vapor from a boiling solution of 6M HC1 is expanded, the detected cluster ions lack HC1 as evidenced by the absence of the characteristic m+2 isotope due to Cl. On the other hand, the observed H+(H20)n distribution is noticeably different than that obtained in the expansion of pure water. For instance, the usual prominent discontinuities at H+(H20)4 anc H+(H20)2 are not evident and a slight local minimum in intensity occurs at H+(H20) 3. The indication is that HC1 is initially present in the neutral cluster but that ionization leads to "boiling off" of HC1. The molecular ions H 5C1+ and H C1+ are observed. Expansion of the vapor over NH S also results in an ionized cluster distribution in which one of the expected components, namely H2S, is absent (7 ). Quite unlike the distribution obtained from the expansion of pure ammonia, a strong local minimum similar to the H+(HN03)(H20)n distribution is present. 

Note that, in general cases, the misfit functional (5.2) may have several minima. We will distinguish three types of minimums strong local minimum, weak local minima, and global minimum. 

Definition 23 The vector m , is a strong local minimum of the functional 0(m), if there exists 6 > 0 such that... 

Figure 5-1 Three types of functional minima strong local minimum, weak local...

A condition for this system to have a strong local minimum at Sp = 0,Sq = 0 is that the Hessian matrix U"(q ) be positive definite. In case the eigenvalues of U"(q ) are all distinct and positive, then the strong local minimum property will also follow for q in relation to the original potential. 

The nonadiabatic hopping probability has been analysed as a function of time and correlated with individual vibrational modes (see Figure 10-14). After approximately 40 fs, a steep rise in the hopping probability of 9Me-keto G is observed marking the transition from the first, planar local minimum structure (a) to the second local minimum (b) from where the system relaxes into the global 5) minimum (c). The strong enhancement of nonadiabatic coupling for 9Me-keto G... 

MP2-FC/6-31G calculations reveal the symmetrically bridged 6-sila-2-norbomyl cation 1 not only to be a local minimum (Fig. 18), but also to be 17.2 kcal mof more stable than the 2-norbomyl cation (Eq. 3) at MP2-FC/6-31G + AZPE(SCF/6-31G 0.89). However, the inherently greater stability of silyl cations contributes to this difference. The Si NMR chemical shift of the bridging silicon atom, ca 1 ppm vs TMS (IGL0/H //MP2-FC/6-31G ), is very strongly shielded in comparison with ca 300 ppm expected for a free RSiH2 species [38]. Thus, the sila congener of the 2-norbomyl carbocation also possesses a nonclassical stracture which is reflected by its stracture as well as its NMR properties. 

Figure 5-1 illustrates the difference between the strong and weak local minima. Actually, a weak local minimum area corresponds to some plateau on the misfit curve (or surface, in general cases). 

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