Semi-stable equilibrium state

It is obvious that the parameter values for which there are only two equilibrium states are bifurcation points because one of the roots of the right-hand side must then be a multiple root (see Figs. 11.2.8(c) and (d) for I3 < 0 and Figs. 11.2.9(c) and (d) for I3 > 0). This root corresponds to a semi-stable equilibrium state that either disappears, or is split into two equilibria, following an arbitrarily small variation in the parameters. 

The amount of available energy which a substance has is relative and depends upon the choice of a dead state. The fundamental dead state is the state that would be attained if each constituent of the substance were reduced to complete stable equilibrium with the components (8,9,10) in the environment—a component-equilibrium dead state. (Thus, one may visualize the available energy as the maximum net work obtainable upon allowing the constituents to come to complete equilibrium with the environment.) The equilibrium is dictated by the dead state temperature T0 and, for ideal gas components, by the dead state partial pressure p-jg of each component j. (The available energy could be completely obtained, say in the form of shaft work, if equilibrium were reached via an ideal process—no dissipations or losses—involving such artifices as perfectly-selective semi-permeable membranes, reversible expanders, etc. (9,10,11).)... 

Theorem 11.4 shows essentially that outside the narrow sector bounded by 1 and 2, the bifurcation behavior does not differ from that of equilibrium states (see Sec. 11.5) fixed points correspond to equilibrium states, and the invariant curves correspond to periodic orbits. However, the transition from the region D2 to the region Dq occurs here in a more complicated way. In the case of equilibrium states the regions D2 and Do are separated by a line on which a stable and an imstable periodic orbits coalesce thereby forming a semi-stable cycle. In the case of invariant closed curves, the existence of a line corresponding to a semi-stable invariant closed curve is possible only in very degenerate cases (for example, when the value of R does not depend on as... 

This bifurcation diagram for the equilibrium state with two zero characteristic exponents had been known for a long time. However, there remained a problem of proving the uniqueness of the limit cycle. In other words, one must prove additionally that there are no other bifurcational curves besides Li,...,L4 (namely, curves corresponding to semi-stable limit cycles). This problem was independently solved by Bogdanov [33] and Takens [146] with whom this bifurcation is often named after. 

Cytisine is a tricyclic quinolizidine alkaloid that binds with high affinity and specificity to nicotinic acetylcholine receptors. In principle, this compound can exist in several conformations, but semi-empirical calculations at the AM 1 and PM3 levels have shown that stmctures 19 and 20 are more stable than other possible conformers by more than 50 kcalmol-1. Both structures differ by 3.7 kcalmol 1 at the AMI level and 2.0 kcalmol 1 at the PM3 level, although this difference is much smaller when ab initio calculations are employed <2001PJC1483>. This conclusion is in agreement with infrared (IR) studies and with H NMR data obtained in CDCI3 solution, which are compatible with an exo-endo equilibrium < 1987JP21159>, although in the solid state cytisine has an exo NH proton (stmcture 19) (see Section 12.01.3.4.2). 

The 17e metal-based radical Ta 3(CO)4(dppe) was formed via hydrogen atom extraction from TaIH(CO)4(dppe).659 In solution, this radical abstacted halogen atoms from many organic halides RX to give TaIX(CO)4(dppe). Ta°(CO)4(dppe) existed in solution as an equilibrium mixture of monomer and dimer [(dppe)(CO)3Ta](/i2-CO)2. The latter is the form stable in the solid state. While no crystal structures are available, DFT calculations indicated a stable pseudo-octahedral stereochemistry for the monomer and several possible (/i2-CO)2 structures for the dimer.656 Each of the latter featured linear, semi-bridging carbonyls supporting a weak, delocalized Ta—Ta interaction. 

The piezoelectricity in amorphous polymers differs from that in semi-crystalline polymers and inorganic crystals in that the polarization is not in a state of thermal equilibrium, but rather a quasi-stable state due to the ffeezing-in of molecular dipoles. As mentioned by Broadhurst and Davis (55), four criteria are essential to make an amorphous polymer exhibit piezoelectric behavior. First, molecular dipoles must be present. As seen in Table 1, these dipoles are typically pendant to the polymer backbone as are the nitrile groups in PAN, PVDCN-VAC, and (p-CN) APB/ODPA. However, the dipoles may also reside within the main chain of the polymer such as the anhydride units in the (P-CN) APB/ODPA polyimide. In addition to a dipole moment X, the dipole concentration N (number of dipoles per unit volume) is also important in determining the ultimate polarization, P , of a polymer. 

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