Interior Equilibrium

Since f p) 0, the constant term is positive, so the Routh-Hurwitz criterion (Appendix A) says that Ec will be asymptotically stable if and only if 

This condition is obviously difficult to verify in general. It will be used later for the special case f p) = e P. 

With our standing hypotheses that all rest points are hyperbolic, Proposition 4.1 implies that if E does not exist then either E2 is asymptotically stable (all eigenvalues of the Jacobian are negative) or Ei exists and is asymptotically stable (all eigenvalues are negative). The results to follow establish that if , i—, 2, is asymptotically stable then it attracts all solutions (is globally asymptotically stable for positive initial conditions). Therefore, when E does not exist, one of the rest points E or E2 attracts all solutions of (3.2). 

By Theorem B.4, Xi(t) and X2 t) can be compared to the solutions of the chemostat system (5.2) of Chapter 1. (See the proof of Proposition 3.2 with the inequalities reversed.) The break-even concentrations are + — 1) and A2. From the comparison, one may conclude that 

Since e may be chosen arbitrarily small, it follows that liminf, Xi(0 1-Aq. 

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Remark. If Ai-I-A2>l then there is no positive solution of (3.2) and hence no interior equilibrium. In this case E2 is a globally asymptotically stable rest point. If A1-I-A2 < 1, there exists a unique interior rest point and E2 is unstable. 

Ci = interior equilibrium concentration, and V = volume of enclosed space. 

Interior equilibrium is the one in which first-order conditions hold for each player. The alternative is boundary equilibrium in which at least one of the players select the strategy on the boundary of his strategy space. 

The other formulation of the principle of virtual work for mechanical systems requires the introduction of virtual loads instead of virtual displacements. Therefore, only those variations of external loads and stress tensor are considered admissible that are compatible with the equations of equilibrium inside the mechanical system and on the boimdary. The interior equilibrium of Eq. (3.14) for the virtual loading leads to the following form ... 

Expansion waves are the mechanism by which a material returns to ambient pressure. In the same spirit as Fig. 2.2, a rarefaction is depicted for intuitive appeal in Fig. 2.7. In this case, the bull has a finite mass, and is free to be accelerated by the collision, leading to a free surface. Any finite body containing material at high pressure also has free surfaces, or zero-stress boundaries, which through wave motion must eventually come into equilibrium with the interior. Expansion waves are also known as rarefaction waves, unloading waves, decompression waves, relief waves, and release waves. Material flow is in the same direction as the pressure gradient, which is opposite to the direction of wave propagation. 

In the third sequence, the diastereomer with a /i-epoxide at the C2-C3 site was targeted (compound 1, Scheme 6). As we have seen, intermediate 11 is not a viable starting substrate to achieve this objective because it rests comfortably in a conformation that enforces a peripheral attack by an oxidant to give the undesired C2-C3 epoxide (Scheme 4). If, on the other hand, the exocyclic methylene at C-5 was to be introduced before the oxidation reaction, then given the known preference for an s-trans diene conformation, conformer 18a (Scheme 6) would be more populated at equilibrium. The A2 3 olefin diastereoface that is interior and hindered in the context of 18b is exterior and accessible in 18a. Subjection of intermediate 11 to the established three-step olefination sequence gives intermediate 18 in 54% overall yield. On the basis of the rationale put forth above, 18 should exist mainly in conformation 18a. Selective epoxidation of the C2-C3 enone double bond with potassium tm-butylperoxide furnishes a 4 1 mixture of diastereomeric epoxides favoring the desired isomer 19 19 arises from a peripheral attack on the enone double bond by er/-butylper-oxide, and it is easily purified by crystallization. A second peripheral attack on the ketone function of 19 by dimethylsulfonium methylide gives intermediate 20 exclusively, in a yield of 69%. 

In the equilibirum capture model, on the other hand, there is a dynamic equilibrium between the growing oligomers and the surface of the particles as well as the possibility of some interchange with the interior of the particles. 

For an ice sheet of thickness H in equilibrium with a climate supplying accumulation at a rate a (thickness of ice per imit time), the vertical velocity near the ice-sheet surface is a and this velocity decreases to zero at the ice-sheet bed. A characteristic time constant for the ice core is H/a. The longest histories are therefore obtained from the thick and dry interiors of the ice sheets (particularly central East Antarctica, where H/a = 2 X 10 yrs). Unfortunately, records from low a sites are also low resolution, so to obtain a high-resolution record a high a site must be used and duration sacrificed (examples are the Antarctic Peninsula (H/a = 10 ) and southern Greenland H/a = 5 x 10 )). 

Suppose this reaction is occurring in a CSTR of fixed volume and throughput. It is desired to find the reaction temperature that maximizes the yield of product B. Suppose Ef > Ef, as is normally the case when the forward reaction is endothermic. Then the forward reaction is favored by increasing temperature. The equilibrium shifts in the desirable direction, and the reaction rate increases. The best temperature is the highest possible temperature and there is no interior optimum. 

At a fixed temperature, a single, reversible reaction has no interior optimum with respect to reaction time. If the inlet product concentration is less than the equilibrium concentration, a very large flow reactor or a very long batch reaction is best since it will give a close approach to equilibrium. If the inlet product concentration is above the equilibrium concentration, no reaction is desired so the optimal time is zero. In contrast, there will always be an interior optimum with respect to reaction time at a fixed temperature when an intermediate product in a set of consecutive reactions is desired. (Ignore the trivial exception where the feed concentration of the desired product is already so high that any reaction would lower it.) For the normal case of bin i , a very small reactor forms no B and a very large reactor destroys whatever B is formed. Thus, there will be an interior optimum with respect to reaction time. 

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