Maximum necessary condition

Resin-based repeUents may be used alone or in combination with durable-press resins. They are widely used as extenders for fluorochemical repeUents. When used alone, several of the resin-based finishes require an acid catalyst and curing at temperatures above 150°C for maximum repeUency and durabUity. When coappUed with durable-press finishes, which themselves require a magnesium chloride catalyst, the catalyst and curing conditions for the durable-press finish provide the necessary conditions for the repeUent. 

These are only necessary conditions, as point ti may be a minimum, maximum, or saddle point. 

Systems are normally designed to work satisfactorily during maximum amhient conditions, and the condenser will he sized for this. In colder weather, the condensing temperature and pressure will fall and the resulting lower pressure difference across a thermostatic expansion valve may lead to malfunction. A drop of pressure difference to half the normal figure may reduce mass flow helow that required, and it will he necessary to prevent the condenser pressure from falling too low. 

The position of the meniscus within the micro-channel defines the type of temperature distribution. In the first case, when the meniscus is near the outlet, the temperature gradient of the vapor region is small. The rate of evaporation is determined mainly by the heat flux in the liquid region. Therefore, the necessary condition of the evaporation consists of the existence of the region (near the meniscus), where the water is overheated (its temperature is higher than the temperature of boiling). The heat losses to the inlet tank cause the existence of the temperature maximum. 

In addressing the issue of postglacial history of S. latifolia (or its progenitor), it is necessary to consider where it existed during the time when ice covered most of northern Europe. The species is not well adapted to survive in cold conditions (Thompson, 1973), which Mastenbroek (1983) pointed out, likely accounts for the absence of this species in the more northerly parts of Europe. He went on to say that even southern Europe may not have provided the necessary conditions for growth, and that the species may have occupied refugia in Northern Africa during maximum ice cover. This issue cannot be resolved on the basis of the data at hand. 

Examine the second term on the right-hand side of Equation (4.4) VTf(x ) Ax. Because Ax is arbitrary and can have both plus and minus values for its elements, we must insist that V/ (x ) = 0. Otherwise the resulting term added to/(x ) would violate Equation (4.5) for a minimum, or Equation (4.6) for a maximum. Hence, a necessary condition for a minimum or maximum of /(x) is that the gradient of/(x) vanishes at x ... 

With the second term on the right-hand side of Equation (4.4) forced to be zero, we next examine the third term (Axr) V2/(x )Ax. This term establishes the character of the stationary point (minimum, maximum, or saddle point). In Figure 4.17b, A and B are minima and C is a saddle point. Note how movement along one of the perpendicular search directions (dashed lines) from point C increases fix), whereas movement in the other direction decreases/(x). Thus, satisfaction of the necessary conditions does not guarantee a minimum or maximum. 

Let x be a local minimum or maximum for the problem (8.15), and assume that the constraint gradients Vhj(x ),j — 1,m, are linearly independent. Then there exists a vector of Lagrange multipliers A = (Af,..., A ) such that (x A ) satisfies the first-order necessary conditions (8.17)-(8.18). 

Examples illustrating what can go wrong if the constraint gradients are dependent at x can be found in Luenberger (1984). It is important to remember that all local maxima and minima of an NLP satisfy the first-order necessary conditions if the constraint gradients at each such optimum are independent. Also, because these conditions are necessary but not, in general, sufficient, a solution of Equations (8.17)-(8.18) need not be a minimum or a maximum at all. It can be a saddle or inflection point. This is exactly what happens in the unconstrained case, where there are no constraint functions hj = 0. Then conditions (8.17)-(8.18) become... 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

There is no path of maximum electron density between the interacting atoms which, according to Cremer-Kraka27 82,83, is a necessary condition for covalent bonding. However, interaction indices derived from the electron density distribution are as large as 30% of the bond order of a normal single bond. 

Interesting chemical properties were discovered in such bicyclic amides as quinuclidin-2-one (3).43-46 In this type of compound the axis of the nitrogen p electrons is orthogonal to the w electrons of the carbonyl group. As a result the necessary condition for conjugation, i.e., parallel axes of n and p electrons with maximum overlap, is not observed. This is why the conjugation of type... 

Condition (22) is identical to the equation of the line (11) in Fig. 7 for large t. Analysis of the working conditions of current furnace chamber constructions falls outside the scope of the present paper. There are a number of indications that only in a small part of their total volume does intensive chemical reaction occur since it is only in this small part that necessary conditions of good mixing of air, fuel and hot reaction products are created. We may expect that, under these conditions, with the thermal intensity referred to the entire volume, the observed maximum and minimum values of the thermal intensity will prove to have been underestimated. 

According to the counter ion condensation theory (Oosawa, 1971 Manning, 1978, 1980, 1985 Dautzenberg et al., 1994) 76% of the DNA phosphate charge is neutralized by territorially bound Na+ ions, in typical aqueous NaCl solutions (less than 1 M). Thus, it was expected that additional neutralization up to 90% is the necessary condition to cause DNA condensation , and that, with divalent cations, the maximum value is about 88%, being insufficient for condensation . As an exception to this theory, Mn2+ induced condensation of super-coiled plasmid DNA was discussed as the facilitating effect in the supercoiled state (Ma and Bloomfield, 1994). Condensation of DNA on a sohd surface inthe presence of divalent cations was also reported (Koltover et al., 2000). 

The criterion locates critical conditions where the values of all objective sensitivities to any model parameter 6 reach a maximum. This is also a necessary condition for runaway, so that fast reactions that do not meet this requirement are not classified by Morbidelli and Varma as proper cases of thermal runaway. 

Stationary points can be a (1) local maximum, (2) local minimum, or (3) saddle point. The existence of a stationary point is a necessary condition for an optimum. 

The surface concentration, size distribution and other properties of metal nanoparticles formed in a dark on the surface of the inert wide-band-gap semiconducting oxides under contact, photocatalytic, or photoelectrochemical deposition depend substantially on the concentration, bulk distribution, and energy characteristics of donor defects in the initial semiconductor substrate. As a rule, the necessary condition for the formation of the smallest-sized particles in the highest surface concentration is the maximum shift of the surface potential of semiconducting matrix from its equilibrium value during metal deposition. This is part of the reason for the experimentally observed fact that the particles formed in the condition of photocatalytic deposition are characterized by less average size and cover superior portion of surface than those obtained under cathodic deposition, all other factors being equal. 

There is a simple method for speeding the calculation of conversion that should be pointed out. After the conditions in the upstream end of a reactor have been calculated, including a region well beyond the point where the temperature is a maximum, the conditions in the rest of the reactor can be calculated accurately enough with only 2 or 3 radial increments, or, if the Biot number is not too large, with the one-dimensional approximation. In this region, the temperature has no intrinsic interest, and it is only necessary to estimate the average conversion. 

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