Weak Sufficient Condition

We will now examine the role of the second variation in identifying the optimum. Let us say we desire to find the minimum of /, and we have determined the minimum from the necessary conditions. Then from Equation (2.33) on p. 52, and noting that SI = 0 the minimum, we get 

The above result implies that /(y, u) — /(y, u) 0 so that /(y, u) is not a minimum We need to impose some condition on 6 1 to ensure its minimmn. 

If we place a condition that the error ratio e2/ u f vanishes with (5u, and (5 7 all Ju p where a is some positive number, then 

If we can find a sufficiently small neighborhood of u such that all Ju therein have norms sufficiently small to render any negative error ratio less than a/2, then 

The above sufficient condition is weak, since it is applicable only in a sufficiently small vicinity of the optimal control. The satisfaction of this condition does not guarantee a global minimum. Moreover, it is not easy to come up with a suitable a. in an optimal control problem. 

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Now let us prove two theorems containing sufficient conditions of weak lower semicontinuity of the functionals. 

Huckel s 4n+2 //-electron rule is a necessary but not a sufficient condition for aromaticity. Coplanarity and electronegativity restrictions of constituent atoms represent the most important restrictions. Phos-phole is a marginally aromatic five-membered heterocycle76 (see further examples and discussion). Mesoionic compounds, mesomeric betaines, and 2H-and 4i+-pyrone have all been considered to be weakly aromatic or non-aromatic, though their conjugated acids are aromatic. Spectroscopic data evidenced the aromaticity of dioxolium and oxathiolium cations 59 (Scheme 28) and mesoionic oxathioles not in the classical sense but by their ring currents and chemical stability.77... 

The carbon nucleophiles in amine-catalyzed reaction conditions are usually rather acidic compounds containing two electron-attracting substituents. Malonic esters, cyanoacetic esters, and cyanoacetamide are examples of compounds which undergo condensation reactions under Knoevenagel conditions.115 Nitroalkanes are also effective nucleophilic reactants. The single nitro group sufficiently activates the a hydrogens to permit deprotonation under the weakly basic conditions. Usually, the product that is isolated is... 

One obvious weakness in the derivation of (4) is that the reacting molecules are represented by spheres with surfaces of uniform reactivity the question of molecular orientation during reaction does not then arise. This is obviously a very crude approximation for the reaction of many complex molecules. Some recent studies (Sole and Stockmayer, 1971, 1973) have employed a more realistic model in which the representation as spheres is retained but with the assumption that only part of the area of the sphere is reactive. The contact of two spheres involving both reactive areas is then taken as a necessary and sufficient condition for reaction. 

Iron Complexes. Iron complexes of tridentate o,o -dihydroxyazo compounds are prepared under weakly acidic conditions at 40-80°C. Both Fe11 and Fe111 salts can serve as iron source. The Fem complexes that result in both cases do not have sufficient stability to dye textile substrates, but the dyeings on leather have good fastness properties [21],... 

Horn and Jackson [50], M. Feinberg [51], Horn [52] and Feinberg and Horn [53] showed that if the scheme of complex conversions is weakly reversible and (most essential) the conditions (169) is fulfilled, the system always has a PCB and hence demonstrates "quasi-thermodynamic behaviour. The weak reversibility condition seems to be less essential since, for its fulfilment, it is sufficient to suggest that all steps are reversible assuming, if necessary, the rate constant for a reversible step to be sufficiently low. Hence the main difficulty for the application of the results of Horn, Jackson and Feinberg to a concrete reaction system can be the impossibility of fulfilling eqn. (169), which cannot be overcome by small additions to the equations. 

The inequality (25), implying a demand for the sufficiently weak reversibility of the adsorption step of B, is a necessary condition for the multiplicity of steady-state solutions. To a necessary and sufficient condition for multiplicity, we must substitute eqn. (24) into the inequality from (23), but the expression obtained will be extremely cumbersome. 

The condition (51), a weak extensive property, is well satisfied for the relevant N by all extended systems of which we are aware. Although it is a sufficient condition, it may not be necessary. 5. The chemical potential n is given by ... 

In its simplest form, this matching of observation and theoretical prediction involves finding diffraction vectors g for which the dislocation is invisible (or very weak) and determining the direction of b by assuming that the condition g-b = 0 is satisfied. However, g b = 0 is a sufficient condition for invisibility only for a pure screw dislocation, and the invisibility of a pure edge dislocation also requires g b x u = 0. Thus, invisibility of a dislocation of mixed character is not expected for any reflection, even though the condition g b = 0 is satisfied, because g b x u 0. The... 

Under weakly acid conditions with sufficient water and free drainage, more silica may be removed, permitting gibbsite to form directly from kaolinite ... 

An elegant application of the oxidation of 2-aminofurans has been described by Nicolaou and co-workers (02JA2190, 02JA2202) in model studies directed towards the total synthesis of CP molecules. In this study, the isolable iminobutenolide 80 is formed by cyclisation of the alkoxide 79 (Method B, Section II.B.l.b). Without a stabilising substituent on the ring the equilibrium favours the imine 80 rather than the amine 81 (Scheme 16). However, it is postulated that there is sufficient 2-ami -nofuran in equilibrium for this to be rapidly oxidised to the hydroperoxide 83. At this stage, the final product is determined by the reaction conditions. In strongly acidic conditions, tautomerism to the amine 85 and hydrolysis rationalises the formation of the isolated anhydride 88. Under weakly acidic conditions, formation of... 

The first condition for the observation of a resonance line has been satisfied. Intense resonance lines have indeed been observed in the M[y y emission spectra. Their size shows that an appreciable amount of overlap exists between the 4/ " orbital and the 3 d shell. Moreover, we have seen that these lines are observable only if the radiative rate is sufficient compared with other processes of de-excitation to ensure that the excited state has a noticeable probability of depopulation by direct radiative decay. In particular, the diffusion of the 4/ supplementary electron to continuum states must remain weak this condition is entirely compatible with the relatively large localization of the 4/ rare earths states mentioned above. 

A necessary but not sufficient condition for efficient extraction by a pure liquid appears to be the presence of an electron donor atom, c.g. a basic oxygen or nitrogen atom, d he possession of an appreciable dielectric constant favors extraction by a liquid even when only weak donor atoms are present. 

Actually, the IR absorption features depend on the chemical short-range order of the lattice, and their appearance is only a necessary (not a sufficient) condition for the existence of the respective phases. There exist some more BN phases with either sp or sp short-range order which show essentially the same IR absorption as hBN and cBN, respectively. However, if used in combination with more direct methods such as electron diffraction, IR absorption is a fast and reliable method for determination of the phase content of BN thin films and has found widespread application. In Fig. 4 the IR absorption spectrum of a BN film is given as an example which shows a distinct absorption near 1050 cm and only weak absorption at about... 

Notice that the sufficient condition for the occurrence of strict Lindemann behaviour, given a strong collision relaxation matrix, is that the d, are constant alternatively, if there is only one grain having a non-zero value of d, the rate is also strict Lindemann, regardless of the form of the assumed relaxation matrix [81.VI]. In the past, it had often been assumed that strict Lindemann behaviour was a strong collision property only, but we now know that near-Lindemann behaviour can often occur in weak collision systems at high temperature, see Chapter 8. 

On the other hand, lipases were found to be more efficient catalysts [400]. Thus, when the hydrolysis was carried out under carefully cOTitroUed reactimi conditions at pH 7.6, Af-benzoyl amino acids of moderate to excellent optical purities were obtained depending on the substituent on C-4. Whereas PPL led to the formatiOTi of L-amino acids, the D-counterparts were obtained with a Upase from Aspergillus niger. Furthermore, the rate of equiUbration of the two cOTifiguraticmaUy unstable substrate antipodes under weakly basic conditions is sufficiently rapid to provide a... 

If f(y) is sufficiently strongly decreasing with increasing y these weakly sloping plateaus can be compensated so that dHjj/dy < 0 which is a sufficient condition for the existence of a critical point. For a wide density-of-sites function this is not possible. The question whether T. > 0, i.e. whether phase separation occurs or not in a disordered metal-hydrogen system has recently been discussed by Griessen ) and Richards ). 

Suppose that at time t = 0, the atoms are in the lower state Ea, which implies that fl(0) = 1 and b 0) = 0. We assume the field amplitude Aq to be sufficiently small so that for times f < T the population of Eb remains small compared with that of Ea, i.e., b t < 7) 2 <3C 1. Under this weak-field condition we can solve (2.68) with an iterative procedure starting with a = 1 and b = 0. Using thermal radiation somces, the field amplitude is generally small enough to make the first iteration step already sufficiently accurate. 

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