Necessary conditions

In any and all cases, desulfurization of diesel fuel is a necessary condition for attaining very low particulate levels such as will be dictated by future regulations (Girard et al., 1993). 

For tire purjDoses of tliis review, a nanocrystal is defined as a crystalline solid, witli feature sizes less tlian 50 nm, recovered as a purified powder from a chemical syntliesis and subsequently dissolved as isolated particles in an appropriate solvent. In many ways, tliis definition shares many features witli tliat of colloids , defined broadly as a particle tliat has some linear dimension between 1 and 1000 nm [1] tire study of nanocrystals may be drought of as a new kind of colloid science [2]. Much of die early work on colloidal metal and semiconductor particles stemmed from die photophysics and applications to electrochemistry. (See, for example, die excellent review by Henglein [3].) However, the definition of a colloid does not include any specification of die internal stmcture of die particle. Therein lies die cmcial distinction in nanocrystals, die interior crystalline stmcture is of overwhelming importance. Nanocrystals must tmly be little solids (figure C2.17.1), widi internal stmctures equivalent (or nearly equivalent) to drat of bulk materials. This is a necessary condition if size-dependent studies of nanometre-sized objects are to offer any insight into die behaviour of bulk solids. 

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

B. The Necessary Condition for Having a Solution for the Adiabatic-to-Diabatic Transformation Matrix... 

A. The Necessary Conditions for Obtaining Single-Valued Diabatic Potentials and the Introduction of the Topological Matrix... 

With these definitions we can now look for the necessary condition(s). Thus, we assume that at each point sq iu configuration space the diabatic potential matrix W(/l) [= W(s,so)] fulfills the condition ... 

XI. THE NECESSARY CONDITIONS FOR A RIGOROUS MINIMAL DIABATIC POTENTIAL MATRIX... 

For example one forms, within a two-dimensional (2D) sub-Hilbert space, a 2x2 diabatic potential matrix, which is not single valued. This implies that the 2D transformation matrix yields an invalid diabatization and therefore the required dimension of the transformation matrix has to be at least three. The same applies to the size of the sub-Hilbert space, which also has to be at least three. In this section, we intend to discuss this type of problems. It also leads us to term the conditions for reaching the minimal relevant sub-Hilbert space as the necessary conditions for diabatization. ... 

When large concentrations of water are added to the solutions, nitration according to a zeroth-order law is no longer observed. Under these circumstances, water competes successfully with the aromatic for the nitronium ions, and the necessary condition for zeroth-order reaction, namely that all the nitronium ions should react with the aromatic as quickly as they are formed, no longer holds. In these strongly aqueous solutions the rates depend on the concentrations and reactivities of the aromatic compound. This situation is reminiscent of nitration in aqueous nitric acid in which partial zeroth-order kinetics could be observed only in the reactions of some extremely reactive compounds, capable of being introduced into the solution in high concentrations ( 2.2.4). 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives... 

A second common approximation is the steady-state condition. That arises in the example if /fy is fast compared with kj in which case [i] remains very small at all times. If [i] is small then d[I] /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). Eor equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equihbrium condition, namely, equations 24a and b. 

Fig. 6. Self-assembled monolayers are formed by immersing a substrate into a solution of the surface-active material. Necessary conditions for the...

The most striking feature of the earth, and one lacking from the neighboring planets, is the extensive hydrosphere. Water is the solvent and transport medium, participant, and catalyst in nearly all chemical reactions occurring in the environment. It is a necessary condition for life and represents a necessary resource for humans. It is an extraordinarily complex substance. Stmctural models of Hquid water depend on concepts of the electronic stmcture of the water molecule and the stmcture of ice. Hydrogen bonding between H2O molecules has an effect on almost every physical property of Hquid water. 

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. 

The necessary conditions for galvanic corrosion are (1) a corrosive interaction of electrochemically dissimilar materials that are (2) exposed to a common conductive fluid and are (3) physically linked so... 

It should be noted that the normality conditions, arising from the work assumption applied to inelastic loading, ensure the existence and uniqueness of solutions to initial/boundary value problems for inelastic materials undergoing small deformations. Uniqueness of solutions is not always desirable, however. Inelastic deformations often lead to instabilities such as localized deformations. It is quite possible that the work assumption, which is essentially a stability postulate, is too strong in these cases. Normality is a necessary condition for the work assumption. Instabilities, while they may occur in real deformations, are therefore likely to be associated with loss of normality and violation of the work assumption. 

Photoluminescence finds its greatest strengths as a qualitative and semiquantitative probe. Quantification based on absolute or relative intensities is difficult, although it is useful in applications where the sample and optical configurations may be carefully controlled. The necessary conditions are most easily met for analytical applica-... 

A word should be said about the weak boundary layer effect and silicone release [40,41]. Studies have shown that having loose silicone oil that can transfer to the PSA will lower release, however subsequent adhesion will likely suffer as well. In most commercial instances using silicone liners, a weak boundary layer is not intentionally employed. Additionally, many low transfer silicone liners are commercially available which provide premium release and show low to no PDMS transfer to PSAs, indicating that PDMS transfer is not a necessary condition for easy release. 

The only doublets consistent with this inequality, with the necessity of identical parity for p and q, and with values of r close to 0.34 nm, are given in Table 1 (from which the doublet (10,0) can be excluded since it is characteristic of a symmetrical, non-helical sheet). Hence, if br — 0, the necessary conditions for two successive helical cylindrical sheets to have strictly identical pitch angles are ... 

Kashiki, I. and Suzuki, A., 1986. Flocculation system as a particulate assemblage A necessary condition for flocculants to be effective. Industrial and Engineering Chemistry Fundamentals, 25, 444-449. 

The reasons for self-assembly and the mechanisms necessary conditions for the aggregation into micelles, mono- or bilayers, structure of aggregates, distribution of aggregation numbers, etc. 

The sufficient and necessary condition is therefore Cb iCa. As a consequence of imposing the more restrictive condition, which is obviously not correct throughout most of the reaction, it is possible for mathematical inconsistencies to arise in kinetic treatments based on the steady-state approximation. (The condition Cb = 0 is exact only at the moment when Cb passes through an extremum and at equilibrium.)... 

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