Homogeneous notions

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

Introduction of the reptation concept by De Gennes [43] led to further essential progress. Proceeding from the notion of a reptile-like motion of the polymer chains within a tube of fixed obstacles, De Gennes [43-45], Doi [46,47] and Edwards [48] were able to confirm Bueche s 3.4-power-law for polymer melts and concentrated polymer solution. This concept has the disadvantage that it is valid only for homogeneous solutions and no statements about flow behaviour at finite shear rates are analysed. 

Central to catalysis is the notion of the catalytic site. It is defined as the catalytic center involved in the reaction steps, and, in Figure 8.1, is the molybdenum atom where the reactions take place. Since all catalytic centers are the same for molecular catalysts, the elementary steps are bimolecular or unimolecular steps with the same rate laws which characterize the homogeneous reactions in Chapter 7. However, if the reaction takes place in solution, the individual rate constants may depend on the nonreactive ligands and the solution composition in addition to temperature. 

Therefore, the results of the present paper showed, that the notion of reactive medium heterogeneity connected with free volume representations, that was expected for diffusion-controlled solid phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of overlapping percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

The notion of homogeneity is not absolute all substances are inhomogeneous upon sufficiently close inspection. Thus, the description of the interaction of an electromagnetic wave with any medium by means of a spatially uniform dielectric function is ultimately statistical, and its validity requires that the constituents—whatever their nature—be small compared with the wavelength. It is for this reason that the optical properties of media usually considered to be homogeneous—pure liquids, for example—are adequately described to first approximation by a dielectric function. There is no sharp distinction between such molecular media and those composed of small particles each of which contains sufficiently many molecules that they can be individually assigned a bulk dielectric function we may consider the particles to be giant molecules with polarizabilities determined by their composition and shape. 

The basic notions of homogeneous catalysis have been amplified and discussed more extensively elsewhere, as have examinations of individual reaction steps. In briefly reviewing these concepts, we are constructing a framework in which to consider the activation of CO, COz, and NO by metal complexes in solution and homogeneously catalyzed conversions of these simple oxides. 

Traditionally, the same overall mechanisms of acid catalysis invoking carben-ium ions have been assumed to prevail both in heterogeneous (2) and in liquid homogeneous (3) systems. But these mechanisms do not adequately take into account the fact that adsorbed, rather than free, carbenium ions are formed in the pores of solid catalysts. Consequently, a quantum-chemical model that demonstrates how the interaction of carbenium ions with the sites of their adsorption can influence the reaction mechanism has been formulated by Kazansky (4), taking double-bond-shift reactions in olefins as a particular example. According to this view, adsorbed carbenium ions are best regarded as transition states rather than reaction intermediates, a notion that had also been proposed earlier by Zhidomirov and one of us (5). 

The mass balance equations of the traditional multicomponent rate-based model (see, e.g., Refs. 57 and 58) are written separately for each phase. In order to give a common description to all three considered RSPs (where it is possible, of course) we will use the notion of two contacting fluid phases. The first one is always the liquid phase, whereas the second fluid phase represents the gas phase for RA, the vapor phase for RD and the liquid phase for RE. Considering homogeneous chemical reactions taking place in the fluid phases, the steady-state balance equations should include the reaction source terms ... 

C to form the dinuclear (butadienyl)zirconium system 120. Treatment with B(C6F5)3 leads to the formation of a mono-addition product (121), even in the presence of excess borane. Complex 121 (Scheme 40) shows only a marginal ethene polymerization activity. This led to the notion that such formation of dimeric zirconium complexes might represent desactivation pathways in homogeneous Ziegler-Natta catalyst chemistry.125... 

When the number of atoms decreases to a few tens of atoms and a fortiori to one atom, the fundamental microscopic and macroscopic concepts of the thermodynamics, valid at tracer level, cannot be applied directly i) Macroscopically, the basic notion of phases disappears since the repartition of the entities E becomes less homogeneous, when the number of... 

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