Functional homogeneous

This complex and structurally related molecules served as a functional homogeneous model system for commercially used heterogeneous catalysts based on chromium (e.g. Cp2Cr on silica - Union Carbide catalyst). The kinetics of the polymerization have been studied to elucidate mechanistic features of the catalysis and in order to characterize the potential energy surface of the catalytic reaction. 

It is obvious that much more time is needed for the activation of functionally heterogeneous groups of cooperating neurons than for the activation of a functionally homogeneous group of neurons. This has already been unequivocally proven in humans (Tibet 1973), although the reason for the observed phenomenon remained unexplained until now. 

Because each B-cell clone produces antibody to a single epitope on an antigen, an opportunity exists for producing chemically and functionally homogeneous antibody in substantial quantity. However, the short B-cell lifetime that is determined by normal apoptosis initially made this impractical. However, the discovery by Kohler and Milstein that B cells could be fused with immortal myeloma cells to create hybridomas that can be maintained almost indef-... 

The synthesis of alkylidenes incorporating late transition metals has resulted in alkene metathesis catalysts having unprecedented functional group tolerance. In particular, the discovery that complexes of Group VIII transition metals were efficient ROMP catalysts introduced several advantages. Relative to their early transition metal counterparts, these classical catalysts functioned well in the presence of a variety of polar and protic functional groups (Table 2), and they functioned homogeneously in water. 

Clearly, homogeneity is concerned with the exponents of the variables in a function. Homogeneity is not always quite so obvious, however. For example... 

This functional expansion series can be reexpressed [24] as the sum of functionals homogeneous of degree 0, 1,2,..., respectively, with respect to the density scaling [22] ... 

Equation (4.66) is also the result of Euler s theorem ( C.2.3) applied to G as a function homogeneous in the first degree in the masses of die conponents. 

This is clearly a (generalized) renewal property. One can easily generalize this formula and obtain analogous formulas for S based models. We will repeatedly use this property, but mainly in a non-explicit way and we will mostly manipulate (restricted) partition functions. Homogeneous Markov and renewal processes are manageable due to their local nature, but, in inhomogeneous frameworks, they may instead display sharply nonlocal features and tools to analyze them go well beyond the tools used for homogeneous systems. 

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