Similarity measures constrained

A definition of the unconstrained scaling-nesting similarity measures is given below the constrained measures can be derived easily from these by applying the appropriate constraint while determining the maximum scaling factor. 

To ensure a constant-flux layer, one can simply move the measurement height closer to the surface. For the eddy correlation method, however, the response time of the instrument must be faster as the measurement height approaches the surface, because high-frequency turbulent eddies then contribute proportionally more to the concentration fluxes than at higher levels. On the other hand, fluxes measured very close to the surface may be less representative of those over the entire area for which the measurement is intended. For the gradient method, the requirement that z/zo 3> 1 (based on the requirements of similarity theory) constrains the minimum measurement height. Under very stable conditions, when turbulence may be intermittent, turbulent fluxes may become very small, and the constant-flux layer may be very shallow. Under conditions such as these, it can be quite difficult to determine the aerodynamic resistance term ra. 

The second, and perhaps most correct, way to compare similarity measures are directly through their similarity matrices. The method, pioneered by Mantel [147], has been applied in a number of areas of biological research [148, 149]. The issue raised by Mantel was the lack of statistical independence among the similarity-based elements of a similarity matrix. Consider, for example, three molecules t,y,and k. Suppose S(i,j) and S(j, k) are both large. Then the value of S(ijc) will be constrained by the values of the first two similarities, and, thus, is dependent to some degree on the other similarity values. Hence, the similarity value of the ordered pair (i,k) is not statistically independent. 

Performing constrained-state DSC measurements on oriented films is more complex than similar measurements on drawn fibers, especially when the film has biaxial orientation it is more difficult to immobilize the total film sample during the DSC measurement, than simply tying together the ends of a single fiber after winding it up on a steel plate. Therefore, various approaches have been suggested to solve this problem. The two most important of these are as follows ... 

Figure 2 The initial dissociative sticking probability for D2 on Cu(l 1 1) extracted from the state selected measurements of desorbing molecules for various vibrational (A) and rotational (B) states of the molecule [19]. Vibrational energy couples effectively to the reaction coordinate, lowering the translational energy requirement for dissociation. Rotational energy initially hinders and then promoted dissociation. Similar effects of rotational energy are predicted in the trajectory calculations shown in (C) for molecules constrained to rotate in a plane perpendicular to the surface [29].

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