Distance range upper-bounds

Following the definition of the continuous symmetry measure (CSM) in Section II, the CSM values are limited to the range 0. .. 1 (where 1 is the normalization scale). The lower bound of the CSM is obvious from the fact that the average of the square of the distances moved by the object points, is necessarily non-negative. The upper bound of the average is limited to 1 since the object is previously... 

Figure 3.34. Distance matrix (a) in which unique interatomic distances for a particular conformation of a molecule are stored. Distance range matrix (b) in which ranges of interatomic distances representing conformational flexibilty of molecule are stored. U = upper bound, L = lower bound.

The goal of the method is to perform a random walk in space. Consider a system consisting of N particles interconnected to form a molecule, and having volume V and temperature T. The end-to-end distance of the molecule ( ) can be discretized into distinct states each state is characterized by its end-to-end distance, in some specified range of interest [, +j and represent a lower and an upper bound, respectively. The partition function 17 of this expanded ensemble is given by... 

Fig. 46. Average pore-water NH4 concentration in the top 0-16 cm at NWC predicted at 22°C by the hollow-cylinder model as a function of (half-distance between individuals) or effective population abundance. Burrow radius r, is fixed at 0.1 (A), 0.5 (A), or 1.0 cm ( ) for each curve. The lower bound is the seawater value 0.0002 mAf and the upper bound is the one-dimensional model [Eq. (6.4)] no burrow case. Typical ranges indicated.

The tunability range depends on the slope of the repulsive potential and on the internuclear distances R and R2 of the classical turning points in the excited vibrational levels. The spectral gain profile is determined by the Franck-Condon factors for bound-free transitions. The corresponding intensity distribution I (jo) of the fluorescence from the upper vibrational levels shows a modulatory structure (see Fig. 2.14) reflecting the R dependence IV vib( ) of the vibrational wave function in these levels [5.197]. 

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