Average crossing number

The writhe of a polygonal curve is the signed average crossing number of the curve, where the average is taken over the observer s positions, located in all space directions. 

If the critical separation is determined for a large number of relative geometries of the electron and molecule it is possible to obtain a three-dimensional picture of the probability of ionization as a function of the orientation of the molecule. Effectively, the idea of an ionization cross section, the area the target molecule presents to the electron, is extended to a three-dimensional object defined by the critical distances, with ionization occurring when the electron penetrates the surface enclosing this volume. The volume enclosed by the electron impact ionization surface may be used to obtain an estimate for the cross section (volume averaged cross section) ... 

Further, one can calculate the distribution of the number and size of spheres intersections, but let us consider just an average number of crossings per sphere, which is n=— 81n(e) and corresponds to an average coordination number for the packing of spheres np. 

In this case the r ate of cross-linkages formation must be proportional to the number of reactive contacts in each particular chain, z- /where j is a number of cross-links/. Assuming the independence of the average contacts number Zj on cross-linkages configuration one can describe the reaction by the following system of... 

The rods, parallel in each region, have an average cross-sectional number density N. The volume fraction for rods of radius a is... 

Figure 4. Micrographs of the cross section of a vanadyl phosphate porous microsphere with 10% amorphous silica obtained by electron probe microanalysis (EPMA). Left backscattered electron image showing average atomic number across the specimen. Right X-ray image showing silicon distribution.

Ab initio and Monte-Carlo calculations. Attempts have appeared in pulse radiolysis to describe the dynamics of free electron production, recombination and solvation on a microscopic scale [31-34]. This requires the knowledge of a number of physical parameters solvated electron and free ion yields, electron and hole mobilities, slowing-down cross-sections, localization and solvation times, etc. The movement and fate of each reactant is examined step by step in a probabilistic way and final results are obtained by averaging a number of calculated individual scenarios. 

The discussions above on the local heat transfer coefficients arc insightful however, they are of limited value in heal transfer calculations since the calculation of heat transfer requires the average heat transfer coefficient over the entire. surface. Of the several such relations available in the literature for the average Nusselt number for cross flow over a cylinder, we present the one proposed by Churchill and Bernstein ... 

Empirical correlations for the average Nusselt number for forced convection over circular and noncircular cylinders in cross flow (from Zukauskas, 1972 and Jakob, 1949)... 

Several correlations, all based on experimental data, have been proposed for the average Nusselt number for cross flow over tube banks. Mote recently, Zukauskas has proposed correlations whose general form is... 

The average Nusselt numbers for cross flow over a cylinder and sphere are... 

From this, the basic equation of the stationary-phase retention process can be derived, a number of assumptions and complex theoretical treatments being required. Taking as example the planetary centrifuge of Type J, the average cross-sectional area of a stationary-phase layer has been estimated for hydrophobic liquid systems, which are characterized by high values of interfacial tension y, low values of viscosity r], and low hydrodynamic equilibrium settling times ... 

Finally, since the intensity of the backscattered electron image used for AIA is highly dependent on the average atomic number of the material in view, pyrite particles with the heavy iron atom cross too often the signal threshold which has been set for a level characterized by clays, thus increasing the possibility of reading more pyrite. A similar bias has been observed previously (9 ), and an empirical factor of 0.75 has been suggested to scale down the inflated pyrite results. 

The ability to determine the pore size and pore size distribution for porous membranes has existed for a number of years (11,40-42). Recent advances have permitted the determination of pore size and distribution for finely porous ultrafiltration membranes (, 1 ). Determination of i j, (the average cross-sectional area of the transport corridor) for the skin layer of reverse osmosis and tight ultrafiltration membranes as well as pervaporatlon membranes at present remains a challenge, although advances are being made in this direction (9,10,48-50). 

Let us consider a tube of arbitrary shape and denote the contour of the cross-section by T. Generally speaking, the Nusselt number varies along the contour T. We define the perimeter-average Nusselt number Nu as follows ... 

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