Dimensionless shape parameter

The scale parameter [i characterizes the overall rate the dimensionless shape parameter a raises the time scale to a power other than 1. While a = 1 represents a mono-exponential, a > 1 describes a sigmoid profile retarded in the beginning, a < 1 represents a profile faster in the beginning but retarded in the tail. Figure 2 illustrates the performance for P = 5 and five differing values of a. All CDF profiles intersect at a point (t = 5,F = 0.632), which closely reflects the mean of the distribution. 

Thus, the dimensionless shape parameter in eqn (4.58) takes the value s = cos2 2/7. Therefore, the bent molecule with a bond angle of 90° has a perfectly bimodal eigenspectrum corresponding to s = 0, as is reflected by its two degenerate levels in Fig. 4.12. The linear molecule with a bond angle of 180° takes a value of s = 1 and lies on the bimodal-unimodal borderline as is reflected by its four evenly spaced levels in Fig. 4.12. Hence, the bent... 

The relative stability of structural types with even-membered rings can often be inferred from the dimensionless shape parameters, s, which from eqn (4.58) is given by... 

Fig. 8.7 Plotting (5+ 1 )/(s + 1) =1 versus local coordination for the pure s and sp cases, where s is the dimensionless shape parameter. The dashed curve gives the Bethe lattice result, eqn (8.23), in which there are no ring contributions. (After Cressoni and Pettifor (1991).)...

Fig. 8.19 The normalized response function ] asa function of the number of valence s electrons per atom for the case of the dimensionless shape parameter, s = 1. (From Pettifor and Aoki (1991).)...

The fibers taken from the same bundle showed a wide range of diameters, which is a typical drawback of natural fibers, justifying the need for the use of a more accurate statistical distribution function. In fact, all the fibers showed wide dispersion of strength with respect to diameter data, indicated by the dimensionless shape parameter of the Weibull equation. By contrast, the advanced statistical approach based on neural network algorithms (PDF estimation technique) mentioned above resulted in asymmetric curves of diameter distributions of four lignocellulosic fibers mentioned above (Fig. 8.7). 

Here i = ujR0I/V0 is the dimensionless current —the control parameter of this model. Furthermore, f(pt) is the dimensionless stationary resistance function of a general shape sketched in Fig. 6.1.3 and k = k/sv 1 is the dimensionless relaxation parameter, assumed large in accordance with Teorell s intuitive ideas. 

The shape of the aggregates can be related to the dimensionless packing parameter (P) defined by Israelachvili (119,126) as the ratio V7o-Z, where V is the apolar volume... 

As discussed by Israelachvili (1992), the shapes of surfactant aggregates can, to a first approximation, be anticipated based on the packing of simple molecular shapes (Tanford 1980). Figure 12-1 from Israelachvili illustrates this principle Conical molecules with bulky head groups attached to slender tails form spherical micelles cylindrical molecules with heads and tails of equal buUdness form bilayers and wedge-shaped molecules with tails bulkier than their heads form inverted micelles containing the heads in their interiors. A simple dimensionless molecular parameter that controls the shape of the aggregates is the molecular shape parameter here v is the volume occupied by the hydrocarbon... 

The most important dimensionless geometric parameter characterizing the cross-section shape is the ratio y/S /V, where V is the cross-section perimeter. In calculations it is convenient to use the shape parameter... 

The dimensionless geometric parameter VA/A is proposed as an alternate parameter for determination of shape factors of complex convex bodies. 

The values for [5vj]rect are obtained from the expressions for the rectangular plates given above. This parameter depends on the cuboid lengths L2 and L3. The proposed expression for cuboids predicts the dimensionless shape factor to within 5 percent. 

Mie Scattering. For systems more complex than very small particles (Rayleigh) or small particles with low refractive indices (Rayleigh-Debye), the scattering from widely separated spherical particles requires solving Maxwells equations. The solution of these boundary-value problems for a plane wave incident upon a particle of arbitrary size, shape, orientation, and index of refraction has not been achieved mathematically, except for spheres via the Mie theory (12,13). Mie obtained a series expression in terms of spherical harmonics for the intensity of scattered light emergent from a sphere of arbitrary size and index of fraction. The coeflBcients of this series are functions of the relative refractive index m and the dimensionless size parameter a = ird/k. 

Which factors determine the packing efhciency of the amphiphiles in the aggregate The dimensionless shape factor or packing parameter (P) provides a relation between the molecular shape of the amphiphile and the preferred morphology of the aggregate in dilute aqueous solution at low ionic strength and at ambient temperatures ... 

Figure il. Variation, of dimensional film shape in a elastohydrodynamically lubricated conjunction for three values of dimensionless load parameters while holding the dimensionless speed and materials parameters fixed at U - 1 x 10 and 0 5007.2. 

Maxwell equation in Eq. (3(Fla) by substituting for k in terms of V, p and A, and multiplying by p to make the operator V x dimensionless. Further parameters are required to account for variations in cross-section, e.g. the eccentricity of an elliptical waveguide, and in profile shape, e.g. the exponent of the power-law profiles of Eq. (1-59). 

For large values of the shape parameter B> and small values of the dimensionless energy a (conditions which are attainable for deep wells), the transmission probability Tp can have a very sharp dependence on a, as indicated in Fig. 9 for a square well with B = 315. 

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