Normalization radius

Hence, the concentration in the particle depends only on the Thiele modulus, s and a normalized radius r/R as illustrated in Fig. 5.34. 

Figure 1.3 Field distributions along the Ag-tip surface and corresponding Ag-tip geometry. z = 0 corresponds to the Au-substrate. r/R is the normalized radius from the pointdirectly beneath the tip (R is the Rayleigh length R = /2n). Reprinted with permission from S. Klein, Electrochemistry, 71, 114 (2003). Copyright 2003, The Electrochemical Society of Japan.

At least for a first approach, the active component in the strain-stress relation may be treated in a simple manner. For some strain emax the active stress aa is maximum, and on both sides the stress decreases almost linearly with e — emax. Moreover, the stress is proportional to the muscle tone xjr. By numerically integrating the passive and active contributions across the arteriolar wall, one can establish a relation among the equilibrium pressure Peq, the normalized radius r, and the activation level xjr [19]. This relation is based solely on the physical characteristics of the vessel wall. However, computation of the relation for every time step of the simulation model is time-consuming. To speed up the process we have used the following analytic approximation [12] ... 

Fig. 12.4 Relation between the equilibrium pressure Peq and the normalized radius r of the afferent arteriole for different values of the muscular activation level [f. The solid curves represent our analytical approximation and the dashed curves represent the exact numerical solution. The area bounded by dotted lines corresponds approximately to the regime of operation for the model.

Figure 12.1. Interpretation of multi point distances for a curve and a surface [19]. (a) Three-point distance for curve. Given any three distinct points, r is the radius of the unique circle that contains the points. When the points are from the same neighborhood on a curve, such as points 1, 2, and 3, r is close to the local radius of curvature. When points, such as 1, 2, and 4, are taken from two different neighborhoods of the curve that are close to intersection, r approximates (half of) the distance of closest approach of the curve to itself, (b) Tangent-point distance for a surface. Given two distinct points 1 and 2 on a surface, p is the radius of the unique sphere that contains both points and is tangent to the surface at point 1. When the points are neighbors on the surface, p approximates the absolute value of the local normal radius of curvature in the direction defined by the two points (not illustrated). When points are taken from different neighborhoods that are close to intersection, p approximates (half of) the distance of closest approach.

The most important of recent theoretical studies on semi-flexible polymers is probably the formulation of Yamakawa and Fuji [2,3] for the steady transport coefficients of the wormlike cylinder. This hydrodynamic model, depicted in Figure 5-2, is a smooth cylinder whose centroid obeys the statistics of wormlike chains. In the figure, r denotes the normal radius vector drawn from a contour... 

In the above equations s is the scaled streamwise coordinate, n the scaled normal coordinate, A=An/An with An denoting the normalized separation between streamlines constituting a streamtube, is the normalized radius of curvature of the streamlines,6 is the direction of flow velocity, and the rest of the variables are the same as those of the previous section. Moreover, for use in nearly frozen flows the momentum theorem. ... 

It is known from Fig. 5 that for a given radius (or normalized radius r = / //c) and sinking depth ff (or normalized depth h = N//c), the position of the three-phase (solid-liquid-air) contact point can be determined from the following relations ... 

In both these compounds the Mn radius deduced is neither the high value found in the decacarbonyl nor the normal radius cited by DAHL and RUNDIE. More detailed discussion is perhaps best deferred until further results are available from structural analyses of related compounds. 

TABLE 16.19 Normalized Radius of a Cell Where the Received Signal Power is Above —116 dB (1 mW) with 90% Probability for S/N = 18 dB, Given That at a Reference Distance from the Base Station (the Cell Border) the Received Mean Signal Power is —102 dB (1 mW)... 

Figure 5. The time dependence of the normalized radius of gyration Rg t) and the fraction of unfolded molecules P (t) for sequence A at T = 0.947/-. Data are averaged over 100 [for Rg t) ] and 600 [for Pu t)] trajectories. P t) decays exponentially with the time scale X/ = 2.07x 10 MCS. The approach of Rg t)) to equilibrium is biexponential with the times scales 0.083x 10 MCS and 0.698 x 10 MCS. The first time scale is due to extremely rapid burst-phase partial collapse. The second time scale, which is associated with the eollapse time x, corresponds to the final compaction. The ratio x//xc is approximately 3.0.

It should be emphasized that D-5 and D-6 are general in effective diffusion coefficient (D), cage radius (p) and normalization radius (rj ). Analysis of the D=0 (f=0) limits was examined first since values for the yield, with no translational motion permitted, are Independent of the connection one chooses between the effective diffusion coefficient of the pair and the macroscopic viscosity of the medium. We have used the Stokes-Einstein equation (D-7) for this purpose where f is fluidity (1/n, cp.) and b is the effective radius for diffusive separation of the pair. 

A second factor in the normalization radius is the effect of the intervening molecules which, when bonded to the radical partners in the starting material, will impart an initial separation (Arjji). For the same cage pair partners, the normalization radius will thus vary with both the structure of the initiator and the solvent (D-9). 

Fig. 14.1 Section of a circularly symmetric fiber, which is unbounded in the r- and z-directions. The core radius is p, the normalized radius R = rjp and n (r) is the refractive-index profile.

If we substitute the Gaussian approximation of Eq. (15-2) into Table 13-2, page 292, we obtain the expressions in Table 15-2 for fundamental-mode quantities on an arbitrary profile. We have generalized the function i/-the fraction of modal power within the core-and define tf(R) to be the fraction of modal power within normalized radius R = r/p. This is a more useful quantity for profiles with no well-defined core-cladding interface. The expressions for Vg and D follow from Eqs. (13-17) and (13-18). If for a particular profile the... 

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