Translational coefficient

The relationships follow from considerations of Brownian motions and thermal fiuctuations which also infiuence the internal motions in flexible objects. is the translation coefficient of the particle s center of mass where the subscript indicates the z-average over the molar mass distribution. The first bracket in Eq. (12) describes the concentration dependence which often is well represented by a linear dependence... 

Solute rejection coefficient of membrane Translation coefficients for translation of a spheroid parallel to its semiaxes, Eqs. (5.1.6), (5.1.7), also any body parallel to its principal axes, Eq. (5.1.10a)... 

When Brownian motion and its attendant randomizing effect is important, an ellipsoid, or colloidal particles of arbitrary shape, will have various orientations. As we have seen, there is a translation coefficient for each orientation. It can be shown that the mean translation coefficient, mean friction coefficient, and mean mobility are given, respectively, by the formulas (Perrin 1936, Happel Brenner 1983)... 

From Eqs. (5.1.6) and (5.1.7) the mean translation coefficient for a prolate or oblate spheroid is simply 2/5, whereas for a sphere it is just the sphere radius. A quantity of interest, sometimes termed the Perrin factor, is the ratio of the mean translation coefficient (or mean friction coefficient) of a prolate or oblate... 

For the prolate spheroid an interesting approximate form of the translation coefficients is obtained for the case when the long semiaxis a is large compared with the short semiaxis b. From Eqs. (5.1.6) to (5.1.8), by expanding for a b> we find that for translation parallel to the a axis... 

In the above equation, W, bj and aj denote the weighting factor, translation coefficient and dilation coefficient, respectively, for each wavelet basis. In Liu s work [25], the wavenumber and transmittance quantities of the IR spectrum were used as the input and target output values, respectively, of the network. Their proposed neural network consisted of a single layer network... 

Y.-L. Xu, Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories. J. Comp. Phys. 139(1), 137-165 (1998a). doi 10.1006/jcph. 1997.5867 Y.-L. Xu, Electromagnetic scattering by an aggregate of spheres asymmetry parameter. Phys. 

In this case the interacting held is the image of the scattered field and the expansion (2.198) can be derived by using the addition theorem for vector spherical wave functions. The elements of the reflection matrix are the translation coefficients, and as a result, the amount of computer time required to solve the scattering problem is significantly reduced. In this regard it should... 

Integral and series representations for the translation coefficients Cmn,m n can be obtained by using the integral representations for the spherical wave... 

Finally, we note the integral representation for the translation coefficients mn m n hc Specific case of axial translation ... 

As in the scalar case, integral and series representations for the translation coefficients can be obtained by using the integral representations for the vector spherical wave functions. First we consider the case of regular vector spherical wave functions. Using the integral representation (B.26), the relation r = 0 + T" ) and the vector spherical wave expansion... 

Inserting the spherical wave expansion of the plane wave expQfe Tq) and integrating over a, we derive the following series representation for the translation coefficients ... 

In (B.70) and (B.71) the integration with respect to the azimuthal angle a can be anal3dically performed and in the case of axial translation, the integral representations for the translation coefficients and read as... 

The translation coefficients for axial translation can be obtained from the above recurrence relations. In general, through rotation of coordinates, the numerical advantages to a common axis can be exploited and a transformation from Oxyz to OxipiZi can be accomplished through three steps [150] ... 

Using the integral representation for the translation coefficients, making the transformation (po Po + changing the variable of integration from (3 to n — p and using the identities (A.24) and (A.25), yields... 

K. T. Kim, The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields, IEEE Trans. Antennas Propagat. 44, 1482 (1996)... 

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