Semi-axis

Let us introduce the variable of displacement x(. which is by semi-axis of conformational ellipsoid the state M(s) appertains to the surface of this ellipsoid [1]... 

If unit cell is orthogonal there are layers lines on oblique texture electron diffraction pattern. These lines occur when certain reciprocal lattice planes lie perpendicular to the texture axis. In this case period c may be more accurately determined by measuring the minor semi-axis R of any ellipse (in the presence of layer lines it is measured directly, since there is a zero line with /=0) and H of any reflection on that ellipse (preferably with a large l) ... 

More generally the following question may be asked. When the values of x are confined to a certain subset / of the real axis, how does this show up in the properties of G If / is the interval — a < x < a it is known that G(k) is analytic in the whole complex k-plane and of exponential type . If I is the semi-axis x 0 the function G(k) is analytic and bounded in the upper half-plane. But no complete answer to the general question is available, although it is important for several problems. 

Since, for the time when c runs through a multitude of positive vectors, Inc moves across the whole of the linear n-dimensional space (In projects a positive real semi-axis to the total axis), the only limitation on if resulting from the existence of a PDE is... 

The sedimentation of isometric particles, such as cubes and octa-hedra, deviates only slightly from the Stokes equation. Significantly anisometric particles that can be approximated by ellipsoids of revolution are amenable to rigorous sizing by centrifugation if the axial ratios q are known. If the major semi-axis is a and the minor semi-axis is c, the sedimentation velocity dx/dt can be written as... 

Fig. 3a, b Plan view of a denser than water a rectangular NAPL pool with dimensions (xy.(y b elliptic NAPL pool with origin at x = x,y = having major semi-axis a and minor semiaxis b. For the special case where a=b=r the elliptic pool becomes a circular pool with radius r... 

For an elliptical hole (a = long semi axis, b = short semi axis) the radius of curvature at the end of the long axis is b2/a so that... 

Perrin [4, 18] has put forth a hydrodjmamic model of the rotational motion of different shaped particles. For an ellipsoid of revolution with semi-axis a> b = c (i.e., prolate ellipsoid), Perrin has shown that the rotational diffusion coefficient is given by... 

Consider a solid spheroid having long semi-axis oj and semi-focal distance a =, which... 

Thus for a flattened ellipsoid, as bja becomes infinite, both relaxation times become infinite, and the corresponding rotary diffusion constants approach zero. When bja is very large, the relaxation times are proportional to the cube of the b semi-axis, and do not depend at all on the length of the a semi-axis. 

The discussion will be restricted to molecules which can be described as ellipsoids of revolution. We shall denote the semi-axis of revolution by a, the equatorial semi-axis by b. The orientation of a molecule may thus be completely described by the orientation of the a semiaxis the frame of reference is shown in Fig. 4. The center of the coordinate system is located in the center of gravity of the ellipsoid. The velocity gradient in the liquid, indicated in Fig. 4, tends to rotate the molecule clockwise. The orientation of the a semi-axis of the molecule is specified by the angles and 0, as defined in the figure and the accompanying legend. 

The proper dynamical formulation of a Levy flight on the semi-infinite interval with an absorbing boundary condition at x = 0, and thus the determination of the first passage time density, has to ensure that in terms of the random walk picture jumps across the sink are forbidden. This objective can be consistently achieved by setting/(x, t) = 0 on the left semi-axis, i.e., actually removing the particle when it crosses the point x = 0. This procedure formally corresponds to the modified dynamical equation... 

Therefore, the best estimate of a solute cavity size can be found by minimizing the difference between the calculated and experimental solvation free energies at different cavity sizes. However, a common scheme for the calculation of the cavity size for different compounds is necessary in order to minimize the number of empirical parameters involved in the theory. For convex cavity shapes (sphere and ellipsoid), it is feasible to assume that the solute cavity is formed by adding the contributions from both the solute and solvent molecules. Such scheme allows to account for the dielectric saturation at the vicinity of the solute molecule and for the change of the cavity size in the solvents with molecules of different size. Thus, in the present study the radius of a spherical cavity was calculated as the sum of the intrinsic radius of the solute molecule, aos, and a constant additive term by solvent (water), aw- In the case of the ellipsoidal cavity, the additive solvent term was added to each semi-axis of the intrinsic ellipsoid of the solute molecule. 

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