Contour tangent

If a is itself the highest point on a previous line of search, there is already an experiment p very near a for the last stage of any onedimensional search always involves two points as close together as possible. Hence in this case only one additional experiment q would be needed to determine the contour tangent. 

Figure 2 shows a yield contour for a particular choice of scales for and x2. The same contour is plotted in Fig. 3 for which the horizontal scale has been doubled. The contour tangent and gradient line at the same point a are given for each choice of horizontal scale. The two gradient lines obviously do not contain the same points. Thus the direction of steepest ascent depends entirely on the relative scales of the... 

Figure 15. Left-. Geometry of the surface 6 = 0 in Eq. (46) with fixed total angular momenta S and N. Properties of the special points A, B, C, and D are listed in Table I. All other permissible classical phase points lie on or inside the surface of the rounded tetrahedron. Right. Critical section at J. Continuous lines are energy contours for y = 0.5 and N/S = 4. Dashed lines are tangents to the section at D. Axes correspond to normalized coordinates, /NS and K JiN + S). Taken from Ref. [2] with permission of Elsevier.

Average projection of the end-to-end vector on the tangent to the chain contour at a chain end in the limit of infinite chain length. 

When a wormlike spherocylinder is in the liquid crystal phase, its tangent vector a at each contour point should align more or less to the preference direction of the phase specified by the director n. This alignment induces the orientational entropy decrease — Sor from the entropy in the isotropic state. Since the orientation of the tangent vector stretches the wormlike spherocylinder, — Sor includes a conformational entropy loss of the spherocylinder. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

Contour plots for n = 2 also help illustrate paths toward minima specified by various minimization methods. The gradient vector is orthogonal to the contour lines. The familiar notion in one dimension that the negative tangent vector at a point x points toward the minimum of a convex quadratic extends naturally to higher dimensions. Thus, if the contour plots are circular... 

An absolute shape characterization is obtained if a molecular contour surface is compared to some standard surface, such as a plane, or a sphere, or an ellipsoid, or any other clo.sed surface selected as standard. For example, if the contour surface is compared to a plane, then the plane can be moved along the contour as a tangent plane, and the local curvature properties of the molecular surface can be compared to the plane. This leads to a subdivision of the molecular contour surface into locally convex, locally concave, and locally saddle-type shape domains. These shape domains are absolute in the above sense, since they are compared to a selected standard, to the plane. A similar technique can be applied when using a different standard. By a topological analysis and characterization of these absolute shape domains, an absolute shape characterization of the molecular surface is obtained. 

Figure 5.3 The shape domains of relative local convexity of a MIDCO surface G(a) of Figure 5.1, relative to a tangent sphere T of curvature b (radius 1/b) are shown. A geometrical interpretation of the classitication of points r of G(a) into locally concave Dq, locally saddle-type D, and locally convex D2 domains relative to b is given when comparing local neighborhoods of the surface to the tangent sphere T. The classification depends on whether at point r the surface G(a) is curved more in all directions, or more in some and less in some other directions, or less in all directions, than the test sphere T of radius 1/b. In the corresponding three types of domains Do(b). Dm,), and D2(b), or in short Dp, D, and D2, the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b.

The case of b=() corresponds to the shape domain subdivision of G(a) in terms of ordinary local convexity 155, 99], Geometrically, this case corresponds to comparing the local regions of the molecular contour surface to a test surface of zero curvature, that is, to a tangent plane. Local convexity and the corresponding classification of points r of G(a) into various domains, in the present case... 

Figure M-2. A subdivision of a molecular contour surface G(m), based on local curvature properties. The contour surface is subdivided into locally concave (Do, ), saddle-type (Di, ), and locally convex (D2, ) domains with respect to the local tangent plane in each point r. The second index k refers to an ordering of these domains according to decreasing surface area.

---