Tangent hull

The condition that a span of the curve might contain a foot-point is that the set of vectors from the target point to the hull of the span should have an intersection with the set of planes perpendicular to the tangent vectors in the hull of the first derivatives. The tangent hull is defined exactly analogously to the hull of points, but using the first difference scheme instead of the original scheme. 

The local equilibrium surface alloy configuration and structure may be found by minimization of the surface free energy, or if several different phases may exist, by finding a convex hull of the lowest free energies of different phases at different alloy compositions (at T=0), or more generally by a common-tangent construction which is completely analogous to the usual treatment of the bulk systems. The procedure is illustrated in Fig. 4. 

From a technical point of view, the big drawback about the tangent cone is that it is non-linear. It is always easier to handle essentially linear objects. The most natural way around this is to study the linear hull of the tangent cone, which we will call the tangent space. Let X be a scheme of finite type over At, and let x X be a closed point. 

Once again, we see that Fxcx is exactly a linear hull of the tangent cone. 

By means of generalized-ensemble computer simulations, an accurate estimate of the density of states g E) can be obtained. For this particular heteropolymer, it turns out that the entropy S E) exhibits a convex region, i.e., a double-tangent with two touching points, at Efoid and E nf > can be constructed. This so-called Gibbs hull is then parametrized... 

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