Convex hull defined

Figure 2.44 The Ti-Zn system. The calculated enthalpy of formation of the intermetallics and the resulting convex hull are shown according to Ghosh et al. (2006). Experimental values of about 22-19 kJ/mol of atoms were measured in the composition range 50-75 at.% Zn. The calculations yield a ground state convex hull defined by the following structures (1) Ti3Zn-Ll2 (2)...

Fig. 1. Geometry concepts tor analyzing protein cavity. The protein bulk is represented in black, probes as little spheres. The convex Hull ot the protein is represented in dash line. The plain vectors emerging trom the probe in grey are pointing towards the bulk solvent, whereas the dash vectors will encounter protein atoms within a radius of 8 A (radius of the influence circle m dot line).Jhe probe in front ot the clett defines the degree of precision in represenfing the molecular surface.

It is important to note that CSTR/PFR/RR extensions can be applied to any convex candidate region, not just the one defined by (P2). The residence time distribution can be used to generate the convex candidates. A sequence of convex hulls can be generated until the conditions for completeness are satisfied (i.e., there are no further extensions). The synthesis flowchart shown in Fig. 4 illustrates these ideas. In the algorithm, we initially check the possibility of global optimality for (P2). If this solution is suboptimal, a more complex model can be solved to give an updated optimal solution. Thus the new or updated convex... 

These three are, of course, in conflict, and there are a number of possible trade-offs between them. Condition (ii) effectively demands that the enclosure should be a convex point-set, and combined with condition (iii) this leads to the use of the convex hull, which is defined to be the convex point set of smallest volume which contains the original set. 

We limit ourselves to plane-faced enclosures. This also helps to satisfy condition (ii). In 3D the convex hull of a finite set of discrete points is plane-faced, whereas the convex hull of a curve can have much more complex shapes. Thus the challenge becomes finding, from the representation of a curve, a finite set of planar half-spaces whose intersection is guaranteed to contain the true convex hull of the curve and thence the curve itself. One way of doing this is to use the convex hull of control points defining the curve. 

Geometric methods where most definitions rest on defining the smallest convex area that covers the training set compounds in descriptor space. This method is also known as the convex hull method (Figure 14.7). 

We choose a Voronoi cell V as a fundamental domain of D. When one hits a three-dimensional boundary of a Voronoi cell, by cutting Eg parallel to the observable space, one projects into the observable space E the dual (three-dimensional) boundary. The result appears as a tile in E. The dual boundary is defined as a convex hull of all lattice points of which Voronoi domains contain the hit boundary. Instead of cutting the six-dimensional Voronoi cells, one can define a procedure on a single projected Voronoi cell y(0) with a hierarchy of all its lower-dimensional boundaries into the orthogonal space E, Vx(0) = W, the window [6]. In case of Z)g, the window W has an outer shape of a triacontahedron see Figure 12-2 top). 

Cluster area is here defined as the area of the 2D convex hull polygon divided by the total sum of cluster areas. A distance between two clusters is defined to be the distance between the centre points of the corresponding convex hull polygons. An overlap between cluster i and J is defined as ... 

To define a local subset of Pareto-optimal solutions dose to y the Delaunay triangulation is calculated for the set Z = ZHf u y. The Delaunay triangidation subdivides the convex hull of a set of points into disjunct simplices. Each simplex consists of d + points whereby d denotes the dimension of the data set. A specific property of the Delaunay triangulation is that for each simplex the circumhypersphere constituted by its points is empty which implies that the Delaunay triangulation is unique. Let D denote the set of Delaunay simplices where each simplex Ak is a set of d+1 points, i.e.di,= ... 

Concentrations Cj, C2,. .., Cj. are members of the set that make up the convex hull of X, conv(X). Recall that points belonging to the convex hull boundary are the unique points of the set. It is clear that the extreme points of the region defined by conv(X) are sufficient to generate the entire set of concentrations that lie in the set X. This is given... 

In order to understand how NBI works, it is necessary to define some terminology. The Convex Hull of Individual Minima (CHIM) is defined as the set of points that are linear combinations of F(xt ) -F for i = l,...,iw, where x, is the global optimal solution of Ffx) and F is the shadow minimum (or utopia point), i.e., the vector containing the individual global minima of the objectives. The pay-off matrix is defined as an m x m matrix whose column is F(Xi ) -F. Given a vector P, d>p defines a point on the CHIM. Mathematically, the so-called NBI subproblem is formulated as ... 

Geometric Methods Convex Hull AD. AD is defined as a convex hull of points in the multidimensional descriptor space (Fechner et al. 2008). 

Figure 8 Convex hull. Rolling a plane over a molecule defines the smallest convex region containing the molecule...

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