Convex envelope

Construction of convex envelopes consists of a system of linear inequalities (it is a typical problem in linear programming see, for example, ref. 31). In the simplest cases a convex envelope can also be constructed directly. This envelope can also be described parametrically without using inequalities. For example, for a system of x1 x2,.. ., xq points, their convex envelope consists of linear combinations l xy +. .. + Xqxq where Xu. . Xq are non-negative values whose sum equals unity... 

For our purposes, however, it is necessary to set a convex envelope (106) by a system of inequalities. Let these inequalities be... 

It is evident that M2 < M, hence e(2V°) = M,. A corresponding level line for G is shown in Fig. 9(a). The points e and e2 are also shown on the figure. Their convex envelope is a vertically hatched tetragon. Horizontal hatches mark its conjunction with a multitude specified by the inequality G(N) =% r,(N°). The entire region of the point N is co-invariant. It is the other parts of the space near N° that is V(N°), i.e. the desired "unattainability region . 

The results of the analysis for a system of three isomers for various E are represented in Fig. 9(a)-(b). Here, a convex envelope for the finite multitude (106) is vertically hatched and its union with the multitude G(N) e( ) is horizontally hatched. The whole of the hatched multitude is co-invariant and the unhatched region is just V(E). This example of only four multitudes makes it possible to construct the "unattainability regions that would not be a union of those for submultitudes. Three multitudes each contain one vertex and a fourth [Fig. 9(d)] includes two vertices, corresponding to the cases when the entire mass is concentrated either in Aj or in A2. 

The other way to describe this property is as follows. A convex envelope for the multitude of stoichiometric vectors for the edges in the graph of predominant directions written as if they were direct reaction steps does not contain zero, i.e. there are no such non-negative Alt. . Aq as... 

It must be noted that, in general, particles do not have a unique surface area, the measured surface depending on the method of measurement that is the degree of discrimination. At a low level of discrimination the area of the convex envelope of the particle is measured at higher levels the areas of concavities in the surface are included. Similarly, unless the particles... 

The convex envelope of these perpendicLilarplanes describes the crystal shape in equilibrium, rte,) = niin ( ) (ny(n e ) ]... 

An occurrence of several critical points for monocomponent fluid leads to complication of binary mixture phase behavior. Following Varchenko s approach", generic phenomena encountered in binary mixtures when the pressure p and the temperature T change, correspond to singularities of the convex envelope (with respect to the x variable) of the front (a multifunction of the variable x) representing the Gibbs potential G(p,T,x). Pressure p and temperature T play the role of external model parameters like ki2. A total... 

The idea of method is to use the interpolated surface for the linear trend approximation not in the entire convex envelope of source point set but only at that zones where a good quality of interpolation can be guaranteed. This approach provides a better quality for the needed linear trend approximation. 

We have seen that the analytic functions p(v), if>(p), and M(p), not reconstructed by the equal-areas and double-tangent rules—i.e., the functions shown as the smooth isothermal curves in Figs 1-8. 3.1, and 3.4—are central to the van der Waals one-density theory and that the analytic functions < (p, t ) and i)(p, < ), not reconstructed by the convex-envelope rule, are likewise central to the (c + l)-density generalization of that theory. We now ask how those functions arise in the theory and what physical quantities they represent. 

But what the pressure p(o), diemical potential M(p), etc., in the constrained system are, depend on the distance L that defines the constraint. If L is very large, the flrrduations within can almost amount to phase separation foe van der Waak loops in p(v) and M(p) would then enclose only small areas, and foe analytic functions p(o), ip(p), etc., would be dose to foe non-analytic functions obtained from them by the equal-areas, double-tangent, or convex-envelope constructions. Tire effect of the constraint with such large L is minimal and in the limit in which L is macroscopic foe thermodynamic properties become those of foe unconstrained fluid. But when L is small, the deviation of p(t>) from the equilibrium pressure in foe unconstrained system at that temperature is considerable, and similarly for foe other thermodynamic functions. 

Buckingham. M. J. in Phase Traiaitioia and Critical Phenomena (ed. C. Domb and M. S. Green) Vol. 2, Chapter 1, Academic Press, London (1972). The convex-envelope construction was used first by Gibbs, J. W. TVans. Cbm. Acad. 2, 382 (1873), reprinted in Collected Works, Vol. 1, p. 33. 

Univariate concave functions are trivially underestimated by their linearization at the lower bound of the variable range. Thus the convex envelope of the concave function ut x) over is the linear function of x ... 

Figure 4. The functions/ (x) are convex underestimators of / (x) over the interval [0, 4]. Note how/+(x) and —f (x) form a convex envelope around/(x).

Figure 6. The interval [0, 4] has been subdivided into [0, 2] and [2, 4]. The convex underestimators (x) for each subinterval, shown above, form a convex envelope around fix). As the intervals get smaller, the envelope gets tighter.

Figure 9. The intervals [0, 2] and [2, 4] have been further subdivided into [0, 1], [1, 2], [2, 3], and [3,4], Shown above are the convex envelopes around/(x) formed by convex underestimators in each of these intervals. Note that the convex envelopes for [0, 1], [1, 2], and [3, 4] intersect the x-axis, but the convex envelope for [2, 3] does not. This will allow us to conclude rigorously that no solutions to/(x) = 0 exist in [2, 3].

Figure 10. The lower bounding problem ftff the interval [2, 3] is solved. Note that the convex envelope must be expanded before it touches the x-axis, resulting in a positive value for This interval will be fathomed. (XnUnjSmn) = (2,+0.479).

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