Convex combination

The 2-RDM/or the radical may be computed from the N + l)-electron density matrix for the dissociated molecule by integrating over the spatial orbital and spin associated with the hydrogen atom and then integrating over N — 2 electrons. Because the radical in the dissociated molecule can exist in a doublet state with its unpaired electron either up or down, that is, M = the 2-RDM for the radical is an arbitrary convex combination... 

The great dofeot of the image produced in the simple camera is, its indistinctness, because the penoils have no foci or point of concourse of the rays. In order to remedy this evil convex lenses, or convex combinations of lenses, are introduced and placed at t. 

Remark 1 A convex combination of two points is in the closed interval of these two points. 

Remark 3 Any point x in the convex hull of a set S in 5ftn can be written as a convex combination of at most n + 1 points in S as demonstrated by the following theorem. 

In this chapter, the basic elements of convex analysis are introduced. Section 2.1 presents the definitions and properties of convex sets, the definitions of convex combination and convex hull along with the important theorem of Caratheodory, and key results on the separation and support of convex sets. Further reading on the subject of convex sets is in the excellent books of Avriel (1976), Bazaraa et al. (1993), Mangasarian (1969), and Rockefellar (1970). 

Proof. The result follows from the fact that Shannon entropy is concave in the space of probability distribution (DeGroot, 1970), and the average entropy is a convex combination of Shannon entropies. 

The set B of all such convex combinations, with reference to a fixed set of points (3.74),... 

At this point, the utility of this property with respect to (P2) deserves attention. A careful look at P2 reveals that the shaded region in the projected space (for example, the X -X space) is exactly the projection on the X -X space of the feasible region of P2. The concave PFR projection defines the concentrations in segregated flow, and the interior is a convex combination of all boundary points created by the residence time distribution function. This gives a new interpretation to the residence time distribution as a convex combiner. For any convex objective function to be maximized, the solution to the segregated flow model will always lead to a boundary point of the AR. 

The main insight in this approach is that the residence time distributions (RTDs) lead to convex combinations and the region enclosed by the segregated flow model is always convex. The aim now is to develop an algorithm by which, given a candidate for an AR, we should be able to check whether it can be extended to our advantage. Here, we restrict ourselves to PFR, RR, and CSTR extensions only. 

Here, J is the value of the objective function at the exit of the recycle reactor 7p7 is the value of the objective obtained from the solution of (P7). Xy is the convex combiner of all points available from the CFR model. The variables X and Re represent the concentrations in the recycle reactor extension and the recycle ratio, respectively. is the vector of exit concentrations from the RR reactor, is a linear combiner of all the concentrations from the plug flow section of the recycle reactor. 

Here, Xn,odei(p) is a constant vector and reflects the concentration at the exit at iteration p in the models previously chosen. A convex combination of this with the cross-flow region described by (P7) gives the fresh feed point for the recycle reactor we are looking for, Xupdate- exit then represents the concentration at the exit of the recycle reactor and if the earlier model chosen... 

Let us start by defining what they are. A pure state may be written as a projection operator p, = mixed state, or so-called density matrix, is defined to be any convex combination of such projectors... 

A convex combination is a linear combination with coefficients cin>0 that satisfy = 1. 

As we will now demonstrate the subset of PS-V-densities of B is not convex. More precisely, we will now show that there are E-V-densities which are not PS-V-densities [1,17]. As any E-V-density is a convex combination of PS-V-densities, this then demonstrates the nonconvexity of the set of PS-V-densities. 

But we also know that FEHK is convex on the set of E-V-densities. This leads to a contradiction and hence we must conclude that n cannot be a pure state density of any potential. The density h is, however, a convex combination of ground state densities corresponding to the same external potential and therefore, by definition, an ensemble v-representable density. We therefore have constructed an E-V-density, which is not a PS-V-density. For an explicit numerical example of such a density we refer to the work of Aryasetiawan and Stott [18],... 

To show this we take the example of a previous section where we presented a density h which did not correspond to a ground state wavefunction. It was a convex combination of 2L + 1 degenerate ground state densities nt with corresponding ground states I F[ni ) for an external potential v. Then we find... 

Let aj < bj < Cj < dj, where [a, is the interval of uncertainty at iteration k and assume that [<3i, di] = [a, d. Since functional evaluations are the most expensive step in the process, the golden section method reduces the amount of overall work by intelligently choosing symmetric points bj and Cj so that they can be reused on successive iterations, as illustrated in Figure 8. This is achieved by using the relationships b = Xa + (1 - X)d and = (1 - A)% + Xdj where A = 0.618. Observe that b and are simply expressed as convex combinations of % and idj A summary of the golden section algorithm follows ... 

Note from (33) that the method actually deflects the current direction of steepest descent by adding on a positive multiple of the direction vector used in the previous step. In fact, it is easily shown that is essentially a convex combination of V/(Xt+i) and... 

Such a property is related to the possibility of constructing approximate atomic and molecular density functions, by means of convex combinations with a basis of structurally simpler functions, which belong to the same VSS a— shell. The ASA described extensively earlier and in Amat et al., " Mestres et al., °° and Carbo-Dorca and Girones is a good illustration of these ideas. Moreover the VSS shell structure can transform density functions into a homothetic construct, a characteristic that is discussed elsewhere. The need to take into account the convex conditions [127] to construct approximate density functions has not been properly performed in the literature, as discussed. 

Jjvj(a) = j/(a) belong by construction to the a— shell S(1 + Na). Thus, convex combinations of the column vectors of a chosen matrix Jn(cx), with a fixed parameter value, a, generate the vectors ... 

The (2,3)-positivity conditions [39] on the 2-RDM may be formed by taking aU convex combinations of the (3,3)-positivity conditions that depend only on the 2-RDM. Here, we consider two important (2,3)-positivity conditions, proposed by Erdahl [12, 60]... 

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