Infimum

Remark 1 If we admit the points oo, then every function f(x) has a supremum and infimum on the set S. 

Remark 2 The minimum (maximum) of a function f x), if it exists, must be finite, and is an attained infimum (supremum) that is,... 

Remark 2 The dual problem consists of (i) an inner minimization problem of the Lagrange function with respect to x 6 X and (ii) on outer maximization problem with respect to the vectors of the Lagrange multipliers (unrestricted A, and ft > 0). The inner problem is parametric in A and fi. For fixed x at the infimum value, the outer problem becomes linear in A and ft. 

Remark 3 The dual function ( A, p) is concave since it is the pointwise infimum of a collection of functions that are linear in A and p. 

Remark 8 If -we assume that the optimum solution of v(y) in (6.4) is bounded for ally Y n V, then we can replace the infimum with a minimum. Subsequently, the... 

Note that the inner problem is written as infimum with respect to x to cover the case of having unbounded solution for a fixed y. Note also that cTy can be taken outside of the infimum since it is independent of x. 

Remark 3 Note that we can replace the infimum with respect to x X with the minimum with respect to X, since fory G fflV existence of solution holds true due to the compactness assumption of X. This excludes the possibility for unbounded solution of the inner problem for fixed y G Y n V. 

Since qi is proper, convex, bounded, and Lipschitzian on X, and X is compact and convex, the infimum in (6-50) is attained, and hence can be replaced by minimum. Similarly the cuts that correspond to infeasible primal problems can be written in terms of q2,94 as follows ... 

For all fuzzy sets, including three-dimensional functions of electron density-like continua provided with suitable membership functions, the differences between the corresponding fuzzy sets can be expressed by a metric based on a generalization of the Hausdorff distance. The basic idea is to take the ordinary Hausdorff distances h a) for the a-cuts of the fuzzy sets for all relevant a values, scale the Hausdorff distance h)a according to the a value, and from the family of the scaled Hausdorff distances, the supremum determines the fuzzy metric distance f A,B) between the fuzzy sets A and B. If, in addition, the relative positions of the fuzzy sets A and B are allowed to change, then the infimum of the f(A, B) values obtained for the various positionings determines a fuzzy metric of the dissimilarities of the intrinsic shapes of the two fuzzy sets. 

Most of these R sets are not unique for a given fuzzy set A hence, the infimum is taken for all such fuzzy sets. ... 

Another measure for the symmetry deficiency of set A, independent of positioning and partitioning, is given by the infimum of ZPA " >taken over all the allowed positionings and partitionings. This measure, h p/ A, A f p), is defined as... 

By taking the infimum for all the allowed choices of P, a symmetry deficiency measure of crisp or fuzzy set A is obtained that is independent of positioning and partitioning. The corresponding df I/l, /Iff.uf,/ )... 

If the relations of Eq. (62) hold, then this functional groupis called an infimtm with respect to the chemical inclusion relation < taken as the partial order. In this case the family/is a lower semilattice. For example, the family/can be chosen so that each member/ contains a common atom, for example, the H atom, where H is also regarded as a functional group. Then the < partial order relation of chemical inclusion implies that family /is a lower semilattice, with /i = H as its infimum element [18]. 

In special cases, both relations of Eqs. (62) and (63) hold, that is, both infimum and supremum exists within family/with respect to the < partial order relation. In such a case the family / of functional groups is a lattice. Lattices and semilattices are important algebraic tools for systematic analyses of various hierarchies. 

We take the infimum over all densities in S on the left hand side. Since we know that there is ground state density matrix corresponding to n0 we obtain from equation... 

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