Supremum

In equation (9.131), sup is short for supremum, which means the final result is the least upper bound. Thus the //qo-optimal controller minimizes the maximum magnitude of the weighted sensitivity function over frequency range uj, or in mathematical terms, minimizes the oo-norm of the sensitivity function weighted by fE(jtj). 

The Kolmogorov-Sinai entropy per unit time is defined in Eq. (89) as the supremum of h over all the possible partitions V. Since we expect that the probability of the nonequilibrium steady state is not time-reversal symmetric, the probability of the time-reversed paths should decay at a different rate, which can be called a time-reversed entropy per unit time [3]... 

Here k(e) is a positive bounded function of e only for briefness of notation the supremum norm has been introduced as... 

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,... 

Using the definition of supremum as the lowest upper bound and introducing a scalar pB we obtain ... 

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. 

The set fi(G )(a),Gg(a)) in definition (22) of g A,B) contains values of ordinary Hausdorff distances that are nonnegative. Consequently, the supremum over this set is also nonnegative. 

Each element in the set a/ifG foXGgfa)) in the definition (43) of the commitment-weighted fuzzy Hausdorff-type distance f(A,B) is nonnegative hence, the supremum over this set is also nonnegative. Consequently,... 

If the fi A, B) supremum in definition (43) is zero, this implies that each a-scaled ordinary Hausdorff distance ahiG ia XGgia )) of a-cuts in the set ah(G (a),GBia)) is zero for any a > 0,... 

Consequently, the supremum f(A,B) in definition (43) is also necessarily symmetric,... 

We assume that the a-cuts G (a), Gg(a), and G(-(a) of three fuzzy sets A, B, and C, respectively, depend at least piecewise continuously on the a parameter from the unit interval [0,1], where continuity is understood within the metric topology of the underlying space X. On the closed interval [0,1], the scaled Hausdorff distance ah GJ a),Ggia)) is an at least piecewise continuous function of the level set value a. This function either attains its maximum ft(G ( a ), Gg a )) at some value a within [0,1] or it converges to its supremum value... 

The properties of the supremum imply that for limits of convergence to any other avalue,... 

A fuzzy set A has the / fuzzy symmetry group G Sg, /3 ) at fuzzy level if has the fuzzy symmetry element R( j8 ) at the fuzzy level j8 of the fuzzy Hausdorff similarity measure for each symmetry operation R of the crisp symmetry group G. The )3 fuzzy symmetry group G(s, at the fuzzy level /3 that is the supremum of the levels j8 of all )3 fuzzy symmetry groups G(Sg, (3 ) of A is the fuzzy symmetry group G s, of the fuzzy set A ... 

We say that a fuzzy set A has the )8 fuzzy symmorphy group at fuzzy level fi if A has the fuzzy symmorphy element 5( )3 ) at the fuzzy level /3 of the FSNSM fs g for each symmorphy operation S of the crisp symmorphy group sph- The fuzzy symmorphy roup G pjj(fs, ) 3t the fuzzy level P that is the supremum of the levels of all fuzzy... 

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. 

The supremum is defined to be the smallest number M such that l/ (r)l M almost everywhere. The term almost everywhere has a precise mathematical meaning for which we refer to the literature [4]. We almost never use it in the remainder of this paper. We therefore conclude that if v L°° then the expectation value of the external potential is finite. To show this we used that dGL1. But we also know that n L3 and if we make use of the Holder inequality... 

If the supremum is attained for some v then n is an E-V-density. This follows because then there is a density matrix D[n that yields density n (see Theorem 5) such that... 

The density matrix D[n must therefore be a ground state density matrix. If n is not an E-V-density then the supremum is not attained for any v. In any case, for every integer k and any density n0 5 we can always find some vk such that... 

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