Specifically, we consider

upper bound constraints dk given as dk =[mathematical expression not reproducible]; see (11).

The following theorem shows an exact expression for the

upper bound [[bar.[rho]].sub.R] in a closed form.

In this section, we evaluate the partition function in the presence of the

upper bound.

In the above formula, the nonlinear shear strength index [[phi].sub.t] can be optimized by the least energy principle when the

upper bound is calculated.

Thus, the deviation as the percent absolute error with respect to solar PV system performance is calculated in both the lower bound and

upper bound sections from the result of the crisp value of solar PV system performance.

Take an arbitrary total ordered subset N [subset] [OMEGA]/ ~, next, to show that the set N has an

upper bound in [OMEGA]/ ~.

The effectiveness of this test depends on the current value of the

upper bound; that is, tighter

upper bound values lead to more effectiveness in the test.

The lower and

upper bounds of sin([[THETA].sup.I.sub.n]) and cos([[THETA].sup.I.sub.n]) are given by

In Figure 5(d), the control measure [w.sub.4] is at

upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time.

The

upper bound of p(q) is [mathematical expression not reproducible] for 0 < r < 0.5, w = ([D.sub.1]/2)/[square root of ([[sigma].sup.2.sub.1]/[n.sub.1] + [[sigma].sup.2.sub.2]/[n.sup.2])], where [mathematical expression not reproducible] is the t-distribution function with [n.sub.1] + [n.sub.2] - 2 degrees of freedom and [mathematical expression not reproducible] is the inverse function of [mathematical expression not reproducible].

Upper Bound. We now obtain an

upper bound of R([t.sub.0], [A.sub.i,j] + [d.sub.i]) of the total amount of processor time demand.

To use a lower-dimensional transformation F for SSDD, we need to find an

upper bound function upper([mathematical expression not reproducible]) that satisfies (1), where [??] and [??] are f-dimensional feature vectors transformed from n-dimensional vectors, [??] and [??], respectively, by the transformation F.