The average and exponential tests have higher power than the

supremum.

We emphasize that this example is closely related to the differentiability of the

supremum norm in C(X).

However, it is worth mentioning that the hybrid numerical scheme proposed by Cen in [1], is an well-known fitted mesh method for solving singularly perturbed BVPs of the form (1.1)-(1.2) with discontinuous convection coefficient and the method is almost second-order accurate throughout the domain [0, 1] provided the perturbation parameter [epsilon] satisfies [epsilon] [much less than] [N.sup.-1], otherwise the method is at worst first-order uniformly convergent with respect to [epsilon] in the discrete

supremum norm (see the detailed discussion in Section 6).

Using the standard argument of the Bertrand-Edgeworth model, we obtain the

supremum and infimum of the price distributions as follows:

Since [mathematical expression not reproducible] and [mathematical expression not reproducible] for all [lambda] [member of] A, by the property of

supremum for any [lambda] [member of] A, there exists a [mathematical expression not reproducible] such that [mathematical expression not reproducible]; that is, [mathematical expression not reproducible], which implies [mathematical expression not reproducible].

All definitions (

supremum, infimum, and magnitude) and operations in IR* are defined the same as IR but computation over IR* is more complicated than IR.

The corresponding physical quantity would be the Yang-Mills action functional, but we get here an average value instead of a

supremum. This questions the choice of the

supremum in the formula

Let [mathematical expression not reproducible], Banach space of real bounded sequences x = {[x.sub.n]} with [mathematical expression not reproducible] and

supremum norm

where [x.sub.1] and [x.sub.s] are called infimum and

supremum, respectively.

where inf{[A.sup.[alpha].sub.1]} and inf{[A.sup.[alpha].sub.2]} are infimum of [A.sub.1] and [A.sub.2], and sup{[A.sup.[alpha].sub.1]} and sup{[A.sup.[alpha].sub.2]} are

supremum of [A.sub.1] and [A.sub.2] (Figure 1).

Therefore g(*) attains its

supremum over [R.sup.d] at some [eta] [member of] [R.sup.d], for which

According to Proposition 4.1 of [6], in the case of the -[DELTA] operator, the

supremum of [mathematical expression not reproducible] is equal to L.