The average and exponential tests have higher power than the

supremum.

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.