For p = 1, every h is completely charaterized by a

real-valued function of a single variable (see Dajczer et al.

We recall that a

real-valued function u on a topological space (X, [tau]) is said to be upper semicontinuous if [u.sup.-1](] - [infinity], [alpha][) = {x [member of] X: u(x) < [alpha]} is an open set for all [alpha] [member of] R.

He proposed that the homogeneous Dirichlet conditions may be satisfied exactly by representing the solution as the product of two functions: (1) an

real-valued function that takes on zero values on the boundary points; and (2) an unknown function that allows to satisfy (exactly or approximately) the differential equation of the problem.

We now define a

real-valued function f by f(t) = [M.sub.F(t)]([[alpha].sup.*]).

where [k.sub.1] is the 1th curvature function and [lambda] is the

real-valued function on [a, b].

Let f be a differentiable

real-valued function defined on Rn.

where [omega] [member of] R and u is

real-valued function. In fact, we find solutions (26) satisfying the boundary condition u([+ or -][infinity]) = 0 and solutions (32) satisfying the boundary condition (u(-[infinity]), u(+[infinity])) = (0,1) or (1, 0).

Note that [L.sub.[rho]] is a

real-valued function; then we get

It can be used to perform the validation, compilation, and evaluation of any

real-valued function. It includes a computation engine as well, which is used to calculate the function values over its definition domain.

where u = u(x, t) is a

real-valued function, a, b are positive real constants with a [greater than or equal to] b, and p > 1.

Let A be a Holder continuous complex-valued function on [GAMMA] with A [not equal to] 0 and [gamma] be a Holder continuous

real-valued function on [GAMMA].

A

real-valued function f (t) (t > 0) is said to be in the space [C.sub.[mu]]([mu] [member of] R) if there exists a real number p > [mu] such that f (t) = [t.sup.p][phi] (t).