If a function u belongs to [U.sup.n.sub.[alpha]] and a twice continuously

differentiable function v : (0, [infinity]) [right arrow] R is

where [f.sub.h2] (x, [[delta].sup.*.sub.e]) is a completely unknown continuously

differentiable function.

Theorem 1: A set of optimal strategies {[p.sup.N.sub.i]|[p.sup.N.sub.i] = [[bar.p].sub.i] - [[[rho].sub.i][summation over (i[member of]N)][u.sub.i]/2[S.sub.i](r + [tau])]} provides the transmit power of n SUs under the condition of grand coalition N, and the continuously

differentiable function W([p.sub.i], y) is expressed as follows

The method allows receiving the mathematical description difficult nonlinear and even piecewise dependence in the form of unitary continuously

differentiable function with any accuracy of approach allowed by data.

Let I(x,y) be a twice

differentiable function in (a, b) x (c, d), given ([x.sub.0], [y.sub.0]) [member of] (a, b) x (c, d), the twice differential of I(x, y) at point ([x.sub.0], [y.sub.0]) is

The rational fractal functions in Figures 5(a)-5(e) are typical fractal functions close to continuous but nowhere

differentiable function. By taking the rational quadratic FIF in Figure 4(a) as the original function, we have calculated the uniform distance between this original function and the rational quadratic FIFs in Figures 4(b) and 4(f) (see Table 3).

By formula (11) we can easily compute the fractal derivative of any

differentiable function using the fractional derivative; for concrete examples see Appendix B.

Let g be a real-valued

differentiable function defined on X, and assume that for each x, y [member of] X, the function F(x, y; *): X [right arrow] R is sublinear.

or, if the matrix A is a

differentiable function of the set of entries [t.sub.ij]

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be two Riemannian manifolds and f a positive

differentiable function on [M.sub.1].

[E.sub.p] can be expressed as a

differentiable function of the output variable [y.sub.k].

which is a variant of Simpson's inequality for first

differentiable function f, [f.sup.'] is integrable and there exist constants [gamma], [GAMMA] [member of] R such that [gamma] [less than or equal to] [f.sup.'](t) [less than or equal to] [GAMMA], t [member of] (a, b).