By invoking the

Implicit Function Theorem, there exists a function f such that the identity [y'.sub.1] - E'Qy = f(t,Qy) holds.

To illustrate that [GAMMA] (V, [[PHI].sup.*.sub.0], [[PHI].sup.*.sub.1]) satisfies the

implicit function theorem, the following theorem is given.

and the

implicit function theorem, there exists a subfamily of variations satisfying restriction equation (43).

(2) In order to handle the nonaffine coupling terms, the

implicit function theorem and the mean value theorem are invoked, respectively.

and using the analytic

implicit function theorem, one solves w in terms of z, [bar.z], and [bar.w] getting a representation of M as

By Propositions 3.1, 4.1 we can apply the

implicit function theorem (see, e.g., [4, Theorem 1.20]) to conclude that F maps a sufficiently small neighborhood of a positive, constant function c in [C.sup.[alpha]][(0, 1).sub.0] homeomorphically onto a neighborhood of [square root of (2c')] in [C.sup.[alpha]+[1/2]] [(0, 1).sub.0].

Advanced topics included vector-valued functions, the

implicit function theorem, extremal problems, matrix-valued holomorphic functions, matrix equations, realization theory, eigenvalue location and zero location problems, convexity, and some special results relating to matrices with nonnegative entries.

The

implicit function theorem is applied in a manner that requires further qualification because the optimization problem is of an unconstrained kind without any redundant variables.

By

implicit function theorem, there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (10) has a unique solution ([lambda], [M.sub.2], [M.sub.3]) satisfying [lambda], [M.sub.2] > 0 and [M.sub.3] > 0.

The continuity of the constant equilibria follows from the

Implicit Function Theorem and the hypothesis of normal hyperbolicity.

We prove that the first-order condition is an implicit function [R.sub.i] = [R.sub.i]([R.sub.j]), except possibly at a point of measure zero using the

Implicit Function Theorem. We first establish two lemmas where we define