A Riemannian metric g is said to be associated with a contact manifold if there exist a (1, 1) tensor ield [phi] and a contravariant global vector field [xi], called the

characteristic vector field of the manifold such that

The projection of eigenvector [u.sup.i] is the largest on X, so when the

characteristic vector of X is the largest, it makes tr([S.sub.x]) the maximum value.

There is a training set D = {([X.sub.1], [y.sub.1]), ([X.sub.2], [y.sub.2]), ..., ([X.sub.n], [y.sub.n])}, where X{ is the

characteristic vector of the training sample and yi is the associated class label.

Hence, the input

characteristic vector for PNN is [[[sigma].sub.[theta]], [[sigma].sub.c], [[sigma].sub.t], [W.sub.et]].

According to the matrix theory, we can judgment that the weight coefficient of each factor is the

characteristic vector w of the judgment matrix.

(3) A warped product space R x [sub.[lambda]][C.sup.n] if k([??], X) < 0; where k([??], X) denotes the sectional curvature of the plane section containing the

characteristic vector field [??] and an arbitrary vector field X.

If matrix A has a unique pair of conjugate purely imaginary characteristic roots [[lambda].sub.1,2] = [+ or -]i[[omega].sub.0] ([[omega].sub.0] > 0), and we denote q as the

characteristic vector corresponding to the characteristic value i[[omega].sub.0] of matrix A, then

Every time, t, in which a j state is input, a

characteristic vector [o.sub.t] is generated, according to the probability density [b.sub.j]([o.sub.t]).

According to the

characteristic vector H = [[H.sub.c], [H.sub.t]] extracted in part 2, the

Clearly the set X = {v: [phi]'(v) = k + 1} is independent in G (since 0 [member of] t(e) for all e [member of] E) and therefore its

characteristic vector p belongs to P.