(Penot, [37]) Given a subset D of a normed vector space X, x [member of] D, v [member of] X, the second-order projective tangent cone to D at (x, v) is the set [T.sub.2] (D, x, v) of pairs (w, r) [member of] X x [R.sub.+] such that there exist sequences {[t.sub.n]}, {[r.sub.n]}, [t.sub.n], [r.sub.n] > 0, with limits 0 and r, respectively, [w.sub.n] [right arrow] w such that [??] and

is the asymptotic second-order tangent cone to D at (x,v).

Penot showed that the second-order projective tangent cone in the finite dimensional case is always nonempty (see Proposition 1, [37]).

Our optimality conditions make use of the contingent cone ([3], [7], [26]) and the second-order projective tangent cones to the constraint set at the extremum point [37].

Penot, [36] and Cambini et al., [8] established second-order necessary optimality conditions for a point to be a local minimum and a local maximimum, respectively, for problem (P) using the theory of second-order projective tangent cones for twice Frechet differentiable objective functions as well as a sufficient optimality condition of the same type.

Then we find the explicit form of the higher-order tangent sets, of the higher-order adjacent sets and of the related higher-order tangent cones in Pavel sense ([32], [33]).

Given a piecewise [C.sup.1] function [gamma] : [0, L] [right arrow] [R.sup.2], one defines the tangent cone of [gamma] at a point s (which is denoted by [T.sub.[gamma]](s)) in terms of the one-sided derivatives.

One extends the tangentially graph-like notion to boundaries that are piecewise [C.sup.1] by defining [partial derivative][OMEGA] to be tangent-cone graph-like (TCGL) at a point [gamma](s) [member of] [partial derivative][OMEGA] if it is graph-like at [gamma](s) for every orientation in the tangent cone of [partial derivative][OMEGA] at s.

The tangent cone is dependent on the direction in which [gamma] traverses [partial derivative][OMEGA] (which by convention was counterclockwise) since an arc-length traversal [??](s, r) = [gamma](L - s, r) would have different tangent cones (namely, w [member of] [T.sub.[gamma]](s) iff -w [member of] [T.sub.[gamma]](s)).

As [partial derivative][OMEGA] is tangent-cone graph-like at q, then [partial derivative][OMEGA] [intersection] C(q, r) is a graph in every orientation in the tangent cone of [partial derivative][OMEGA] at q.

If we can find s [member of] [[s.sub.1], [s.sub.2]] and w in the tangent cone of [partial derivative][OMEGA] at [gamma](s) satisfying <w, p - [gamma](s)> = 0, we will contradict that [partial derivative][OMEGA] is tangent-cone graph-like.

Topics of the remaining 11 papers include minimal free resolutions of lexsegment ideals,

tangent cones of numerical semigroup rings, topological Cohen-Macaulay criteria for monomial ideals, and the type of the base ring associated to a transversal polymatroid.