Since [h.sup.1](N, [I.sub.Z[intersection]N,N](t)) = 0 for every hyperplane N [subset] [P.sup.r], Lemma
1 gives [h.sup.1]([I.sub.Z](t - 1)) [greater than or equal to] r + [h.sup.1]([I.sub.Z](t)) = r + 1.
9, let u = u in bounded domain M and u = 0 in [R.sup.n] M; we can easily obtain the following lemma
2.4, there exists some [beta] > 0 such that
12 and linear systems (16), (18), and (19), we have
3, we have [b.sub.5] (G) = -4n + 8- 2[c.sub.5] (G).
8, for any [epsilon] > 0, there is C([epsilon]) > 0 such that
3, [k.sub.w] [member of] [F.sub.[alpha]] and [mathematical expression not reproducible].
The first one is obtained by combining Lemma
3 with the previous theorem.
Again apply Lemma
2.3 (3) to [B.sub.11] -[B.sub.12] + [B.sub.21], then a", b", c", and d" [member of] R.
Due to the above lemma
, the ellipsoidal approximation of the central path H([mu]) is mathematically defined as follows:
Then, by using Lemma
1, we get S(RL] = (SS](.RL] [subset or equal to] (SS * RL] = (SR * SL] [subset or equal to] (SR * (SL]] = (SR * L] = ((SS]R * L] [subset or equal to] ((SS)R * L] = ((RS)S * L] [subset or equal to] ((RS]S * L] [subset or equal to] (RL], which shows that (RL] is a left ideal of S.
In the next lemma
we find the asymptotics of [V.sup.*].