By R. T Rockafellar

Offers a comparatively short advent to conjugate duality in either finite- and infinite-dimensional difficulties. An emphasis is put on the basic value of the techniques of Lagrangian functionality, saddle-point, and saddle-value. basic examples are drawn from nonlinear programming, approximation, stochastic programming, the calculus of diversifications, and optimum keep an eye on

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On the other hand the sum of the last n columns of"( is 0 and therefore Pf(,,() = O. e). We now prove the lemma. Consider the alternate (n x n)-matrix (3 = ((ai 0 aj)a;aj)~j=l over S. It is clear that Pf((3(i)) = Pf(O(i)) Ilk#iat, for i = 1, ... ,n. 1. Determinant and Pfaflian for alternate trilinear forms 35 As we know, the sum of the columns of (J is O. e), for any i, j = 1, ... ,n. Since the annihilator of a'k in S is zero, this implies that (_I)i Pf(O(i)) aj = (-I)j Pf(O(j)) ai for any i,j.

Then a direct computation shows that [T](M,e) = T(M, e) + t K / 2 v E Z[H]. 5. 5 27 The Alexander polynomials We show in this section that the torsion dominates the Alexander polynomials of 3-manifolds. 1. The Alexander polynomials of groups. Consider a finitely generated group 7r. Set H = HI (rr), G = HI TorsH, and apply the projection pr : Z[H] --t Z[G] to the ideal E(rr) C Z[H]. Since Z[G] is a unique factorization domain, one can consider the greatest common divisor of the elements of pr(E(rr)) C Z[G].

It was first observed by Milnor [Mil] that for a compact connected 3-manifold M with non-void boundary consisting of tori, the polynomial ~(M) can be interpreted as a torsion of M. This was extended to closed 3-manifolds in [Tul]. In Section 5 we generalize these results to the twisted Alexander polynomials. -S. Lin, M. J. C. Wang, T. Kitano, P. Kirk, and C. Livingston. 5 goes back to [Fol], Prop. 4 at least in the case TorsH = 0. Chapter III The Torsion versus the Cohomology Rings In this chapter we compute the "leading term" of the torsion of a closed oriented 3-manifold M in terms of the trilinear form on I-cohomology induced by the cupproduct.