By Gottlieb E. S.
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Extra resources for (1,k)-configuration facets for the generalized assignment problem - Elsie Sterbin Gottlieb and M. R. Rao
Our idea is to consider the image of the real axis β(R), under a mapping class β ∈ Bn . Of course there are choices here, but there is a unique “canonical form” in which (roughly speaking) R∩β(R) has the fewest number of components. Now declare a braid β to be positive if (going from left to right) the ﬁrst departure of the canonical curve β(R) from R itself is into the upper half of the complex plane. Amazingly, this simple idea works, and gives exactly the same ordering as Dehornoy’s combinatorial deﬁnition.
These rings also play a role in the theory of Vassiliev invariants. A basic property of a ring would be whether it has (nontrivial) zero divisors. If a group G has an element g of ﬁnite order, say g p = 1, but no smaller power of g is the identity, then we can calculate in ZG (1 − g)(1 + g + g 2 + · · · + g p−1 ) = 1 − g p = 0 and we see that both terms of the left-hand side of the equation are (nonzero) divisors of zero. A long-standing question of algebra is whether the group ring ZG can have zero divisors if G is torsion-free.
Proof . n−1 n i=0 j=0 (−1)i di ∂n−1 ◦ ∂n = (−1)j dj (−1)i+j di dj + = 0≤i
(1,k)-configuration facets for the generalized assignment problem - Elsie Sterbin Gottlieb and M. R. Rao by Gottlieb E. S.