We give an explicit simplicial model for the Hopf map S3 → S2. For this purpose, we construct a model of S3 as a principal twisted cartesian product K×ηS, where K is a simplicial model for S1 acting by left multiplication on itself, S2 is given the simplest simplicial model and the twisting map is η : ( S2 ) n → (K)n−1. We construct a Kan complex for the simplicial model K of S1. The simplicial...

This paper considers classical online scheduling problems on uniform machines. We show the tight competitive ratio of LS for any combinations of speeds of three machines. We prove that LS is optimal when s3 ≥ s2 ≥ s1 = 1 and s3 ≥ s2 + s2s3 + s2, where s1, s2, s3 are the speeds of three machines. On the other hand, LS can not be optimal for all combinations of machine speeds, even restricted to ...

This note is a supplement to [S2]. The main point is to prove Theorem 2.1.1, which is a special case of [S2, Theorem 3.3.6], without the use of Lie superalgebras or Koszul duality. The focus here is on the case of the symplectic group, though the proofs given here could be adapted to the other cases from [S2]. The proof we give here was found before the one in [S2] and we believe that it is of ...

Proposition 1. Consider two stocking policies s1 and s2 where for some j , s2 j = s1 j − 1, s2 j−1 = s1 j−1 + 1, and s2 i = s1 i , ∀ i = j j − 1. We have that: (i) ILi s2 i s 2 N ≥ ILi s1 i s1 N for i ∈ 1 j − 1 , and (ii) ∑j i=1 Bi i s 2 i s 2 N ≥ ∑j i=1 Bi i s 1 i s 1 N . Proof. From (6) and (7) we find that ILj s2 j s 2 N = ILj s1 j − 1 s1 j+1 s1 N = ILj s1 j s1 N − 1. Thus from (8) and (9), ...

A bisection of a graph G is a bipartition S1, S2 of V (G) such that −1 ≤ |S1|− |S2| ≤ 1. It is NP-hard to find a bisection S1, S2 of a graph G maximizing e(S1, S2) (respectively, minimizing max{e(S1), e(S2)}), where e(S1, S2) denotes the number of edges of G between S1 and S2, and e(Si) denotes the number of edges of G with both ends in Si. There has been algorithmic work on bisections, but ver...