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...

An anticipatory system S2 is one which contains a model of a system S1 with which it interacts. This model is a predictive model; its present states provide information about future states of S1. Further, the present state of the model causes a change of state in other subsystems of S2; these subsystems are (a) involved in the interaction of S2 with S1, and (b) they do not affect (that is, are ...

the value of good X (x-axis) and such that the response is menu dependent (red when X is offered against Y, blue when X is offered against Z). Transitivity violations (i.e., cases where x-axis orderings are not conserved on the y-axis) are possible, as for X1, X2, X3. However, if the representation is menu invariant (i.e., if red and blue lines coincide), transitivity violations cannot occur. I...

Let M be a closed 4-manifold with π = π1(M) 6= 1 and π2(M) ∼= Z, and let u : π → Aut(π2(M)) be the natural action. If π ∼= Ker(u) × Z/2Z then M is homotopy equivalent to the total space of an RP -bundle over an aspherical surface. We show here that if π is not such a product then M is homotopy equivalent to the total space of an S-orbifold bundle over a 2orbifold B. There are at most two such o...

[1] The principal solar semidiurnal tide (S2) in the ocean is forced by the pressure loading of the atmospheric thermal tide as well as by the gravitational tidal potential. This paper examines the effects of adding the atmospheric S2 forcing to a forward tide model. When the model is forced only by the gravitational potential, the S2 relative elevation error with respect to pelagic tide gauges...