We investigate properties of exposed points in the unit ball of the Hardy space H 1 in connection with the well known conjecture by D. Sarason. We construct a counterexample for the remaining part of the conjecture.

Grigni and Hung [10] conjectured that H-minor-free graphs have (1 + )-spanners that are light, that is, of weight g(|H|, ) times the weight of the minimum spanning tree for some function g. This conjecture implies the efficient polynomial-time approximation scheme (PTAS) of the traveling salesperson problem in H-minor free graphs; that is, a PTAS whose running time is of the form 2f( )nO(1) for...

We give two proofs of a conjecture of Neumann [N1] that a reduced algebraic plane curve is regular at infinity if and only if its link at infinity is a regular toral link. This conjecture has also been proved by Ha H. V. [H] using Lojasiewicz numbers at infinity. Our first proof uses the polar invariant and the second proof uses linear systems of plane curve singularities. The second approach a...

We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace KΘ = H 2 ⊖ ΘH of the Hardy space H, where Θ is an inner function. First, we verify the Feichtinger conjecture for the kernels k̃λn = kλn/‖kλn‖ under the assumption that sup n |Θ(λn)| < 1. Secondly, we prove the Feichtinger conjecture in the case where Θ is a one-compone...

in this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. we obtain one results, which can give a negative answer to an uniqueness question relate to the bruck conjecture dealt by liu and yang. meanwhile, we also establish a difference analogue of the bruck conjecture for entire functions of order less than 2, ...

A conjecture by Bollobás and Komlós states the following: For every γ > 0 and integers r ≥ 2 and ∆, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ( r−1 r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most ∆, then G contains a copy of H. This conjecture generalises s...

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

We present previously unpublished elementary proofs by Dekker and Ottens (1991) and Boyce (private communication) of a special case of the Dinitz conjecture. We prove a special case of a related basis conjecture by Rota, and give a reformulation of Rota's conjecture using the Nullstellensatz. Finally we give an asymptotic result on a related Latin square conjecture.

Let hom(H,G) denote the number of homomorphisms from a graph H to a graph G. Sidorenko’s conjecture asserts that for any bipartite graph H, and a graph G we have hom(H,G) > v(G) ( hom(K2, G) v(G)2 )e(H) , where v(H), v(G) and e(H), e(G) denote the number of vertices and edges of the graph H and G, respectively. In this paper we prove Sidorenko’s conjecture for certain special graphs G: for the ...

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. An H-minor is a minor isomorphic to H. The Hadwiger number of G, denoted by h(G), is the maximum integer t such that G contains a Kt-minor, where Kt is the complete graph with t vertices. Hadwiger [8] conjectured that every graph that is not (t−1)-colourable contains a Kt-mino...