For a polynomial p of degree n<N we compare two norms: ‖p‖ := sup{|p(z)| : z ∈ C; |z| = 1} and ‖p‖N := sup {∣∣p (zj )∣∣ : j = 0, . . . , N − 1} ; zj = e2 i j N . We show that there exist universal constants C1 and C2 such that 1+ C1 log ( N N − n ) sup { ‖p‖ ‖p‖N : p ∈ Pn } C2 log ( N N − n ) + 1. © 2005 Elsevier Inc. All rights reserved.

It is unknown if an Artinian level O-sequence of codimension 3 and type r (≥ 2) is unimodal, while it is known that any Gorenstein O-sequence of codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence of codimension 3 cannot be level. We also find another non-level case: if some Artinian algebra A of codimension 3 has the Hilbert function H : h0 h1 · · · hd−1 hd · · · hd }...

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose thatH is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than ( n−1 2 )

We study the problem of constructing a (near) random proper q-colouring of a simple k-uniform hypergraph with n vertices and maximum degree ∆. (Proper in that no edge is mono-coloured and simple in that two edges have maximum intersection of size one). We give conditions on q,∆ so that if these conditions are satisfied, Glauber dynamics will converge in O(n log n) time from a random (improper) ...

Let H be a 3-partite 3-uniform hypergraph with each partition class of size n, that is, a 3-uniform hypergraph such that every edge intersects every partition class in exactly one vertex. We determine the Dirac-type vertex degree thresholds for perfect matchings in 3-partite 3-uniform hypergraphs.

In this paper, we characterize all extremal trees with the largest Laplacian spectral radius in the set of all trees with a given degree sequence. Consequently, we also obtain all extremal trees with the largest Laplacian spectral radius in the sets of all trees of order n with the largest degree, the leaves number and the matching number, respectively. © 2007 Elsevier B.V. All rights reserved....

Here we denote a diameter six tree by (c; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where c is the center of the tree; ai, i = 1, 2, . . . ,m, bj, j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to c; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful lab...

Two studies were performed to test whether abstract concepts are grounded in experience and activate introspective/linguistic information. In Study 1, four groups of participants, each with different expertise in the domain of safety and security at the workplace (S&S), defined abstract concepts belonging to the S&S domain and differing in degree of abstractness. The definitions included mainly...

We give a sharp refinement of a result of Alon, Ben-Shimon and Krivelevich. This gives a sufficient condition for a finite sequence of positive integers to be the vertex degree list of both parts of a bipartite graph. The condition depends only on the length of the sequence and its largest and smallest elements.