In this paper, we determine the structures of zero-divisor semigroups whose graph is Kn + 1, the complete graph Kn together with an end vertex. We also present a formula to calculate the number of non-isomorphic zero-divisor semigroups corresponding to the complete graph Kn, for all positive integer n.

S = k[T1, T2, . . . , Tn](= k[T ] for short) as k-valued functions on the space kn: if v = (v1, . . . , vn) ∈ kn and f = f(T1, . . . , Tn) then f(v) = f(v1, . . . , vn). We call v ∈ kn a zero of f if f(v) = 0. Given an ideal I ⊂ S, let V (I) denote the set of all common zeros of all functions in I, i.e. V (I) = {v ∈ k | f(v) = 0 for all f ∈ I}. Given a subset X ⊂ kn, let I(X) denote the ideal o...

For S a subordinator and Πn an independent Poisson process of intensity ne−x, x > 0, we are interested in the number Kn of gaps in the range of S that are hit by at least one point of Πn. Extending previous studies in [7, 10, 11] we focus on the case when the tail of the Lévy measure of S is slowly varying. We view Kn as the terminal value of a random process Kn, and provide an asymptotic analy...

Given a graph G, let fk be the number of forests of cardinality k in G. Then the sequence (fk) has been conjectured to be unimodal for any graph G. In this paper we confirm this conjecture for Kn and Kn,n by showing that the sequence for Kn is strictly increasing (when n ≥ 4) and the sequence for Kn,n is strictly increasing except for the very last term. As a corollary we also confirm the conje...

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph G is denoted γID(G). It was recently shown by Gravier, Moncel and Semri that γ(Kn Kn) = ⌊ 3n 2 ⌋. Letting n,m ≥ 2 be any integers, we consider identifying codes o...

Hedgehog (Hh) plays an important role in Drosophila wing patterning by inducing expression of Dpp, which serves to organize the wing globally across the A-P axis. We show here how Hh signalling also plays a direct role in patterning the medial wing through the activation of the Hh-target gene, knot (kn). kn is expressed in Hh-responsive cells near the A-P compartment boundary, where its express...

Kirby and Lickorish [3] have previously obtained the second result using a different construction. This answers problem 13 in Gordon [2]. Results of Schubert [6] show that the knots constructed are prime. In the second half of this paper we will generalize his results to characterize prime "generalized" cable knots. If JSΓX is a knot in S x B and K2 is a knot in S , we can form the Kx cable of ...