Given positive integers m, k and s with m > ks, let Dm,k,s represent the set {1, 2, · · · ,m} − {k, 2k, · · · , sk}. The distance graph G(Z,Dm,k,s) has as vertex set all integers Z and edges connecting i and j whenever |i − j| ∈ Dm,k,s. The chromatic number and the fractional chromatic number of G(Z,Dm,k,s) are denoted by χ(Z,Dm,k,s) and χf (Z,Dm,k,s), respectively. For s = 1, χ(Z,Dm,k,1) was s...

We prove that a well-distributed subset of R2 can have a distance set ∆ with #(∆ ∩ [0, N ]) ≤ CN3/2− only if the distance is induced by a polygon K. Furthermore, if the above estimate holds with = 1/2, then K can have only finitely many sides.

In a recent paper, Petrov and Pohoata developed new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave simple proof Bannai-Bannai-Stanton bound on size $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ determine at most $s$ different distances). this paper we extend t...

The reference to ethnicity in the 2012 National Suicide Prevention Strategy for England (1) is mainly with regard to promoting mental health in Black and Minority Ethnic (BME) groups. However, there is little detailed information about any specific strategy to reduce suicide rates in BME groups. The Suicide Prevention Strategy and Action Plan for Northern Ireland has only one brief reference re...

This derivation of (2) is not valid, because this duality applies to inferences and theorems only, not to definitions. In fact, (1) should be treated as an axiom. Accordingly, the duality of Lemma 3.3.1 would not be valid until the dual (2) of (1) is proved independently, as we do below. Our new proof will make use of Proposition 3.3.3, the original proof of which is based on duality. So, we mu...

Answering a recent question posed by Gregori, Morillas and Sapena (“On a class of completable fuzzy metric spaces”, Fuzzy Sets and Systems, 161 (2010), 2193–2205) we present two examples of non strong fuzzy metrics (in the sense of George and Veeramani).

We provide an extension of the notion of chain-valued frame introduced by Pultr and Rodabaugh in [Category theoretic aspects of chain-valued frames: parts I and II, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558] by relaxing the assumption that L be a complete chain. As a result of this investigation we formulate the category L-Frm of L-frames under the weaker assumption that L is a comp...