For integers k ⩾ 2 $k\geqslant 2$ and N + 1 $N\geqslant 2k+1$ there are ! $k!2^k$ canonical orderings of the edges complete $k$ -uniform hypergraph with vertex set [ ] = { , ⋯ } $[N] \lbrace 1,2,\dots N\rbrace$ . These exactly property that any two subsets A B ⊆ $A, B\subseteq [N]$ same size induce isomorphic suborderings. We study associated canonisation problem to estimate, given n $n$ least ...

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime candidates are obtained in terms of the first perfect number. The asymptotic behaviour is also considered. We obtain for the first time an explicit relation for g...

To the degree that citizens have participated in, or derived benefits from, social institutions that have helped cause serious, life-threatening, or rights-threatening environmental injustice (EIJ), this article argues that they have duties either to stop their participation in these institutions or to compensate for it by helping to reform them. (EIJ occurs whenever children, poor people, mino...

The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2 . . . xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in...

let $g$ be a non-abelian group of order $p^n$‎, ‎where $nleq 5$ in which $g$ is not extra special of order $p^5$‎. ‎in this paper we determine the maximal size of subsets $x$ of $g$‎ ‎with the property that $xyneq yx$ for any $x,y$ in $x$ with‎ ‎$xneq y$‎.

We study the determination of finite subsets of the integer lattice Zn, n ≥ 2, by X-rays. In this context, an X-ray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory a...

The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2 . . . xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in...

Let f k (n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f"(n) < f"(n + 1) < with equalities holding iff there exists a Steiner-system Y(t, 2t-1, n). The bounds are approximately best possible for k < 6 and of correct order of magnitude for k > 7, as well, even if the corresponding Steiner-systems do not exist. Exponential lower and upper ...