Tightening interatomic distance bounds, obtained from NMR experiments, is a critical step in solving the molecular conformation problem. Tetrangle inequality (resulting from the Cayley-Menger determinant for the 3D embedding) is applied to quadruples of atoms in order to tighten the distance bounds. Complete graph Kn is used to model an n-atom molecule, where the nodes represent the atoms. Only...

Philip Gibbs Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of ...

In this paper, we show that there does not exist a quadruple of positive integers a1 < a2 < a3 < a4 such that aiaj + 1 (i 6= j) are all members of the Tribonacci sequence (Tn)n≥0.

We further develop the corresp0ndenc.e established in Part I of this paper to characterize optimal soluticns to the gossip problem in which no one hears his own information. To reiterate, these are graphs on n vertices with 2n -4. edges (n even), for which the edges c:ln bc linearly ordered to produce an increasing path from each vertex to every other, without having an imzzeasing path from any...

In an Apollonian circle packing, the curvatures a, b, c, d of any four mutually tangent circles satisfy Descartes’ equation, 2(a + b + c + d) = (a+ b+ c+ d). For an equilateral triangle and a point P , if a, b, c, d denote the squares of the lengths of the sides of the triangle and the distances from P to the vertices of the triangle, then a, b, c, d, satisfy 3(a+ b+ c+d) = (a+ b+ c+d). Define ...

A set of data supposed to give possible axioms for spacetimes. It is hoped that such a proposal can serve to become a testing ground on the way to a general formulation. At the moment, the axioms are known to be sufficient for cases with a sufficient number of symmetries, in particular for 1+1 de Sitter spacetime. A major shortcoming of applications of noncommutative geometry to high energy phy...

In this paper, we study the existence of Diophantine quadruples with the property D(z) in the ring Z[ √−2 ]. We find several new polynomial formulas for Diophantine quadruples with the property D(a+ b √−2 ), for integers a and b satisfying certain congruence conditions. These formulas, together with previous results on this subject by Abu Muriefah, Al-Rashed and Franušić, allow us to almost com...