[M(bpy)3]q+ cations (bpy = 2,2′-bipyridine) are archetypical coordination entities containing chelating bidentate N,N′-donor ligands. Each propeller-shaped cation is chiral, existing as a Δ or Λ enantiomer. The supramolecular chemistry of in the crystal dominated by cation-anion interactions and, to lesser extent, weaker non-covalent interactions. Analysis data for salts Cambridge Structural Da...

Host−guest interactions in porous supramolecular structures have been studied on surfaces using scanning tunneling microscopy, with anticipation of biochemical and sensor applications, but limited to cases of van der Waals interactions and hydrogen bonds. Here, we studied the intermolecular structures of 4,4′′-dibromo-p-terphenyl molecules self-caged in porous supramolecular structures with hal...

This paper presents a hexagon-based Q-leaning algorithm to find a hidden target object with multiple robots. An experimental environment was designed with five small mobile robots, obstacles, and a target object. Robots went in search of a target object while navigating in a hallway where obstacles were strategically placed. This experiment employed two control algorithms: an area-based action ...

Abstract. Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n − 1, 2n − 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides a, a and b. We prove explicit formulas for the number of lozenge tilings of the...

Pictures of the G2(2) hexagon and its dual are presented. A way to obtain these pictures is discussed. 1. Introduction This paper presents pictures of the two classical generalised hexagons over the eld with two elements. Information on these generalised hexagons is employed so that the pictures make some properties of these hexagons obvious. Familiarity with the theory of generalised polygons ...

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonome...

In Aristotelian logic, the predominant view has always been that there are only two kinds of quantities: universal and particular. For this reason, philosophers have struggled with singular propositions (e.g., “Socrates is running”). One modern approach to this problem, as first proposed in 1955 by Tadeusz Czeżowski, is to extend the traditional Square of Opposition to a Hexagon of Opposition. ...

In 1988, Proctor presented a result on plane partitions which implied a formula for the number of lozenge tilings of a hexagon with side-lengths a, b, c, a, b, c after removing a “maximal staircase.” More recently, Ciucu proved a weighted version of Proctor’s result. Here we present weighted and unweighted formulas for a similar region which has an additional unit triangle removed. We use Kuo’s...