A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper, we present a counterpart of this formula, corresponding to the exterior of a concave hexagon obtained by turning 120° after drawing each side (MacMahon’s hexagon is obtained by turning 60° af...

Abstract The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines and 21 flags, whereas those of the latter set having their counte...

We prove that the near hexagon Q(5,2)× L3 has a non-abelian representation in the extra-special 2-group 21+12 + and that the near hexagon Q(5,2)⊗Q(5,2) has a non-abelian representation in the extra-special 2-group 21+18 − . The description of the non-abelian representation of Q(5,2)⊗Q(5,2) makes use of a new combinatorial construction of this near hexagon.

We review recent computational results for hexagon patterns in nonBoussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a verti...

In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent hexagons sharing a side, respectively. Simulation results show the accuracy of the obtained closed-form distance distribution functions, which are important in a wi...

A plane, passing through the center and orthogonal to a diagonal, slices cube into two identical halves each having three triangles, pentagons hexagon. If you label ten vertices of half-cube with numbers, then face is said receive an induced given by sum all around it. Label digits 0 9 so that labels triangles constitute six consecutive multiples fixed number, or more generally numbers in arith...

The C60 ions C^" and C ^ are discussed comparatively on the basis of ab initio calculations. The basis set employed is of 3-21 G* quality. By analogy with the neutral molecule the geometry of the anion is icosahedral (Ih). Both Ih systems differ, however, in the bondlength alternation. In the neutral molecule the hexagon-hexagon bondlength exceeds the hexagon-pentagon value; vice versa in the a...

The aim of this study was to evaluate the integrity of the external hexagon of an implant system with internal and external hexagons but with prosthetic connection through the external hexagon (Internal Torque, IT) in comparison with that of an implant system with external hexagon with mount (External Hexagon, EH). A device was made to measure the rotational freedom angles between implant and a...