2 9 D ec 2 00 6 Complex Tilings ∗

2 00 7 Self - Dual Tilings with Respect to Star - Duality

The concept of star-duality is described for self-similar cut-and-project tilings in arbitrary dimensions. This generalises Thurston's concept of a Galois-dual tiling. The dual tilings of the Penrose tilings as well as the Ammann-Beenker tilings are calculated. Conditions for a tiling to be self-dual are obtained.

ar X iv : h ep - p h / 03 12 27 6 v 1 1 9 D ec 2 00 3 The a 0 K + K − - vertex in light cone QCD sum rules

ar X iv : h ep - e x / 01 12 02 6 v 1 1 9 D ec 2 00 1 Results on CP Violation from Belle

I describe the recent measurement of the CP violating parameter sin 2φ 1 = (0.99 ± 0.14 ± 0.05) from the Belle experiment at KEK.

2 4 D ec 1 99 7 AN ALGEBRAIC INVARIANT FOR SUBSTITUTION TILING SYSTEMS

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.

2 9 D ec 2 00 8 On potentially K 6 − C 5 graphic sequences ∗

For given a graph H , a graphic sequence π = (d1, d2, · · · , dn) is said to be potentially H-graphic if there exists a realization of π containing H as a subgraph. In this paper, we characterize the potentially K6 − C5 -graphic sequences .