In this article we consider examples of Fredholm modules over the group C*-algebras of the infinite dihedral and discrete Heisenberg groups. In the first case we exhibit sufficient Fredholm modules to generate the entire K-homology of the algebra. We then turn our attention to the six term cyclic exact sequence in K-homology for a crossed product by Z dual to the Pimsner-Voiculescu sequence for...

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop ...

The title mol-ecule, C(17)H(15)ClN(2)O(5), contains a quinolyl unit linked to a functionalized oxirane system with a 2,3-trans arrangement of the substituents (ester group versus quinol-yl). The structure can be described as being built up from zigzag layers parallel to (1[Formula: see text]0). The heterocyclic ring of the quinolyl unit forms a dihedral angle of 60.05 (1)° with the oxirane plan...

The mol-ecule of the title compound, C(21)H(26)N(2)O(6), which lies across a crystallographic inversion centre, crystallizes with two unique half-molecules in the symmetric unit and adopts a linear configuration and the imino group is coplanar with the aromatic ring, making a dihedral angle of 3.26 (3)°. Strong intra-molecular O-H⋯N and weak inter-molecular O-H⋯O and C-H⋯O hydrogen bonds and we...

In the title compound, C(29)H(24)N(4)O(4)S(3), the two N-tosyl-benzimidazolyl unit are connected through a -S-CH(2)- fragment, the dihedral angle between the benzimidazole rings being 76.09 (5)°. The methyl-thio group is disordered with respect to exchange of the S and C atoms in a 0.547 (4):0.453 (4) ratio. In the crystal, C-H⋯O and C-H⋯π inter-actions connect adjacent mol-ecules into infinite...

The study of perfect state transfer on graphs has attracted a great deal attention during the past ten years because its applications to quantum information processing and computation. Perfect is understood be rare phenomenon. This paper establishes necessary sufficient conditions for bi-Cayley graph having over any given finite abelian group. As corollaries, many known new results are obtained...

The two fused five- and six-membered rings in the mol-ecule of the title compound, C(8)H(6)BrN(3)O(2), are approximately coplanar, the largest deviation from the mean plane being 0.011 (3) Å at the NH atom. The acetyl group is slightly twisted with respect to the imidazo[4,5-b]pyridine system, making a dihedral angle of 2.7 (2)°. In the crystal, adjacent mol-ecules are linked by inter-molecular...

In the heterocyclic title compound, C(17)H(19)N(5)O(3), the quinazolinone ring system forms a dihedral angle of 67.22 (7)° with the triazole ring. The butyl acetate group has a non-linear conformation, with an alternation of synclinal and anti-periplanar torsion angles [N-C-C-C = 58.5 (2)°, C-C-C-C = 170.72 (19)° and C-C-C-O = -65.9 (3)°]. The crystal structure features inter-molecular C-H⋯N an...