In the title compound, C(31)H(29)NO(5), the methyl carboxyl-ate and dimethyl-amino groups on the naphtho-pyran group are almost coplanar with the naphtho-pyran ring system [r.m.s. deviations = 0.08 (2) and 0.161 (2) Å, respectively]. The dihedral angle between the methyl carboxyl-ate and dimethyl-amino groups is 4.9 (1)°. The pyran ring has an envelope conformation with the quaternary C atom ou...

We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group-a group of twisted automor-phisms of the corresponding infinite dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2, C) Lax operators with simple poles and corresponding integrable equations. In the last section we give...

In the title compound, C(21)H(16)F(2)N(2), the dihedral angle between the fluoro-phenyl groups is 66.34 (8)°, and the dihedral angle between the envelope-configured pyrazole group (N/N/C/C/C) and the benzene ring is 11.50 (9)°. The dihedral angles between the benzene and the two fluoro-substituted phenyl groups are 77.7 (6) and 16.7 (5)°. Weak C-H⋯π interactions contribute to the stability of t...

Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \mathbb Z_2$-equivariant degree theory, where $O(2)$ is related reversing symmetry, $\Gamma$ reflects symmetric character coupling in corresponding network $\mathbb Z_2$ oddness right-hand side. Abstract...

In the title salt, C(23)H(26)N(4) (2+)·2C(6)H(2)N(3)O(7) (-), the dihedral angle between the imidazolium rings in the dication is 69.9 (1)°. The aromatic ring of the benzyl group is almost perpendicular to the N-heterocyclic ring that is directly connected to it [dihedral angles = 83.2 (2) and 77.3 (3)°].

The title compound, C(20)H(16)N(2), has been synthesized by the reaction of benzaldehyde with o-phenyl-endiamine and l-proline. The benzimidazole group makes a dihedral angle of 29.04 (1)° with the attached benzene ring, and is approximately perpendicular to the plane of the benzyl group [dihedral angle = 88.9 (1)°] The crystal packing exhibits no unusually short inter-molecular contacts.

We examine two particular constructions that derive from a 2-group G = G(·) another 2-group G(∗) for the case when G(·) is one of D2n , SD2n , Q2n . The constructions (the cyclic one and the dihedral one) have the property that x ∗ y = x · y for exactly 3/4 of all pairs (x, y) ∈ G × G. If G(◦) and G(∗) are such 2-groups that x ◦ y 6= x ∗ y for less then a quarter of all pairs (x, y) ∈ G × G, th...

In the crystal structure of the title compound, C(12)H(10)N(2)O(2)S, the benzene and the 2-nitro-thio-phene rings make a dihedral angle of 7.47 (12)°. The dihedral angle between the nitro group and the attached ring is 1.9 (6)°.

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...