In the title chromenone derivative, C(10)H(8)O(2), the two fused six-membered rings are coplanar, with a mean deviation of 0.0261 (1) Å from the plane through the non-H atoms of the rings. The carbonyl and methyl substituents of the pyran ring also lie close to that plane, with the O and C atoms deviating by 0.0557 (1) and 0.1405 (1) Å, respectively. In the crystal, weak C-H⋯O contacts form cha...

In the title compound C(33)H(30)S(6), the three pendant methyl-ene benzodithio-ate groups lie to one side of the central benzene ring in a cis-cis-cis 'tripod' arrangement. The dihedral angles between the central benzene ring and the three pendant rings are 72.54 (4), 89.68 (4) and 86.74 (4)°. In the crystal structure, one of the benzene rings is disordered over two orientations in a 0.559 (13)...

The asymmetric unit of the title compound, C(34)H(28)S(8), contains two crystallographically independent half-mol-ecules. The mol-ecules lie on centers of inversion. The four benzene rings of each mol-ecule are substantially twisted from the planes of the 1,3-dithiole rings, forming dihedral angles of 43.6 (2) and 61.4 (1)° in one mol-ecule and 54.2 (1) and 65.2 (1)° in the other.

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An add...

In the title compound, C(16)H(15)N(3)O(2)·H(2)O, the two benzene rings and the triazole ring lie almost in the same plane, the triazole ring forming dihedral angles of 5.07 (9) and 5.80 (8)° with the benzene rings. In the crystal, there are three relatively strong inter-molecular O-H⋯N and N-H⋯O hydrogen bonds, which lead to the formation of a one-dimensional double chain running parallel to th...

Except in the most hindered of cases, N,N'-diaryl N,N'-dimethyl ureas adopt a conformation with the two aryl rings disposed cis to one another. Variable temperature NMR studies reveal the rate at which the Ar-N bonds rotate as well as the conformational preference of ortho disubstituted ureas in which more than one cis orientation is possible. In general, a conformation in which the aryl rings ...

In the crystal structure of the title compound, C(16)H(19)O(6)P, inter-molecular O-H⋯O inter-actions link the mol-ecules into chains parallel to the b axis. These chains are linked by C-H⋯π and π-π inter-actions [centroid-centroid distance = 3.7307 (29) Å] into a three-dimensional network. The dihedral angle between the benzene rings is 73.5 (1)°. The C and O atoms of all four methoxy groups li...

In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Then we generalize it to a result in the language of tensor algebra, which can be applied to other contexts.

In the title compound, C(21)H(18)O(5)S(2), the two sulfur-bound phenyl rings lie on opposite sides of the meth-oxy-phenyl group, making dihedral angles of 77.58 (8) and 87.45 (8)°with it. The dihedral angle between the sulfur-bound phenyl rings is 57.31 (8)°. In the crystal, π-π stacking is observed between the two sulfur-bound phenyl rings, with a centroid-centroid distance of 3.878 (1) Å and ...

The Nagano-Yagi-Goldmann theorem states that on the torus T, every affine (or projective) structure is invariant or is constructed on the basis of some Goldmann rings [N-Y]. It shows the interest to study the invariant affine structure on the torus T or on abelian Lie groups. Recently, the works of Kim [K] and Dekempe-Ongenae [D-O] precise the number of non equivalent invariant affine structure...