We classify integral rootless lattices which are sums of pairs of EE8-lattices (lattices isometric to √ 2 times the E8-lattice) and which define dihedral groups of orders less than or equal to 12. Most of these may be seen in the Leech lattice. Our classification may help understand Miyamoto involutions on lattice type vertex operator algebras and give a context for the dihedral groups which oc...

K. Cho and S. Nelson introduced the notion of a quandle coloring quiver, which is quiver-valued link invariant, cocycle quiver an enhancement by assigning to each vertex weight computed using 2-cocycle. In this paper, we study quivers dihedral quandles introduce notions shadow are versions quiver. We show that, when use prime order, equivalent numbers order Mochizuki's 3-cocycle, invariants.

The Fourier transforms x̂j = ∑ k xkω , k, j = 0, 1, . . . , n− 1 of a data vector x with scalar components xk indexed by n-fold planar rotations ω k define a n × n Hermitian matrix F with entries ωjk/√n, so that x̂ = √n Fx and the inverse Fourier transform is simply x = F∗x̂/√n. Here, the frequencies ω and their harmonics ω fully describe the set (Ĝ) of all n irreducible representations of the Abe...

Dihedral Golay sequences are introduced and found for lengths 7,9,15, and 19. Applications include new classes of signed group Hadamard matrices and 19 real Hadamard matrices of orders 2p, p S 4169.

In rhe title compound, C(28)H(20)N(2)S(4), the 1,3-dithiol-2-yl-idene and 4,5-diaza-fluoren-9-one (dafone) groups are almost coplanar, making a dihedral angle of only 5.65 (4)°. The two benzyl groups are on different sides of the 1,3-dithiol-2-yl-idene ring, forming a dihedral angle of 35.54 (2)°.

The title compound, C(18)H(7)BrCl(2)F(3)N(5)O(2), is an L-shaped tricyclic imine. The pyrazole ring is essentially coplanar with the nitro-substituted benzene ring [dihedral angle = 3.6 (2)°] and approximately perpendicular to the trifluoro-methyl-substituted ring [dihedral angle = 88.5 (2)°].

In the title compound, C(23)H(16)N(2)O, the bond angle at the O atom that connects the benzene ring and the quinoline ring system is 116.0 (2)°. The quinoline ring system make a dihedral angle of 16.5 (2)° with the adjacent benzene ring. The dihedral angle between the biphenyl benzene rings is 70.8 (2)°.

The title compound, C(37)H(26)N(2)O(2), was synthesized from 1,1'-methyl-enedinaphthalen-2-ol and 3-(bromo-methyl)-benzo-nitrile. The two naphthyl systems are almost perpendicular to each other [dihedral angle 83.3 (9)°] and the two cyano-benz-yloxy rings approximately parallel to each other [dihedral angle 15.5 (2)°].

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and g...

In the crystal structure of the title compound, C(13)H(10)ClNO(3)S, the conformation of the N-H bond in the C-SO(2)-NH-C(O) segment is anti to the C=O bond. The dihedral angle between the two aromatic rings is 68.6 (1)°. The mol-ecule is twisted at the S atom with a dihedral angle of 75.7 (1)° between the sulfonyl benzene ring and the -SO(2)-NH-C-O segment; the dihedral angle between the latter...