1. Abstract characterization of Dn The group Dn has two generators r and s with orders n and 2 such that srs−1 = r−1. We will show any group with a pair of generators like r and s (except for their order) admits a homomorphism onto it from Dn, and is isomorphic to Dn if it has the same size as Dn. Theorem 1.1. Let G be generated by elements a and b where an = 1 for some n ≥ 3, b2 = 1, and bab−1...

Given a combinatorial description C of a polyhedron having E edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize C is generally not a convex subset of R [9]. If C has five or more faces, Andreev’s Theorem states that the corresponding space of dihedral angles AC obtained by restricting to non-obtuse angles is a convex polytope. In this paper we explain why Andre...

in this paper we consider the group algebra $r(c_2times‎ ‎d_infty)$‎. ‎it is shown that $r(c_2times d_infty)$ can be‎ ‎represented by a $4times 4$ block circulant matrix‎. ‎it is also‎ ‎shown that $mathcal{u}(mathbb{z}_2(c_2times d_infty))$ is‎ ‎infinitely generated‎.

Photocatalytic hydrogen evolution over semiconducting niobate nanosheets is synergistically improved by coexisting photochemically inactive additives of clay particles and sodium chloride without precise nanoscopic structural regulation. In addition, the Pt cocatalyst loaded on the clay particles works better than that photodeposited on the photocatalytic nanosheets.

The cyclic sieving phenomenon of Reiner, Stanton, and White characterizes the stabilizers group actions on finite sets using q-analogue polynomials. Eu Fu demonstrated a generalized cluster complexes every type q-Catalan numbers. In this paper, we exhibit dihedral phenomenon, introduced for odd n by Rao Suk, clusters type. A case, show that Raney numbers count both reflection-symmetric k-angula...