No abstract

A polytope plays a central role in different areas of mathematics. It is used quite heavily in applied fields of mathematics, such as medical imaging and robotics, geometric modeling.A polytope has many users in modern science such as computer graphics, optimization, and search engine. It is intended for a broad audience of mathematically inclined. Therefore in this paper, we shall take a polytope with one kind of application, which is known as a Stasheff polytope. it has been applied in: moduli spaces, Erhart polynomial, physics-chemistry, and Hopf algebras.In this paper, a finite graph G with tubing is taken, the nodes of the graph are reduce to points in and the convex hull for them are simplex, permutahedron and associahedron (Stasheff polytope ) are studied. Also, a fan graph to the painted tree is also taken and reduces its nodes to points in . The converse of this result is also given with different examples to consult our results.

In this paper, the commuting and centralizing of symmetric reverse ∗- -derivation on Lie ideal are studied and the commutativity of prime ∗-ring with the concept of symmetric reverse ∗- -derivations are proved under certain conditions.

In this paper, the commutativity of *-ring and some related results are obtained by introducing the new concept which is called skew left *-derivations.

This paper investigate the possibility of inheriting the properties of the ring