This paper presents a result of approximation an arc circles by using a quartic Bezier curve. Based on the barycentric coordinates of two and three combination of control points, the interior control points are determined by forcing the curvature at median point as similar as the given curvature at end points. Hausdorff distance is used to investigate the order of accuracy compare to the actual arc circles through central angle of 0 T S d. We found that the optimal approximation order is eight which is somewhat similar to preceding methods in the literatures.

Abstract. The generalized presentation of a Bieberbach group with cyclic point group of order two can be obtained from the fact that any Bieberbach group of dimension n is a direct product of the group of the smallest dimension with a free abelian group. In this paper, by using the group presentation, the homological functor of a Bieberbach group a with cyclic point group of order two of dimension n is found.

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Let R be an associative ring with identity. In this paper we extend the J.H. Maynes results, which he treatised in [28]. In particular, we prove that if R is a δ-prime ring with charR non equal 2 and I is a nonzero δ-ideal of R, where   δ ∈ D, c ∈ R and [c, δ(c)] in the center of R, then R is commutative.

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