In this paper, we establish several weighted ${{L}^{p}}\left( 1\,<\,p\,\infty \right)$ Hardy type inequalities related to the generalized Greiner operator by improving the method of Kombe. Then the best constants in inequalities are discussed by introducing new polar coordinates.

This paper presents a polar-space kinematics control method to achieve simultaneous tracking and stabilization for an omnidirectional wheeled mobile robot with three independent driving omnidirectional wheels equally spaced at 120° from one another. The kinematic model of the robot in polar coordinates is presented. With the kinematic model, a kinematic control method based on feedback linearization is proposed in order to achieve simultaneous tracking and stabilization. The proposed method is easily extended to address the path following problem. Computer simulations and experimental results are presented to show the effectiveness and usefulness of the proposed control method at slow speeds.

The results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.