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In this paper, our main aim is to study periodic and potent elements of a ring. We specially study periodic elements of graded rings and generalize some classical results related to idempotent of polynomial rings. We show that a (von Neumann) quasi-inverse of a potent element is a root of unity. We study the isomorphism of potent elements and analyze some closure properties of the set Pot(R) of potent elements of a ring R. The potent elements of the endomorphism ring of a Fitting module are described, and we apply this to matrices over division rings. In the case of matrices over finite fields, we connect features of potent elements with the exponent of their minimal polynomials. [ABSTRACT FROM AUTHOR]

An element a of a ring R is called a weak zero-divisor if there exists a nonzero x of R such that xa is nilpotent. As a new generalization of nil clean rings, we define a ring R to be weak zero-clean if each of its elements is a sum of a weak zero-divisor and an idempotent. We give some examples and properties of weak zero-clean rings and discuss the relationships between weak zero-clean rings and related rings. Some extensions of weak zero-clean rings are studied. Also, we define and study uniquely weak zero-clean rings and give some characterizations of such rings. [ABSTRACT FROM AUTHOR]

This paper discusses the robust, real-time detection of stationary and moving pedestrians utilizing a single car-mounted monochrome camera. First, the system detects potential pedestrians above the ground plane by combining conventional Inverse Perspective Mapping (IPM)-based obstacle detection with the vertical 1D profile evaluation of the IPM detection result. Usage of the vertical profile increases the robustness of detection in low-contrast images as well as the detection of distant pedestrians significantly. A fast digital image stabilization algorithm is used to compensate for erroneous detections whenever the flat ground plane assumption is an inaccurate model of the road surface. Finally, a low-level pedestrian-oriented segmentation and fast symmetry search on the leg region of pedestrians is also presented. A novel approach termed Pedestrian Detection Strip (PDS) is used to improve the calculation time by a factor of six compared to conventional approaches. [ABSTRACT FROM AUTHOR]