In the paper, we consider some types of vectorial saddle-point problems. We present some existence results of vectorial saddle-point problems. After that we consider a generalized vector equilibrium problem as an application.

M. Tanaka ([2]) determined the radius of the ball which consists of all poles in a von Mangoldt surface of revolution. The purpose of the present paper is to give an alternative proof and a geometrical meaning of the radius. Furthermore, we estimate the radius of the maximal ball consisting of poles in a complete surface homeomorphic to the plane under a certain condition.

In this paper, we show that the fundamental surgeries of the graph representation for real rational functions can be achieved by classical interior variations essentially due to Schiffer. As an application we give a constructive proof of the main theorem of Natanzon, Shapiro and Vainshtein in [2].

In this note, we study in detail the remark in the appendix of Danzer [6]. We find that planer Danzer tilings have many different aspects than Penroze tilings. For e.g., we observe that Danzer tiling with 7-fold symmetry does not belong to the topological closure of tilings generated by up-down generation.