In this paper, we consider a generalized triangle inequality of the following type: \begin{equation*} \Vert a_1 x_1+\cdots + a_1 x_n \Vert ^p \leq \Vert x_1\Vert^p +\cdots +\Vert x_n\Vert ^p \ (x_1,\ldots, x_n \in X ), \end{equation*} where $(X, \Vert \cdot \Vert)$ is a normed space, $(a_1, \ldots, a_n) \in \Bbb C^n$ and $p>0$. By using generalized $\ell_p$-spaces, we present a characterization of above inequality for infinite sequences $\{x_n\}_{n=1}^{\infty} \subset X$.

We give a condition on commutativity of a pair of normal operators with respect to the continuous functional calculus. We show a generalization of the theorem of Fuglede and Putnum with respect to the continuous functional calculus.

The purpose of this paper is to investigate periodic points of discontinuous maps. For some discontinuous maps, we establish a characterization of periodic points.

Multiple fibers of the simplest kind in a pencil of algebraic curves are studied, in order to clarify the influence on the gonality and the base locus of the canonical linear system.

Let $S_n$ be the unit sphere with respect to the operator norm of the algebra of $n\times n$ complex matrices. We give a complete description of the form of surjections on $S_n$ which preserve the metric induced by a unitarily invariant strictly convex norm.

Given a non-trivial automorphism (resp. anti-automorphism) of a semigroup, we study its homomorphic (resp. anti-homomorphic) extensions to a larger semigroup by considering the images of the absorbing elements. Then we exhibit some examples to show the application of the results obtained.

In the paper, we propose Ricceri type theorem on Fan-Takahashi minimax inequality for set-valued maps by using the scalarization method proposed by Kuwano, Tanaka and Yamada based on a certain type of the set-relations.