H. Glöckner proved that an operator-valued positive definite function on an open convex subset of $\boldsymbol Q^N$ is a restriction of the Laplace transform of an operator-valued measure on $\boldsymbol R^N$. We generalize this result to a function on an open convex subset of a certain subsemigroup of $\boldsymbol Q^2$.

We provide straight-chains with the tetrahedral bond angle $\cos^{-1}(-1/3)$ as a mathematical model of $n$-membered straight-chain hydrocarbon molecules. We study the squared chain length function on the configuration space of the model and determine the critical points with planar configurations when $n = 5, 6$.

We show a condition that a Galois covering $\pi : V \longrightarrow \mathbb P^n$ is induced by a Galois embedding. Then we consider the Galois embedding for $\mathbb P^n$. If the Galois group $G$ is abelian, then $G \cong \bigoplus\limits^{n} Z_d$ and the projection $\pi$ can be expressed as $\pi(X_0:X_1: \cdots :X_n)=({X_0}^d:{X_1}^d:\cdots:{X_n}^d)$.

In 2013, Basha, Shahzad and Jeyaraj proved two existence theorems of best proximity points for Kannan-type and Chatterjea-type mappings. In this paper, in order to clarify the mathematical structure of these theorems, we improve these theorems in the aspects of both statements and proofs. Indeed, we give very simple proofs of these theorems. We also discuss the best possibility on the numbers that appear in these theorems.

This paper mainly considers cyclic vectors in the Fock-type spaces $ L_{a,\alpha}^{p,s }(\mathbb{C} )$ $(\alpha>0,p\geq 1, s>0)$ which consists of all entire functions $f$ such that $|f|^p$ is integrable with respect to the measure $\exp(-\alpha |z|^s) dA(z).$ The case of $s$ not being an integer was done in [9], where cyclic vectors are exactly those non-vanishing entire functions in $ L_{a,\alpha}^{p,s }(\mathbb{C} )$. In this paper it is shown that for each positive integer $s$, a function $f$ is cyclic in $ L_{a,\alpha}^{p,s }(\mathbb{C} )$ if and only if $f$ is non-vanishing and $f \mathcal{C} \subseteq L_{a,\alpha}^{p,s }(\mathbb{C} )$, where $ \mathcal{C} $ denotes the polynomial ring. Moreover, the condition that $f \mathcal{C} \subseteq L_{a,\alpha}^{p,s }(\mathbb{C})$ can not be dropped.

In this paper, we study the von Neumann-Jordan constant of $\pi/2$-rotation invariant norms on $\mathbb{R}^2$. We give some estimations of the constant and have a relationship between the constant and a ratio of two certain functions. These results are an extension of existing results of a unitary version of the von Neumann-Jordan constant.

In this research, we propose computational methods to evaluate scalarizing functions, which are defined via set-relations. In recent years, many theoretical results of the scalarizing functions for sets have been published. The aim of this paper is to show that each value of the scalarizing functions can be computed and to introduce computational algorithms of them for convex polytopes in a finite dimensional space.