We prove that a bounded affine vector field on a complete Finsler manifold is a Killing vector field. This generalizes the analogous result of Hano for Riemannian manifolds [3]. Even though our result is more general, the proof is significantly simpler.

For square contingency tables with ordered categories, Tahata and Tomizawa (2010) considered the double linear diagonals-parameter symmetry (DLDPS) model. The present paper proposes the generalized DLDPS model, which implies the structure of both asymmetry with represent to the main diagonal and with respect to the reverse diagonal in the table. Also, the double symmetry (DS) model is separated into the generalized DLDPS model and the moment equality model. Also it is shown that the test statistic for the DS model is asymptotically equivalent to the sum of those for separated models. An example is given.

A function $f$ is called an $F$-geometric mean labeling of a graph $G=(V,E)$ with $p$ vertices and $q$ edges if $f:V\to \{1,2,3,\dots ,q+1\}$ is injective and the induced function $f^{*}:E\to \{1,2,3,\dots ,q\}$ defined as

$$f^{*}(uv)=\lfloor \sqrt{f(u)f(v)}\rfloor ,\text{ for all }uv\in E,$$

is bijective. A graph that admits an $\text{F}$-geometric mean labeling is called an F-geometric mean graph. In this paper we discuss the $\text{F}$-geometric meanness of graphs obtained by duplicating any edge of some graphs.

We consider roots of a generalized Lens polynomial $L(z,\overline{z})={\overline{z}}^{m}q(z)–p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\overline{z})=r(\overline{z})q(z)–p(z)$ with $\text{deg }q\left(z\right)=n$, $\text{deg }p(z)\le n$ and $\text{deg }r(\overline{z})=m$. We have shown that there exists a harmonically splitting polynomial $r(\overline{z})q(z)–p(z)$ which takes $5n+m–6$ roots, using a bifurcation family of polynomial. In this note, we show that this number of roots can be taken by a generalized Lens polynomial ${\overline{z}}^{m}q(z)–p(z)$ after a slight modification of the bifurcation family of a Rhie polynomial.

For a bound quiver algebra satisfying the condition that the every oriented cycles in the quiver are vanished in the algebra, Fernández and Platzeck determined the bound quiver algebra which is isomorphic to the trivial extension algebra. In this paper, we consider a Hochschild extension algebra which is a generalization of a trivial extension algebra. The purpose of this paper is to determine the bound quiver algebras which are isomorphic to Hochschild extension algebras of some finite dimensional self-injective Nakayama algebras.