Let $G$ be a graph and $f:V\left(G\right)\to \{1,2,3,\dots ,p+q\}$ be an injection. For each edge $uv$, the induced edge labeling $f^{*}$ is defined as $f^{*}\left(uv\right)=\left[\sqrt{f\left(u\right)f\left(v\right)}\right]$. Then $f$ is called a super geometric mean labeling if $f\left(V\left(G\right)\right){\cup }^{\text{}}\left\{f^{*}\left(uv\right);uv\in E\left(G\right)\right\}=\{1,2,3,\dots ,p+q\}$. A graph that admits a super geometric mean labeling is called a super geometric mean graph. In this paper, we discuss the super geometric meanness of union of any paths, union of any cycles of order $\ge 5$, the graph $P_n\odot S_m$ for $m\le 3$, square graph, total graph, the $H$-graph, the graph $G\odot S_1$ and $G\odot S_2$ for any $H$-graph $G$, subdivision of $K_{1,3}$ and some chain graphs.

A graph $G$ with $p$ vertices and $q$ edges is said to have an even vertex odd mean labeling if there exists an injective function $f:V\left(G\right)\to \{0,2,4,\dots ,2q-2,2q\}$ such that the induced map $f^{*}E\left(G\right)\to \{1,3,5,\dots ,2q-1\}$ defined by $f^{*}\left(uv\right)=\frac{f\left(u\right)+f(v)}{2}$ is a bijection. A graph that admits an even vertex odd mean labeling is called an even vertex odd mean graph. In this paper, we prove that every $T_p$-tree $T$, $T@P_n$, $T@2P_n$, $T\odot \overline{K_n}$, $T@c_n$ and $T\widehat{○}{C}_n$ are even vertex odd mean graphs.

The markaracter table of a finite group $G$ is a matrix obtained from the mark table of $G$ in which we select rows and columns corresponding to cyclic subgroups of $G$. This concept was introduced by a Japanese chemist Shinsaku Fujita in the context of stereochemistry and enumeration of molecules. In this note, the markaracter table of generalized quaternion groups and finite groups of order $pqr$, $p$, $q$ and $r$ are prime numbers and $p\ge q\ge r$, are computed.

We discuss approximations for the distribution of eigenvalues of the ratio of Wishart matrices when the population eigenvalues are infinitely dispersed. The first approximation is expressed as the $F$ distribution with suitable parameters, and the second is expressed by the product of $F$ distributions. Numerical examples show that the proposed approximations are more accurate than the known asymptotic expansions of the normal distribution.

We determine the ring structure of the Hochschild cohomology $H{H}^{*}(\text{Γ})$ of an integral cyclic algebra $\text{Γ}$ by giving a projective bimodule resolution of $\text{Γ}$ and calculating cup product by means of a diagonal approximation map.

In this paper, we show the existence of the wave operators for the Schrödinger equation with time-dependent variable coefficients by using the method introduced by the author and $\text{K}$. Kato [14] and give characterizations of their ranges by wave packet transform similar to those in [14].

Generalized linear models of binary data including a logistic regression model and a probit model are considered. For testing the null hypothesis that the considered model is correct, the $\varphi$-divergence family of goodness-of-fit test statistics $C_{\varphi {\varphi }^{*}}$ that is based on a minimum ${\varphi }^{*}$-divergence estimator is considered. The family of statistics $C_{\varphi {\varphi }^{*}}$ includes a power divergence family of statistics $R^{a,b}$ that is based on a minimum power divergence estimator. The derivation of an expression of a continuous term of asymptotic expansion for the distribution of $C_{\varphi {\varphi }^{*}}$ under the null hypothesis is shown. Using the expression, a transformed $C_{\varphi {\varphi }^{*}}$ statistic that improves the speed of convergence to the chi-square limiting distribution of $C_{\varphi {\varphi }^{*}}$ is obtained. In the case of $R^{a,b}$, it is numerically shown that the transformed statistics usually perform better than the original statistics with respect to speed of convergence to the chi-square limiting distribution and it is also numerically shown that the power of the transformed statistics is almost the same as that of the original statistics.