In this paper we introduce a new type of labeling known as even vertex odd mean labeling. 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 $f:V(G)\to \{0,2,4,\dots ,2q-2,2q\}$ such that the induced map $f^{\ast }:E(G)\to \{1,3,5,\dots ,2q-1\}$ defined by $f^{\ast }(uv)=\frac{f(u)+f(v)}{2}$ is a bijection. A graph that admits an even vertex odd mean labeling is called an even vertex odd mean graph. Here we investigate the even vertex odd mean behaviour of some standard graphs.

The AIC and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework and a high-dimensional framework by Satoh, Kobayashi and Fujikoshi [9] and Fujikoshi, Enomoto and Sakurai [4], respectively. They note that the AIC and its modifications have no consistency property. In this paper we consider asymptotic properties of the AIC and its modification when the number q of groups or explanatory variables and the sample size n are large. First we show that the AIC has a consistency property under a large-$(q,n)$ framework such that $q/n\to d\in [0,1)$, under a condition on the noncentrality matrix, but the dimension p is fixed. Next we propose a modification of the AIC (denoted by MAIC) which is an asymptotic unbiased estimator of the risk under the asymptotic framework. It is shown that MAIC has a consistency property under a condition on the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation.

The present paper deals with a study of certain transformations on a Kenmotsu manifold. We study an infinitesimal $CL$-transformation on a Kenmotsu manifold. We also study $CL$-transformation on a Kenmotsu manifold and obtain a new tensor field which is invariant under such a transformation. Finally we study $CL$-semisymmetric Kenmotsu manifold and prove that it is a manifold of constant curvature $-1$, from which we obtain some equivalent conditions as characterization of such a manifold.

For the analysis of square contingency tables, many kinds of symmetry models have proposed. The present paper proposes a new kind of symmetry model, and gives a decomposition of the new model by introducing an extended model of it. Moreover, it shows the orthogonality of statistic for testing goodness-of-fit of the new model. Two unaided vision data analyses are also shown.

Some geometrical aspects of the Kerr-Newman black hole and its special cases have been studied. It is seen that the Gaussian curvature of the two or three dimensional induced metrics on some hypersurfaces outside of these black holes can be expressed in terms of the eigen values of the characteristic equation and depend upon the physical parameters which describe these black holes.

We build a framework for $R(p,q)$-deformed calculus, which provides a method of computation for deformed $R(p,q)$-derivative and integration, generalizing known deformed derivatives and integrations of analytic functions defined on a complex disc as particular cases corresponding to conveniently chosen meromorphic functions. Under prescribed conditions, we define the $R(p,q)$-derivative and integration. Relevant examples are also given.