A graph $G(V,E)$ is $(a,d)$-edge antimagic total if there exists a bijection $f:V(G)\cup E(G)\to \{1,2,\dots |V(G)|+|E(G)\left|\right\}$ such that the edge-weights $\text{Λ}(uv)=f(u)+f(uv)+f(v),uv\in E(G)$ form an arithmetic progression with first term $a$ and common difference $d$. It is said to be a super $(a,d)$-edge antimagic total if f $(V(G))=\{1,2,\dots ,|V(G)\left|\right\}$. In this paper, we have obtained a relation between a super $(a,0)$-edge antimagic total labeling and a super $(a,2)$- edge antimagic total labeling of any graph. Also we study the super $(a,d)$-edge antimagic total labeling of fan graphs and two special classes of star graphs namely bi-star and extended bi-star.

We consider the problem of testing for multivariate mean vector when the data have two-step monotone pattern missing observations. We obtain two test statistics for this problem: a test statistic similar to Hotelling’s $T^2$ test statistic and the likelihood ratio test statistic. We propose the approximate upper percentiles of these statistics. The accuracy of the approximation is investigated by Monte Carlo simulation. A test statistic for the components of mean vector is outlined. Approximate simultaneous confidence intervals are obtained and the proposed method is illustrated using an example.

This paper is concerned with the testing problem about the covariance matrix under growth curve model. The testing problems treated in this paper are the following problems: (i) the problem of testing that a covariance matrix is equal to a specified positive definite matrix, (ii) the problem of testing for sphericity of the covariance matrix, and (iii) the problem of intraclass model for the covariance matrix. We give asymptotic distributions of the null distributions for the likelihood ratio statistics under an asymptotic framework that the total sample size $n$ go to infinity, the number of the observation points p go to infinity, and $p/n$ go to a constant $c\in (0,1)$. Simulation reveals that the proposed approximations have good accuracies compared with the classical chi-square approximations.

In this paper we study the relation between polycyclic codes and sequential codes over finite fields. It is shown that, for a sequential code $C\subseteq {\text{F}}^n$, $C$ is realized as an ideal in the quotient ring of the polynomial ring. Furthermore, we characterize the dual codes of polycyclic codes.

This paper provides an asymptotic expansion for the distribution of the Studentized linear discriminant function with $k$-step monotone missing training data. It turns out to be a certain generalization of the results derived by Anderson [1] and Shutoh and Seo [12]. Furthermore we also derive the cut-off point constrained by a conditional probability of misclassification using the idea of McLachlan [8]. Finally we perform Monte Carlo simulation to evaluate our results.

The object of the present paper is to obtain a necessary and sufficient condition for a 3-dimensional quasi-Sasakian manifold to be an $\eta$−Einstein manifold. An example is given to verify the theorem. Finally Ricci solitons and gradient Ricci solitons have been studied.

For two-way contingency tables, Tomizawa (1985b) gave a theorem that the double symmetry $(DS)$ model holds if and only if both the quasi $DS$ and the marginal $DS$ models hold. The present paper proposes, for multiway tables, the $DS$, some quasi $DS$ and marginal $DS$ models, and extends Tomizawa℉s theorem into multi-way tables. It also shows that for multi-way tables the likelihood ratio statistic for testing goodness of fit of the $DS$ model is asymptotically equivalent to the sum of those for testing the quasi $DS$ model with some order and the marginal $DS$ with the corresponding order. An example is given.

In this paper, we consider the multivariate normality test based on the sample measures of multivariate skewness and kurtosis defined by Srivastava [11]. Koizumi et al. [4] proposed test statistics $M_1$ and $M_2$ using Srivastava’s sample skewness and kurtosis, which are asymptotically distributed as ${\chi }^2$-distribution. We propose a new test statistic $M_3$ by taking account of the variance of $M_2$ under the normality. In order to evaluate the accuracy of the proposed test statistic, the numerical results by a Monte Carlo simulation for some selected values of parameters are presented.