Six-dimensional planar Hermitian submanifolds of the Octonian are considered. It is proved that if such a submanifold of the octave algebra satisfies the $1$-cosymplectic hypersurfaces axiom, then it is Kählerian.

In this paper, we construct a new family of $q$−Hermite polynomials denoted by $H_n\left(x,s|q\right)$. Main properties and relations are established and proved. In addition, is deduced a sequence of novel polynomials, ${ℒ}_n\left(\cdot ,\cdot |q\right),$ which appear to be connected with well known $(q,n)$−exponential functions $E_{q,n}(\cdot )$ introduced by Ernst in his work entitled: A New Method for $q$−calculus, (Uppsala Dissertations in Mathematics, Vol. 25, 2002). Relevant results spread in the literature are retrieved as particular cases. Fourier integral transforms are explicitly computed and discussed. A $(q;p)$−extension of the $H_n\left(x,s|q\right)$ is also provided.

In this paper, we introduce the notion of radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions $D_1$, $D_2$ and $RadTM$ on radical transversal screen pseudo-slant lightlike submanifolds of an indefinite Kaehler manifold have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds.

We prove strong instability (instability by blowup) of standing waves for some nonlinear Schrödinger equations with double power nonlinearity.

This paper is concerned with consistency properties of some criteria for selecting row vectors of a $k\times p$ design matrix within individuals in the growth curve model, based on a sample of size $n$. Recently Enomoto, Sakurai and Fujikoshi (2013) showed that $\text{AIC}$ and its modification have a consistency property for selecting hierarchical models of the row vectors under a condition on the order of the noncentrality matrix, assuming a large-$(q,n)$ asymptotic framework such that $q/n\to d\in [0,1)$. We extend the result to a family of log-likelihood-based information criteria including $\text{AIC}$ and $\text{BIC}$, and $\text{Cp}$. Further, their consistency properties are also obtained under a new condition on the order of the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation.

In this paper, we consider the problem of simultaneous testing of the mean vector and the covariance matrix when the data have a two-step monotone pattern that is missing observations. We give the likelihood ratio test (LRT) statistic and propose an approximate upper percentile of the null distribution using linear interpolation based on an asymptotic expansion of the modified LRT statistic in the case of a complete data set. As another approach, we give the modified LRT statistics with a two-step monotone missing data pattern using the coefficient of the modified LRT statistic with complete data. Finally, we investigate the asymptotic behavior of the upper percentiles of these test statistics by Monte Carlo simulation.

For square contingency tables with ordered categories, Tomizawa (1995a) considered the generalized marginal homogeneity model. The present paper proposes a measure which represents the degree of departure from generalized marginal homogeneity. The proposed measure is expressed by the weighted sum of the Cressie-Read power-divergence or Patil-Taillie diversity index. This measure would be useful for measuring the degree of departure from generalized marginal homogeneity toward the maximum departure from it. The relationship between the measure and the further generalized marginal homogeneity model is shown by simulation studies.

We consider a canonical $S^1$ action on $S^3$ which is defined by $(\rho ,(z_1,z_2))\mapsto (z_1{\rho }^p,z_2{\rho }^q)$ for $\rho \in S^1$ and $(z_1,z_2)\in S^3\subset {ℂ}^2$. We consider a link consisting of finite orbits of this action, where some of the orbits are reversely oriented. Such a link appears as a link of a certain type of mixed polynomials. We study the space of such links and show smooth degeneration relations.