The object of the present paper is to study some curvature conditions on Kenmotsu manifolds. Also, we classify Kenmotsu manifolds which satisfy $P\cdot \tilde{C}=0,\tilde{C}\cdot \tilde{C}=0,\tilde{Z}\cdot \tilde{C}=0,\tilde{C}\cdot \tilde{Z}=0$ and $C\cdot \tilde{C}=0$, where $P$ is the projective curvature tensor, $\tilde{Z}$ is the concircular curvature tensor, $\tilde{C}$ is the quasi-conformal curvature tensor and $C$ is the conformal curvature tensor.

The object of this paper is to study $\text{K}$-contact manifolds with quasi-conformal curvature tensor. We characterised $\text{K}$-contact manifolds satisfying certain curvature conditions on the quasi-conformal curvature tensor.

In this paper, the problem of classifying a new observation vector into one of the two normal populations for high-dimensional data is considered. High-dimensional data means that the total number of observation vectors from the two groups is less than the dimension of the observation vectors. Recently, linear discriminant analysis (LDA) for high-dimensional data such as microarray data has been considered. A simple way is to use the Moore-Penrose inverse when the sample covariance matrix is singular. In this paper, we suggest another type LDA approach for high-dimensional data. This method is based on a ridge type estimator of covariance matrix which was proposed by Srivastava and Kubokawa (2008). In addition, we derive asymptotic approximation of EPMC for this method in the situation of .

Let $\mathit{y}=(y_1,\dots ,y_p{)}^{\prime }$ be a $p$-dimensional random vector measurable on the individuals drawn from each of $k$ $p$-dimensional normal populations ${\prod }_i:N_p({\mu }_i,\Sigma ),i=1,\dots ,k$. In this paper we consider the growth curve model which has a mean structure as follows: ${\mathit{\mu }}_i=X{\mathit{\theta }}_i,i=1,\dots ,k$, where $X$ is a $p\times q$ given matrix with rank $q$ and ${\mathit{\theta }}_i$’s are unknown parameter vectors. First we derive an $\text{LR}$ test for a parallelism hypothesis $H_{\text{1}}:X{\mathit{\theta }}_i-X{\mathit{\theta }}_k={\gamma }_i{\mathbf{1}}_p,i=1,\dots ,k-1$, where ${\gamma }_i$’s are unknown parameters, and ${\mathbf{1}}_p$ is the $p$-dimensional vector with all the elements 1. Next we obtain the MLE of $\mathit{\gamma }=({\gamma }_1,\dots ,{\gamma }_{k-1}{)}^{\prime }$ and its distribution, and propose a simultaneous confidence interval for linear combinations of $\mathit{\gamma }$.

We consider the following the $p$-Laplacian equation in a bounded domain $\mathrm{\Omega }$:

We treat the case of nonlinearity term $f$ satisfying the following conditions

for constants $a_0,b_0,a$ and $b$. We prove the existence of a positive solution or a negative solution in the case of $(a_0-{\lambda }_1)(a-{\lambda }_1)=0$ or $(b_0-{\lambda }_1)(b-{\lambda }_1)=0$ respectively, where ${\lambda }_1$ is the first eigenvalue of $-{\mathrm{\Delta }}_p$.

The object of the present paper is to study $\varphi$-Ricci symmetric $N(k)$-contact metric manifolds and $\varphi$-Ricci symmetric $N(k)$-contact metric manifolds of dimension three. It is proved that a $\varphi$-Ricci symmetric $N(k)$-contact metric manifold is Sasakian. Next we prove that a three dimensional $N(k)$-contact metric manifold is locally $\varphi$-Ricci symmetric if and only if the scalar curvature is constant. Finally we give examples of $\varphi$-Ricci symmetric $N(k)$-contact metric manifolds.