This paper is concerned with profile analysis of $k$ $p$-dimensional normal populations ${\mathrm{\Pi }}_i:N_p({\mu }_i,\mathrm{\Sigma }),i=1,\dots ,k$, when the mean vectors are expressed as ${\mu }_i=X{\theta }_i,i=1,\cdots ,k$, where $X$ is a $p\times q$ given matrix with rank $q$ and ${\theta }_i$’s are unknown parameter vectors. The model with such a mean structure is applied to growth curve data. Fujikoshi (2009) studied a likelihood ratio statistic for a parallelism hypothesis. In this paper we derive likelihood ratio statistics for level and flatness hypotheses under the parallelism hypothesis. Their null distributions are obtained. We also give an example.

In this article, we study $(ϵ)$-para Sasakian manifolds with conformal curvature tensor. In this context, we consider conformally flat, quasi-conformally flat and quasi-conformally semi-symmetric $(ϵ)$-para Sasakian manifolds. It is proved that $(ϵ)$-para Sasakian manifold is conformally flat if and only if it is locally isometric to a pseudo hyperbolic space $H_v^n(1)$ or to a pseudo sphere $S_v^n(1)$.

Let $G$ be a $(p,q)$ graph and $f:V(G)\to \{0,1,2,3,\dots ,q\}$ be an injection. For each edge $e=uv$, let $f^{*}(e)=\lceil \frac{f(u)+f(v)}{2}\rceil$ Then $f$ is called a mean labeling if $\{f^{*}(e):e\in E(G)\}=\{1,2,3,\dots ,q\}$. A graph that admits a mean labeling is called a $\text{m}\text{e}\text{a}\text{n}$ $\text{g}\text{r}\text{a}\text{p}\text{h}$. In this paper, we prove $TôC_n,TõC_n,T@P_n,Tⓒ2P_n$ where $T$ is a $T_p$-tree, are mean graphs.

Decaying properties of the local energy for the dissipative wave equations with the Dirichlet boundary conditions in exterior domains are discussed. For the dissipation coefficient, natural conditions ensuring that waves trapped by obstacles may lose their energy are considered. Under this setting, the cases that the dissipation coefficient does not have a compact support, but has some regularities are treated. Further, the cases that the dissipation coefficients decay sufficiently fast at infinity are also discussed.

Let $G=(V,E)$ be a connected graph. For a set $S\subseteq V$, the differential of $S$, denoted by $\partial (S)$, is defined to be $\partial (S)=\left|B(S)\right|-\left|S\right|$, where $B(S)=N(S)\cap (V-S)$. A set $S\subseteq V$ is said to be a positive differential set if $\partial (S)\ge 0$. A partition $\{V_1,V_2,\dots ,V_k\}$ of $V$ is said to be a positive differential chromatic partition of $G$ if each $V_i$ is both independent and positive differential. The minimum order of a positive differential chromatic partition of $G$ is called the differential chromatic number of $G$ and is denoted by ${\chi }_{\partial }(G)$. In this paper we initiate a study of this parameter.

In this paper, we determine the wave front sets of solutions to the Schrödinger equations of a free particle and of a harmonic oscillator by using the representation of the Schrödinger evolution operator of a free particle introduced by Kato, Kobayashi and Ito (2011) and a new representation of the evolution operator of a harmonic oscillator via wave packet transform (short time Fourier transform).

Given two $m$-sectorial operators in a reflexive Banach space $X$, a sufficient condition is presented for the family $\{T+\kappa A;\kappa \in {\mathrm{\Sigma }}^c\}$ to be holomorphic of type (A), where $\mathrm{\Sigma }$ is a closed convex region in the $\kappa$-plane such that $\partial \mathrm{\Sigma }$ is the left branch of a hyperbola. The abstract formulation is almost identical with Kato’s when $X$ is a Hilbert space. However, $\partial \mathrm{\Sigma }$ is typically reduced to a parabola when $X$ is a Hilbert space. The $m$-sectoriality is a generalized notion of the nonnegative selfadjointness so that Schrödinger operators with singular potentials in the $L^p$-spaces are typical examples of the abstract theory. In this connection a detailed analysis is given in the case of $-\Delta +\kappa |x{|}^{-2}$. This application is an $L^p$-generalization of Kato’s.

We show the nonexistence of a global solution of the Dirichlet problem of a quasilinear hyperbolic equation.