In this paper, we consider the effect of non-normality on the distribution of the sample fourth order cumulant in elliptical distributions. Asymptotic properties of the distribution of the sample fourth order cumulant is presented under non-normality, especially, the class of elliptical populations in detail. Asymptotic expansion formulas for the distributions are derived by a perturbation method. Finally, the numerical results are obtained.

Assume that two media are laying in a half-space and the interface wall is parallel to the boundary of the half-space. We can directly observe the data near the boundary of the half-space, but we cannot directly observe inside the half-space. In this situation, we try to identify these unknown things by creating artificial explosions and observing on the boundary the waves generated by the explosions. Here, the waves are described by the elastic equation.

In this paper, we describe the structures of the left $B^e$-modules ${\tau }_{B^e}^i(B)$ and ${\mathcal{N}}_{B^e}^i(B)$ for $i⩾0$, where $B$ is a certain finite dimensional selfinjective Nakayama algebra, $B^e$ is the enveloping algebra of $B$, ${\tau }_{B^e}$ is the Auslander-Reiten translation in the category mod $(B^e)$ of finitely generated left $B^e$-modules and ${\mathcal{N}}_{B^e}:\text{mod}(B^e)\to \text{mod}(B^e)$ is the Nakayama functor. Moreover, we compute the ${\tau }_{B^e}$-period and the ${\mathcal{N}}_{B^e}$-period of $B$.

We study the initial-boundary value problem for a general class of nonlinear dissipative equations on a half-line

(0.1)$$\{\begin{array}{l}{u}_t+N(u,u_x)+{\mathbb{K}}_u=f,(x,t)\in R^{+}\times R^{+}\hfill \\ u(x,0)=u_0(x),x\in R^{+},\hfill \\ {\partial }_x^{j-1}u(0,t)=h_j(t)\text{for}j=1,\dots ,M,\hfill \end{array}$$

where the nonlinear term $\mathbb{N}(u,u_x)$ depends on the unknown function $u$ and its derivative $u_x$ and satisfies the estimate

$$|\mathbb{N}(u,\upsilon )|\le C\left|u{|}^{\rho }\right|\upsilon {|}^{\sigma }$$

with $\rho ,\sigma \ge 0$ and the linear operator $\mathbb{K}(u)$ is defined as follows

$$\mathbb{K}(u)={\alpha }_n{\partial }_x^n+{\alpha }_m{\partial }_x^m,$$

where the constants ${\alpha }_n$, ${\alpha }_m\in \mathbf{R}$, $n$,$m$ are integers, , $n\le M+1,n$ is an even integer.

The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1). We find the main term of the asymptotic representation of solutions.

Let $k$, $n$ be integers with $k\ge 3$ and $n\ge 3k$, and let $G$ be a bipartite graph having partite sets $V_1$, $V_2$ with $\left|V_1\right|=\left|V_2\right|=n$. We show that if $d_G(u)+d_G(\upsilon )\ge n+2k-1$ for any $u\in V_1$ and $\upsilon \in V_2$ with $u\upsilon \cancel{\in }E(G)$, then for any vertex disjoint paths $P_1,P_2,\dots ,P_k$ of order 3, $G$ contains vertex disjoint cycles $H_1,H_2,\mathrm{...},H_k$ such that ${{\displaystyle \cup }}_{1\le i\le k}V(H_i)=V(G)$ and $H_i$ passes through $P_i$ for each $i$ with $1\le i\le k$.

We study the propagation of singularities for nonlinear wave equation $\square u=F(u,Du)$. Our main result in this paper is Theorem 1.1, which is an extension of Theorem 2.7 in [2]. When the nonlinearity $F(u,Du)$ satisfies the null condition, we improve a condition with respect to regularity of solutions $u$.

The approximate upper percentile of Hotelling’s $T^2$-type statistic is derived in order to construct simultaneous confidence intervals for pairwise multiple comparisons and comparisons with a control under elliptical populations. The accuracy and conservativeness of the first and the modified second order Bonferroni approximations are evaluated via a Monte Carlo simulation study.