This paper studies the following fourth order quasilinear elliptic equation
\begin{equation*}
\left\{\begin{aligned}
&\triangle^{2} u-\triangle u+V(x)u-\frac{1}{2}u\triangle (u^{2})=f(u),&x\in \mathbb{R}^{N},\\
&u\in H^{2}(\mathbb{R}^{N}),
\end{aligned}
\right.
\end{equation*}
where △2:=△(△) is the biharmonic operator, 2<N≤6. We prove that the equation admits a ground state solution of the Nehari-Poho\u{z}aev type.
Let γ∗(D) denote the twin domination number of digraph D and let Cm◻Cn denote the Cartesian product of the directed cycle Cm and Cn, for m,n≥2. In this paper, we give a lower bound for γ∗(Cm◻Cn) and we determine the exact values of γ∗(Cm◻Cn) when m, n≡0 (mod 3) and when m≡2 (mod 3).
Let p be an odd prime. In this paper, we obtain the number of (connected)
Cayley graphs on the dicyclic groups T4p=⟨a,b | ap=b4=1,b−1ab=a−1⟩ up to isomorphism by using the P\'{o}lya enumeration theorem.
In this paper, a complete classification of 11-valent symmetric graphs of 4p order is given, where p is a prime.
According to the results of the classification, there is only one the 11-valent graph of order 4p, which is a complete graph K12.
Since the classical ARMA residual control chart is susceptible to outlier values, this paper first establishes a robust ARMA model based on the Institute of Geodesy & Geophysics III IGGIII weight function to obtains an independent residual distribution sequence. Then a robust residual control chart is constructed based on the estimations of the mean and standard deviation by the weighted three mean and average absolute deviation, respectively. Simulations and empirical tests show that the robust control chart can resist the outlier interferences better, and the monitoring effect is better than that of the traditional control chart.