In this paper, the Dirichlet boundary value problems of the nonlinear beam equation $u_{tt} + ∆^2u + αu + ϵϕ(t)(u + u^3 ) = 0, α > 0$ in the dimension one is considered, where $u(t, x)$ and $ϕ(t$) are analytic quasi-periodic functions in $t$, and $ϵ$ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.

We characterize the boundedness and compactness of weighted composition operators among some Fock-Sobolev spaces. We also estimate the norm and essential norm of these operators. Furthermore, we discuss the duality spaces of Fock-Sobolev spaces $\mathcal{F}^{p,m}_s$ when $0 < p < ∞$.

In recent study the bank of real square integrable functions that have nonlinear phases and admit a well-behaved Hilbert transform has been constructed for adaptive representation of nonlinear signals. We first show in this paper that the available basic functions are adequate for establishing an ideal adaptive decomposition algorithm. However, we also point out that the best approximation algorithm, which is a common strategy in decomposing a function into a sum of functions in a prescribed class of basis functions, should not be considered as a candidate for the ideal algorithm.

In this paper, we consider the problem of determining the order of INAR($q$) model on the basis of the Bayesian estimation theory. The Bayesian estimator for the order is given with respect to a squared-error loss function. The consistency of the estimator is discussed. The results of a simulation study for the estimation method are presented.

Let $γ^∗ (D)$ denote the twin domination number of digraph $D$ and let $D_1 ⊗ D_2$ denote the strong product of $D_1$ and $D_2$. In this paper, we obtain that the twin domination number of strong product of two directed cycles of length at least $2$. Furthermore, we give a lower bound of the twin domination number of strong product of two digraphs, and prove that the twin domination number of strong product of the complete digraph and any digraph $D$ equals the twin domination number of $D$.

In this article, we introduce and study the concept of $n$-Gorenstein injective (resp., $n$-Gorenstein flat) modules as a nontrivial generalization of Gorenstein injective (resp., Gorenstein flat) modules. We investigate the properties of these modules in various ways. For example, we show that the class of $n$-Gorenstein injective (resp., $n$-Gorenstein flat) modules is closed under direct sums and direct products for $n ≥ 2$. To this end, we first introduce and study the notions of $n$-injective modules and $n$-flat modules.

In this paper, we establish two optimal the double inequalities for Sándor-Yang type mean and one-parameter mean.

Let $G$ be a simple graph. A total coloring $f$ of $G$ is called an E-total coloring if no two adjacent vertices of $G$ receive the same color, and no edge of $G$ receives the same color as one of its endpoints. For an E-total coloring $f$ of a graph $G$ and any vertex $x$ of $G$, let $C(x)$ denote the set of colors of vertex $x$ and of the edges incident with , we call $C(x)$ the color set of $x$. If $C(u)≠C(v)$ for any two different vertices $u$ and $v$ of $V (G)$, then we say that $f$ is a vertex-distinguishing E-total coloring of $G$ or a VDET coloring of $G$ for short. The minimum number of colors required for a VDET coloring of $G$ is denoted by $χ^e_{vt}(G)$ and is called the VDET chromatic number of $G$. The VDET coloring of complete bipartite graph $K_{7,n} (7 ≤ n ≤ 95)$ is discussed in this paper and the VDET chromatic number of $K_{7,n} (7 ≤ n ≤ 95)$ has been obtained.

For a handlebody $H$ with $∂H = S$, let $F ⊂ S$ be an essential connected subsurface of $S$. Let $\mathcal{C}(S)$ be the curve complex of $S$, $\mathcal{AC}(F)$ be the arc and curve complex of $F$, $\mathcal{D}(H) ⊂ \mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ⊂ \mathcal{AC}(F)$ be the image of $\mathcal{D}(H)$ in $\mathcal{AC}(F)$. We introduce the definition of subsurface 1-distance between the 1-simplices of $\mathcal{AC}(F)$ and show that under some hypothesis, $π_F (\mathcal{D}(H))$ comes within subsurface 1-distance at most 4 of every 1-simplex of $\mathcal{AC}(F)$.