In this article, the complete convergence for weighted sums of arrays of rowwise extended negatively dependent random variables is obtained by using the Rosenthal type moment inequality. Some sufficient conditions to prove the complete convergence are provided. In addition, a necessary condition to prove the complete convergence is presented. The results obtained in the paper extended some corresponding ones for independent sequences and some dependent sequences.

Let $\{X_k;k\ge1\}$ be a moving average process defined by $X_k=\sum_{i=0}^\infty a_i\varepsilon_{k-i},$ where $\{\varepsilon_i;-\infty<i<\infty\}$ is a doubly infinite sequence of identically distributed $\varphi$-mixing random variables, $a_i\sim i^{-\alpha}l(i)$ and $l(i)$ is a slowly varying function. When $1/2<\alpha<1,$ $\{X_k;k\ge1\}$ is a long memory process. Under the assumption that $\text{E}\varepsilon_0^2$ may be infinite, a general strong approximation theorem for partial sums of the long memory process is derived.

When experiments are carried out in a sequential of time, the response may be subject to the uncontrollable time trends. In this paper, magic cubes and hypo-magic cubes are used in these trend free plans with multi-factors, which show that all main effects and two-factor interaction effects are both linear trend free. Moreover, the proposed method can be applied for non prime power levers.

Based on discretely observed sample of local stationary diffusion model, the weighted least squares estimations of time-varying drift parameters are obtained by using local linear fitting. The consistency, asymptotic normality and convergence rate of the proposed estimations are discussed. Moreover, it is shown that the estimations are effective through a simulation study.

An $(l-1)$-outpath of an arc $x_1x_2$ in a multipartite tournament is a path $x_1x_2\cdots x_l$ of length $l-1$ starting with $x_1x_2$, such that either $x_l$ and $x_1$ are in the same partite set or $x_l$ dominates $x_1$. Specially, $x_1x_2\cdots x_lx_1$ is a Hamilton cycle when $l=|V(D)|$ and $x_l$ dominates $x_1$. Guo (Discrete Appl Math 95 (1999) 273-277) proved that every arc of a regular $c$-partite tournament with $c\geq 3$ has a $(k-1)$-outpath for each $k\in \{3, 4, \cdots, c\}$. As a generalization, the paper proves that every arc in a regular $c$-partite tournament with $c\geq 5$ has a $(k-1)$-outpath for each $k\in \{3, 4, \cdots, |V(D)|\}$ in this article. Furthermore, using the method of path-contracting, the paper also proves the following result: Let $D$ be a regular $c$-partite tournament. If $c\geq 8$ and there are two vertices in every partite set, then each arc in $D$ is contained in a Hamilton cycle. This result gives a partial support to the conjecture posed by Volkmann and Yeo (Discrete Math 281 (2004) 267-276) that each arc of a regular multipartite tournament is contained in a Hamilton cycle.

A $k$-cyclic graph is a connected graph in which the number of edges equals the number of vertices plus $k+1$. This paper determines the maximal signless Laplacian spectral radius together with the corresponding extremal graph among all triangle-free $k$-cyclic graphs of order $n$. Moreover, this paper gives the first five triangle-free unicyclic graphs on $n \,(n\geq 8)$ vertices, and the first eight triangle-free bicyclic graphs on $n \,(n\geq 12)$ vertices according to the signless Laplacian spectral radius. Finally, the authors of this paper show that the results obtained in this paper also hold for Laplacian spectral radius of triangle-free $k$-cyclic graphs of order $n$.

The problem of determining whether $K_n$ (for $n$ odd) or $K_n$ minus a 1-factor (for $n$ even) has a 2-factorization is called Oberwolfach problem. The notation OP$(m_1^{\alpha_1},m_2^{\alpha_2},\cdots,m_t^{\alpha_t})$ represents the case in which each 2-factor consists of exactly $\alpha_i$ cycles of length $m_i$ for $i=1,2,\cdots,t$. Proved that the OP$(4^{a},s^b)$ with $a\geq 0$, $b=2,3$, $s=3,5,6$ and $(a,s,b)\not=(0,3,2)$ have solutions.

A quantity discount problem between a manufacturer and two retailers was considered in this paper. For the demand of customer is uncertain, manufacturer adopted quantity discount policy to encourage retailers to increase the single order quantity thereby reducing inventory costs in decisionmaking problems. Combined with the retailer’s bargaining power, the cooperation situation between the two retailers and non-cooperative situation from the perspective of online algorithm and competitive analysis were considered. The corresponding equilibrium strategy is designed. The conclusion that the strategy is the optimal strategy is proved.

Based on the original second Bianchi identity, the authors derive the extended second Bianchi identities via the second or third covariant derivatives. These extended identities have some applications in the evolution equations for the Riemannian or algebraic curvature tensors along two special geometric flows, i.e. the Ricci flow and the hyperbolic geometric flow. Some examples are given to illustrate the related applications.

A echinococcosis transmission dynamical model with stochastic environmental fluctuation and two discrete time delays is investigated. When basic reproductive number $R_0>1$ and the threshold of noise intensity $R_0^N<1$, the stability of the infectious equilibrium point $E^*$ in probability is proved. Furthermore, the effect of white noise and delay on the control of echinococcosis transmission is discussed.

Using the Schaefer fixed point theorem, the existence of solutions to two-point boundary value problems of fractional p-Laplacian systems is studied. Under certain growth conditions of the nonlinearity, a sufficient condition is obtained for the existence of at least one solution by converting the system into an operator equation. Finally, the application of the obtained results is illustrated.

A circle pattern is a configuration of circles with prescribed intersection angles in the complex plane $\mathbf{C}$. Given a analytic function with a finite critical points defined on a bounded simply connected region ${\mathit\Omega}\subset\mathbf{C}$, the techniques of branched circle patterns is used to construct the approximating solutions of $F$. It is proved that the sequence of approximating solutions converges uniformly on compact subsets of ${\mathit\Omega}$ to the analytic function $F$. This provides a new numeral method of computing analytic functions with critical points.

In this paper, the authors prove the boundedness of the commutator $\mathcal{M}^{\rho}_{b}$ generated by the parameter Marcinkiewicz integral $\mathcal{M}^{\rho}$ with Lipschitz function $b$. Under the assumption that the kernel of $\mathcal{M}$ satisfies certain condition, the authors prove that $\mathcal{M}^{\rho}_{b}$ is bounded from the Lebesgue space $L^{\frac{n}{n-\beta}}(\mu)$ to the Hardy space $H^{1}(\mu)$, and from the Lebesgue space $L^{\frac{n}{\beta}}(\mu)$ to the space $\mathrm{RBMO}(\mu)$.

The concept of quasi C-continuity for posets is introduced. In terms of quasi Ccontinuity, it is proved that quasi-continuity of a dcpo is equivalent to the quasi-continuity of its Scott-closed lattice. It is obtained that the Scott-closed lattice of a dcpo satisfying the property M is C-algebraic. A new sufficient condition is given for the isomorphism of two dcpos with isomorphic Scott-closed lattices.