In this article, we present a third-order weighted essentially non-oscillatory (WENO) method for generalized Rosenau-KdV-RLW equation. The third order finite difference WENO reconstruction and central finite differences are applied to discrete advection terms and other terms, respectively, in spatial discretization. In order to achieve the third order accuracy both in space and time, four stage third-order L-stable SSP Implicit-Explicit Runge-Kutta method (Third-order SSP EXRK method and third-order DIRK method) is applied to temporal discretization. The high order accuracy and essentially non-oscillatory property of finite difference WENO reconstruction are shown for solitary wave and shock wave for Rosenau-KdV and Rosenau-KdV-RLW equations. The efficiency, reliability and excellent SSP property of the numerical scheme are demonstrated by several numerical experiments with large CFL number.

Our aim in this paper is to study a coupled Cahn–Hilliard system for copolymer/homopolymer mixtures. We prove the existence, uniqueness and regularity of solutions. We then prove the existence of finite dimensional global attractors.

We study the ill-posedness degree of the reconstruction processes of missing boundary data or initial states in the transient heat conduction. Both problems are severely ill-posed. This is a powerful indicator about the way the instabilities will affect the computations in the numerical recovery methods. We provide rigorous proofs of this result where the conductivites are space dependent. The theoretical work is concerned with the unsteady heat equation in one dimension even though most of the results obtained here are readily extended to higher dimensions.

In this paper, we introduce and study $s$-sequence-covering mappings and 1-$s$-sequence-covering mappings, obtain some characterizations of $s$-sequence-covering and compact images of metric spaces, and prove that every $s$-sequence-covering and compact mapping in first-countable spaces is a 1-$s$-sequence-covering mapping.

Let $G$ be a finite group. An irreducible character $\chi$ of $G$ is said to be primitive if $\chi \neq \vartheta^{G}$ for any character $\vartheta$ of a proper subgroup of $G$. In this paper, we consider about the zeros of primitive characters. Denote by ${\rm Irr_{pri} }(G)$ the set of all irreducible primitive characters of $G$. We proved  that  if $g\in G$ and the order of $gG'$ in the factor group $G/G'$ does not divide $|{\rm Irr_{pri}}(G)|$, then there exists $\varphi \in {\rm Irr_{pri}}(G)$ such that $\varphi(g)=0$.

In this article, we will consider the Dirichlet problem for special Lagrangian equation on $\Omega\subset M$, where $(M,J)$ is a compact almost complex manifold. Under the existence of $C^{2}$-smooth strictly $J$-plurisubharmonic subsolution $\underline{u}$, in the supercritical phase case, we  obtain a uniform global gradient estimate.

In this paper, we find all repdigits expressible as difference of two Fibonacci numbers in base $b$ for $2\leq b\leq10.$ The largest repdigits in base $b$, which can be written as difference of two Fibonacci numbers are \begin{align*}&F_{9}-F_{4}=34-3=31=(11111)_{2},~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(111111)_{3},\\&F_{14}-F_{7}=377-13=364=(222)_{4},~~ \text{ }F_{9}-F_{4}=34-3=31=(111)_{5},\\&F_{11}-F_{4}=89-3=86=(222)_{6},~~~~~~~~\text{ }F_{13}-F_{5}=233-5=228=(444)_{7},\\&F_{10}-F_{2}=55-1=54=(66)_{8},~~~~~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(444)_{9},\end{align*} and $$F_{15}-F_{10}=610-55=555=(555)_{10}.$$

As a result, it is shown that the largest Fibonacci number which can be written as a sum of a repdigit and a Fibonacci number is $F_{15}=610=555+55=555+F_{10}.$

In this paper, by using the Hermitian metric and Chern connection, we study the case of a strictly pseudoconvex domain $G$ with non-smooth boundaries in a complex manifold. By constructing a new integral kernel, we obtain a new Koppelman-Leray-Norguet formula of type $(p,q)$ on $G$, and get the continuous solutions of $\bar{\partial}$-equations on $G$ under a suitable condition. The new formula doesn't involve integrals on the boundary, thus one can avoid complex estimations of the boundary integrals, and the density of integral may be not defined on the boundary but only in the domain. As some applications, we discuss the Koppelman-Leray-Norguet formula of type $(p,q)$ for general strictly pseudoconvex polyhedrons (unnecessarily non-degenerate) on Stein manifolds, also get the continuous solutions of $\bar{\partial}$-equations under a suitable condition.