Nearly cloaking the elastic wave fields

In this work, we develop a general mathematical framework on regularized approximate cloaking of elastic waves governed by the Lam\'e system via the approach of transformation elastodynamics. Our study is rather comprehensive. We first provide a rigorous justification of the transformation elastodynamics. Based on the blow-up-a-point construction, elastic material tensors for a perfect cloak are derived and shown to possess singularities. In order to avoid the singular structure, we propose to regularize the blow-up-a-point construction to be the blow-up-a-small-region construction. However, it is shown that without incorporating a suitable lossy layer, the regularized construction would fail due to resonant inclusions. In order to defeat the failure of the lossless construction, a properly designed lossy layer is introduced into the regularized cloaking construction . We derive sharp asymptotic estimates in assessing the cloaking performance. The proposed cloaking scheme is capable of nearly cloaking an arbitrary content with a high accuracy

Long properly colored cycles in edge colored complete graphs

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A properly colored cycle (path) in $K_{n}^{c}$ is a cycle (path) in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s (1976) proposed the following conjecture: if $\Delta^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if $\Delta^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored cycle of length at least $\lceil \frac{n+2}{3}\rceil+1$. In this paper, we improve the bound to $\lceil \frac{n}{2}\rceil + 2$.Comment: 8 page

List version of (pp,1)-total labellings

The ($p$,1)-total number $\lambda_p^T(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and the edges of $G$ such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least $p$. In this paper we consider the list version. Let $L(x)$ be a list of possible colors for all $x\in V(G)\cup E(G)$. Define $C_{p,1}^T(G)$ to be the smallest integer $k$ such that for every list assignment with $|L(x)|=k$ for all $x\in V(G)\cup E(G)$, $G$ has a ($p$,1)-total labelling $c$ such that $c(x)\in L(x)$ for all $x\in V(G)\cup E(G)$. We call $C_{p,1}^T(G)$ the ($p$,1)-total labelling choosability and $G$ is list $L$-($p$,1)-total labelable. In this paper, we present a conjecture on the upper bound of $C_{p,1}^T$. Furthermore, we study this parameter for paths and trees in Section 2. We also prove that $C_{p,1}^T(K_{1,n})\leq n+2p-1$ for star $K_{1,n}$ with $p\geq2, n\geq3$ in Section 3 and $C_{p,1}^T(G)\leq \Delta+2p-1$ for outerplanar graph with $\Delta\geq p+3$ in Section 4.Comment: 11 pages, 2 figure

Oscillation of Nonlinear Delay Partial Difference Equations

In this paper, we consider certain nonlinear partial difference equations$$aA_{m+1,n}+bA_{m, n+1}-cA_{m,n}+\sum\limits_{i=1}^{u} p_{i}(m,n)A_{m-\sigma_{i},n-\tau_{i}}=0$$where $a,b,c\in(0,\infty )$, $u$ is a positive integer, $p_{i}(m,n),~(i=0,1,2,\cdots u)$ are positive real sequences. $\sigma_i,\tau_i\in N_{0}=\{1,2,\cdots \},~i=1,2,\cdots,u$. A new comparison theorem for oscillation of the above equation is obtained

Nonoscillation Theorems for a Class of Fourth Order Quasilinear Dynamic Equations on Time Scales

In this paper,some sufficient and necessary conditions for nonoscillation of the fourth order quasilinear dynamic equations on time scales T are established. Our results as special case when T = R and T = N,involve and improve some known results