Traveling Wave Solutions of a Reaction-Diffusion Equation with State-Dependent Delay

This paper is concerned with the traveling wave solutions of a reaction-diffusion equation with state-dependent delay. When the birth function is monotone, the existence and nonexistence of monotone traveling wave solutions are established. When the birth function is not monotone, the minimal wave speed of nontrivial traveling wave solutions is obtained. The results are proved by the construction of upper and lower solutions and application of the fixed point theorem

Unstable Galaxy Models

The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jean's theorem, all the spherically symmetric steady galaxy models are given by a distribution of {\Phi}(E,L), where E is the particle energy and L the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem, the galaxy model {\Phi}(E,L) is stable if the distribution {\Phi} is monotonically decreasing with respect to the particle energy E. On the other hand, the stability of {\Phi}(E,L) remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin, we constuct examples of unstable galaxy models of f(E,L) and f(E) in which f fails to be monotone in E

Set Representations of Linegraphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A family $\mathcal{S}$ of nonempty sets $\{S_1,\ldots,S_n\}$ is a set representation of $G$ if there exists a one-to-one correspondence between the vertices $v_1, \ldots, v_n$ in $V(G)$ and the sets in $\mathcal{S}$ such that $v_iv_j \in E(G)$ if and only if S_i\cap S_j\neq \es. A set representation $\mathcal{S}$ is a distinct (respectively, antichain, uniform and simple) set representation if any two sets $S_i$ and $S_j$ in $\mathcal{S}$ have the property $S_i\neq S_j$ (respectively, $S_i\nsubseteq S_j$, $|S_i|=|S_j|$ and $|S_i\cap S_j|\leqslant 1$). Let $U(\mathcal{S})=\bigcup_{i=1}^n S_i$. Two set representations $\mathcal{S}$ and $\mathcal{S}'$ are isomorphic if $\mathcal{S}'$ can be obtained from $\mathcal{S}$ by a bijection from $U(\mathcal{S})$ to $U(\mathcal{S}')$. Let $F$ denote a class of set representations of a graph $G$. The type of $F$ is the number of equivalence classes under the isomorphism relation. In this paper, we investigate types of set representations for linegraphs. We determine the types for the following categories of set representations: simple-distinct, simple-antichain, simple-uniform and simple-distinct-uniform

The peculiar filamentary HI structure of NGC 6145

In this paper, we report the peculiar HI morphology of the cluster spiral galaxy NGC 6145, which has a 150 kpc HI filament on one side that is nearly parallel to its major axis. This filament is made up of several HI clouds and the diffuse HI gas between them, with no optical counterparts. We compare its HI distribution with other one-sided HI distributions in the literature, and find that the overall HI distribution is very different from the typical tidal and ram-pressure stripped HI shape, and its morphology is inconsistent with being a pure accretion event. Only about 30% of the total HI gas is anchored on the stellar disk, while most of HI gas forms the filament in the west. At a projected distance of 122 kpc, we find a massive elliptical companion (NGC 6146) with extended radio emission, whose axis points to an HI gap in NGC 6145. The velocity of the HI filament shows an overall light-of- sight motion of 80 to 180 km/s with respect to NGC 6145. Using the long-slit spectra of NGC 6145 along its major stellar axis, we find that some outer regions show enhanced star formation, while in contrast, almost no star formation activities are found in its center (less than 2 kpc). Pure accretion, tidal or ram-pressure stripping is not likely to produce the observed HI filament. An alternative explanation is the jet-stripping from NGC 6146, although direct evidence for a jet-cold gas interaction has not been found.Comment: 12 pages, 6 figures; Accepted for publication in A