Narrow Results

This review is an elementary introduction to the concepts of the recently developed asymptotic methods and new developments. Particular attention is paid throughout the paper to giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameter-expansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-infinity theory in high energy physics, Kleiber's 3/4 law in biology, possible mechanism in spider-spinning process and fractal approach to carbon nanotube are briefly introduced. Bubble-electrospinning for mass production of nanofibers is illustrated. There are in total more than 280 references.

We reconsider the link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local theory of [2].

We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory.

Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the homology of the (2,n) torus knots.

The $p$-competition graph $C_p(D)$ of a digraph $D=(V,A)$ is a graph with $V(C_p(D))=V(D)$, where for distinct vertices $x$ and $y$, $xy\in E(C_p(D))$ if and only if there exist distinct $p$ vertices $v_1,v_2,\dots ,v_p\in V(D)$ such that $x\to v_i$, $y\to v_i$ are arcs of the digraph $D$ for each $i=1,2,\dots ,p$. In this paper, we show that spiders $S_{m,n}$ are $p$-competition graphs, where $p\le n-2$ and $m$ is even.

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A signed Roman dominating function (SRDF) of $G$ is a function $f:V\mapsto \{-1,1,2\}$ satisfying the conditions that (i) ${\sum }_{x^{\prime }\in N[x]}f(x^{\prime })\ge 1$ for each $x\in V$, where $N[x]$ is the set, called closed neighborhood of $x$, consists of $x$ and the vertex of $V$ adjacent to $x$ (ii) every vertex $x\in V$ for which $f(x)=-1$ is adjacent to at least one vertex $y\in V$ for which $f(y)=2$. The weight of a SRDF $f$ is $\omega (f)={\sum }_{x\in V}f(x)$. The signed Roman domination number ${\gamma }_{\mathrm{\text{sR}}}(G)$ of $G$ is the minimum weight of a SRDF of $G$. In this paper, we determine the exact values of signed Roman domination number of spider and double star. Specially, one of them generalizes the known result.