titania nanotubes (tnt) prepared by anodization of ti foils were used for water splitting in a standalone cell. the concentration polarization between the anode side (1m naoh) and cathode side (0.5 m h2so4) ensured that the water splitting reaction could take place with no external bias and separate h2 and o2 evolution could be achieved. the destruction of tnt structures under 365 nm uv irradia...

We start with a given modular invariantM of a two dimensional ŝu(n)k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE ...

 Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m in T(M)$. In this paper we study the domination number of $T(Gamma(M))$ a...

let g(v,e) be a graph with p vertices and q edges. a graph g is said to have an odd mean labeling if there exists a function f : v (g) → {0, 1, 2,...,2q - 1} satisfying f is 1 - 1 and the induced map f* : e(g) → {1, 3, 5,...,2q - 1} defined by f*(uv) = (f(u) + f(v))/2 if f(u) + f(v) is evenf*(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is a bijection. a graph that admits an odd mean lab...

Applications of graph and hypergraph decomposition techniques to SAT solving have been studied under various conditions. Theoretical research focused on the direction given by tree decompositions, while practical approaches either integrated tree decomposition guidance into existing SAT solver, or studied splitting techniques using hypergraph cuts. In this article we present a uniform and gener...

Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the complement has a transitive orientation). The operation of “vertex splitting”, introduced in [3], f...

We generalize the JSJ-splitting of Rips and Sela to give decompositions of nitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually ZZ groups. The approach, called \track zipping", is relatively elementary, and diiers from the Rips-Sela approach in that it does not rely on the th...

a fullerene graph is a 3−connected planar graph whose faces are pentagons and hexagons.the clar number of a fullerene is the maximum size of sextet patterns, the sets of disjointhexagons which are all m-alternating for a kekulé structure m of f. an exact formula of clarnumber of some fullerene graphs and a class of carbon nanocones are obtained in this paper.