This paper continues the investigation of the rst author begun in part one. The hereditary properties of n-homomorphism amenability for Banach algebras are investigated and the relations between n-homomorphism amenability of a Banach algebra and its ideals are found. Analogous to the character amenability, it is shown that the tensor product of two unital Banach algebras is n-homomorphism amena...

Inspired by the recently remarkable successes of Sparse Representation (SR), Collaborative Representation (CR) and sparse graph, we present a novel hypergraph model named Regression-based Hypergraph (RH) which utilizes the regression models to construct the high quality hypergraphs. Moreover, we plug RH into two conventional hypergraph learning frameworks, namely hypergraph spectral clustering ...

Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...

Let H = (V, E) be a 3-uniform linear hypergraph with one hypercycle C3. We consider a blow-up hypergraph B[H]. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph B[H] of the hypergraph H, with hyperedge densities satisfying special conditions, such that the hypergraph H appears in a blow-up hypergraph as a transversal. We present an efficient...

The cyclicity of a hypergraph is an e ciently computable integer that extends the notion of the cyclomatic number of a graph The formula for the cyclicity is suggested by the join invariant of an acyclic hypergraph which is the multiset of all joining sets in any of its join trees Once we gure out how the multiplicity of a joining set depends on the structure of the acyclic hypergraph we de ne ...

A hypergraph is d-degenerate if every subhypergraph has a vertex of degree at most d. A greedy algorithm colours every such hypergraph with at most d+ 1 colours. We show that this bound is tight, by constructing an r-uniform d-degenerate hypergraph with chromatic number d + 1 for all r ≥ 2 and d ≥ 1. Moreover, the hypergraph is triangle-free, where a triangle in an r-uniform hypergraph consists...