A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...

Since the inception of the notion of fuzzy topology much attention was paid to the possible means of interaction between fuzzy and crisp topological settings, in order to see whether fuzzy topology was doing anything new. In particular, different functors relating the categories Top of topological spaces and L-Top of Ltopological spaces (L being a suitable complete lattice) appeared in the lite...

As a specific kind of non-Euclidean metric lies in projective space, Cayley-Klein metric has been recently introduced in metric learning to deal with the complex data distributions in computer vision tasks. In this paper, we extend the original Cayley-Klein metric to the multiple Cayley-Klein metric, which is defined as a linear combination of several Cayley-Klein metrics. Since Cayley-Klein is...

The classic Cayley identity states that det(∂) (detX) = s(s+ 1) · · · (s+ n− 1) (detX) where X = (xij) is an n× n matrix of indeterminates and ∂ = (∂/∂xij) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterio...

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 ...

A C-coloring of a hypergraph H = (X, E) is a vertex coloring φ : X → N such that each edge E ∈ E has at least two vertices with a common color. The related parameter χ(H), called the upper chromatic number of H, is the maximum number of colors can be used in a C-coloring of H. A hypertree is a hypergraph which has a host tree T such that each edge E ∈ E induces a connected subgraph in T . Notat...

Let H be a hypergraph with vertices V(H) and hyperedges E(H). The total graph of H, T(H), is the simple graph with vertex set V(H)U E(H) where vertices x and y of T(H) are adjacent if and only if x is contained in, contains or is adjacent to y in H. We give a simple characterisation of those graphs which are the total graphs of some hypergraphs. We show that the total graph uniquely defines a l...

I published several papers with similar titles. One of my latest ones [13] (also see [16] and the yearly meetings at Boca Raton or Baton Rouge) contains, in the introduction, many references to my previous papers . I discuss here as much as possible new problems, and present proofs in only one case. I use the same notation as used in my previous papers . G(' )(n ;1) denotes an r-graph (uniform ...

These notes were produced in response to a question from John Cannon: what issues are involved in adding facilities to CAYLEY for doing finite geometry? More specifically, what facilities would geometers want? At first sight, the idea of geometry in CAYLEY seems very natural. Groups and geometry have been intimately linked since long before the time of Klein. Many of the groups with which CAYLE...

Table 1: Comparison of sum of cluster densities for MDS Peeling+DPRP and for the best of 10 Density-FM runs. Results for avq.small benchmark using Density-FM would require many days of SPARC-10 time. Bounded Size Maximum Density Subhyper-graph (BMDS) problem : Given a hypergraph H(V; E) and an integer B, nd the subhypergraph of H with maximum density and size B. While MDS was polynomial time so...