In a graph G, the distance from an edge e to a set F ⊆ E(G) is the vertex distance from e to F in the line graph L(G). For a decomposition of E(G) into k sets, the distance vector of e is the k-tuple of distances from e to these sets. The decomposition dimension dec(G) of G is the smallest k such that G has a decomposition into k sets so that the distance vectors of the edges are distinct. For ...

Sufficient dimension reduction (SDR) using distance covariance (DCOV) was recently proposed as an approach to dimension-reduction problems. Compared with other SDR methods, it is model-free without estimating link function and does not require any particular distributions on predictors (see Sheng Yin, 2013, 2016). However, the DCOV-based method involves optimizing a nonsmooth nonconvex objectiv...

Medial Axis (MA), also known as Centres of Maximal Disks, is a useful representation of a shape for image description and analysis. MA can be computed on a distance transform, where each point is labelled to its distance to the background. Recent algorithms allow one to compute Squared Euclidean Distance Transform (SEDT) in linear time in any dimension. While these algorithms provide exact meas...

Let G = (V,E) be a graph. A k-neighborhood in G is a set of vertices consisting of all the vertices at distance at most k from some vertex of G. The hypergraph on vertex set V which edge set consists of all the k-neighborhoods of G for all k is the neighborhood hypergraph of G. Our goal in this paper is to investigate the complexity of a graph in terms of its neighborhoods. Precisely, we define...

We propose a new hierarchical clustering method for high dimension, low sample size (HDLSS) data. The method utilizes the fact that each individual data vector accounts for exactly one dimension in the subspace generated by HDLSS data. The linkage that is used for measuring the distance between clusters is the orthogonal distance between affine subspaces generated by each cluster. The ideal imp...

We propose a new method to construct a four parameter family of quantum-mechanical point interactions in one dimension, which is known as all possible self-adjoint extensions of the symmetric operator T = −∆⌈C 0 (R\{0}). It is achieved in the small distance limit of equally spaced three neighboring Dirac’s δ potentials. The strength for each δ is appropriately renormalized according to the dist...