A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. A graph is distance-regular if, for any two vertices u, v at each distance i, the number of neighbours of v at each possible distance from u (i.e. i−1, i or i...

The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Carathéodory distances and prove that this also fails f...

If the proper dissimilarity measures are provided, some sets of objects, e.g. curves or blobs, may be better described by using representation sets instead of features. The dissimilarity matrix of such a set is the base for further analysis. The question arises how from a given dissimilarity matrix can be judged whether the size of the training set is sufficient to describe the peculiarities of...

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Eu...

We study random bipartite planar maps defined by assigning non-negative weights to each face of a map. We proof that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marke...

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked...

In this paper we consider positively 1-homogeneous supremal functionals of the type F (u) := supΩ f(x,∇u(x)). We prove that the relaxation F̄ is a difference quotient, that is F̄ (u) = RF (u) := sup x,y∈Ω, x6=y u(x)− u(y) dF (x, y) for every u ∈ W (Ω), where dF is a geodesic distance associated to F . Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respe...

Similarity measure is a very important problem in fuzzy soft set theory. In this paper, seven similarity measures of fuzzy soft sets are introduced, which are based on the normalized Hamming distance, the normalized Euclidean distance, the generalized normalized distance, the Type-2 generalized normalized distance, the Type-2 normalized Euclidean distance, the Hausdorff distance and the Chebysh...

An `2 metric is a metric ρ such that √ ρ can be embedded isometrically into R endowed with Euclidean norm, and the minimal possible d is the dimension associated with ρ. A dimension reduction of an `2 metric ρ is an embedding of ρ into another ` 2 2 metric μ so that distances in μ are similar to those in ρ and moreover, the dimension associated with μ is small. Much of the motivation in investi...

Received (received date) Revised (revised date) Communicated by (Name) The article analyzes similarity of closed polygonal curves with respect to the Fréchet metric, which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications. An algorithm is described that determines whether the Fréchet distance between two closed polygonal curves with m and n ...