Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...

The power graphPG of a finite group G is the graph with the vertex set G, where two distinct vertices are adjacent if one is a power of the other. We first show that PG has a transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups of G (under usual inclusion) to characterize the structure ofPG. Finally, a closed formula fo...

We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the ...

Minkowski dimension is one of the most widely used dimensions. Its popularity is largely due to its relative ease of mathematical calculation. The definition goes back at least to the 1930’s and it has been variously termed Kolmogorov entropy, metric dimension, information dimension, ... etc. Let F be any non-empty bounded subset of R and let Nγ(F ) be the smallest number of sets of diameter at...

An embedding of one metric space (X, d) into another (Y, ρ) is an injective map f : X → Y . The central genre of problems in the area of metric embedding is finding such maps in which the distances between points do not change “too much”. Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, st...

Felsner, Li and Trotter showed that the dimension of adjacency poset an outerplanar graph is at most 5, gave example whose has 4. We improve their upper bound to 4, which then best possible.

We study a family of unbounded polyhedra arising in the study of uncapacitated lot-sizing problems with Wagner-Whitin costs. With n the number of periods, we completely characterize the bounded faces of maximal dimension, and derive an O(n) algorithm to express any point within the polyhedron as a convex combination of extreme points and extreme rays. We also study adjacency on the polyhedra, a...

In this paper, we study the problem of comparing similarity and dissimilarity between two distinct vehicular trajectories by proposing an adjacency-based metric. This approach has a broad application in building truthfulness vehicles evaluating paths hazardous materials transportation. Given sequences $A$ $B$ Point Interests (POIs) visited road/transportation network, is to delete some nodes fr...

Let , be a graph with vertex set and edge set . Let then is said to be a local metric basis of , if for any two adjacent vertices , ⁄ , there exists a such that , , . The minimum cardinality of local metric basis is called the local metric dimension (lmd) of graph G. In this paper we investigate the local metric basis and local metric dimension of Cyclic Split Graph .