Notions of denseness

On Ultralimits of Sparse Graph Classes

The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. Our goal is to demonstrate that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, ...

On the denseness of the invertible group in uniform Frechet algebras

Denseness for norm attaining operator-valued functions

In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss’ famous result on the denseness of norm attaining operators. Specifically, we show given any A ∈ L(H) there is a sequence of rank-1 operators Kn such that A+Kn is norm attaining for each n and Kn converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operato...

Classification of Breast Mass Abnormalities using Denseness and Architectural Distortion

This paper presents an electronic second opinion system for the classification of mass abnormalities in mammograms into benign and malignant categories. This system is designed to help radiologists to reduce the number of benign breast cancer biopsies. Once a mass abnormality is detected and marked on a mammogram by a radiologist, two textural features, named denseness and architectural distort...

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized ...