Connectedness of complete metric groups

Unsupervised Clustering using Metric Space Connectedness

In metric space theory connectedness can be described in terms of a mapping of sets onto the real axis. This function is essentially a labelling function which clustering methods approximate. With the use of metric space theory a proposition and a conjecture are given which are implemented to perform unsupervised clustering which lacks many of the limitations of other, model-based, methods.

Connectedness at Infinity of Complete Kähler Manifolds

One of the main purposes of this paper is to prove that on a complete Kähler manifold of dimension m, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum λ1(M) ≥ m2, then it must either be connected at infinity or isometric to R×N with a specialized metric, with N being compact. Generalizations to complete Kähler manifolds satisfying a weighted Poincaré...

Connectedness of Degree Graphs of Nonsolvable Groups

Complete Metric Spaces

Complete Generalized Metric Spaces

The well-known Banach’s fixed point theorem asserts that ifD X, f is contractive and X, d is complete, then f has a unique fixed point inX. It is well known that the Banach contraction principle 1 is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong 2 introduced the notion ofΦ-contraction. A mapping f : X → X on a metric space is called Φ-contraction if there exists...