New K-means Clustering Method Using Minkowski’s Distance as its Metric

Cluster analysis is an unsupervised learning method that classifies data points, usually multidimensional into groups (called clusters) such that members of one cluster are more similar (in some sense) to each other than those in other clusters. In this paper, we propose a new k-means clustering method that uses Minkowski’s distance as its metric in a normed vector space which is the generalization of both the Euclidean distance and the Manhattan distance. The k-means clustering methods discussed in this paper are Forgy’s method, Lloyd’s method, MacQueen’s method, Hartigan and Wong’s method, Likas’ method and Faber’s method which uses the usual Euclidean distance. It was observed that the new k-means clustering method performed favourably in comparison with the existing methods in terms of minimization of the total intra-cluster variance using simulated data and real-life data sets.

On the stability of radical septic functional equations

This paper deals with the approximate solution of the following functional equation fx7+y77=f(x)+f(y), where f is a mapping from R into a normed vector space. We show stability results of this equation in quasi-β-Banach spaces and (β,p)-Banach spaces. We also prove the nonstability of the previous functional equation in a relevant case.

Conical geodesic bicombings on subsets of normed vector spaces

We establish existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a convex geodesic bicombing that is not consistent. Furthermore, we show that under a mild geometric assumption on the norm a conical geodesic bicombing on an open subset of a normed vector space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan–Hadamard type result that if a closed convex subset of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by the linear segments.

Order-Lipschitz mappings restricted with linear bounded mappings in normed vector spaces without normalities of involving cones via methods of upper and lower solutions

In this paper, without assuming the normalities of cones, we prove some new fixed point theorems of order-Lipschitz mappings restricted with linear bounded mappings in normed vector space in the framework of w-convergence via the method of upper and lower solutions. It is worth mentioning that the unique existence result of fixed points in this paper, presents a characterization of Picard-completeness of order-Lipschitz mappings.

On differentiability of mappings with finite dilation

We study mappings f : (a,b) → Y with finite dilation having Lebesgue integrable majorant, where Y is a real normed vector space. We construct Lipschitz mapping f : (a,b) → Y, dim Y = ∞ , which is nowhere differentiable but its graph has everywhere trivial contingent. We show that if the contingent of the graph of a mapping with finite dilation is a nontrivial space, then f is almost everywhere differentiable.

Uniform Domains and Uniform Domain Decomposition Property in Real Normed Vector Spaces

Let $E$ be a real normed vector space with $\dim(E)\geq 2$, $D$ a proper subdomain of $E$. In this paper we characterize uniform domains in $E$ in terms of the uniform domain decomposition property. In addition, we discuss the relation between quasiballs and domains with the quasiball decomposition property in $\mathsf{R}^n$.