Normality in terms of distances and contractions

The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of $\gamma$-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of $\gamma$-separated sets by some Urysohn contractive map, (2) to Kat\v{e}tov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space. The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Kat\v{e}tov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered

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Benchmarking of localization solutions : guidelines for the selection of evaluation points

Indoor localization solutions are key enablers for next-generation indoor navigation and track and tracing solutions. As a result, an increasing number of different localization algorithms have been proposed and evaluated in scientific literature. However, many of these publications do not accurately substantiate the used evaluation methods. In particular, many authors utilize a different number of evaluation points, but they do not (i) analyze if the number of used evaluation points is sufficient to accurately evaluate the performance of their solutions and (ii) report on the uncertainty of the published results. To remedy this, this paper evaluates the influence of the selection of evaluation points. Based on statistical parameters such as the standard error of the mean value, an estimator is defined that can be used to quantitatively analyze the impact of the number of used evaluation points on the confidence interval of the mean value of the obtained results. This estimator is used to estimate the uncertainty of the presented accuracy results, and can be used to identify if more evaluations are required. To validate the proposed estimator, two different localization algorithms are evaluated in different testbeds and using different types of technology, showing that the number of required evaluation points does indeed vary significantly depending on the evaluated solution. (C) 2017 Elsevier B.V. All rights reserved