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1

Correcting theorem 1 from "When lines go bad in hyperspace"

100%

Schlicker S. , Bay C. , Lembcke A.

Demonstratio Mathematica

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2009

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tom Vol. 42, nr 2

237-240

EN

This is in regards to the paper "When Lines go bad in hyperspace" by Christopher Bay, Amber Lembcke, and Steven Schlicker which appears in Demonstratio Mathematica, No. 3, Volume 38 (2005), p. 689-701. It has recently been brought to our attention that Theorem 1 from this paper is not correct. Please note that the main conclusions of the paper do not depend at all on this theorem. However, as the authors we feel it is our responsibility to bring this erroneous theorem to your attention. As stated in the paper, Theorem 1 intends to demonstrate that there can be infinitely many elements at a given location between two sets A and B.

2

Tangent Lines and Lipschitz Differentiability Spaces

100%

Cavalletti F. , Rajala T.

Analysis and Geometry in Metric Spaces

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2016

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tom 4

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nr 1

EN

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces.We show that any tangent space of a Lipschitz differentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space.

3

On the Hausdorff Dimension of CAT(κ) Surfaces

100%

Constantine D. , Lafont J.-F.

Analysis and Geometry in Metric Spaces

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2016

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tom 4

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nr 1

EN

We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.

4

Low Distortion Embedding of the Hamming Space into a Sphere with Quadrance Metric and k-means Clustering of Nominal-continuous Data

88%

Denisiuk A. , Grabowski M.

Fundamenta Informaticae

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2017

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tom Vol. 153, nr 3

221--233

EN

In this article we propose a new clustering algorithm for combinations of continuous and nominal data. The proposed algorithm is based on embedding of the nominal data into the unit sphere with a quadrance metrics, and adaptation of the general k-means clustering algorithm for the embedding data. It is also shown that the distortion of new embedding with respect to the Hamming metrics is less than that of other considered possibilities. A series of numerical experiments on real and synthetic datasets show that the proposed algorithm provide a comparable alternative to other clustering algorithms for combinations of continuous and nominal data.

5

Mereological foundations of point-free geometry via multi-valued logic

75%

Coppola C. , Gerla G.

Logic and Logical Philosophy

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2015

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tom 24

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nr 4 Mereology and Beyond

535-553

EN

We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science.

6

Isometric Embeddings of Pro-Euclidean Spaces

75%

Minemyer B.

Analysis and Geometry in Metric Spaces

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2015

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tom 3

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nr 1

EN

In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].