On the complexity of optimal homotopies

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves $\gamma_1$ and $\gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $\gamma_1$ and $\gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting

Flexibility of industrial product service systems: An assessment based on concept modelling

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Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.Comment: 16 pages, 6 figure

Socio‐economic impact classification of alien taxa (SEICAT)

1 Many alien taxa are known to cause socio‐economic impacts by affecting the different constituents of human well‐being (security; material and non‐material assets; health; social, spiritual and cultural relations; freedom of choice and action). Attempts to quantify socio‐economic impacts in monetary terms are unlikely to provide a useful basis for evaluating and comparing impacts of alien taxa because they are notoriously difficult to measure and important aspects of human well‐being are ignored. 2 Here, we propose a novel standardised method for classifying alien taxa in terms of the magnitude of their impacts on human well‐being, based on the capability approach from welfare economics. The core characteristic of this approach is that it uses changes in peoples' activities as a common metric for evaluating impacts on well‐being. 2 Impacts are assigned to one of five levels, from Minimal Concern to Massive, according to semi‐quantitative scenarios that describe the severity of the impacts. Taxa are then classified according to the highest level of deleterious impact that they have been recorded to cause on any constituent of human well‐being. The scheme also includes categories for taxa that are not evaluated, have no alien population, or are data deficient, and a method for assigning uncertainty to all the classifications. To demonstrate the utility of the system, we classified impacts of amphibians globally. These showed a variety of impacts on human well‐being, with the cane toad (Rhinella marina) scoring Major impacts. For most species, however, no studies reporting impacts on human well‐being were found, i.e. these species were data deficient. 2 The classification provides a consistent procedure for translating the broad range of measures and types of impact into ranked levels of socio‐economic impact, assigns alien taxa on the basis of the best available evidence of their documented deleterious impacts, and is applicable across taxa and at a range of spatial scales. The system was designed to align closely with the Environmental Impact Classification for Alien Taxa (EICAT) and the Red List, both of which have been adopted by the International Union of Nature Conservation (IUCN), and could therefore be readily integrated into international practices and policies

Ariel - Volume 6 Number 2

Editors Mark Dembert J.D. Kanofsky Frank Chervenak John Lammie Curt Cummings Entertainment Robert Breckenridge Joe Conti Gary Kaskey Photographer Larry Glazerman Overseas Editor Mike Sinason Humorist Jim McCann Staff Ken Jaffe Bob Skarloff Halley Faust Jim Burk

Advisor: Y.Usenko TU/e HG 5.71 Customer: C. Plevier Dutch Space

Ivo van der Linden 054763

SkyMapper Southern Survey: First Data Release (DR1)

We present the first data release (DR1) of the SkyMapper Southern Survey, a hemispheric survey carried out with the SkyMapper Telescope at Siding Spring Observatory in Australia. Here, we present the survey strategy, data processing, catalogue construction and database schema. The DR1 dataset includes over 66,000 images from the Shallow Survey component, covering an area of 17,200 deg$^2$ in all six SkyMapper passbands $uvgriz$, while the full area covered by any passband exceeds 20,000 deg$^2$. The catalogues contain over 285 million unique astrophysical objects, complete to roughly 18 mag in all bands. We compare our $griz$ point-source photometry with PanSTARRS1 DR1 and note an RMS scatter of 2%. The internal reproducibility of SkyMapper photometry is on the order of 1%. Astrometric precision is better than 0.2 arcsec based on comparison with Gaia DR1. We describe the end-user database, through which data are presented to the world community, and provide some illustrative science queries.Comment: 31 pages, 19 figures, 10 tables, PASA, accepte

Approximation properties for noncommutative Lp-spaces associated with lattices in Lie groups

In 2010, Lafforgue and de la Salle gave examples of noncommutative Lp-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on Sp and proved that for p 4 the groups SL(n,Z), with n \geq 3, do not have it. Since for 1 < p < \infty the property of completely bounded approximation by Schur multipliers on Sp is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also the first examples of exact groups without the AP. Recently, Haagerup and the author proved that also the group Sp(2,R) does not have the AP, without using the property of completely bounded approximation by Schur multipliers on Sp. In this paper, we prove that Sp(2,R) does not have the property of completely bounded approximation by Schur multipliers on Sp for p 12. It follows that a large class of noncommutative Lp-spaces does not have the OAP or CBAP.Comment: Version 2, 20 pages. Minor corrections, builds on results from arXiv:1201.125