scholarly journals Local topology in deformation spaces of hyperbolic 3-manifolds II

2019 ◽  
Vol 13 (3) ◽  
pp. 767-793
Author(s):  
Jeffrey Brock ◽  
Kenneth Bromberg ◽  
Richard Canary ◽  
Cyril Lecuire ◽  
Yair Minsky
Keyword(s):  
Deformation Spaces ◽  
Local Topology

scholarly journals Local topology in deformation spaces of hyperbolic 3–manifolds

2011 ◽  
Vol 15 (2) ◽  
pp. 1169-1224 ◽  
Author(s):  
Jeffrey F Brock ◽  
Kenneth W Bromberg ◽  
Richard D Canary ◽  
Yair N Minsky
Keyword(s):  
Deformation Spaces ◽  
Local Topology

scholarly journals Distortion Effects in Equal Area Unit Maps

2021 ◽  
Author(s):  
Jochen Schiewe
Keyword(s):  
Hot Spots ◽  
Extreme Values ◽  
Generation Process ◽  
Equal Area ◽  
Topological Relationships ◽  
Area Unit ◽  
Task Oriented ◽  
Local Topology ◽  
Selection Of ◽  
Local Extreme

AbstractMaps that correctly represent the geographic size and shape of regions, taking into account scaling and generalization, have the disadvantage that small regions can easily be overlooked or not seen at all. Hence, for some map use tasks where small regions are of importance, alternative map types are needed. One option is the so-called equal area unit maps (EAUMs), where every enumeration unit has the same area size, possibly also the same basic shape such as squares or hexagons. The geometrical distortion of EAUMs, however, leads to a more difficult search for regions as well as a falsification of topological relationships and spatial patterns. To describe these distortions, a set of analytical measures is proposed. But it turns out that the expressiveness of these measures is rather limited. To better understand and to model the influence of distortions, two user studies were conducted. The study on the search in EAUMs (also with the aim of reconstruct the search strategies of the users) revealed how important it is to consider the local topology (e.g. corner or border positions of regions) during the generation process. With regard to pattern identification, it could be shown that EAUMs significantly increase the detection rate of local extreme values. On the other hand, global lateral gradients or geostatistical hot spots often get blurred or even lost. As a consequence, a task-oriented selection of map types and further developments are recommended.

scholarly journals A dichotomy for bounded displacement equivalence of Delone sets

2021 ◽  
pp. 1-18
Author(s):  
YOTAM SMILANSKY ◽  
YAAR SOLOMON
Keyword(s):  
Compact Space ◽  
Equivalence Classes ◽  
Natural Topology ◽  
Fell Topology ◽  
Local Complexity ◽  
Substitution Tilings ◽  
Local Topology ◽  
Delone Sets

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

A Mapping Theorem in “Local” Topology

1953 ◽  
Vol s1-28 (3) ◽  
pp. 269-278
Author(s):  
H. B. Griffiths
Keyword(s):  
Local Topology
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