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Bajnok, Béla; Damelin, Steven B.; Mullen, Gary L.

doi:10.1007/s00607-001-1434-9

Cited by 9 publications

(34 citation statements)

References 0 publications

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“…-The maps Φ in (*) at the same time agree with certain Euclidean motions "globally away from" the sets E and sometimes also with Euclidean motions locally at each point in E. Sections (2,(5)(6)(7)(8)(9)(10)(11)(12)(14)(15).…”

Section: Mathematical Worldsmentioning

confidence: 92%

“…That is the maps Φ agree with the maps φ on the sets E. Here, c, c are small enough and depend on D and possibly in addition other quantities. c depends on c. Sections (2,(5)(6)(7)(8)(9)(10)(11)(12)(14)(15).…”

Section: Mathematical Worldsmentioning

confidence: 99%

“…We urge the reader to look at our central problems, Problems (1)(2)(3)(4)(5)(6)(7)(8) in Sections (2.3.2) and (2.3.3) before reading (a-d) below. Let us suppose now the following regarding the above:…”

Section: 13mentioning

confidence: 99%

“…We guess that Problems (1)(2)(3)(4)(5)(6)8) in Section (2.3.2) might be interesting in order to explore (ad) for our coregistration or shape matching problem above. See also Problem (7) in Section (2.3.3).…”

Section: 13mentioning

confidence: 99%

“…This expository paper details work of the author et al taken from the papers [7,12,24,28,29,30,31,32,33,34,35,36,37,38,45,47,50,51,52,53,79,100,43,44,133,39,40,41,42] and which lies in the intersection of several mathematics subjects: algebraic-differential geometry, analysis on manifolds, complex-harmonic-analysis, data science, partial differential equations, optimization and probability. The paper describes fascinating connections between several mathematical problems covering these topics which on the surface do not seem to be strongly connected but are in many deep ways.…”

mentioning

confidence: 99%

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On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

Damelin¹

2021

Preprint

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This paper is an exposition of work of the author et al. detailing fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic-differential geometry, analysis on manifolds, complex-harmonic analysis, data science, partial differential equations, optimization and probability. A significant portion of the work is based on joint research with Charles Fefferman in the papers [39,40,41,42]. The topics of this work include (a) The space of maps of bounded mean oscillation (BMO) in R D , D ≥ 2. (b) The labeled and unlabeled near alignment and Procrustes problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (c) The Whitney near isometry extension problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (d) Partitions and clustering of compact sets and point sets with certain geometries in R D , D ≥ 2 and analysis on certain manifolds in R D , D ≥ 2. Many open problems for future research are given.

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