Polarization and Greedy Energy on the Sphere
We investigate the behavior of a greedy sequence on the sphere \(\mathbb{S}^d\) defined so that at each step the point that minimizes the Riesz \(s\)-energy is added to the existing set of points. We show that for \(0
Gespeichert in:
| Veröffentlicht in: | arXiv.org 2023-07 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | |
|---|---|
| container_issue | |
| container_start_page | |
| container_title | arXiv.org |
| container_volume | |
| creator | Bilyk, Dmitriy Mastrianni, Michelle Matzke, Ryan W Steinerberger, Stefan |
| description | We investigate the behavior of a greedy sequence on the sphere \(\mathbb{S}^d\) defined so that at each step the point that minimizes the Riesz \(s\)-energy is added to the existing set of points. We show that for \(0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2780574196</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2780574196</sourcerecordid><originalsourceid>FETCH-proquest_journals_27805741963</originalsourceid><addsrcrecordid>eNpjYuA0MjY21LUwMTLiYOAtLs4yMDAwMjM3MjU15mTQCcjPSSzKrEosyczPU0jMS1FwL0pNTalUcM1LLUqvVAAKlmSkKgQXZKQWpfIwsKYl5hSn8kJpbgZlN9cQZw_dgqL8wtLU4pL4rPzSojygVLyRuYWBqbmJoaWZMXGqAGX1MVE</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2780574196</pqid></control><display><type>article</type><title>Polarization and Greedy Energy on the Sphere</title><source>Free E- Journals</source><creator>Bilyk, Dmitriy ; Mastrianni, Michelle ; Matzke, Ryan W ; Steinerberger, Stefan</creator><creatorcontrib>Bilyk, Dmitriy ; Mastrianni, Michelle ; Matzke, Ryan W ; Steinerberger, Stefan</creatorcontrib><description>We investigate the behavior of a greedy sequence on the sphere \(\mathbb{S}^d\) defined so that at each step the point that minimizes the Riesz \(s\)-energy is added to the existing set of points. We show that for \(0<s<d\), the greedy sequence achieves optimal second-order behavior for the Riesz \(s\)-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz \(s\)-kernels is of order \(N^{s/d}\) in the same range \(0<s<d\). Furthermore, using the Stolarsky principle relating the \(L^2\)-discrepancy of a point set with the pairwise sum of distances (Riesz energy with \(s=-1\)), we also obtain a simple upper bound on the \(L^2\)-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Polarization ; Spherical caps ; Upper bounds</subject><ispartof>arXiv.org, 2023-07</ispartof><rights>2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>780,784</link.rule.ids></links><search><creatorcontrib>Bilyk, Dmitriy</creatorcontrib><creatorcontrib>Mastrianni, Michelle</creatorcontrib><creatorcontrib>Matzke, Ryan W</creatorcontrib><creatorcontrib>Steinerberger, Stefan</creatorcontrib><title>Polarization and Greedy Energy on the Sphere</title><title>arXiv.org</title><description>We investigate the behavior of a greedy sequence on the sphere \(\mathbb{S}^d\) defined so that at each step the point that minimizes the Riesz \(s\)-energy is added to the existing set of points. We show that for \(0<s<d\), the greedy sequence achieves optimal second-order behavior for the Riesz \(s\)-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz \(s\)-kernels is of order \(N^{s/d}\) in the same range \(0<s<d\). Furthermore, using the Stolarsky principle relating the \(L^2\)-discrepancy of a point set with the pairwise sum of distances (Riesz energy with \(s=-1\)), we also obtain a simple upper bound on the \(L^2\)-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.</description><subject>Polarization</subject><subject>Spherical caps</subject><subject>Upper bounds</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><recordid>eNpjYuA0MjY21LUwMTLiYOAtLs4yMDAwMjM3MjU15mTQCcjPSSzKrEosyczPU0jMS1FwL0pNTalUcM1LLUqvVAAKlmSkKgQXZKQWpfIwsKYl5hSn8kJpbgZlN9cQZw_dgqL8wtLU4pL4rPzSojygVLyRuYWBqbmJoaWZMXGqAGX1MVE</recordid><startdate>20230730</startdate><enddate>20230730</enddate><creator>Bilyk, Dmitriy</creator><creator>Mastrianni, Michelle</creator><creator>Matzke, Ryan W</creator><creator>Steinerberger, Stefan</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope></search><sort><creationdate>20230730</creationdate><title>Polarization and Greedy Energy on the Sphere</title><author>Bilyk, Dmitriy ; Mastrianni, Michelle ; Matzke, Ryan W ; Steinerberger, Stefan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_27805741963</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Polarization</topic><topic>Spherical caps</topic><topic>Upper bounds</topic><toplevel>online_resources</toplevel><creatorcontrib>Bilyk, Dmitriy</creatorcontrib><creatorcontrib>Mastrianni, Michelle</creatorcontrib><creatorcontrib>Matzke, Ryan W</creatorcontrib><creatorcontrib>Steinerberger, Stefan</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bilyk, Dmitriy</au><au>Mastrianni, Michelle</au><au>Matzke, Ryan W</au><au>Steinerberger, Stefan</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>Polarization and Greedy Energy on the Sphere</atitle><jtitle>arXiv.org</jtitle><date>2023-07-30</date><risdate>2023</risdate><eissn>2331-8422</eissn><abstract>We investigate the behavior of a greedy sequence on the sphere \(\mathbb{S}^d\) defined so that at each step the point that minimizes the Riesz \(s\)-energy is added to the existing set of points. We show that for \(0<s<d\), the greedy sequence achieves optimal second-order behavior for the Riesz \(s\)-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz \(s\)-kernels is of order \(N^{s/d}\) in the same range \(0<s<d\). Furthermore, using the Stolarsky principle relating the \(L^2\)-discrepancy of a point set with the pairwise sum of distances (Riesz energy with \(s=-1\)), we also obtain a simple upper bound on the \(L^2\)-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | EISSN: 2331-8422 |
| ispartof | arXiv.org, 2023-07 |
| issn | 2331-8422 |
| language | eng |
| recordid | cdi_proquest_journals_2780574196 |
| source | Free E- Journals |
| subjects | Polarization Spherical caps Upper bounds |
| title | Polarization and Greedy Energy on the Sphere |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-26T20%3A23%3A26IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest&rft_val_fmt=info:ofi/fmt:kev:mtx:book&rft.genre=document&rft.atitle=Polarization%20and%20Greedy%20Energy%20on%20the%20Sphere&rft.jtitle=arXiv.org&rft.au=Bilyk,%20Dmitriy&rft.date=2023-07-30&rft.eissn=2331-8422&rft_id=info:doi/&rft_dat=%3Cproquest%3E2780574196%3C/proquest%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2780574196&rft_id=info:pmid/&rfr_iscdi=true |