The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects

This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-05
Main Author: Kajino, Naotaka
Format: Article
Language:English
Subjects:
Asymptotic properties
Carpets
Eigenvalues
Fractals
Riemann manifold
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites
container_end_page
container_issue
container_start_page
container_title arXiv.org
container_volume
creator Kajino, Naotaka
description This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on the Riemann sphere \(\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\)), including the classical Apollonian gasket and some round Sierpi\'{n}ski carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.
format article
fullrecord <record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2343361590</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2343361590</sourcerecordid><originalsourceid>FETCH-proquest_journals_23433615903</originalsourceid><addsrcrecordid>eNqNjb0KwjAUhYMgKOo7XHBwKtTE-reJKA6OgmO5xButpEnNTYX69GbwAZzOGb7vnJ4YSqXm2Xoh5UBMmJ95nsvlShaFGorP5UFwxsairtCBd8C-JmCyJtPeGR9qtGAC6oiWAd0NrtTZWarc1U30sdIMCYMqMlB1J_dG2xJvYQfchjd14A3E9IIObZfwZDakI49F36RNmvxyJKbHw2V_yprgX2khlk_fhiRxKdVCqeW82OTqP-oL_gtOag</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2343361590</pqid></control><display><type>article</type><title>The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects</title><source>Access via ProQuest (Open Access)</source><creator>Kajino, Naotaka</creator><creatorcontrib>Kajino, Naotaka</creatorcontrib><description>This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on the Riemann sphere \(\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\)), including the classical Apollonian gasket and some round Sierpi\'{n}ski carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Asymptotic properties ; Carpets ; Eigenvalues ; Fractals ; Riemann manifold</subject><ispartof>arXiv.org, 2021-05</ispartof><rights>2021. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (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>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.proquest.com/docview/2343361590?pq-origsite=primo$$EHTML$$P50$$Gproquest$$Hfree_for_read</linktohtml><link.rule.ids>780,784,25753,37012,44590</link.rule.ids></links><search><creatorcontrib>Kajino, Naotaka</creatorcontrib><title>The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects</title><title>arXiv.org</title><description>This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on the Riemann sphere \(\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\)), including the classical Apollonian gasket and some round Sierpi\'{n}ski carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.</description><subject>Asymptotic properties</subject><subject>Carpets</subject><subject>Eigenvalues</subject><subject>Fractals</subject><subject>Riemann manifold</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>PIMPY</sourceid><recordid>eNqNjb0KwjAUhYMgKOo7XHBwKtTE-reJKA6OgmO5xButpEnNTYX69GbwAZzOGb7vnJ4YSqXm2Xoh5UBMmJ95nsvlShaFGorP5UFwxsairtCBd8C-JmCyJtPeGR9qtGAC6oiWAd0NrtTZWarc1U30sdIMCYMqMlB1J_dG2xJvYQfchjd14A3E9IIObZfwZDakI49F36RNmvxyJKbHw2V_yprgX2khlk_fhiRxKdVCqeW82OTqP-oL_gtOag</recordid><startdate>20210506</startdate><enddate>20210506</enddate><creator>Kajino, Naotaka</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>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20210506</creationdate><title>The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects</title><author>Kajino, Naotaka</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_23433615903</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Asymptotic properties</topic><topic>Carpets</topic><topic>Eigenvalues</topic><topic>Fractals</topic><topic>Riemann manifold</topic><toplevel>online_resources</toplevel><creatorcontrib>Kajino, Naotaka</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</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 (Proquest) (PQ_SDU_P3)</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Access via ProQuest (Open Access)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kajino, Naotaka</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects</atitle><jtitle>arXiv.org</jtitle><date>2021-05-06</date><risdate>2021</risdate><eissn>2331-8422</eissn><abstract>This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on the Riemann sphere \(\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\)), including the classical Apollonian gasket and some round Sierpi\'{n}ski carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.</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, 2021-05
issn 2331-8422
language eng
recordid cdi_proquest_journals_2343361590
source Access via ProQuest (Open Access)
subjects Asymptotic properties
Carpets
Eigenvalues
Fractals
Riemann manifold
title The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-03T15%3A27%3A48IST&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=The%20Laplacian%20on%20some%20self-conformal%20fractals%20and%20Weyl's%20asymptotics%20for%20its%20eigenvalues:%20A%20survey%20of%20the%20analytic%20aspects&rft.jtitle=arXiv.org&rft.au=Kajino,%20Naotaka&rft.date=2021-05-06&rft.eissn=2331-8422&rft_id=info:doi/&rft_dat=%3Cproquest%3E2343361590%3C/proquest%3E%3Cgrp_id%3Ecdi_FETCH-proquest_journals_23433615903%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2343361590&rft_id=info:pmid/&rfr_iscdi=true