On nodal sets and nodal domains on S^2 and R^2

Name: Work Video:
Duration: 3 min 30 s
Description: We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S^2. We also construct a solution of the equation Delta u=u in R^2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
Alexandre Erëmenko; Dmitry Jakobson; Nikolaï Nadirashvili
{'id': 'https://openalex.org/W4298227826', 'doi': 'https://doi.org/10.48550/arxiv.math/0611627', 'title': 'On nodal sets and nodal domains on S^2 and R^2', 'display_name': 'On nodal sets and nodal domains on S^2 and R^2', 'publication_year': 2006, 'publication_date': '2006-01-01', 'ids': {'openalex': 'https://openalex.org/W4298227826', 'doi': 'https://doi.org/10.48550/arxiv.math/0611627'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/math/0611627', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I205783295', 'host_organization_name': 'Cornell University', 'host_organization_lineage': ['https://openalex.org/I205783295'], 'host_organization_lineage_names': ['Cornell University'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': None, 'is_accepted': False, 'is_published': False}, 'type': 'preprint', 'type_crossref': 'journal-article', 'indexed_in': ['arxiv', 'datacite'], 'open_access': {'is_oa': True, 'oa_status': 'green', 'oa_url': 'https://arxiv.org/abs/math/0611627', 'any_repository_has_fulltext': True}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5012326758', 'display_name': 'Alexandre Erëmenko', 'orcid': 'https://orcid.org/0000-0002-6681-7211'}, 'institutions': [], 'countries': [], 'is_corresponding': False, 'raw_author_name': 'Eremenko, Alexandre', 'raw_affiliation_strings': [], 'affiliations': []}, {'author_position': 'middle', 'author': {'id': 'https://openalex.org/A5002765072', 'display_name': 'Dmitry Jakobson', 'orcid': None}, 'institutions': [], 'countries': [], 'is_corresponding': False, 'raw_author_name': 'Jakobson, Dmitry', 'raw_affiliation_strings': [], 'affiliations': []}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5112441955', 'display_name': 'Nikolaï Nadirashvili', 'orcid': None}, 'institutions': [], 'countries': [], 'is_corresponding': False, 'raw_author_name': 'Nadirashvili, Nikolai', 'raw_affiliation_strings': [], 'affiliations': []}], 'institution_assertions': [], 'countries_distinct_count': 0, 'institutions_distinct_count': 0, 'corresponding_author_ids': [], 'corresponding_institution_ids': [], 'apc_list': None, 'apc_paid': None, 'fwci': None, 'has_fulltext': False, 'cited_by_count': 4, 'citation_normalized_percentile': {'value': 0.841153, 'is_in_top_1_percent': False, 'is_in_top_10_percent': False}, 'cited_by_percentile_year': {'min': 74, 'max': 76}, 'biblio': {'volume': None, 'issue': None, 'first_page': None, 'last_page': None}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T12100', 'display_name': 'Advanced Mathematical Modeling in Engineering', 'score': 0.9985, 'subfield': {'id': 'https://openalex.org/subfields/1703', 'display_name': 'Computational Theory and Mathematics'}, 'field': {'id': 'https://openalex.org/fields/17', 'display_name': 'Computer Science'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, 'topics': [{'id': 'https://openalex.org/T12100', 'display_name': 'Advanced Mathematical Modeling in Engineering', 'score': 0.9985, 'subfield': {'id': 'https://openalex.org/subfields/1703', 'display_name': 'Computational Theory and Mathematics'}, 'field': {'id': 'https://openalex.org/fields/17', 'display_name': 'Computer Science'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T10194', 'display_name': 'Nonlinear Partial Differential Equations', 'score': 0.9957, 'subfield': {'id': 'https://openalex.org/subfields/2604', 'display_name': 'Applied Mathematics'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T11205', 'display_name': 'Numerical methods in inverse problems', 'score': 0.9944, 'subfield': {'id': 'https://openalex.org/subfields/2610', 'display_name': 'Mathematical Physics'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}], 'keywords': [{'id': 'https://openalex.org/keywords/modified-nodal-analysis', 'display_name': 'Modified nodal analysis', 'score': 0.8021024}, {'id': 'https://openalex.org/keywords/eigenfunction', 'display_name': 'Eigenfunction', 'score': 0.7687715}, {'id': 'https://openalex.org/keywords/nodal-analysis', 'display_name': 'Nodal analysis', 'score': 0.6626918}], 'concepts': [{'id': 'https://openalex.org/C83330619', 'wikidata': 'https://www.wikidata.org/wiki/Q21124872', 'display_name': 'NODAL', 'level': 2, 'score': 0.9577302}, {'id': 'https://openalex.org/C60643732', 'wikidata': 'https://www.wikidata.org/wiki/Q6889414', 'display_name': 'Modified nodal analysis', 'level': 3, 'score': 0.8021024}, {'id': 'https://openalex.org/C128803854', 'wikidata': 'https://www.wikidata.org/wiki/Q1307821', 'display_name': 'Eigenfunction', 'level': 3, 'score': 0.7687715}, {'id': 'https://openalex.org/C105473148', 'wikidata': 'https://www.wikidata.org/wiki/Q1203451', 'display_name': 'Nodal analysis', 'level': 2, 'score': 0.6626918}, {'id': 'https://openalex.org/C138395690', 'wikidata': 'https://www.wikidata.org/wiki/Q376733', 'display_name': 'Cardinal point', 'level': 2, 'score': 0.5927799}, {'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.5492393}, {'id': 'https://openalex.org/C3768446', 'wikidata': 'https://www.wikidata.org/wiki/Q877100', 'display_name': 'Spherical harmonics', 'level': 2, 'score': 0.49055117}, {'id': 'https://openalex.org/C134306372', 'wikidata': 'https://www.wikidata.org/wiki/Q7754', 'display_name': 'Mathematical analysis', 'level': 1, 'score': 0.40293598}, {'id': 'https://openalex.org/C184720557', 'wikidata': 'https://www.wikidata.org/wiki/Q7825049', 'display_name': 'Topology (electrical circuits)', 'level': 2, 'score': 0.38407755}, {'id': 'https://openalex.org/C121332964', 'wikidata': 'https://www.wikidata.org/wiki/Q413', 'display_name': 'Physics', 'level': 0, 'score': 0.28977823}, {'id': 'https://openalex.org/C114614502', 'wikidata': 'https://www.wikidata.org/wiki/Q76592', 'display_name': 'Combinatorics', 'level': 1, 'score': 0.23666814}, {'id': 'https://openalex.org/C158693339', 'wikidata': 'https://www.wikidata.org/wiki/Q190524', 'display_name': 'Eigenvalues and eigenvectors', 'level': 2, 'score': 0.23071337}, {'id': 'https://openalex.org/C120665830', 'wikidata': 'https://www.wikidata.org/wiki/Q14620', 'display_name': 'Optics', 'level': 1, 'score': 0.07372454}, {'id': 'https://openalex.org/C71924100', 'wikidata': 'https://www.wikidata.org/wiki/Q11190', 'display_name': 'Medicine', 'level': 0, 'score': 0.0}, {'id': 'https://openalex.org/C62520636', 'wikidata': 'https://www.wikidata.org/wiki/Q944', 'display_name': 'Quantum mechanics', 'level': 1, 'score': 0.0}, {'id': 'https://openalex.org/C105702510', 'wikidata': 'https://www.wikidata.org/wiki/Q514', 'display_name': 'Anatomy', 'level': 1, 'score': 0.0}], 'mesh': [], 'locations_count': 3, 'locations': [{'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/math/0611627', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I205783295', 'host_organization_name': 'Cornell University', 'host_organization_lineage': ['https://openalex.org/I205783295'], 'host_organization_lineage_names': ['Cornell University'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': None, 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'http://arxiv.org/abs/math/0611627', 'pdf_url': 'http://arxiv.org/pdf/math/0611627', 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I205783295', 'host_organization_name': 'Cornell University', 'host_organization_lineage': ['https://openalex.org/I205783295'], 'host_organization_lineage_names': ['Cornell University'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': False, 'landing_page_url': 'https://api.datacite.org/dois/10.48550/arxiv.math/0611627', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4393179698', 'display_name': 'DataCite API', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I4210145204', 'host_organization_name': 'DataCite', 'host_organization_lineage': ['https://openalex.org/I4210145204'], 'host_organization_lineage_names': ['DataCite'], 'type': 'metadata'}, 'license': None, 'license_id': None, 'version': None}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/math/0611627', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I205783295', 'host_organization_name': 'Cornell University', 'host_organization_lineage': ['https://openalex.org/I205783295'], 'host_organization_lineage_names': ['Cornell University'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': None, 'is_accepted': False, 'is_published': False}, 'sustainable_development_goals': [{'display_name': 'Affordable and clean energy', 'score': 0.84, 'id': 'https://metadata.un.org/sdg/7'}], 'grants': [], 'datasets': [], 'versions': [], 'referenced_works_count': 0, 'referenced_works': [], 'related_works': ['https://openalex.org/W638538785', 'https://openalex.org/W4298227826', 'https://openalex.org/W4293262929', 'https://openalex.org/W2803993633', 'https://openalex.org/W2656397757', 'https://openalex.org/W2377292611', 'https://openalex.org/W2359005373', 'https://openalex.org/W2015631438', 'https://openalex.org/W1896353054', 'https://openalex.org/W1540684046'], 'abstract_inverted_index': {'We': [0, 20], 'discuss': [1], 'possible': [2], 'topological': [3], 'configurations': [4], 'of': [5, 12, 25, 44], 'nodal': [6, 36], 'sets,': [7], 'in': [8, 30, 41], 'particular': [9], 'the': [10, 26, 42], 'number': [11], 'their': [13], 'components,': [14], 'for': [15], 'spherical': [16], 'harmonics': [17], 'on': [18], 'S^2.': [19], 'also': [21], 'construct': [22], 'a': [23], 'solution': [24], 'equation': [27, 39], 'Delta': [28], 'u=u': [29], 'R^2': [31], 'that': [32], 'has': [33], 'only': [34], 'two': [35], 'domains.': [37], 'This': [38], 'arises': [40], 'study': [43], 'high': [45], 'energy': [46], 'eigenfunctions.': [47]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W4298227826', 'counts_by_year': [{'year': 2017, 'cited_by_count': 1}, {'year': 2016, 'cited_by_count': 1}, {'year': 2015, 'cited_by_count': 1}], 'updated_date': '2025-01-04T19:32:45.168304', 'created_date': '2022-10-01'}
Publication Information

Basic Information

Access and Citation

AI Researcher Chatbot

Primary Location

Authors

Topics

Keywords

Related Works
