{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:OSGGBEMJYSFK7UTUCHCC32QLWY","short_pith_number":"pith:OSGGBEMJ","schema_version":"1.0","canonical_sha256":"748c609189c48aafd27411c42dea0bb6150f42d6b1e63d34d869679a07457f79","source":{"kind":"arxiv","id":"1908.10354","version":1},"attestation_state":"computed","paper":{"title":"Energy on spheres and discreteness of minimizing measures","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MG","math.MP"],"primary_cat":"math.CA","authors_text":"Alexey Glazyrin, Dmitriy Bilyk, Josiah Park, Oleksandr Vlasiuk, Ryan Matzke","submitted_at":"2019-08-27T17:51:03Z","abstract_excerpt":"In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.10354","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2019-08-27T17:51:03Z","cross_cats_sorted":["math-ph","math.MG","math.MP"],"title_canon_sha256":"afa651e1febcb3cefb15543bf802771669e102db686d7ddcd74b761c87738933","abstract_canon_sha256":"e1617fee3f11ca424199d520d6f33768079d39c00aa05d93509f89896146fdcb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:00:11.325792Z","signature_b64":"tZNnZX4QhHj0i1ji+y1UN9B0UY2rsmQwONliHzqiCqX/dOPE3NRvOLfQ+7rTrxJeGPBn5V+xXn/J4i6Bi18+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"748c609189c48aafd27411c42dea0bb6150f42d6b1e63d34d869679a07457f79","last_reissued_at":"2026-07-05T00:00:11.325331Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:00:11.325331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Energy on spheres and discreteness of minimizing measures","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MG","math.MP"],"primary_cat":"math.CA","authors_text":"Alexey Glazyrin, Dmitriy Bilyk, Josiah Park, Oleksandr Vlasiuk, Ryan Matzke","submitted_at":"2019-08-27T17:51:03Z","abstract_excerpt":"In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.10354","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.10354/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.10354","created_at":"2026-07-05T00:00:11.325390+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.10354v1","created_at":"2026-07-05T00:00:11.325390+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.10354","created_at":"2026-07-05T00:00:11.325390+00:00"},{"alias_kind":"pith_short_12","alias_value":"OSGGBEMJYSFK","created_at":"2026-07-05T00:00:11.325390+00:00"},{"alias_kind":"pith_short_16","alias_value":"OSGGBEMJYSFK7UTU","created_at":"2026-07-05T00:00:11.325390+00:00"},{"alias_kind":"pith_short_8","alias_value":"OSGGBEMJ","created_at":"2026-07-05T00:00:11.325390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.00885","citing_title":"Optimal measures for p-frame energies on spheres","ref_index":11,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY","json":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY.json","graph_json":"https://pith.science/api/pith-number/OSGGBEMJYSFK7UTUCHCC32QLWY/graph.json","events_json":"https://pith.science/api/pith-number/OSGGBEMJYSFK7UTUCHCC32QLWY/events.json","paper":"https://pith.science/paper/OSGGBEMJ"},"agent_actions":{"view_html":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY","download_json":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY.json","view_paper":"https://pith.science/paper/OSGGBEMJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.10354&json=true","fetch_graph":"https://pith.science/api/pith-number/OSGGBEMJYSFK7UTUCHCC32QLWY/graph.json","fetch_events":"https://pith.science/api/pith-number/OSGGBEMJYSFK7UTUCHCC32QLWY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY/action/storage_attestation","attest_author":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY/action/author_attestation","sign_citation":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY/action/citation_signature","submit_replication":"https://pith.science/pith/OSGGBEMJYSFK7UTUCHCC32QLWY/action/replication_record"}},"created_at":"2026-07-05T00:00:11.325390+00:00","updated_at":"2026-07-05T00:00:11.325390+00:00"}