{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:VYZVBZVU2PFN5GX4B4QWGT5Y32","short_pith_number":"pith:VYZVBZVU","schema_version":"1.0","canonical_sha256":"ae3350e6b4d3cade9afc0f21634fb8de958cadfa3c21173b99de218660a64fd9","source":{"kind":"arxiv","id":"1605.03840","version":2},"attestation_state":"computed","paper":{"title":"Generating Point Configurations via Hypersingular Riesz Energy With an External Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CA","authors_text":"D. P. Hardin, E. B. Saff, O. V. Vlasiuk","submitted_at":"2016-05-10T18:32:17Z","abstract_excerpt":"For a compact $ d $-dimensional rectifiable subset of $ \\mathbb{R}^{p} $ we study asymptotic properties as $ N\\to\\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ s\\geq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure suppo"},"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":"1605.03840","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-05-10T18:32:17Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"103a1bf46a37416c4aaaa626f6813fbb03552601f759b27b5ff7072fd27ddb5f","abstract_canon_sha256":"d6ea4c75111637d49fdcb94ea5ac6bd3e338c125ff7bd900867cca3faac932af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:30.609937Z","signature_b64":"9KkFqv5oAw763KngD3jkmsF5VdrtIN0yqdh5AiDIAZcbjiDc0Lcjk+uNOhBWDBI1mQVsbVX9bJafoaqIvadRBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae3350e6b4d3cade9afc0f21634fb8de958cadfa3c21173b99de218660a64fd9","last_reissued_at":"2026-05-18T01:02:30.609329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:30.609329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generating Point Configurations via Hypersingular Riesz Energy With an External Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CA","authors_text":"D. P. Hardin, E. B. Saff, O. V. Vlasiuk","submitted_at":"2016-05-10T18:32:17Z","abstract_excerpt":"For a compact $ d $-dimensional rectifiable subset of $ \\mathbb{R}^{p} $ we study asymptotic properties as $ N\\to\\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ s\\geq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure suppo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.03840","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1605.03840","created_at":"2026-05-18T01:02:30.609412+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.03840v2","created_at":"2026-05-18T01:02:30.609412+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.03840","created_at":"2026-05-18T01:02:30.609412+00:00"},{"alias_kind":"pith_short_12","alias_value":"VYZVBZVU2PFN","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VYZVBZVU2PFN5GX4","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VYZVBZVU","created_at":"2026-05-18T12:30:48.956258+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32","json":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32.json","graph_json":"https://pith.science/api/pith-number/VYZVBZVU2PFN5GX4B4QWGT5Y32/graph.json","events_json":"https://pith.science/api/pith-number/VYZVBZVU2PFN5GX4B4QWGT5Y32/events.json","paper":"https://pith.science/paper/VYZVBZVU"},"agent_actions":{"view_html":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32","download_json":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32.json","view_paper":"https://pith.science/paper/VYZVBZVU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.03840&json=true","fetch_graph":"https://pith.science/api/pith-number/VYZVBZVU2PFN5GX4B4QWGT5Y32/graph.json","fetch_events":"https://pith.science/api/pith-number/VYZVBZVU2PFN5GX4B4QWGT5Y32/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32/action/storage_attestation","attest_author":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32/action/author_attestation","sign_citation":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32/action/citation_signature","submit_replication":"https://pith.science/pith/VYZVBZVU2PFN5GX4B4QWGT5Y32/action/replication_record"}},"created_at":"2026-05-18T01:02:30.609412+00:00","updated_at":"2026-05-18T01:02:30.609412+00:00"}