{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:J7NN3ZEDHHMA3ZCYT7GBCBJPH7","short_pith_number":"pith:J7NN3ZED","schema_version":"1.0","canonical_sha256":"4fdadde48339d80de4589fcc11052f3ffcc79995344261669567f527546533d3","source":{"kind":"arxiv","id":"1310.2765","version":1},"attestation_state":"computed","paper":{"title":"Riesz external field problems on the hypersphere and optimal point separation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Edward B. Saff, Johann S. Brauchart, Peter D. Dragnev","submitted_at":"2013-10-10T10:49:00Z","abstract_excerpt":"We consider the minimal energy problem on the unit sphere $\\mathbb{S}^d$ in the Euclidean space $\\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \\leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\\mathbb{S}^d$. When $Q$ is the Riesz $s$"},"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":"1310.2765","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-10-10T10:49:00Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"cc7af0dcd1d61f8c316a671c67a80c4651b888c79ad59635c65549489bcb9b83","abstract_canon_sha256":"0efa2b674b4d00d77d0ed73a0d0a6a39e75ac198b7acdbe1d99648e187b0a0d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:51.020940Z","signature_b64":"91UO8oUuuPSeNw1iWAEFJAmz4Zg/2qlLsnGCyPKmzM5ksjMQXd3DwYAkC5bAzxjXm1j5+80El/m1h6t+2PbQDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fdadde48339d80de4589fcc11052f3ffcc79995344261669567f527546533d3","last_reissued_at":"2026-05-18T01:23:51.020361Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:51.020361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Riesz external field problems on the hypersphere and optimal point separation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Edward B. Saff, Johann S. Brauchart, Peter D. Dragnev","submitted_at":"2013-10-10T10:49:00Z","abstract_excerpt":"We consider the minimal energy problem on the unit sphere $\\mathbb{S}^d$ in the Euclidean space $\\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \\leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\\mathbb{S}^d$. When $Q$ is the Riesz $s$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2765","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":""},"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":"1310.2765","created_at":"2026-05-18T01:23:51.020460+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.2765v1","created_at":"2026-05-18T01:23:51.020460+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2765","created_at":"2026-05-18T01:23:51.020460+00:00"},{"alias_kind":"pith_short_12","alias_value":"J7NN3ZEDHHMA","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"J7NN3ZEDHHMA3ZCY","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"J7NN3ZED","created_at":"2026-05-18T12:27:49.015174+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/J7NN3ZEDHHMA3ZCYT7GBCBJPH7","json":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7.json","graph_json":"https://pith.science/api/pith-number/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/graph.json","events_json":"https://pith.science/api/pith-number/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/events.json","paper":"https://pith.science/paper/J7NN3ZED"},"agent_actions":{"view_html":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7","download_json":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7.json","view_paper":"https://pith.science/paper/J7NN3ZED","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.2765&json=true","fetch_graph":"https://pith.science/api/pith-number/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/graph.json","fetch_events":"https://pith.science/api/pith-number/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/action/storage_attestation","attest_author":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/action/author_attestation","sign_citation":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/action/citation_signature","submit_replication":"https://pith.science/pith/J7NN3ZEDHHMA3ZCYT7GBCBJPH7/action/replication_record"}},"created_at":"2026-05-18T01:23:51.020460+00:00","updated_at":"2026-05-18T01:23:51.020460+00:00"}