{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:BQRRMJ55R3CJDG7A6RPBPJKZGD","short_pith_number":"pith:BQRRMJ55","schema_version":"1.0","canonical_sha256":"0c231627bd8ec4919be0f45e17a55930d3ebf5df586594e688bd3724cc2730a7","source":{"kind":"arxiv","id":"0906.3382","version":2},"attestation_state":"computed","paper":{"title":"Scattering for the focusing ${\\dot H}^{1/2}$-critical Hartree equation with radial data","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Yanfang Gao","submitted_at":"2009-06-18T09:00:23Z","abstract_excerpt":"We investigate the focusing $\\dot H^{1/2}$-critical nonlinear Schr\\\"{o}dinger equation (NLS) of Hartree type $i\\partial_t u + \\Delta u = -(|\\cdot|^{-3} \\ast |u|^2)u$ with $\\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\\sup_{t}\\big\\||\\nabla|^{{1/2}}u\\big\\|_2 < \\frac{\\sqrt{6}}{3} \\big\\||\\nabla|^{{1/2}}Q\\big\\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\\ref{e14}). Then the solution is global and scatters."},"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":"0906.3382","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2009-06-18T09:00:23Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"71f25c7b1b6bcfba6edf073e31d06a8517e9ec20a8104075056c02e441e74d7e","abstract_canon_sha256":"504f765ad4d5cec1fa9ac90c68873bc0810f569e1977b9d17c725e38f296646e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:45:58.512570Z","signature_b64":"RzCXT/nv9003dKzlnpXA7bGvsCOCXpP8QkMOLgcR0SjHJPYdOpDyekBh5apthuun0DIQZheb4sYbl7IhRjaiBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c231627bd8ec4919be0f45e17a55930d3ebf5df586594e688bd3724cc2730a7","last_reissued_at":"2026-07-04T15:45:58.512208Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:45:58.512208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering for the focusing ${\\dot H}^{1/2}$-critical Hartree equation with radial data","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Yanfang Gao","submitted_at":"2009-06-18T09:00:23Z","abstract_excerpt":"We investigate the focusing $\\dot H^{1/2}$-critical nonlinear Schr\\\"{o}dinger equation (NLS) of Hartree type $i\\partial_t u + \\Delta u = -(|\\cdot|^{-3} \\ast |u|^2)u$ with $\\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\\sup_{t}\\big\\||\\nabla|^{{1/2}}u\\big\\|_2 < \\frac{\\sqrt{6}}{3} \\big\\||\\nabla|^{{1/2}}Q\\big\\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\\ref{e14}). Then the solution is global and scatters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.3382","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0906.3382/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":"0906.3382","created_at":"2026-07-04T15:45:58.512269+00:00"},{"alias_kind":"arxiv_version","alias_value":"0906.3382v2","created_at":"2026-07-04T15:45:58.512269+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.3382","created_at":"2026-07-04T15:45:58.512269+00:00"},{"alias_kind":"pith_short_12","alias_value":"BQRRMJ55R3CJ","created_at":"2026-07-04T15:45:58.512269+00:00"},{"alias_kind":"pith_short_16","alias_value":"BQRRMJ55R3CJDG7A","created_at":"2026-07-04T15:45:58.512269+00:00"},{"alias_kind":"pith_short_8","alias_value":"BQRRMJ55","created_at":"2026-07-04T15:45:58.512269+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/BQRRMJ55R3CJDG7A6RPBPJKZGD","json":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD.json","graph_json":"https://pith.science/api/pith-number/BQRRMJ55R3CJDG7A6RPBPJKZGD/graph.json","events_json":"https://pith.science/api/pith-number/BQRRMJ55R3CJDG7A6RPBPJKZGD/events.json","paper":"https://pith.science/paper/BQRRMJ55"},"agent_actions":{"view_html":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD","download_json":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD.json","view_paper":"https://pith.science/paper/BQRRMJ55","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0906.3382&json=true","fetch_graph":"https://pith.science/api/pith-number/BQRRMJ55R3CJDG7A6RPBPJKZGD/graph.json","fetch_events":"https://pith.science/api/pith-number/BQRRMJ55R3CJDG7A6RPBPJKZGD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD/action/storage_attestation","attest_author":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD/action/author_attestation","sign_citation":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD/action/citation_signature","submit_replication":"https://pith.science/pith/BQRRMJ55R3CJDG7A6RPBPJKZGD/action/replication_record"}},"created_at":"2026-07-04T15:45:58.512269+00:00","updated_at":"2026-07-04T15:45:58.512269+00:00"}