{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:43FF232LSAF6CZNDNSGCPK3GSO","short_pith_number":"pith:43FF232L","schema_version":"1.0","canonical_sha256":"e6ca5d6f4b900be165a36c8c27ab669383d1a8105107512540510a9364ecb3cd","source":{"kind":"arxiv","id":"1203.6375","version":2},"attestation_state":"computed","paper":{"title":"Remark on the periodic mass critical nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nobu Kishimoto","submitted_at":"2012-03-28T20:43:04Z","abstract_excerpt":"We consider the mass critical NLS on $\\mathbb{T}$ and $\\mathbb{T}^2$. In the $\\mathbb{R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $L^6$ Strichartz on $\\mathbb{T}$ and the $L^4$ Strichartz on $\\mathbb{T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $\\mathbb{T}$ and $\\mat"},"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":"1203.6375","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-03-28T20:43:04Z","cross_cats_sorted":[],"title_canon_sha256":"c6a13b1105faf78d670448902d7bc3315cb047ce57e185c6bc927bfa50fa65b8","abstract_canon_sha256":"b2b486f92e17a6ee47ae9629bd97a154cd11596e94940122e1d24c55eaaba459"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:13.553274Z","signature_b64":"ghbUFf27pW5MJLwcLlzf2tBugNEf228TlbOwHdvVHzsW8VvIMQjxMREVknDpytkdh7Jwylociq8yLY3AoOgDCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6ca5d6f4b900be165a36c8c27ab669383d1a8105107512540510a9364ecb3cd","last_reissued_at":"2026-05-18T03:44:13.552900Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:13.552900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Remark on the periodic mass critical nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nobu Kishimoto","submitted_at":"2012-03-28T20:43:04Z","abstract_excerpt":"We consider the mass critical NLS on $\\mathbb{T}$ and $\\mathbb{T}^2$. In the $\\mathbb{R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $L^6$ Strichartz on $\\mathbb{T}$ and the $L^4$ Strichartz on $\\mathbb{T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $\\mathbb{T}$ and $\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6375","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":"1203.6375","created_at":"2026-05-18T03:44:13.552958+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.6375v2","created_at":"2026-05-18T03:44:13.552958+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6375","created_at":"2026-05-18T03:44:13.552958+00:00"},{"alias_kind":"pith_short_12","alias_value":"43FF232LSAF6","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"43FF232LSAF6CZND","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"43FF232L","created_at":"2026-05-18T12:26:53.410803+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/43FF232LSAF6CZNDNSGCPK3GSO","json":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO.json","graph_json":"https://pith.science/api/pith-number/43FF232LSAF6CZNDNSGCPK3GSO/graph.json","events_json":"https://pith.science/api/pith-number/43FF232LSAF6CZNDNSGCPK3GSO/events.json","paper":"https://pith.science/paper/43FF232L"},"agent_actions":{"view_html":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO","download_json":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO.json","view_paper":"https://pith.science/paper/43FF232L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.6375&json=true","fetch_graph":"https://pith.science/api/pith-number/43FF232LSAF6CZNDNSGCPK3GSO/graph.json","fetch_events":"https://pith.science/api/pith-number/43FF232LSAF6CZNDNSGCPK3GSO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO/action/storage_attestation","attest_author":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO/action/author_attestation","sign_citation":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO/action/citation_signature","submit_replication":"https://pith.science/pith/43FF232LSAF6CZNDNSGCPK3GSO/action/replication_record"}},"created_at":"2026-05-18T03:44:13.552958+00:00","updated_at":"2026-05-18T03:44:13.552958+00:00"}