{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:WMAXIYDN6YX2WINGYQHYPDKEZW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4e8f01d6a59652906a8d0dd83c19d7d292f18e34ebdf18e6565ce867af998267","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-20T13:28:32Z","title_canon_sha256":"e4b941713d923785db48c0487e596dd6f315050b55d5592e0560f2fb84b6f84a"},"schema_version":"1.0","source":{"id":"2411.13312","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.13312","created_at":"2026-07-05T09:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"2411.13312v1","created_at":"2026-07-05T09:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.13312","created_at":"2026-07-05T09:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"WMAXIYDN6YX2","created_at":"2026-07-05T09:38:08Z"},{"alias_kind":"pith_short_16","alias_value":"WMAXIYDN6YX2WING","created_at":"2026-07-05T09:38:08Z"},{"alias_kind":"pith_short_8","alias_value":"WMAXIYDN","created_at":"2026-07-05T09:38:08Z"}],"graph_snapshots":[{"event_id":"sha256:940a13ca88f2fa6247ffc5ed7f6f7f4c35e9d95e8b8ef0c892a5853886c97cf9","target":"graph","created_at":"2026-07-05T09:38:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.13312/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations on torus. The normal form is complete up to arbitrary finite order. The proof is based on a valid non-resonant condition and a suitable norm of Hamiltonian function. Then as two examples, we apply this theorem to nonlinear wave equation in one dimension and nonlinear Schr\\\"{o}dinger equation in high dimension. Consequently, the polynomially long time stability is proved in Sobolev spaces $H^s$ with the index $s$ being much smaller than before. Further, by taking the iterative steps depending on the","authors_text":"Duohui Xiang, Jianjun Liu","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-20T13:28:32Z","title":"A Birkhoff Normal Form Theorem for Partial Differential Equations on torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13312","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:30bebd0c73eac25fffb8aa1d58551735787c89480b8b1848e94ad7bed80d3fec","target":"record","created_at":"2026-07-05T09:38:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4e8f01d6a59652906a8d0dd83c19d7d292f18e34ebdf18e6565ce867af998267","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-20T13:28:32Z","title_canon_sha256":"e4b941713d923785db48c0487e596dd6f315050b55d5592e0560f2fb84b6f84a"},"schema_version":"1.0","source":{"id":"2411.13312","kind":"arxiv","version":1}},"canonical_sha256":"b30174606df62fab21a6c40f878d44cd8a12343780b470cb6e6c0a5bcdbc0130","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b30174606df62fab21a6c40f878d44cd8a12343780b470cb6e6c0a5bcdbc0130","first_computed_at":"2026-07-05T09:38:08.639388Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:38:08.639388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vxAlrZsQh0dA9m9YaqGMhO7df3+FbZZn+i5XPBBc1+sw9+zdq4Xk2THSNIIquAwo/p+yB8IwpXrn8bJ9AzhQDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:38:08.639819Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.13312","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:30bebd0c73eac25fffb8aa1d58551735787c89480b8b1848e94ad7bed80d3fec","sha256:940a13ca88f2fa6247ffc5ed7f6f7f4c35e9d95e8b8ef0c892a5853886c97cf9"],"state_sha256":"f4eee5eab15dc0b36ff892de969273043aed578af6f4149332ec0eda02a09164"}