{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TPB2N4VKELSFK6BXV6QZDI7NUO","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":"a8c75092e1ed7ef9e5a61ae30e339e752e15b406cd40b893f26a8a91cb42ca32","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-09T04:09:17Z","title_canon_sha256":"68c55b1d5e0a2001d4620505aef785b1651d72d2caee6c925d9886966b8e419d"},"schema_version":"1.0","source":{"id":"1605.02412","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.02412","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"arxiv_version","alias_value":"1605.02412v1","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02412","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"pith_short_12","alias_value":"TPB2N4VKELSF","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"TPB2N4VKELSFK6BX","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"TPB2N4VK","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:637d1aa725dc9d480f385949f166ff14c4a444b40ab9656851cf7bd8b5de8f32","target":"graph","created_at":"2026-05-18T01:15:21Z","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"},"paper":{"abstract_excerpt":"In this paper, we investigate the Cauchy problem for the shallow water type equation \\begin{eqnarray*} u_{t}+\\partial_{x}^{2j+1}u + \\frac{1}{2}\\partial_{x}(u^{2})+ \\partial_{x}(1-\\partial_{x}^{2})^{-1}\\left[u^{2}+\\frac{1}{2}u_{x}^{2}\\right]=0 \\end{eqnarray*} with low regularity data in the periodic settings. Himonas and Misiolek (Communications in Partial Differential Equations, 23(1998), 123-139.) have proved that the problem is locally well-posed for small initial data in H^{s}(\\mathbf{T}) with s\\geq-\\frac{j}{2}+1,j\\in N^{+} with the aid of the standard Fourier restriction norm method. To th","authors_text":"Wei Yan, Xiaoping Zhai, Yimin Zhang, Yongsheng Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-09T04:09:17Z","title":"The Cauchy problem for higher-order modified Camassa-Holm equations on the circle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02412","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":"sha256:53464f80ad332b77a3655790f50f91c6ae23dd78798584cc9c5632ec841f0d32","target":"record","created_at":"2026-05-18T01:15:21Z","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":"a8c75092e1ed7ef9e5a61ae30e339e752e15b406cd40b893f26a8a91cb42ca32","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-09T04:09:17Z","title_canon_sha256":"68c55b1d5e0a2001d4620505aef785b1651d72d2caee6c925d9886966b8e419d"},"schema_version":"1.0","source":{"id":"1605.02412","kind":"arxiv","version":1}},"canonical_sha256":"9bc3a6f2aa22e4557837afa191a3eda397c0de0c565ccffb04c8eee5c60a4f34","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9bc3a6f2aa22e4557837afa191a3eda397c0de0c565ccffb04c8eee5c60a4f34","first_computed_at":"2026-05-18T01:15:21.079309Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:21.079309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/wgOep3QbQRvbPAo4EY9NU+S/svd8yPmLlYzBp+hq2NZ8wHsM2T8QVike++YwMYXCThZFvqjzmja36AxMvSRAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:21.079865Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.02412","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53464f80ad332b77a3655790f50f91c6ae23dd78798584cc9c5632ec841f0d32","sha256:637d1aa725dc9d480f385949f166ff14c4a444b40ab9656851cf7bd8b5de8f32"],"state_sha256":"58b6988f39a4c1e8b7d2d17e1db6c7b231962b4e664614ea1edc86b3dfe50db6"}