{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:DG3GIFVDYGBFDUAP7T3WOV2ZEA","short_pith_number":"pith:DG3GIFVD","canonical_record":{"source":{"id":"1412.0051","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-28T23:59:57Z","cross_cats_sorted":[],"title_canon_sha256":"eb6144673eb75ff8352910492b05a0a21bd839e429512b80da2888b92ba69b1a","abstract_canon_sha256":"a8417b45f5c291213b443d4a5397a46107e1e5d9dd230c0a9eabe6c83493761a"},"schema_version":"1.0"},"canonical_sha256":"19b66416a3c18251d00ffcf7675759203b5b4054a65651bb4d6f5c937a8d565c","source":{"kind":"arxiv","id":"1412.0051","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.0051","created_at":"2026-05-18T02:32:13Z"},{"alias_kind":"arxiv_version","alias_value":"1412.0051v2","created_at":"2026-05-18T02:32:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.0051","created_at":"2026-05-18T02:32:13Z"},{"alias_kind":"pith_short_12","alias_value":"DG3GIFVDYGBF","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DG3GIFVDYGBFDUAP","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DG3GIFVD","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:DG3GIFVDYGBFDUAP7T3WOV2ZEA","target":"record","payload":{"canonical_record":{"source":{"id":"1412.0051","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-28T23:59:57Z","cross_cats_sorted":[],"title_canon_sha256":"eb6144673eb75ff8352910492b05a0a21bd839e429512b80da2888b92ba69b1a","abstract_canon_sha256":"a8417b45f5c291213b443d4a5397a46107e1e5d9dd230c0a9eabe6c83493761a"},"schema_version":"1.0"},"canonical_sha256":"19b66416a3c18251d00ffcf7675759203b5b4054a65651bb4d6f5c937a8d565c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:13.513598Z","signature_b64":"BnwLb42TKizHNWwZLx+3RQSjex2mFae/NidOBHkBpdC8XO30O2/mFtxbjyJxmHdjGahaNLJrKZ5hPI/rZv3FBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19b66416a3c18251d00ffcf7675759203b5b4054a65651bb4d6f5c937a8d565c","last_reissued_at":"2026-05-18T02:32:13.513113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:13.513113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.0051","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BDiPIF3teuxtC8BWHAHBnt+L3mqqSz+Kd4v3zjuIBmzoLYYej7MXAJDtcIoK/JyFAfEtcjgBrcAu8NSOdtdBAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T08:16:34.405634Z"},"content_sha256":"39a8498f3f8ec547b36fcccb56bdf540ee79a94ea06403faaec3cbad2208692c","schema_version":"1.0","event_id":"sha256:39a8498f3f8ec547b36fcccb56bdf540ee79a94ea06403faaec3cbad2208692c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:DG3GIFVDYGBFDUAP7T3WOV2ZEA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An inverse approach to the center-foci problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Rafael Ram\\'irez, Valent\\'in Ram\\'irez","submitted_at":"2014-11-28T23:59:57Z","abstract_excerpt":"The classical Center-Focus Problem posed by H. Poincar\\'e in 1880's is concerned on the characterization of planar polynomial vector fields $X=(-y+P(x,y))\\dfrac{\\partial}{\\partial x}+(x+Q(x,y))\\dfrac{\\partial}{\\partial y},$ with $P(0,0)=Q(0,0)=0,$ such that all their integral trajectories are closed curves whose interiors contain a fixed point called center or such that all their integral trajectories are spirals called foci. In this paper we state and study the inverse problem to the Center-Foci Problem i.e., we require to determine the analytic planar vector fields $X$ in such a way that for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0051","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CVr5PJ1CT7jnkCOBujJahTIOMJaSg5THu2xKqhlwyyb2oypg05S2ry6rClvKmWB2kFCMQc2HQYcPZqUPYulnAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T08:16:34.405986Z"},"content_sha256":"5766ebffcf904c80a3668f74fdc5aa3d7b2020b929bf78c954ce173490d30478","schema_version":"1.0","event_id":"sha256:5766ebffcf904c80a3668f74fdc5aa3d7b2020b929bf78c954ce173490d30478"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA/bundle.json","state_url":"https://pith.science/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T08:16:34Z","links":{"resolver":"https://pith.science/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA","bundle":"https://pith.science/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA/bundle.json","state":"https://pith.science/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DG3GIFVDYGBFDUAP7T3WOV2ZEA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:DG3GIFVDYGBFDUAP7T3WOV2ZEA","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":"a8417b45f5c291213b443d4a5397a46107e1e5d9dd230c0a9eabe6c83493761a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-28T23:59:57Z","title_canon_sha256":"eb6144673eb75ff8352910492b05a0a21bd839e429512b80da2888b92ba69b1a"},"schema_version":"1.0","source":{"id":"1412.0051","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.0051","created_at":"2026-05-18T02:32:13Z"},{"alias_kind":"arxiv_version","alias_value":"1412.0051v2","created_at":"2026-05-18T02:32:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.0051","created_at":"2026-05-18T02:32:13Z"},{"alias_kind":"pith_short_12","alias_value":"DG3GIFVDYGBF","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DG3GIFVDYGBFDUAP","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DG3GIFVD","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:5766ebffcf904c80a3668f74fdc5aa3d7b2020b929bf78c954ce173490d30478","target":"graph","created_at":"2026-05-18T02:32:13Z","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":"The classical Center-Focus Problem posed by H. Poincar\\'e in 1880's is concerned on the characterization of planar polynomial vector fields $X=(-y+P(x,y))\\dfrac{\\partial}{\\partial x}+(x+Q(x,y))\\dfrac{\\partial}{\\partial y},$ with $P(0,0)=Q(0,0)=0,$ such that all their integral trajectories are closed curves whose interiors contain a fixed point called center or such that all their integral trajectories are spirals called foci. In this paper we state and study the inverse problem to the Center-Foci Problem i.e., we require to determine the analytic planar vector fields $X$ in such a way that for","authors_text":"Rafael Ram\\'irez, Valent\\'in Ram\\'irez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-28T23:59:57Z","title":"An inverse approach to the center-foci problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0051","kind":"arxiv","version":2},"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":"sha256:39a8498f3f8ec547b36fcccb56bdf540ee79a94ea06403faaec3cbad2208692c","target":"record","created_at":"2026-05-18T02:32:13Z","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":"a8417b45f5c291213b443d4a5397a46107e1e5d9dd230c0a9eabe6c83493761a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-28T23:59:57Z","title_canon_sha256":"eb6144673eb75ff8352910492b05a0a21bd839e429512b80da2888b92ba69b1a"},"schema_version":"1.0","source":{"id":"1412.0051","kind":"arxiv","version":2}},"canonical_sha256":"19b66416a3c18251d00ffcf7675759203b5b4054a65651bb4d6f5c937a8d565c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"19b66416a3c18251d00ffcf7675759203b5b4054a65651bb4d6f5c937a8d565c","first_computed_at":"2026-05-18T02:32:13.513113Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:13.513113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BnwLb42TKizHNWwZLx+3RQSjex2mFae/NidOBHkBpdC8XO30O2/mFtxbjyJxmHdjGahaNLJrKZ5hPI/rZv3FBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:13.513598Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.0051","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:39a8498f3f8ec547b36fcccb56bdf540ee79a94ea06403faaec3cbad2208692c","sha256:5766ebffcf904c80a3668f74fdc5aa3d7b2020b929bf78c954ce173490d30478"],"state_sha256":"58cdcd2a157b0bd20360bbcf90256dfbf54705bd345ae7b101699c3f8d35e8f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/yK412B9ijgFjJtgeBp4AY9nsPw0FiGlZvf683ZjfFzl76ahVu5ESKCBghuLf+bqI4OtOiSNRNVFjAhDftQAAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T08:16:34.407823Z","bundle_sha256":"76e1d691fd02a4a19c35e1737c1490df10e8d8f78680d12bc64d571fbad765aa"}}