{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:5NICDSCLHCQS3WFLD7AS3MVZXH","short_pith_number":"pith:5NICDSCL","canonical_record":{"source":{"id":"1302.4309","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-02-18T15:17:21Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"adb962c07d61e24788056ffe5e495b06ef33ab30b1f02b3f23cff29f17187c48","abstract_canon_sha256":"b6996efc226cfadf614db64ed910091c7c113dcef25b1765e71cedeb18378e5b"},"schema_version":"1.0"},"canonical_sha256":"eb5021c84b38a12dd8ab1fc12db2b9b9f4ccf42dffc57d3835696d4ef9eb1a37","source":{"kind":"arxiv","id":"1302.4309","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.4309","created_at":"2026-05-18T03:33:23Z"},{"alias_kind":"arxiv_version","alias_value":"1302.4309v1","created_at":"2026-05-18T03:33:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4309","created_at":"2026-05-18T03:33:23Z"},{"alias_kind":"pith_short_12","alias_value":"5NICDSCLHCQS","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"5NICDSCLHCQS3WFL","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"5NICDSCL","created_at":"2026-05-18T12:27:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:5NICDSCLHCQS3WFLD7AS3MVZXH","target":"record","payload":{"canonical_record":{"source":{"id":"1302.4309","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-02-18T15:17:21Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"adb962c07d61e24788056ffe5e495b06ef33ab30b1f02b3f23cff29f17187c48","abstract_canon_sha256":"b6996efc226cfadf614db64ed910091c7c113dcef25b1765e71cedeb18378e5b"},"schema_version":"1.0"},"canonical_sha256":"eb5021c84b38a12dd8ab1fc12db2b9b9f4ccf42dffc57d3835696d4ef9eb1a37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:23.438124Z","signature_b64":"0whGUOWSj+InEuF2C3Vjn1gzzI9W2OIIw0500zMVeP629g7ljdz1xRsuQvY4/rJS6MHoln87eeDFHFeyl2p3Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb5021c84b38a12dd8ab1fc12db2b9b9f4ccf42dffc57d3835696d4ef9eb1a37","last_reissued_at":"2026-05-18T03:33:23.437179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:23.437179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.4309","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:33:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TYRmFJRntUrc00/x+RFbJpcykuI3Yt1paxOd9l1PTaT0xW+3JdzEjnAPsjwgrvT1cS8/JlN09UgzgYWD0HriBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T21:24:37.324630Z"},"content_sha256":"324a01fce7d646e62ee517ac37697f906ba866ceda2de4b0e7d128dd388fb662","schema_version":"1.0","event_id":"sha256:324a01fce7d646e62ee517ac37697f906ba866ceda2de4b0e7d128dd388fb662"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:5NICDSCLHCQS3WFLD7AS3MVZXH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Subharmonic solutions for nonautonomous sublinear first order Hamiltonian systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"A. Raouf Chouikha, Mohsen Timoumi","submitted_at":"2013-02-18T15:17:21Z","abstract_excerpt":"In this paper, the existence of subharmonic solutions for a class of non-autonomous first-order Hamiltonian systems is investigated. We also study the minimality of periods for such solutions. Our results which extend and improve many previous results will be illustrated by specific examples. Our main tools are the minimax methods in critical point theory and the least action principle. {\\bf Key words.} Hamiltonian systems. Critical point theory. Least action principle. Subharmonic solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4309","kind":"arxiv","version":1},"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":"2026-05-18T03:33:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PI4TEK1KzBUaCxq3FIWH9taGpWx0Bz5V7p8j3BwYAboKf0ruoAK9PgSbUEEXZZX0MHxwFMY/PU+98VN/b9GnCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T21:24:37.324976Z"},"content_sha256":"e26ce40d7363f25b26e54022100b04fe0ba9183b97900e6599a174b9e28a2894","schema_version":"1.0","event_id":"sha256:e26ce40d7363f25b26e54022100b04fe0ba9183b97900e6599a174b9e28a2894"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5NICDSCLHCQS3WFLD7AS3MVZXH/bundle.json","state_url":"https://pith.science/pith/5NICDSCLHCQS3WFLD7AS3MVZXH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5NICDSCLHCQS3WFLD7AS3MVZXH/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":"2026-06-30T21:24:37Z","links":{"resolver":"https://pith.science/pith/5NICDSCLHCQS3WFLD7AS3MVZXH","bundle":"https://pith.science/pith/5NICDSCLHCQS3WFLD7AS3MVZXH/bundle.json","state":"https://pith.science/pith/5NICDSCLHCQS3WFLD7AS3MVZXH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5NICDSCLHCQS3WFLD7AS3MVZXH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:5NICDSCLHCQS3WFLD7AS3MVZXH","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":"b6996efc226cfadf614db64ed910091c7c113dcef25b1765e71cedeb18378e5b","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-02-18T15:17:21Z","title_canon_sha256":"adb962c07d61e24788056ffe5e495b06ef33ab30b1f02b3f23cff29f17187c48"},"schema_version":"1.0","source":{"id":"1302.4309","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.4309","created_at":"2026-05-18T03:33:23Z"},{"alias_kind":"arxiv_version","alias_value":"1302.4309v1","created_at":"2026-05-18T03:33:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4309","created_at":"2026-05-18T03:33:23Z"},{"alias_kind":"pith_short_12","alias_value":"5NICDSCLHCQS","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"5NICDSCLHCQS3WFL","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"5NICDSCL","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:e26ce40d7363f25b26e54022100b04fe0ba9183b97900e6599a174b9e28a2894","target":"graph","created_at":"2026-05-18T03:33:23Z","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, the existence of subharmonic solutions for a class of non-autonomous first-order Hamiltonian systems is investigated. We also study the minimality of periods for such solutions. Our results which extend and improve many previous results will be illustrated by specific examples. Our main tools are the minimax methods in critical point theory and the least action principle. {\\bf Key words.} Hamiltonian systems. Critical point theory. Least action principle. Subharmonic solutions.","authors_text":"A. Raouf Chouikha, Mohsen Timoumi","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-02-18T15:17:21Z","title":"Subharmonic solutions for nonautonomous sublinear first order Hamiltonian systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4309","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":"sha256:324a01fce7d646e62ee517ac37697f906ba866ceda2de4b0e7d128dd388fb662","target":"record","created_at":"2026-05-18T03:33:23Z","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":"b6996efc226cfadf614db64ed910091c7c113dcef25b1765e71cedeb18378e5b","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-02-18T15:17:21Z","title_canon_sha256":"adb962c07d61e24788056ffe5e495b06ef33ab30b1f02b3f23cff29f17187c48"},"schema_version":"1.0","source":{"id":"1302.4309","kind":"arxiv","version":1}},"canonical_sha256":"eb5021c84b38a12dd8ab1fc12db2b9b9f4ccf42dffc57d3835696d4ef9eb1a37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb5021c84b38a12dd8ab1fc12db2b9b9f4ccf42dffc57d3835696d4ef9eb1a37","first_computed_at":"2026-05-18T03:33:23.437179Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:33:23.437179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0whGUOWSj+InEuF2C3Vjn1gzzI9W2OIIw0500zMVeP629g7ljdz1xRsuQvY4/rJS6MHoln87eeDFHFeyl2p3Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:33:23.438124Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.4309","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:324a01fce7d646e62ee517ac37697f906ba866ceda2de4b0e7d128dd388fb662","sha256:e26ce40d7363f25b26e54022100b04fe0ba9183b97900e6599a174b9e28a2894"],"state_sha256":"5af8ff46b22a485e08bc32b67a74837a61fb2a0ccdf0d524118adb231486caa5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bujo01snk8c2vXvxZH6l+eapmlaLMZNeWe6OKJQlboe0hZmIPNPSNTA+Phv8BZG32KIrEl9bVvHLtsmHtU1OCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T21:24:37.326999Z","bundle_sha256":"44a0bcad45e27f4aae071d4ad4b8cfb9c9432c74daa2c381750e11b005bf9d60"}}