{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:B7FRPPYEBXZGC7DNT6W42OFWT5","short_pith_number":"pith:B7FRPPYE","canonical_record":{"source":{"id":"1308.1137","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-05T22:53:28Z","cross_cats_sorted":[],"title_canon_sha256":"6562988326a096f5a35a0632806168e42aa30916413ae190267b760b2dd82046","abstract_canon_sha256":"5e001ad9aa96f1fd7933c6432083138b2550345c2fe130bdc4c6466478a05448"},"schema_version":"1.0"},"canonical_sha256":"0fcb17bf040df2617c6d9fadcd38b69f47e854cbaae64ea50e6a7ba9cc4cf193","source":{"kind":"arxiv","id":"1308.1137","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1137","created_at":"2026-05-18T03:13:09Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1137v3","created_at":"2026-05-18T03:13:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1137","created_at":"2026-05-18T03:13:09Z"},{"alias_kind":"pith_short_12","alias_value":"B7FRPPYEBXZG","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"B7FRPPYEBXZGC7DN","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"B7FRPPYE","created_at":"2026-05-18T12:27:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:B7FRPPYEBXZGC7DNT6W42OFWT5","target":"record","payload":{"canonical_record":{"source":{"id":"1308.1137","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-05T22:53:28Z","cross_cats_sorted":[],"title_canon_sha256":"6562988326a096f5a35a0632806168e42aa30916413ae190267b760b2dd82046","abstract_canon_sha256":"5e001ad9aa96f1fd7933c6432083138b2550345c2fe130bdc4c6466478a05448"},"schema_version":"1.0"},"canonical_sha256":"0fcb17bf040df2617c6d9fadcd38b69f47e854cbaae64ea50e6a7ba9cc4cf193","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:09.626972Z","signature_b64":"ietDnhfM679HeSomkp20OXu3CjOPBGzxaeVOPCrCvQfumMswChsM6rbM67n6rp+fM2DJZU7KQyuhnaLdNE3iAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fcb17bf040df2617c6d9fadcd38b69f47e854cbaae64ea50e6a7ba9cc4cf193","last_reissued_at":"2026-05-18T03:13:09.626518Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:09.626518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.1137","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WJsYaQhEl6GP7Ek0f5VvFB2bkihKR/TabtIHdXroMbB0MeltPwpr4B24nxPiMnzP1KwM2whIhx2y1SLNsBMKAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T08:23:02.500777Z"},"content_sha256":"188b369928a55f96b443cb0c60657ea80c912124ccc59ca3e2a57f5f3034e30d","schema_version":"1.0","event_id":"sha256:188b369928a55f96b443cb0c60657ea80c912124ccc59ca3e2a57f5f3034e30d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:B7FRPPYEBXZGC7DNT6W42OFWT5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Products of 2X2 matrices related to non autonomous Fibonacci difference equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Henrique Oliveira, Rafael Lu\\'is","submitted_at":"2013-08-05T22:53:28Z","abstract_excerpt":"A technique to compute arbitrary products of a class of Fibonacci $2\\times2$ square matrices is proved in this work. General explicit solutions for non autonomous Fibonacci difference equations are obtained from these products. In the periodic non autonomous Fibonacci difference equations the monodromy matrix, the Floquet multipliers and the Binet's formulas are obtained. In the periodic case explicit solutions are obtained and the solutions are analyzed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1137","kind":"arxiv","version":3},"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":"2026-05-18T03:13:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qAqV3zTdtqLQeu2yHmS9fvwh/DJbkGrQcjsWqfi3VEPDahGJE1jhQj+KzSw+ATSm+bnLPn+o3ErYsbyLSRzQAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T08:23:02.501118Z"},"content_sha256":"07d1e946bdb16ebbdcf46c018a8359b3b6aeff72bf0700562f90a44e746b6cff","schema_version":"1.0","event_id":"sha256:07d1e946bdb16ebbdcf46c018a8359b3b6aeff72bf0700562f90a44e746b6cff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B7FRPPYEBXZGC7DNT6W42OFWT5/bundle.json","state_url":"https://pith.science/pith/B7FRPPYEBXZGC7DNT6W42OFWT5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B7FRPPYEBXZGC7DNT6W42OFWT5/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":"2026-06-23T08:23:02Z","links":{"resolver":"https://pith.science/pith/B7FRPPYEBXZGC7DNT6W42OFWT5","bundle":"https://pith.science/pith/B7FRPPYEBXZGC7DNT6W42OFWT5/bundle.json","state":"https://pith.science/pith/B7FRPPYEBXZGC7DNT6W42OFWT5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B7FRPPYEBXZGC7DNT6W42OFWT5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:B7FRPPYEBXZGC7DNT6W42OFWT5","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":"5e001ad9aa96f1fd7933c6432083138b2550345c2fe130bdc4c6466478a05448","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-05T22:53:28Z","title_canon_sha256":"6562988326a096f5a35a0632806168e42aa30916413ae190267b760b2dd82046"},"schema_version":"1.0","source":{"id":"1308.1137","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1137","created_at":"2026-05-18T03:13:09Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1137v3","created_at":"2026-05-18T03:13:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1137","created_at":"2026-05-18T03:13:09Z"},{"alias_kind":"pith_short_12","alias_value":"B7FRPPYEBXZG","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"B7FRPPYEBXZGC7DN","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"B7FRPPYE","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:07d1e946bdb16ebbdcf46c018a8359b3b6aeff72bf0700562f90a44e746b6cff","target":"graph","created_at":"2026-05-18T03:13:09Z","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":"A technique to compute arbitrary products of a class of Fibonacci $2\\times2$ square matrices is proved in this work. General explicit solutions for non autonomous Fibonacci difference equations are obtained from these products. In the periodic non autonomous Fibonacci difference equations the monodromy matrix, the Floquet multipliers and the Binet's formulas are obtained. In the periodic case explicit solutions are obtained and the solutions are analyzed.","authors_text":"Henrique Oliveira, Rafael Lu\\'is","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-05T22:53:28Z","title":"Products of 2X2 matrices related to non autonomous Fibonacci difference equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1137","kind":"arxiv","version":3},"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":"sha256:188b369928a55f96b443cb0c60657ea80c912124ccc59ca3e2a57f5f3034e30d","target":"record","created_at":"2026-05-18T03:13:09Z","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":"5e001ad9aa96f1fd7933c6432083138b2550345c2fe130bdc4c6466478a05448","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-05T22:53:28Z","title_canon_sha256":"6562988326a096f5a35a0632806168e42aa30916413ae190267b760b2dd82046"},"schema_version":"1.0","source":{"id":"1308.1137","kind":"arxiv","version":3}},"canonical_sha256":"0fcb17bf040df2617c6d9fadcd38b69f47e854cbaae64ea50e6a7ba9cc4cf193","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fcb17bf040df2617c6d9fadcd38b69f47e854cbaae64ea50e6a7ba9cc4cf193","first_computed_at":"2026-05-18T03:13:09.626518Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:09.626518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ietDnhfM679HeSomkp20OXu3CjOPBGzxaeVOPCrCvQfumMswChsM6rbM67n6rp+fM2DJZU7KQyuhnaLdNE3iAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:09.626972Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.1137","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:188b369928a55f96b443cb0c60657ea80c912124ccc59ca3e2a57f5f3034e30d","sha256:07d1e946bdb16ebbdcf46c018a8359b3b6aeff72bf0700562f90a44e746b6cff"],"state_sha256":"a9e4e4c90278e13fab1ea7e935b3c5faadd1b32b1a4c6361e4307e52bed6a223"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JwoagMBi54wIG46XpbYtOxfnBduQ/zjvkM8jdxII10HPoTpxt/vgs+szm2v7i70pkLGOwQ8sebcoeV9Nplm4Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T08:23:02.503000Z","bundle_sha256":"0487e832ed280fd2b4722439e6348e4a60672d9d3fd2df52b101b558ce599709"}}