{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:O4KZKXRXLSTBEUBM7Y6P65ZXKZ","short_pith_number":"pith:O4KZKXRX","canonical_record":{"source":{"id":"1202.1068","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-02-06T08:46:14Z","cross_cats_sorted":[],"title_canon_sha256":"b543d1d2a000d941d213ac957bf9c36b29bc29a3978dd4d7f1cd30bde6b0ac4a","abstract_canon_sha256":"11b50555ee9f85b2956e3cc1ef240a7e8d5597716a36f8e62577b62c199701c3"},"schema_version":"1.0"},"canonical_sha256":"7715955e375ca612502cfe3cff773756455e84de956c73bd6d9c1f1b9a54d145","source":{"kind":"arxiv","id":"1202.1068","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1068","created_at":"2026-05-18T04:03:05Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1068v1","created_at":"2026-05-18T04:03:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1068","created_at":"2026-05-18T04:03:05Z"},{"alias_kind":"pith_short_12","alias_value":"O4KZKXRXLSTB","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"O4KZKXRXLSTBEUBM","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"O4KZKXRX","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:O4KZKXRXLSTBEUBM7Y6P65ZXKZ","target":"record","payload":{"canonical_record":{"source":{"id":"1202.1068","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-02-06T08:46:14Z","cross_cats_sorted":[],"title_canon_sha256":"b543d1d2a000d941d213ac957bf9c36b29bc29a3978dd4d7f1cd30bde6b0ac4a","abstract_canon_sha256":"11b50555ee9f85b2956e3cc1ef240a7e8d5597716a36f8e62577b62c199701c3"},"schema_version":"1.0"},"canonical_sha256":"7715955e375ca612502cfe3cff773756455e84de956c73bd6d9c1f1b9a54d145","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:05.196046Z","signature_b64":"Nf0aGeJFSDLQw9DeWb5Y1jbbpkFqYDMD5JAvhH4nmkqrjijlxGDI/IagRgCjnCZZiBh2+nFAKmK9QdW7zrzuAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7715955e375ca612502cfe3cff773756455e84de956c73bd6d9c1f1b9a54d145","last_reissued_at":"2026-05-18T04:03:05.195299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:05.195299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.1068","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":"2026-05-18T04:03:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yn+8ZnHhKAaGFbRWHi/5fFMXPtlCxHZ2ENPU7uH1XcVIt7A6RTILODz2opG5mrDMnDDFsBfEcgosxw+pdH3qCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:30:12.770351Z"},"content_sha256":"86af38f690941a49db81c61b41ec97877f7882bbbbedf897794bfbb1cdc2bba4","schema_version":"1.0","event_id":"sha256:86af38f690941a49db81c61b41ec97877f7882bbbbedf897794bfbb1cdc2bba4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:O4KZKXRXLSTBEUBM7Y6P65ZXKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Determinants and Inverses of Circulant Matrices with a General Number Sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Durmu\\c{s} Bozkurt","submitted_at":"2012-02-06T08:46:14Z","abstract_excerpt":"The generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\\inZ and n\\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish some useful formulas for the determinants and inverses of W_{n} using the nice properties of the number sequences. Matrix decompositions are derived for W_{n} in order to obtain the results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1068","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":"2026-05-18T04:03:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zgWUjkK25tPDKQ5et7HG8A8dBvTAMJn2VOl6LvOq5fSVYxwuGm12+pOQbTKTwP/yBGDZyqG8zWiwdfXBoyEDDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:30:12.770698Z"},"content_sha256":"f226044e618006a6be51e275c2a781ad30e866f4b62cad1fa51554473415ecf5","schema_version":"1.0","event_id":"sha256:f226044e618006a6be51e275c2a781ad30e866f4b62cad1fa51554473415ecf5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ/bundle.json","state_url":"https://pith.science/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ/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":"2026-06-22T21:30:12Z","links":{"resolver":"https://pith.science/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ","bundle":"https://pith.science/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ/bundle.json","state":"https://pith.science/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O4KZKXRXLSTBEUBM7Y6P65ZXKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:O4KZKXRXLSTBEUBM7Y6P65ZXKZ","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":"11b50555ee9f85b2956e3cc1ef240a7e8d5597716a36f8e62577b62c199701c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-02-06T08:46:14Z","title_canon_sha256":"b543d1d2a000d941d213ac957bf9c36b29bc29a3978dd4d7f1cd30bde6b0ac4a"},"schema_version":"1.0","source":{"id":"1202.1068","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1068","created_at":"2026-05-18T04:03:05Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1068v1","created_at":"2026-05-18T04:03:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1068","created_at":"2026-05-18T04:03:05Z"},{"alias_kind":"pith_short_12","alias_value":"O4KZKXRXLSTB","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"O4KZKXRXLSTBEUBM","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"O4KZKXRX","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:f226044e618006a6be51e275c2a781ad30e866f4b62cad1fa51554473415ecf5","target":"graph","created_at":"2026-05-18T04:03:05Z","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 generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\\inZ and n\\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish some useful formulas for the determinants and inverses of W_{n} using the nice properties of the number sequences. Matrix decompositions are derived for W_{n} in order to obtain the results.","authors_text":"Durmu\\c{s} Bozkurt","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-02-06T08:46:14Z","title":"On the Determinants and Inverses of Circulant Matrices with a General Number Sequence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1068","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":"sha256:86af38f690941a49db81c61b41ec97877f7882bbbbedf897794bfbb1cdc2bba4","target":"record","created_at":"2026-05-18T04:03:05Z","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":"11b50555ee9f85b2956e3cc1ef240a7e8d5597716a36f8e62577b62c199701c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-02-06T08:46:14Z","title_canon_sha256":"b543d1d2a000d941d213ac957bf9c36b29bc29a3978dd4d7f1cd30bde6b0ac4a"},"schema_version":"1.0","source":{"id":"1202.1068","kind":"arxiv","version":1}},"canonical_sha256":"7715955e375ca612502cfe3cff773756455e84de956c73bd6d9c1f1b9a54d145","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7715955e375ca612502cfe3cff773756455e84de956c73bd6d9c1f1b9a54d145","first_computed_at":"2026-05-18T04:03:05.195299Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:05.195299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nf0aGeJFSDLQw9DeWb5Y1jbbpkFqYDMD5JAvhH4nmkqrjijlxGDI/IagRgCjnCZZiBh2+nFAKmK9QdW7zrzuAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:05.196046Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.1068","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86af38f690941a49db81c61b41ec97877f7882bbbbedf897794bfbb1cdc2bba4","sha256:f226044e618006a6be51e275c2a781ad30e866f4b62cad1fa51554473415ecf5"],"state_sha256":"861d4c1bdc67c99b575b435aa7fe96f5ea3c179006ec96f724aa8f0b936088c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ntijfplfPozcCPnQNmjrYYRbXiTW28OrGp4H99OlV4m+SM70SuxKKEIPYg8wqBvxa0a6DQXh3mQzOKsH+X8xAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T21:30:12.772634Z","bundle_sha256":"4a745080158f639356b174c60e0218356d9ec7de110573367be4428ac24bc851"}}