{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4XQ5FEG34XXX3K5BAPSKV5E4Z5","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":"c768408f1a3c2413cddd2882602551cba461e2106575256bb0c80e3c50dbcf73","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-07-11T13:41:12Z","title_canon_sha256":"6d10dba7efa9a0112b0fb7a54e85a65e4641f9b9a920680022627b0a0cdbf0d0"},"schema_version":"1.0","source":{"id":"1307.3108","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.3108","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1307.3108v1","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3108","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"4XQ5FEG34XXX","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4XQ5FEG34XXX3K5B","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4XQ5FEG3","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:c483ed496eba275f75d8b331eedb7e8b071faad8b847cc68263c0dac4ac0935b","target":"graph","created_at":"2026-05-18T03:18:41Z","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":"By using one of the definitions of the Bernoulli numbers, we prove that they solve particular odd and even lower triangular Toeplitz (l.t.T.) systems of equations. In a paper Ramanujan writes down a sparse lower triangular system solved by Bernoulli numbers; we observe that such system is equivalent to a sparse l.t.T. system. The attempt to obtain the sparse l.t.T. Ramanujan system from the l.t.T. odd and even systems, has led us to study efficient methods for solving generic l.t.T. systems. Such methods are here explained in detail in case n, the number of equations, is a power of b, b=2,3 an","authors_text":"C. Di Fiore, F. Tudisco, P. Zellini","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-07-11T13:41:12Z","title":"Bernoulli, Ramanujan, Toeplitz and the triangular matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3108","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":"sha256:9c678097d612112b3fcae14af736ba5395612e131ec9d8c827e902c3f3611834","target":"record","created_at":"2026-05-18T03:18:41Z","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":"c768408f1a3c2413cddd2882602551cba461e2106575256bb0c80e3c50dbcf73","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-07-11T13:41:12Z","title_canon_sha256":"6d10dba7efa9a0112b0fb7a54e85a65e4641f9b9a920680022627b0a0cdbf0d0"},"schema_version":"1.0","source":{"id":"1307.3108","kind":"arxiv","version":1}},"canonical_sha256":"e5e1d290dbe5ef7daba103e4aaf49ccf4bb2c5a0a1b4ffd033217b6c985bf517","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5e1d290dbe5ef7daba103e4aaf49ccf4bb2c5a0a1b4ffd033217b6c985bf517","first_computed_at":"2026-05-18T03:18:41.173881Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:41.173881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xqfvqUsaBdV/5R/zpuQfmkRoH2pf9qfB9KRqIyf9i/IXUtI36XxtNFQRePbjgDHrkGdg5OSvQS72Oa1PV7kvBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:41.174656Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.3108","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c678097d612112b3fcae14af736ba5395612e131ec9d8c827e902c3f3611834","sha256:c483ed496eba275f75d8b331eedb7e8b071faad8b847cc68263c0dac4ac0935b"],"state_sha256":"8ae9fc80ce0be877daae69de7f018b6da8f57e7df6ddf3b0e5bac8bde54befb0"}