{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:QKFHQYIOFSDJZ2EYKEC6IM2HLA","short_pith_number":"pith:QKFHQYIO","schema_version":"1.0","canonical_sha256":"828a78610e2c869ce8985105e433475831fb5c390c7450b657773aaad71c1dd5","source":{"kind":"arxiv","id":"1908.09179","version":1},"attestation_state":"computed","paper":{"title":"Commutators, matrices and an identity of Copeland","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Darij Grinberg","submitted_at":"2019-08-24T18:12:08Z","abstract_excerpt":"Given two elements $a$ and $b$ of a noncommutative ring, we express $\\left( ba\\right)^n$ as a \"row vector times matrix times column vector\" product, where the matrix is the $n$-th power of a matrix with entries $\\dbinom{i}{j}\\operatorname{ad}_a^{i-j}\\left( b\\right)$. This generalizes a formula by Tom Copeland used in the study of Pascal-style matrices."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.09179","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.RA","submitted_at":"2019-08-24T18:12:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"67ba7807191d46badffb85a4eb6a919a7dbde809454a482a0d291ab73ae78ce0","abstract_canon_sha256":"be223a6a31df3a0e80524b867476e2e05754a7c0e324eb82021dc6de5bfe7fb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:59:37.368977Z","signature_b64":"YMCzpLGtZ5sTqQIuZpw+S4QRP7He+E/nuIXuq9ZnLAuWL2wLuVxHEC9ta0yKlfzxFIwjALSP3R0B3cy0YHE5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"828a78610e2c869ce8985105e433475831fb5c390c7450b657773aaad71c1dd5","last_reissued_at":"2026-07-04T23:59:37.368570Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:59:37.368570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutators, matrices and an identity of Copeland","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Darij Grinberg","submitted_at":"2019-08-24T18:12:08Z","abstract_excerpt":"Given two elements $a$ and $b$ of a noncommutative ring, we express $\\left( ba\\right)^n$ as a \"row vector times matrix times column vector\" product, where the matrix is the $n$-th power of a matrix with entries $\\dbinom{i}{j}\\operatorname{ad}_a^{i-j}\\left( b\\right)$. This generalizes a formula by Tom Copeland used in the study of Pascal-style matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09179","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09179/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.09179","created_at":"2026-07-04T23:59:37.368628+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.09179v1","created_at":"2026-07-04T23:59:37.368628+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09179","created_at":"2026-07-04T23:59:37.368628+00:00"},{"alias_kind":"pith_short_12","alias_value":"QKFHQYIOFSDJ","created_at":"2026-07-04T23:59:37.368628+00:00"},{"alias_kind":"pith_short_16","alias_value":"QKFHQYIOFSDJZ2EY","created_at":"2026-07-04T23:59:37.368628+00:00"},{"alias_kind":"pith_short_8","alias_value":"QKFHQYIO","created_at":"2026-07-04T23:59:37.368628+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA","json":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA.json","graph_json":"https://pith.science/api/pith-number/QKFHQYIOFSDJZ2EYKEC6IM2HLA/graph.json","events_json":"https://pith.science/api/pith-number/QKFHQYIOFSDJZ2EYKEC6IM2HLA/events.json","paper":"https://pith.science/paper/QKFHQYIO"},"agent_actions":{"view_html":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA","download_json":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA.json","view_paper":"https://pith.science/paper/QKFHQYIO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.09179&json=true","fetch_graph":"https://pith.science/api/pith-number/QKFHQYIOFSDJZ2EYKEC6IM2HLA/graph.json","fetch_events":"https://pith.science/api/pith-number/QKFHQYIOFSDJZ2EYKEC6IM2HLA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA/action/storage_attestation","attest_author":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA/action/author_attestation","sign_citation":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA/action/citation_signature","submit_replication":"https://pith.science/pith/QKFHQYIOFSDJZ2EYKEC6IM2HLA/action/replication_record"}},"created_at":"2026-07-04T23:59:37.368628+00:00","updated_at":"2026-07-04T23:59:37.368628+00:00"}