{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MJICISNAAEOKTTK3AAAA2W5ADG","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":"a2ac7946a8f7a084125c659add5d9b69ebf445431210d8ac5bce93536ade132c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-17T02:19:47Z","title_canon_sha256":"df8274b2ba8b14e404f2c1c0cd2bbf030f383bed7718c23435bc46306b82c4de"},"schema_version":"1.0","source":{"id":"1509.05105","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05105","created_at":"2026-05-18T01:32:49Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05105v1","created_at":"2026-05-18T01:32:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05105","created_at":"2026-05-18T01:32:49Z"},{"alias_kind":"pith_short_12","alias_value":"MJICISNAAEOK","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MJICISNAAEOKTTK3","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MJICISNA","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:35bddf77cb7ab9ed134a5fa93b310189117e7b89d64443a656ba7e110d532a49","target":"graph","created_at":"2026-05-18T01:32:49Z","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":"Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a straight-forward generalization to the case of matrix coefficient differential operators that applies even in the case that the leading coefficient is singular.","authors_text":"Alex Kasman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-17T02:19:47Z","title":"Factorization of a Matrix Differential Operator Using Functions in its Kernel"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05105","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":"sha256:14b8a4a69f9463d0814856c3e0478e40296fc8f7b909977027ac1a3e5e3a03f1","target":"record","created_at":"2026-05-18T01:32:49Z","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":"a2ac7946a8f7a084125c659add5d9b69ebf445431210d8ac5bce93536ade132c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-17T02:19:47Z","title_canon_sha256":"df8274b2ba8b14e404f2c1c0cd2bbf030f383bed7718c23435bc46306b82c4de"},"schema_version":"1.0","source":{"id":"1509.05105","kind":"arxiv","version":1}},"canonical_sha256":"62502449a0011ca9cd5b00000d5ba01999543e33eb8efbd68c259ce762ff8cf4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62502449a0011ca9cd5b00000d5ba01999543e33eb8efbd68c259ce762ff8cf4","first_computed_at":"2026-05-18T01:32:49.903789Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:49.903789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9wx+AOC/DGQYB/heW5XShjF0zMhRd55gQLKh0S/9hFz+nrYkcQnr/ZFwdFBsTXbip9yWIYQOKeRNTyFxCYToBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:49.904462Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05105","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14b8a4a69f9463d0814856c3e0478e40296fc8f7b909977027ac1a3e5e3a03f1","sha256:35bddf77cb7ab9ed134a5fa93b310189117e7b89d64443a656ba7e110d532a49"],"state_sha256":"d289e022ecd49f08245ed8ee0db638336aca98aa77bc1149cf4792c6702b6c96"}