{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:UZNSRNLSUTHROEP73PFVQEF7FE","short_pith_number":"pith:UZNSRNLS","canonical_record":{"source":{"id":"1901.01947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-01-07T17:58:11Z","cross_cats_sorted":["math.NT","math.RA"],"title_canon_sha256":"815b45065b0e7a311ef23eca480882fc982362e0bcb858abb8a5c4eaa209be5b","abstract_canon_sha256":"f5bbf16fb0207d829f7405414e8a35ccba7722b7d784c04e3dfff13338c7cef3"},"schema_version":"1.0"},"canonical_sha256":"a65b28b572a4cf1711ffdbcb5810bf293ef199a2e2fbc3fe086e5c4dd5d797b3","source":{"kind":"arxiv","id":"1901.01947","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01947","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01947v1","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01947","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"pith_short_12","alias_value":"UZNSRNLSUTHR","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"UZNSRNLSUTHROEP7","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"UZNSRNLS","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:UZNSRNLSUTHROEP73PFVQEF7FE","target":"record","payload":{"canonical_record":{"source":{"id":"1901.01947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-01-07T17:58:11Z","cross_cats_sorted":["math.NT","math.RA"],"title_canon_sha256":"815b45065b0e7a311ef23eca480882fc982362e0bcb858abb8a5c4eaa209be5b","abstract_canon_sha256":"f5bbf16fb0207d829f7405414e8a35ccba7722b7d784c04e3dfff13338c7cef3"},"schema_version":"1.0"},"canonical_sha256":"a65b28b572a4cf1711ffdbcb5810bf293ef199a2e2fbc3fe086e5c4dd5d797b3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:48.998695Z","signature_b64":"H1IkSrbTWf74Xhg4Gf67ZNrQmBBNT11kO1oYO5JDBdg6+rwkM5ZXafIYOFhRvfEXqtazDZxEd20us+HOdsLyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a65b28b572a4cf1711ffdbcb5810bf293ef199a2e2fbc3fe086e5c4dd5d797b3","last_reissued_at":"2026-05-17T23:56:48.998220Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:48.998220Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.01947","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":"2026-05-17T23:56:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BrI/1EoU3A99DY2pK7/FRYc0fAf9D4pagvi9n4X5e05cyaxCS52wTOrPMPSe3LGnbSkW/MUKtAbZzpgU36LYBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:23:04.285823Z"},"content_sha256":"1bce9c64f35351ac7035e2f0b39939dbfbd2f19b59f9725d1c49966f73bfcbc4","schema_version":"1.0","event_id":"sha256:1bce9c64f35351ac7035e2f0b39939dbfbd2f19b59f9725d1c49966f73bfcbc4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:UZNSRNLSUTHROEP73PFVQEF7FE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Total nonnegativity of GCD matrices and kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.RA"],"primary_cat":"math.CA","authors_text":"Dominique Guillot, Jiaru Wu","submitted_at":"2019-01-07T17:58:11Z","abstract_excerpt":"Let $X = (x_1,\\dots,x_n)$ be a vector of distinct positive integers. The $n \\times n$ matrix $S = S(X) := (\\gcd(x_i,x_j))_{i,j=1}^n$, where $\\gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest common divisor (GCD) matrix on $X$. By a surprising result of Beslin and Ligh [Linear Algebra and Appl. 118], all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when $n \\geq 3$, and is TN"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01947","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":"2026-05-17T23:56:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8fyK/LDudkKprkxDUdVdCBRkfl4nIXx2GLim20Oi1rH7zGT+OXAkSg41eiXsopTZVbZGy++WqDXBvoSK/A0OAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:23:04.286164Z"},"content_sha256":"4a3428b1cf382ae4f47f845ad656672ec881b1ce29777369d8536e2eae4e5081","schema_version":"1.0","event_id":"sha256:4a3428b1cf382ae4f47f845ad656672ec881b1ce29777369d8536e2eae4e5081"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UZNSRNLSUTHROEP73PFVQEF7FE/bundle.json","state_url":"https://pith.science/pith/UZNSRNLSUTHROEP73PFVQEF7FE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UZNSRNLSUTHROEP73PFVQEF7FE/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":"2026-06-24T18:23:04Z","links":{"resolver":"https://pith.science/pith/UZNSRNLSUTHROEP73PFVQEF7FE","bundle":"https://pith.science/pith/UZNSRNLSUTHROEP73PFVQEF7FE/bundle.json","state":"https://pith.science/pith/UZNSRNLSUTHROEP73PFVQEF7FE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UZNSRNLSUTHROEP73PFVQEF7FE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:UZNSRNLSUTHROEP73PFVQEF7FE","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":"f5bbf16fb0207d829f7405414e8a35ccba7722b7d784c04e3dfff13338c7cef3","cross_cats_sorted":["math.NT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-01-07T17:58:11Z","title_canon_sha256":"815b45065b0e7a311ef23eca480882fc982362e0bcb858abb8a5c4eaa209be5b"},"schema_version":"1.0","source":{"id":"1901.01947","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01947","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01947v1","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01947","created_at":"2026-05-17T23:56:48Z"},{"alias_kind":"pith_short_12","alias_value":"UZNSRNLSUTHR","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"UZNSRNLSUTHROEP7","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"UZNSRNLS","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:4a3428b1cf382ae4f47f845ad656672ec881b1ce29777369d8536e2eae4e5081","target":"graph","created_at":"2026-05-17T23:56:48Z","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":"Let $X = (x_1,\\dots,x_n)$ be a vector of distinct positive integers. The $n \\times n$ matrix $S = S(X) := (\\gcd(x_i,x_j))_{i,j=1}^n$, where $\\gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest common divisor (GCD) matrix on $X$. By a surprising result of Beslin and Ligh [Linear Algebra and Appl. 118], all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when $n \\geq 3$, and is TN","authors_text":"Dominique Guillot, Jiaru Wu","cross_cats":["math.NT","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-01-07T17:58:11Z","title":"Total nonnegativity of GCD matrices and kernels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01947","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":"sha256:1bce9c64f35351ac7035e2f0b39939dbfbd2f19b59f9725d1c49966f73bfcbc4","target":"record","created_at":"2026-05-17T23:56:48Z","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":"f5bbf16fb0207d829f7405414e8a35ccba7722b7d784c04e3dfff13338c7cef3","cross_cats_sorted":["math.NT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-01-07T17:58:11Z","title_canon_sha256":"815b45065b0e7a311ef23eca480882fc982362e0bcb858abb8a5c4eaa209be5b"},"schema_version":"1.0","source":{"id":"1901.01947","kind":"arxiv","version":1}},"canonical_sha256":"a65b28b572a4cf1711ffdbcb5810bf293ef199a2e2fbc3fe086e5c4dd5d797b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a65b28b572a4cf1711ffdbcb5810bf293ef199a2e2fbc3fe086e5c4dd5d797b3","first_computed_at":"2026-05-17T23:56:48.998220Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:48.998220Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H1IkSrbTWf74Xhg4Gf67ZNrQmBBNT11kO1oYO5JDBdg6+rwkM5ZXafIYOFhRvfEXqtazDZxEd20us+HOdsLyBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:48.998695Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.01947","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1bce9c64f35351ac7035e2f0b39939dbfbd2f19b59f9725d1c49966f73bfcbc4","sha256:4a3428b1cf382ae4f47f845ad656672ec881b1ce29777369d8536e2eae4e5081"],"state_sha256":"74c700576e878591bf012786627555be13d275d491526eb3f58341bb78111132"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xFJGdGC3Be9Mak31gJhHISKl5xSJhu6wUbKjRKeMIDyvSdlJ9OGRpSZcZbf3NxPIkSMEkWKhzDGTAOhQGq+LDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T18:23:04.288013Z","bundle_sha256":"9bbac03de11f40e0b7faf61b242da1a18557e16ba162e235a2f7fc7c5abeec9b"}}