{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:VLBMJWL7ORLM4XFPQEMPVC242D","short_pith_number":"pith:VLBMJWL7","canonical_record":{"source":{"id":"1903.05183","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-03-12T20:07:35Z","cross_cats_sorted":[],"title_canon_sha256":"fd77742cb1e9514810f0c80f2bd2d633ddffc26de01674c3823760f768c06f04","abstract_canon_sha256":"ee587549b67e28ad9ef3d2f15e3df3fa63cf6f9eea32d2dbea5acfe5a879e72b"},"schema_version":"1.0"},"canonical_sha256":"aac2c4d97f7456ce5caf8118fa8b5cd0dc8d8b0520a1d7989d68529132e4a05b","source":{"kind":"arxiv","id":"1903.05183","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.05183","created_at":"2026-05-17T23:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"1903.05183v2","created_at":"2026-05-17T23:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.05183","created_at":"2026-05-17T23:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"VLBMJWL7ORLM","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VLBMJWL7ORLM4XFP","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VLBMJWL7","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:VLBMJWL7ORLM4XFPQEMPVC242D","target":"record","payload":{"canonical_record":{"source":{"id":"1903.05183","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-03-12T20:07:35Z","cross_cats_sorted":[],"title_canon_sha256":"fd77742cb1e9514810f0c80f2bd2d633ddffc26de01674c3823760f768c06f04","abstract_canon_sha256":"ee587549b67e28ad9ef3d2f15e3df3fa63cf6f9eea32d2dbea5acfe5a879e72b"},"schema_version":"1.0"},"canonical_sha256":"aac2c4d97f7456ce5caf8118fa8b5cd0dc8d8b0520a1d7989d68529132e4a05b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:09.432877Z","signature_b64":"Mn1ytSOxPzPzDzxoC9xkYfZm0nGWgWWlRnQh4iHBgplcmKQ29c9EhdGWNb3tfDoGii7AeSqWN0A9P9Lbwy2hCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aac2c4d97f7456ce5caf8118fa8b5cd0dc8d8b0520a1d7989d68529132e4a05b","last_reissued_at":"2026-05-17T23:41:09.432263Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:09.432263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.05183","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eO977IhoOvaNAHtn2ZyHvSJECszC9kYw50Iih9pPrOR3tVgdWYIQMTLRgAx0Ik2kYk0tS+I6x/nDPoYOR1ZlCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:47:07.689774Z"},"content_sha256":"c3295e72b999cec99752540d704122d78f86ee3e20ecefe0ef516ad770578879","schema_version":"1.0","event_id":"sha256:c3295e72b999cec99752540d704122d78f86ee3e20ecefe0ef516ad770578879"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:VLBMJWL7ORLM4XFPQEMPVC242D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singularities of Base Polynomials and Gau-Wu Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ilya M. Spitkovsky, Kristin A. Camenga, Louis Deaett, Patrick X. Rault, Rebekah B. Johnson Yates, Tsvetanka Sendova","submitted_at":"2019-03-12T20:07:35Z","abstract_excerpt":"In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an $n\\times n$ matrix $A$, which counts the maximal number of orthonormal vectors $\\textbf x_j$ such that the scalar products $\\langle A\\textbf x_j,\\textbf x_j\\rangle$ lie on the boundary of the numerical range $W(A)$. We refer to $k(A)$ as the Gau--Wu number of the matrix $A$. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify $k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.05183","kind":"arxiv","version":2},"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":"2026-05-17T23:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JAPwwgbzDSwg73XQlf/ZMWV801BOUfDv0pQuk3bDWSWmeE/pqBciDHEWPxsrE5NKWSD1eY+PWlGwSJP4U4inDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:47:07.690388Z"},"content_sha256":"90f4ab4e496961a1d96404b07937c386f23a47f01d1342edb6269e9a6c5e7d13","schema_version":"1.0","event_id":"sha256:90f4ab4e496961a1d96404b07937c386f23a47f01d1342edb6269e9a6c5e7d13"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VLBMJWL7ORLM4XFPQEMPVC242D/bundle.json","state_url":"https://pith.science/pith/VLBMJWL7ORLM4XFPQEMPVC242D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VLBMJWL7ORLM4XFPQEMPVC242D/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":"2026-05-27T15:47:07Z","links":{"resolver":"https://pith.science/pith/VLBMJWL7ORLM4XFPQEMPVC242D","bundle":"https://pith.science/pith/VLBMJWL7ORLM4XFPQEMPVC242D/bundle.json","state":"https://pith.science/pith/VLBMJWL7ORLM4XFPQEMPVC242D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VLBMJWL7ORLM4XFPQEMPVC242D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VLBMJWL7ORLM4XFPQEMPVC242D","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":"ee587549b67e28ad9ef3d2f15e3df3fa63cf6f9eea32d2dbea5acfe5a879e72b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-03-12T20:07:35Z","title_canon_sha256":"fd77742cb1e9514810f0c80f2bd2d633ddffc26de01674c3823760f768c06f04"},"schema_version":"1.0","source":{"id":"1903.05183","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.05183","created_at":"2026-05-17T23:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"1903.05183v2","created_at":"2026-05-17T23:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.05183","created_at":"2026-05-17T23:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"VLBMJWL7ORLM","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VLBMJWL7ORLM4XFP","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VLBMJWL7","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:90f4ab4e496961a1d96404b07937c386f23a47f01d1342edb6269e9a6c5e7d13","target":"graph","created_at":"2026-05-17T23:41:09Z","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":"In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an $n\\times n$ matrix $A$, which counts the maximal number of orthonormal vectors $\\textbf x_j$ such that the scalar products $\\langle A\\textbf x_j,\\textbf x_j\\rangle$ lie on the boundary of the numerical range $W(A)$. We refer to $k(A)$ as the Gau--Wu number of the matrix $A$. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify $k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu nu","authors_text":"Ilya M. Spitkovsky, Kristin A. Camenga, Louis Deaett, Patrick X. Rault, Rebekah B. Johnson Yates, Tsvetanka Sendova","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-03-12T20:07:35Z","title":"Singularities of Base Polynomials and Gau-Wu Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.05183","kind":"arxiv","version":2},"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:c3295e72b999cec99752540d704122d78f86ee3e20ecefe0ef516ad770578879","target":"record","created_at":"2026-05-17T23:41:09Z","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":"ee587549b67e28ad9ef3d2f15e3df3fa63cf6f9eea32d2dbea5acfe5a879e72b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-03-12T20:07:35Z","title_canon_sha256":"fd77742cb1e9514810f0c80f2bd2d633ddffc26de01674c3823760f768c06f04"},"schema_version":"1.0","source":{"id":"1903.05183","kind":"arxiv","version":2}},"canonical_sha256":"aac2c4d97f7456ce5caf8118fa8b5cd0dc8d8b0520a1d7989d68529132e4a05b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aac2c4d97f7456ce5caf8118fa8b5cd0dc8d8b0520a1d7989d68529132e4a05b","first_computed_at":"2026-05-17T23:41:09.432263Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:09.432263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mn1ytSOxPzPzDzxoC9xkYfZm0nGWgWWlRnQh4iHBgplcmKQ29c9EhdGWNb3tfDoGii7AeSqWN0A9P9Lbwy2hCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:09.432877Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.05183","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3295e72b999cec99752540d704122d78f86ee3e20ecefe0ef516ad770578879","sha256:90f4ab4e496961a1d96404b07937c386f23a47f01d1342edb6269e9a6c5e7d13"],"state_sha256":"5c03819d93a9113522577262009610203cc29b40d504d194eee93764e808751c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1bFd0WqwTx/uKoYIQrNqsbzLp9nUI4Uw/4Bc2OVXyjsAHaXcu4aZgjWPBBWUxxdHZSZdV9PS07nyXO4rPW+eCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T15:47:07.693710Z","bundle_sha256":"b25a322d1f3ccf501112a42b7727473b30e099046331848ad03c986c982e2c59"}}