{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:MOLZDPTWAFMFLB5CIS4MVGL3IR","short_pith_number":"pith:MOLZDPTW","canonical_record":{"source":{"id":"1310.4952","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-18T08:54:43Z","cross_cats_sorted":[],"title_canon_sha256":"cd5e58bda9ff4edec32aa0e901a370af6dbc215a249c748b9b583f2df8444152","abstract_canon_sha256":"efff7159a4c3eaf241b48f8d6808c4c239e55c416c4c25bd9b32753fa32af049"},"schema_version":"1.0"},"canonical_sha256":"639791be7601585587a244b8ca997b44512b87b9a18a4532b5dfd86eaff99caf","source":{"kind":"arxiv","id":"1310.4952","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.4952","created_at":"2026-05-18T03:10:01Z"},{"alias_kind":"arxiv_version","alias_value":"1310.4952v1","created_at":"2026-05-18T03:10:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.4952","created_at":"2026-05-18T03:10:01Z"},{"alias_kind":"pith_short_12","alias_value":"MOLZDPTWAFMF","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MOLZDPTWAFMFLB5C","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MOLZDPTW","created_at":"2026-05-18T12:27:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:MOLZDPTWAFMFLB5CIS4MVGL3IR","target":"record","payload":{"canonical_record":{"source":{"id":"1310.4952","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-18T08:54:43Z","cross_cats_sorted":[],"title_canon_sha256":"cd5e58bda9ff4edec32aa0e901a370af6dbc215a249c748b9b583f2df8444152","abstract_canon_sha256":"efff7159a4c3eaf241b48f8d6808c4c239e55c416c4c25bd9b32753fa32af049"},"schema_version":"1.0"},"canonical_sha256":"639791be7601585587a244b8ca997b44512b87b9a18a4532b5dfd86eaff99caf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:01.786802Z","signature_b64":"tQUlITjxNeNIbHWpwJOPpO1r1frGSxki7WNBLjqg4qFfneIzV2HU5fudYdU9B2RCLRc/h1oYMxi8DeE21d5mBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"639791be7601585587a244b8ca997b44512b87b9a18a4532b5dfd86eaff99caf","last_reissued_at":"2026-05-18T03:10:01.786227Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:01.786227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.4952","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":"2026-05-18T03:10:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tas58VbbZ9H/F/c1waTkoltlt5/vtxn4Em1wg/kOCqzIrYkoaVxkSduC++pksMgMyVxEOGmk1KosfP7m4QdCBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T16:57:31.190187Z"},"content_sha256":"c0375168ac237d7dbe44e72222c06572ea9ae24415b9b11dd8ac6e906f24a5c7","schema_version":"1.0","event_id":"sha256:c0375168ac237d7dbe44e72222c06572ea9ae24415b9b11dd8ac6e906f24a5c7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:MOLZDPTWAFMFLB5CIS4MVGL3IR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structures and Numerical Ranges of Power Partial Isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Pei Yuan Wu","submitted_at":"2013-10-18T08:54:43Z","abstract_excerpt":"We derive a matrix model, under unitary similarity, of an $n$-by-$n$ matrix $A$ such that $A, A^2, \\ldots, A^k$ ($k\\ge 1$) are all partial isometries, which generalizes the known fact that if $A$ is a partial isometry, then it is unitarily similar to a matrix of the form ${\\scriptsize\\left[\\begin{array}{cc} 0 & B 0 & C\\end{array}\\right]}$ with $B^*B+C^*C=I$. Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \\ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4952","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":"2026-05-18T03:10:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jjMPwpD+n9XwFcbEXPK3nN0hV+N9oDyG8FKqPC5BtTh3hmw4h8ncqtoIHXYEeLMDG8dTRcSeuEHXeFbkx7ncBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T16:57:31.190907Z"},"content_sha256":"bcc5f18c89feb6845dc374ba2e9b6cca40527c06ff525c366744c9a32f352dbd","schema_version":"1.0","event_id":"sha256:bcc5f18c89feb6845dc374ba2e9b6cca40527c06ff525c366744c9a32f352dbd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/bundle.json","state_url":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/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":"2026-06-11T16:57:31Z","links":{"resolver":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR","bundle":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/bundle.json","state":"https://pith.science/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MOLZDPTWAFMFLB5CIS4MVGL3IR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:MOLZDPTWAFMFLB5CIS4MVGL3IR","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":"efff7159a4c3eaf241b48f8d6808c4c239e55c416c4c25bd9b32753fa32af049","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-18T08:54:43Z","title_canon_sha256":"cd5e58bda9ff4edec32aa0e901a370af6dbc215a249c748b9b583f2df8444152"},"schema_version":"1.0","source":{"id":"1310.4952","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.4952","created_at":"2026-05-18T03:10:01Z"},{"alias_kind":"arxiv_version","alias_value":"1310.4952v1","created_at":"2026-05-18T03:10:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.4952","created_at":"2026-05-18T03:10:01Z"},{"alias_kind":"pith_short_12","alias_value":"MOLZDPTWAFMF","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MOLZDPTWAFMFLB5C","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MOLZDPTW","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:bcc5f18c89feb6845dc374ba2e9b6cca40527c06ff525c366744c9a32f352dbd","target":"graph","created_at":"2026-05-18T03:10:01Z","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":"We derive a matrix model, under unitary similarity, of an $n$-by-$n$ matrix $A$ such that $A, A^2, \\ldots, A^k$ ($k\\ge 1$) are all partial isometries, which generalizes the known fact that if $A$ is a partial isometry, then it is unitarily similar to a matrix of the form ${\\scriptsize\\left[\\begin{array}{cc} 0 & B 0 & C\\end{array}\\right]}$ with $B^*B+C^*C=I$. Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \\ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct","authors_text":"Hwa-Long Gau, Pei Yuan Wu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-18T08:54:43Z","title":"Structures and Numerical Ranges of Power Partial Isometries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4952","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:c0375168ac237d7dbe44e72222c06572ea9ae24415b9b11dd8ac6e906f24a5c7","target":"record","created_at":"2026-05-18T03:10:01Z","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":"efff7159a4c3eaf241b48f8d6808c4c239e55c416c4c25bd9b32753fa32af049","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-18T08:54:43Z","title_canon_sha256":"cd5e58bda9ff4edec32aa0e901a370af6dbc215a249c748b9b583f2df8444152"},"schema_version":"1.0","source":{"id":"1310.4952","kind":"arxiv","version":1}},"canonical_sha256":"639791be7601585587a244b8ca997b44512b87b9a18a4532b5dfd86eaff99caf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"639791be7601585587a244b8ca997b44512b87b9a18a4532b5dfd86eaff99caf","first_computed_at":"2026-05-18T03:10:01.786227Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:01.786227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tQUlITjxNeNIbHWpwJOPpO1r1frGSxki7WNBLjqg4qFfneIzV2HU5fudYdU9B2RCLRc/h1oYMxi8DeE21d5mBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:01.786802Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.4952","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c0375168ac237d7dbe44e72222c06572ea9ae24415b9b11dd8ac6e906f24a5c7","sha256:bcc5f18c89feb6845dc374ba2e9b6cca40527c06ff525c366744c9a32f352dbd"],"state_sha256":"695a462cfbd6bf911c608ac0d152630980c5ce176bc7d3ae47ac96d965873df4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6QAaekvCUoHBmIJDgGs24ygsNtKxNly3i1+OrgSjThFTRG+jFzbuiHL1GAqVPzH5JIHCAK2uSAvTiI6T5JTICw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T16:57:31.195172Z","bundle_sha256":"a2daffc9c2f84d92f22d3c4506a82a521fc50f7dc55ec5fb87774c906bdc049c"}}