{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:V2F56WFOVZBTE6YEWKEK3ROH4X","short_pith_number":"pith:V2F56WFO","canonical_record":{"source":{"id":"1808.00397","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-08-01T16:12:15Z","cross_cats_sorted":[],"title_canon_sha256":"6d3dabd2ed8d736e65d764834ccfae94327b015f3193a02f9e6fafab714488e6","abstract_canon_sha256":"7c5c00fd842980a4ec1438f345e3e7f380d98cc9742ba35280807109faeb0ab5"},"schema_version":"1.0"},"canonical_sha256":"ae8bdf58aeae43327b04b288adc5c7e5f02bfdf69c0c85864ccb183669d7bd7c","source":{"kind":"arxiv","id":"1808.00397","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.00397","created_at":"2026-05-18T00:09:06Z"},{"alias_kind":"arxiv_version","alias_value":"1808.00397v1","created_at":"2026-05-18T00:09:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.00397","created_at":"2026-05-18T00:09:06Z"},{"alias_kind":"pith_short_12","alias_value":"V2F56WFOVZBT","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V2F56WFOVZBTE6YE","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V2F56WFO","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:V2F56WFOVZBTE6YEWKEK3ROH4X","target":"record","payload":{"canonical_record":{"source":{"id":"1808.00397","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-08-01T16:12:15Z","cross_cats_sorted":[],"title_canon_sha256":"6d3dabd2ed8d736e65d764834ccfae94327b015f3193a02f9e6fafab714488e6","abstract_canon_sha256":"7c5c00fd842980a4ec1438f345e3e7f380d98cc9742ba35280807109faeb0ab5"},"schema_version":"1.0"},"canonical_sha256":"ae8bdf58aeae43327b04b288adc5c7e5f02bfdf69c0c85864ccb183669d7bd7c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:06.209562Z","signature_b64":"6MuT9gnNYVk0PlcrEBfNInGruOBTZBMowjscR8tEzwFAk6WBZLcBgsTv6j7r9Nu+L1wXPDrRcsFVJcOPzFHzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae8bdf58aeae43327b04b288adc5c7e5f02bfdf69c0c85864ccb183669d7bd7c","last_reissued_at":"2026-05-18T00:09:06.208908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:06.208908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1808.00397","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":"2026-05-18T00:09:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7K15HlxIN2xO4Qim7k5VVYcrtMcn3IdUU4wQgI2oLCl+5NoefN78nObAqc9y5jKUkgBMpO/wowRHNh2gRSjoDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T00:40:58.038337Z"},"content_sha256":"cec87cd4e27dbae195f49685410ebfc3903afc0b1aeecf0fb2df68b01a8508aa","schema_version":"1.0","event_id":"sha256:cec87cd4e27dbae195f49685410ebfc3903afc0b1aeecf0fb2df68b01a8508aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:V2F56WFOVZBTE6YEWKEK3ROH4X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Hao Zhu","submitted_at":"2018-08-01T16:12:15Z","abstract_excerpt":"In this paper, we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear Hamiltonian systems are equal, where both endpoints are in the limit circle cases. The proof is fundamental and is given for both continuous and discrete Hamiltonian systems. The method used in this paper also works for both endpoints are regular, or one endpoint is regular and the other is in the limit circle case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.00397","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":"2026-05-18T00:09:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uVQUV+0M3+9TANxa2q2kZ4LCUkOYEnzaYnrBzf7A9nNnWJxR1QRfmaL53go4x/0SOwtDyDX/as3IeJapWBWIBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T00:40:58.038688Z"},"content_sha256":"d468d27e9494405993fc0071069f04d7b49c1779986d59a9439d07488a116c98","schema_version":"1.0","event_id":"sha256:d468d27e9494405993fc0071069f04d7b49c1779986d59a9439d07488a116c98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V2F56WFOVZBTE6YEWKEK3ROH4X/bundle.json","state_url":"https://pith.science/pith/V2F56WFOVZBTE6YEWKEK3ROH4X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V2F56WFOVZBTE6YEWKEK3ROH4X/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":"2026-06-03T00:40:58Z","links":{"resolver":"https://pith.science/pith/V2F56WFOVZBTE6YEWKEK3ROH4X","bundle":"https://pith.science/pith/V2F56WFOVZBTE6YEWKEK3ROH4X/bundle.json","state":"https://pith.science/pith/V2F56WFOVZBTE6YEWKEK3ROH4X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V2F56WFOVZBTE6YEWKEK3ROH4X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:V2F56WFOVZBTE6YEWKEK3ROH4X","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":"7c5c00fd842980a4ec1438f345e3e7f380d98cc9742ba35280807109faeb0ab5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-08-01T16:12:15Z","title_canon_sha256":"6d3dabd2ed8d736e65d764834ccfae94327b015f3193a02f9e6fafab714488e6"},"schema_version":"1.0","source":{"id":"1808.00397","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.00397","created_at":"2026-05-18T00:09:06Z"},{"alias_kind":"arxiv_version","alias_value":"1808.00397v1","created_at":"2026-05-18T00:09:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.00397","created_at":"2026-05-18T00:09:06Z"},{"alias_kind":"pith_short_12","alias_value":"V2F56WFOVZBT","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V2F56WFOVZBTE6YE","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V2F56WFO","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:d468d27e9494405993fc0071069f04d7b49c1779986d59a9439d07488a116c98","target":"graph","created_at":"2026-05-18T00:09:06Z","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 this paper, we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear Hamiltonian systems are equal, where both endpoints are in the limit circle cases. The proof is fundamental and is given for both continuous and discrete Hamiltonian systems. The method used in this paper also works for both endpoints are regular, or one endpoint is regular and the other is in the limit circle case.","authors_text":"Hao Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-08-01T16:12:15Z","title":"A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.00397","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:cec87cd4e27dbae195f49685410ebfc3903afc0b1aeecf0fb2df68b01a8508aa","target":"record","created_at":"2026-05-18T00:09:06Z","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":"7c5c00fd842980a4ec1438f345e3e7f380d98cc9742ba35280807109faeb0ab5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-08-01T16:12:15Z","title_canon_sha256":"6d3dabd2ed8d736e65d764834ccfae94327b015f3193a02f9e6fafab714488e6"},"schema_version":"1.0","source":{"id":"1808.00397","kind":"arxiv","version":1}},"canonical_sha256":"ae8bdf58aeae43327b04b288adc5c7e5f02bfdf69c0c85864ccb183669d7bd7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ae8bdf58aeae43327b04b288adc5c7e5f02bfdf69c0c85864ccb183669d7bd7c","first_computed_at":"2026-05-18T00:09:06.208908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:06.208908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6MuT9gnNYVk0PlcrEBfNInGruOBTZBMowjscR8tEzwFAk6WBZLcBgsTv6j7r9Nu+L1wXPDrRcsFVJcOPzFHzAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:06.209562Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.00397","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cec87cd4e27dbae195f49685410ebfc3903afc0b1aeecf0fb2df68b01a8508aa","sha256:d468d27e9494405993fc0071069f04d7b49c1779986d59a9439d07488a116c98"],"state_sha256":"7431f279451820705b93dff6d716319d58ab64d3d5d47640dbcf931d40d55e6f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9rMpHJneJXIyjUNP9aTgU8OKWWWmPYsIUNi/+IZHD4LYSKRfzQ6g5gSo+EM5V+iTMttDzQd+FbqSLTYN5WbnDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T00:40:58.040594Z","bundle_sha256":"7f163c2da3971f79b6a30c474becef52a3a01c9333cf6c270711b07f32483b71"}}