{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:WC54VOSEXUOQIBD35FICK7TTAW","short_pith_number":"pith:WC54VOSE","canonical_record":{"source":{"id":"1207.3240","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-07-13T13:32:32Z","cross_cats_sorted":["math.FA","math.SP"],"title_canon_sha256":"3b53c7a26c4bcd7177474d13662fe13262ebde5847f7e1c9066add7d6f65da1c","abstract_canon_sha256":"d359acc67229de2e76d229a36e646494cf261ba2bdc691a57a9802271c19c800"},"schema_version":"1.0"},"canonical_sha256":"b0bbcaba44bd1d04047be950257e7305829e28c10a7388a651946ccb2f97da80","source":{"kind":"arxiv","id":"1207.3240","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3240","created_at":"2026-05-18T01:01:52Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3240v2","created_at":"2026-05-18T01:01:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3240","created_at":"2026-05-18T01:01:52Z"},{"alias_kind":"pith_short_12","alias_value":"WC54VOSEXUOQ","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WC54VOSEXUOQIBD3","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WC54VOSE","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:WC54VOSEXUOQIBD35FICK7TTAW","target":"record","payload":{"canonical_record":{"source":{"id":"1207.3240","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-07-13T13:32:32Z","cross_cats_sorted":["math.FA","math.SP"],"title_canon_sha256":"3b53c7a26c4bcd7177474d13662fe13262ebde5847f7e1c9066add7d6f65da1c","abstract_canon_sha256":"d359acc67229de2e76d229a36e646494cf261ba2bdc691a57a9802271c19c800"},"schema_version":"1.0"},"canonical_sha256":"b0bbcaba44bd1d04047be950257e7305829e28c10a7388a651946ccb2f97da80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:52.710479Z","signature_b64":"YnuMcwlI5J6lQ8iZ7AcPb8ARA6Abj9EkZXe7+saGWqKWipjdu9wba70sv7CkRUBTDC/LX5JIQ14u7ViO987+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0bbcaba44bd1d04047be950257e7305829e28c10a7388a651946ccb2f97da80","last_reissued_at":"2026-05-18T01:01:52.709870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:52.709870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.3240","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":"2026-05-18T01:01:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mcnr8NnM3r8CP+/71bXg1H2t5Xr81oDHyCnS78GFYkmzElA6O4h9+aS7CST8nnFaps1NM9XjYOlHFA833ucgCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:49:26.916491Z"},"content_sha256":"4a535f6a49db3793a5dc8b9a5442405800897ebb0bd41982d6d187a7a3665dc6","schema_version":"1.0","event_id":"sha256:4a535f6a49db3793a5dc8b9a5442405800897ebb0bd41982d6d187a7a3665dc6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:WC54VOSEXUOQIBD35FICK7TTAW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds for the Rayleigh quotient and the spectrum of self-adjoint operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.SP"],"primary_cat":"math.NA","authors_text":"Andrew V. Knyazev, Merico E. Argentati, Peizhen Zhu","submitted_at":"2012-07-13T13:32:32Z","abstract_excerpt":"The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If $x$ is an eigenvector of a self-adjoint bounded operator $A$ in a Hilbert space, then the RQ of the vector $x$, denoted by $\\rho(x)$, is an exact eigenvalue of $A$. In this case, the absolute change of the RQ $|\\rho(x)-\\rho(y)|$ becomes the absolute error in an eigenvalue $\\rho(x)$ of $A$ approximated by the RQ $\\rho(y)$ on a given vector $y.$ There are three traditional kinds of bounds of the eigenvalue error: a priori bounds via the angle between ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3240","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":"2026-05-18T01:01:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y1uBhEe38SZSE47OJJA2+KDrEi900LnyiaMRA3h7Jn371Yb8oaxIrWXR/ZJhI5C2nBhAF6/Rh53Mon7VcZsADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:49:26.917058Z"},"content_sha256":"5dd7535d12d41fa4abf19038bfb09b6983a95f79895566e997489d5cb3191125","schema_version":"1.0","event_id":"sha256:5dd7535d12d41fa4abf19038bfb09b6983a95f79895566e997489d5cb3191125"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WC54VOSEXUOQIBD35FICK7TTAW/bundle.json","state_url":"https://pith.science/pith/WC54VOSEXUOQIBD35FICK7TTAW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WC54VOSEXUOQIBD35FICK7TTAW/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":"2026-05-27T16:49:26Z","links":{"resolver":"https://pith.science/pith/WC54VOSEXUOQIBD35FICK7TTAW","bundle":"https://pith.science/pith/WC54VOSEXUOQIBD35FICK7TTAW/bundle.json","state":"https://pith.science/pith/WC54VOSEXUOQIBD35FICK7TTAW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WC54VOSEXUOQIBD35FICK7TTAW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WC54VOSEXUOQIBD35FICK7TTAW","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":"d359acc67229de2e76d229a36e646494cf261ba2bdc691a57a9802271c19c800","cross_cats_sorted":["math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-07-13T13:32:32Z","title_canon_sha256":"3b53c7a26c4bcd7177474d13662fe13262ebde5847f7e1c9066add7d6f65da1c"},"schema_version":"1.0","source":{"id":"1207.3240","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3240","created_at":"2026-05-18T01:01:52Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3240v2","created_at":"2026-05-18T01:01:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3240","created_at":"2026-05-18T01:01:52Z"},{"alias_kind":"pith_short_12","alias_value":"WC54VOSEXUOQ","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WC54VOSEXUOQIBD3","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WC54VOSE","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:5dd7535d12d41fa4abf19038bfb09b6983a95f79895566e997489d5cb3191125","target":"graph","created_at":"2026-05-18T01:01:52Z","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":"The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If $x$ is an eigenvector of a self-adjoint bounded operator $A$ in a Hilbert space, then the RQ of the vector $x$, denoted by $\\rho(x)$, is an exact eigenvalue of $A$. In this case, the absolute change of the RQ $|\\rho(x)-\\rho(y)|$ becomes the absolute error in an eigenvalue $\\rho(x)$ of $A$ approximated by the RQ $\\rho(y)$ on a given vector $y.$ There are three traditional kinds of bounds of the eigenvalue error: a priori bounds via the angle between ve","authors_text":"Andrew V. Knyazev, Merico E. Argentati, Peizhen Zhu","cross_cats":["math.FA","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-07-13T13:32:32Z","title":"Bounds for the Rayleigh quotient and the spectrum of self-adjoint operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3240","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":"sha256:4a535f6a49db3793a5dc8b9a5442405800897ebb0bd41982d6d187a7a3665dc6","target":"record","created_at":"2026-05-18T01:01:52Z","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":"d359acc67229de2e76d229a36e646494cf261ba2bdc691a57a9802271c19c800","cross_cats_sorted":["math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-07-13T13:32:32Z","title_canon_sha256":"3b53c7a26c4bcd7177474d13662fe13262ebde5847f7e1c9066add7d6f65da1c"},"schema_version":"1.0","source":{"id":"1207.3240","kind":"arxiv","version":2}},"canonical_sha256":"b0bbcaba44bd1d04047be950257e7305829e28c10a7388a651946ccb2f97da80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0bbcaba44bd1d04047be950257e7305829e28c10a7388a651946ccb2f97da80","first_computed_at":"2026-05-18T01:01:52.709870Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:52.709870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YnuMcwlI5J6lQ8iZ7AcPb8ARA6Abj9EkZXe7+saGWqKWipjdu9wba70sv7CkRUBTDC/LX5JIQ14u7ViO987+Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:52.710479Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.3240","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a535f6a49db3793a5dc8b9a5442405800897ebb0bd41982d6d187a7a3665dc6","sha256:5dd7535d12d41fa4abf19038bfb09b6983a95f79895566e997489d5cb3191125"],"state_sha256":"6b6390db1f6133807f1a6103f7e5a95b97439425817db07492dda08a2c6ad5d5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/16UICMsHK0VPuG3zO7+LnT13nCybWrpdkfTv8Dzs85oUhiP610XJQc58bvAnJ0Y3UqkLXbHvGNtJl4p/MPxCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T16:49:26.920986Z","bundle_sha256":"b438616a7d642de6d7befe061c19457b800ca66f25ae6c983ec2704f695ed143"}}