{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:L4Q55RL5IE55BXACE5MPUB7FPM","short_pith_number":"pith:L4Q55RL5","canonical_record":{"source":{"id":"1005.2954","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2010-05-17T15:29:51Z","cross_cats_sorted":[],"title_canon_sha256":"19de6123cfe3de082546f349a9e0efba4aea28eea152fcaf1bd2326f431873b3","abstract_canon_sha256":"2dbef37f47fc43c6a9fa85aed1f7e6c9691f9829fba1857c20a2c930e73f16d9"},"schema_version":"1.0"},"canonical_sha256":"5f21dec57d413bd0dc022758fa07e57b1e2d7db0e90cdcea1a8589dda02b3832","source":{"kind":"arxiv","id":"1005.2954","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.2954","created_at":"2026-05-18T04:41:15Z"},{"alias_kind":"arxiv_version","alias_value":"1005.2954v3","created_at":"2026-05-18T04:41:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.2954","created_at":"2026-05-18T04:41:15Z"},{"alias_kind":"pith_short_12","alias_value":"L4Q55RL5IE55","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"L4Q55RL5IE55BXAC","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"L4Q55RL5","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:L4Q55RL5IE55BXACE5MPUB7FPM","target":"record","payload":{"canonical_record":{"source":{"id":"1005.2954","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2010-05-17T15:29:51Z","cross_cats_sorted":[],"title_canon_sha256":"19de6123cfe3de082546f349a9e0efba4aea28eea152fcaf1bd2326f431873b3","abstract_canon_sha256":"2dbef37f47fc43c6a9fa85aed1f7e6c9691f9829fba1857c20a2c930e73f16d9"},"schema_version":"1.0"},"canonical_sha256":"5f21dec57d413bd0dc022758fa07e57b1e2d7db0e90cdcea1a8589dda02b3832","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:15.082368Z","signature_b64":"7sax7aqocY/MNGBEGe3FrussfEVuc6UnZjdMV2zpTdwp84xuT9at4ajr/s+6R/1vmsswhG03Quc9xvWhWdv/BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f21dec57d413bd0dc022758fa07e57b1e2d7db0e90cdcea1a8589dda02b3832","last_reissued_at":"2026-05-18T04:41:15.081832Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:15.081832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.2954","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BzbTBLXXfBzDsly6S7lAUtkrp4vGSxZp1v25nfqbTdms8WnB/X0K6csH/CSowluDaUo1bQZuBPIUWJjZIhKWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:06:40.680149Z"},"content_sha256":"377afd6ce58861304035f0d568261a093564c1984f90c6bb8443f107cffc27de","schema_version":"1.0","event_id":"sha256:377afd6ce58861304035f0d568261a093564c1984f90c6bb8443f107cffc27de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:L4Q55RL5IE55BXACE5MPUB7FPM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operator","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Changyu Xia, Daguang Chen, Qiaoling Wang, Qing-Ming Cheng","submitted_at":"2010-05-17T15:29:51Z","abstract_excerpt":"Let $\\om $ be a bounded domain in an $n$-dimensional Euclidean space $\\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations:\n  $$  \\{\\aligned &\\Delta {\\mathbf u}+ \\alpha{\\rm grad}(\\text{div}{\\mathbf u})=-\\sigma {\\mathbf u}, \\ \\text{in $\\Omega$},\n  &{\\mathbf u}|_{\\partial \\Omega}={\\mathbf 0}. \\aligned . $$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the $(k+1)^{\\text{th}}$ eigenvalue $\\sigma_{k+1}$. We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.2954","kind":"arxiv","version":3},"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":"2026-05-18T04:41:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E2ElXl3VQ4gYWP/b6KaGugOjXZsN3XwtoDe+QdPXCjmxFPDebVwS5SGsADXMrgsQPaWBnJwDBDIuekd9sk6kDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:06:40.680847Z"},"content_sha256":"11af2b22594aa5eb3608e37fe2012f0669613c09f04751680c5bb1ec4e0c6e6b","schema_version":"1.0","event_id":"sha256:11af2b22594aa5eb3608e37fe2012f0669613c09f04751680c5bb1ec4e0c6e6b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L4Q55RL5IE55BXACE5MPUB7FPM/bundle.json","state_url":"https://pith.science/pith/L4Q55RL5IE55BXACE5MPUB7FPM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L4Q55RL5IE55BXACE5MPUB7FPM/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":"2026-05-25T16:06:40Z","links":{"resolver":"https://pith.science/pith/L4Q55RL5IE55BXACE5MPUB7FPM","bundle":"https://pith.science/pith/L4Q55RL5IE55BXACE5MPUB7FPM/bundle.json","state":"https://pith.science/pith/L4Q55RL5IE55BXACE5MPUB7FPM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L4Q55RL5IE55BXACE5MPUB7FPM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:L4Q55RL5IE55BXACE5MPUB7FPM","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":"2dbef37f47fc43c6a9fa85aed1f7e6c9691f9829fba1857c20a2c930e73f16d9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2010-05-17T15:29:51Z","title_canon_sha256":"19de6123cfe3de082546f349a9e0efba4aea28eea152fcaf1bd2326f431873b3"},"schema_version":"1.0","source":{"id":"1005.2954","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.2954","created_at":"2026-05-18T04:41:15Z"},{"alias_kind":"arxiv_version","alias_value":"1005.2954v3","created_at":"2026-05-18T04:41:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.2954","created_at":"2026-05-18T04:41:15Z"},{"alias_kind":"pith_short_12","alias_value":"L4Q55RL5IE55","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"L4Q55RL5IE55BXAC","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"L4Q55RL5","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:11af2b22594aa5eb3608e37fe2012f0669613c09f04751680c5bb1ec4e0c6e6b","target":"graph","created_at":"2026-05-18T04:41:15Z","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 $\\om $ be a bounded domain in an $n$-dimensional Euclidean space $\\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations:\n  $$  \\{\\aligned &\\Delta {\\mathbf u}+ \\alpha{\\rm grad}(\\text{div}{\\mathbf u})=-\\sigma {\\mathbf u}, \\ \\text{in $\\Omega$},\n  &{\\mathbf u}|_{\\partial \\Omega}={\\mathbf 0}. \\aligned . $$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the $(k+1)^{\\text{th}}$ eigenvalue $\\sigma_{k+1}$. We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problem","authors_text":"Changyu Xia, Daguang Chen, Qiaoling Wang, Qing-Ming Cheng","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2010-05-17T15:29:51Z","title":"Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.2954","kind":"arxiv","version":3},"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":"sha256:377afd6ce58861304035f0d568261a093564c1984f90c6bb8443f107cffc27de","target":"record","created_at":"2026-05-18T04:41:15Z","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":"2dbef37f47fc43c6a9fa85aed1f7e6c9691f9829fba1857c20a2c930e73f16d9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2010-05-17T15:29:51Z","title_canon_sha256":"19de6123cfe3de082546f349a9e0efba4aea28eea152fcaf1bd2326f431873b3"},"schema_version":"1.0","source":{"id":"1005.2954","kind":"arxiv","version":3}},"canonical_sha256":"5f21dec57d413bd0dc022758fa07e57b1e2d7db0e90cdcea1a8589dda02b3832","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f21dec57d413bd0dc022758fa07e57b1e2d7db0e90cdcea1a8589dda02b3832","first_computed_at":"2026-05-18T04:41:15.081832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:15.081832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7sax7aqocY/MNGBEGe3FrussfEVuc6UnZjdMV2zpTdwp84xuT9at4ajr/s+6R/1vmsswhG03Quc9xvWhWdv/BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:15.082368Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.2954","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:377afd6ce58861304035f0d568261a093564c1984f90c6bb8443f107cffc27de","sha256:11af2b22594aa5eb3608e37fe2012f0669613c09f04751680c5bb1ec4e0c6e6b"],"state_sha256":"4b0554136c22b08f5aa6deb76f392e70c7e924d4849e4ab14dfc536b6ba18c3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FSdw0atyvMlYFEBYSXJ+fpqC+lx++P4iNUbjCuDlFnf6yZuj0qGvJ3spA2BTrxGM0JXsQ8pZjILcYR7uZ2Y/Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T16:06:40.684384Z","bundle_sha256":"8af85f58a466fe425df946126a23463db69247efae9687905b9373c8ec72a53b"}}