{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:S7GU7FCGX5VXVACHEKRGE222XP","short_pith_number":"pith:S7GU7FCG","schema_version":"1.0","canonical_sha256":"97cd4f9446bf6b7a804722a2626b5abbce42e92164d4e17af1099db4d1b21b4e","source":{"kind":"arxiv","id":"1905.13373","version":1},"attestation_state":"computed","paper":{"title":"Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongge Chen, Hua Chen","submitted_at":"2019-05-31T01:34:52Z","abstract_excerpt":"Let $\\Omega$ be a bounded connected open subset in $\\mathbb{R}^n$ with smooth boundary $\\partial\\Omega$. Suppose that we have a system of real smooth vector fields $X=(X_{1},X_{2},$ $\\cdots,X_{m})$ defined on a neighborhood of $\\overline{\\Omega}$ that satisfies the H\\\"{o}rmander's condition. Suppose further that $\\partial\\Omega$ is non-characteristic with respect to $X$. For a self-adjoint sub-elliptic operator $\\triangle_{X}= -\\sum_{i=1}^{m}X_{i}^{*} X_i$ on $\\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by $\\lambda_k$. We will provide an uniform upper bound for the sub-elliptic Dirich"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.13373","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-05-31T01:34:52Z","cross_cats_sorted":[],"title_canon_sha256":"2ec5fed1f757f40601da8005c49c61ac47b0648422c04fa9f8a1e8c6caacc65a","abstract_canon_sha256":"552c75578d8fda22459d8d990dfab281cea6d5ab3045ada4e91d4df3c4fe1314"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:35.126932Z","signature_b64":"inlJy16CPaeQoNR3PWS+M6dee3N88sJwIzhj8quOyEntYAk3LDoOR/KP7jHs8otzOKpTqDmHjf9PMO6Ld7A6Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97cd4f9446bf6b7a804722a2626b5abbce42e92164d4e17af1099db4d1b21b4e","last_reissued_at":"2026-05-17T23:44:35.126334Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:35.126334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongge Chen, Hua Chen","submitted_at":"2019-05-31T01:34:52Z","abstract_excerpt":"Let $\\Omega$ be a bounded connected open subset in $\\mathbb{R}^n$ with smooth boundary $\\partial\\Omega$. Suppose that we have a system of real smooth vector fields $X=(X_{1},X_{2},$ $\\cdots,X_{m})$ defined on a neighborhood of $\\overline{\\Omega}$ that satisfies the H\\\"{o}rmander's condition. Suppose further that $\\partial\\Omega$ is non-characteristic with respect to $X$. For a self-adjoint sub-elliptic operator $\\triangle_{X}= -\\sum_{i=1}^{m}X_{i}^{*} X_i$ on $\\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by $\\lambda_k$. We will provide an uniform upper bound for the sub-elliptic Dirich"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.13373","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1905.13373","created_at":"2026-05-17T23:44:35.126439+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.13373v1","created_at":"2026-05-17T23:44:35.126439+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.13373","created_at":"2026-05-17T23:44:35.126439+00:00"},{"alias_kind":"pith_short_12","alias_value":"S7GU7FCGX5VX","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_16","alias_value":"S7GU7FCGX5VXVACH","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_8","alias_value":"S7GU7FCG","created_at":"2026-05-18T12:33:27.125529+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP","json":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP.json","graph_json":"https://pith.science/api/pith-number/S7GU7FCGX5VXVACHEKRGE222XP/graph.json","events_json":"https://pith.science/api/pith-number/S7GU7FCGX5VXVACHEKRGE222XP/events.json","paper":"https://pith.science/paper/S7GU7FCG"},"agent_actions":{"view_html":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP","download_json":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP.json","view_paper":"https://pith.science/paper/S7GU7FCG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.13373&json=true","fetch_graph":"https://pith.science/api/pith-number/S7GU7FCGX5VXVACHEKRGE222XP/graph.json","fetch_events":"https://pith.science/api/pith-number/S7GU7FCGX5VXVACHEKRGE222XP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP/action/storage_attestation","attest_author":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP/action/author_attestation","sign_citation":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP/action/citation_signature","submit_replication":"https://pith.science/pith/S7GU7FCGX5VXVACHEKRGE222XP/action/replication_record"}},"created_at":"2026-05-17T23:44:35.126439+00:00","updated_at":"2026-05-17T23:44:35.126439+00:00"}