{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:P2I6IYUP42TARDXYTT3BC4VOWE","short_pith_number":"pith:P2I6IYUP","schema_version":"1.0","canonical_sha256":"7e91e4628fe6a6088ef89cf61172aeb132b44f66b8825011f259b5b452cb4e4b","source":{"kind":"arxiv","id":"1801.02114","version":1},"attestation_state":"computed","paper":{"title":"Measure Upper Bounds of Nodal Sets of Robin Eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fang Liu, Long Tian, Xiaoping Yang","submitted_at":"2018-01-07T02:32:20Z","abstract_excerpt":"In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e.,\n  \\begin{equation*} {\\left\\{\\begin{array}{l}\n  \\triangle u+\\lambda u=0,\\quad in\\quad \\Omega,\\\\ u_{\\nu}+\\mu u=0,\\quad on\\quad\\partial\\Omega, \\end{array} \\right.} \\end{equation*} where the domain $\\Omega\\subseteq\\mathbb{R}^n$, $u_{\\nu}$ means the derivative of $u$ along the outer normal direction of $\\partial\\Omega$. We show that, if $\\Omega$ is bounded and analytic, and the corresponding eigenvalue $\\lambda$ is large enough,then the measure upper bounds "},"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":"1801.02114","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-07T02:32:20Z","cross_cats_sorted":[],"title_canon_sha256":"2973918b01e2f112e77435a2ad3a3622253b930420b0a93791b4cdca57b0f428","abstract_canon_sha256":"4c0164f6efad53d772166e56e00caa11e5440c374661183f272b52642ba25cb3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:34.840867Z","signature_b64":"tnoCxeVSfTtcwBwTCCfvM1+deKVYMA+PTXdJn4NI7E9sUt7rmxRJYUEjUe61P9iVKPpiQVOm6T6Ji+bbJ3UeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e91e4628fe6a6088ef89cf61172aeb132b44f66b8825011f259b5b452cb4e4b","last_reissued_at":"2026-05-18T00:26:34.840198Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:34.840198Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Measure Upper Bounds of Nodal Sets of Robin Eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fang Liu, Long Tian, Xiaoping Yang","submitted_at":"2018-01-07T02:32:20Z","abstract_excerpt":"In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e.,\n  \\begin{equation*} {\\left\\{\\begin{array}{l}\n  \\triangle u+\\lambda u=0,\\quad in\\quad \\Omega,\\\\ u_{\\nu}+\\mu u=0,\\quad on\\quad\\partial\\Omega, \\end{array} \\right.} \\end{equation*} where the domain $\\Omega\\subseteq\\mathbb{R}^n$, $u_{\\nu}$ means the derivative of $u$ along the outer normal direction of $\\partial\\Omega$. We show that, if $\\Omega$ is bounded and analytic, and the corresponding eigenvalue $\\lambda$ is large enough,then the measure upper bounds "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02114","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":"1801.02114","created_at":"2026-05-18T00:26:34.840326+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.02114v1","created_at":"2026-05-18T00:26:34.840326+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.02114","created_at":"2026-05-18T00:26:34.840326+00:00"},{"alias_kind":"pith_short_12","alias_value":"P2I6IYUP42TA","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"P2I6IYUP42TARDXY","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"P2I6IYUP","created_at":"2026-05-18T12:32:43.782077+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/P2I6IYUP42TARDXYTT3BC4VOWE","json":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE.json","graph_json":"https://pith.science/api/pith-number/P2I6IYUP42TARDXYTT3BC4VOWE/graph.json","events_json":"https://pith.science/api/pith-number/P2I6IYUP42TARDXYTT3BC4VOWE/events.json","paper":"https://pith.science/paper/P2I6IYUP"},"agent_actions":{"view_html":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE","download_json":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE.json","view_paper":"https://pith.science/paper/P2I6IYUP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.02114&json=true","fetch_graph":"https://pith.science/api/pith-number/P2I6IYUP42TARDXYTT3BC4VOWE/graph.json","fetch_events":"https://pith.science/api/pith-number/P2I6IYUP42TARDXYTT3BC4VOWE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE/action/storage_attestation","attest_author":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE/action/author_attestation","sign_citation":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE/action/citation_signature","submit_replication":"https://pith.science/pith/P2I6IYUP42TARDXYTT3BC4VOWE/action/replication_record"}},"created_at":"2026-05-18T00:26:34.840326+00:00","updated_at":"2026-05-18T00:26:34.840326+00:00"}