{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:PVGPNXWLMGZILJAX324CEXUJ5Z","short_pith_number":"pith:PVGPNXWL","schema_version":"1.0","canonical_sha256":"7d4cf6decb61b285a417deb8225e89ee5a170a5de2cf0dba18d4d746a4c45419","source":{"kind":"arxiv","id":"1207.4236","version":3},"attestation_state":"computed","paper":{"title":"Critical sets of elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Daniele Valtorta, Jeff Cheeger","submitted_at":"2012-07-17T23:54:28Z","abstract_excerpt":"Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\\Cr(u)\\equiv \\{x:|\\nabla u|(x)=0\\}$. The results are new even for harmonic functions on $\\dR^n$. Given such a $u$, the standard {\\it first order} stratification $\\{\\cS^k\\}$ of $u$ separates points $x$ based on the degrees of symmetry of the leading order polynomial of $u-u(x)$. In this paper we give a quantitative stratification $\\{\\cS^k_{\\eta,r}\\}$ of $u$, which separates points based on the number of {\\it almost} s"},"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":"1207.4236","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-07-17T23:54:28Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"a2b5e4ed75a55a5730c3be786549e1299b99c6dced983c7d321df789ffcc28c5","abstract_canon_sha256":"eff987922cb375f97157c0a9cbd5d0a35acdd17975185f8a1fee5d4a92a972cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:29.171850Z","signature_b64":"tiXX7g+SGyEi+Lk7T7hnF/u36aG2LKo4FLWfm8JIJ6WlqDrU4V7FUUVrddntkhDzNWtP0u0djip+57y0jyx9Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d4cf6decb61b285a417deb8225e89ee5a170a5de2cf0dba18d4d746a4c45419","last_reissued_at":"2026-05-18T03:16:29.171128Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:29.171128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical sets of elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Daniele Valtorta, Jeff Cheeger","submitted_at":"2012-07-17T23:54:28Z","abstract_excerpt":"Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\\Cr(u)\\equiv \\{x:|\\nabla u|(x)=0\\}$. The results are new even for harmonic functions on $\\dR^n$. Given such a $u$, the standard {\\it first order} stratification $\\{\\cS^k\\}$ of $u$ separates points $x$ based on the degrees of symmetry of the leading order polynomial of $u-u(x)$. In this paper we give a quantitative stratification $\\{\\cS^k_{\\eta,r}\\}$ of $u$, which separates points based on the number of {\\it almost} s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.4236","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1207.4236","created_at":"2026-05-18T03:16:29.171240+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.4236v3","created_at":"2026-05-18T03:16:29.171240+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.4236","created_at":"2026-05-18T03:16:29.171240+00:00"},{"alias_kind":"pith_short_12","alias_value":"PVGPNXWLMGZI","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PVGPNXWLMGZILJAX","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PVGPNXWL","created_at":"2026-05-18T12:27:18.751474+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/PVGPNXWLMGZILJAX324CEXUJ5Z","json":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z.json","graph_json":"https://pith.science/api/pith-number/PVGPNXWLMGZILJAX324CEXUJ5Z/graph.json","events_json":"https://pith.science/api/pith-number/PVGPNXWLMGZILJAX324CEXUJ5Z/events.json","paper":"https://pith.science/paper/PVGPNXWL"},"agent_actions":{"view_html":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z","download_json":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z.json","view_paper":"https://pith.science/paper/PVGPNXWL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.4236&json=true","fetch_graph":"https://pith.science/api/pith-number/PVGPNXWLMGZILJAX324CEXUJ5Z/graph.json","fetch_events":"https://pith.science/api/pith-number/PVGPNXWLMGZILJAX324CEXUJ5Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z/action/storage_attestation","attest_author":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z/action/author_attestation","sign_citation":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z/action/citation_signature","submit_replication":"https://pith.science/pith/PVGPNXWLMGZILJAX324CEXUJ5Z/action/replication_record"}},"created_at":"2026-05-18T03:16:29.171240+00:00","updated_at":"2026-05-18T03:16:29.171240+00:00"}