{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ZYJHLKCQ3PZUS23BU7VWTGCYLS","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":"1fb31d0aa2634c43ae3603f2e641f6723881e24dafd6e85d6cc6dfbaae84a549","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-03T20:00:02Z","title_canon_sha256":"1a244799aaaea492e29985b3cff107c9958e01c8f6f20b80fef06afef16f097c"},"schema_version":"1.0","source":{"id":"1306.0565","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.0565","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"arxiv_version","alias_value":"1306.0565v1","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0565","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"pith_short_12","alias_value":"ZYJHLKCQ3PZU","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZYJHLKCQ3PZUS23B","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZYJHLKCQ","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:901806c9f52fd3dc4c47255c0aeea433d24d81371b75cc3390a705a4e558ac70","target":"graph","created_at":"2026-05-18T01:25:39Z","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 u, v be two harmonic functions in the disk of radius two which have exactly the same set Z of zeros. We observe that the gradient of \\log |u/v| is bounded in the unit disk by a constant which depends on Z only. In case Z is empty this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives H\\\"older estimates on \\log |u/v|.","authors_text":"Dan Mangoubi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-03T20:00:02Z","title":"A gradient estimate for harmonic functions sharing the same zeros"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0565","kind":"arxiv","version":1},"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":"sha256:40b3da5effc38657906bc7a33d18d24a8d1bdfce9ade64ccdabf5c0bdf8d601c","target":"record","created_at":"2026-05-18T01:25:39Z","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":"1fb31d0aa2634c43ae3603f2e641f6723881e24dafd6e85d6cc6dfbaae84a549","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-06-03T20:00:02Z","title_canon_sha256":"1a244799aaaea492e29985b3cff107c9958e01c8f6f20b80fef06afef16f097c"},"schema_version":"1.0","source":{"id":"1306.0565","kind":"arxiv","version":1}},"canonical_sha256":"ce1275a850dbf3496b61a7eb6998585cad82d26efddd9b22f2e786900ffcbdbe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce1275a850dbf3496b61a7eb6998585cad82d26efddd9b22f2e786900ffcbdbe","first_computed_at":"2026-05-18T01:25:39.987782Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:39.987782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4J5BFncpvRNZ9UbrA4UFFxN8WDY1yci+cYkGGpu+1KlHwhk7Ns3ix+HOsdq+5uSDYGpKm29vq2EDocYkVzVLDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:39.988373Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.0565","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40b3da5effc38657906bc7a33d18d24a8d1bdfce9ade64ccdabf5c0bdf8d601c","sha256:901806c9f52fd3dc4c47255c0aeea433d24d81371b75cc3390a705a4e558ac70"],"state_sha256":"5b33bdc05f314c307fff0984d3605ac75100a459ee759d45e680e7bc46545cd9"}