{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:KYCQ2LEQKXY352OG47JNR5U44X","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":"b41627d6a64bb536a6eaba3816b559a555e00b0ef3f95d0975da6b37f444cd68","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-02-12T16:38:55Z","title_canon_sha256":"5b42c578e3f583ee6e96d4528057124c349734268ed9b299f2cc7227a18a8be1"},"schema_version":"1.0","source":{"id":"1402.2888","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.2888","created_at":"2026-05-18T02:25:26Z"},{"alias_kind":"arxiv_version","alias_value":"1402.2888v2","created_at":"2026-05-18T02:25:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.2888","created_at":"2026-05-18T02:25:26Z"},{"alias_kind":"pith_short_12","alias_value":"KYCQ2LEQKXY3","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KYCQ2LEQKXY352OG","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KYCQ2LEQ","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:761550c78a89dc0c7686b8a231e5e840195e0c3f61a9e741f7b8c09acc5ee585","target":"graph","created_at":"2026-05-18T02:25:26Z","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 $u$ and $v$ be harmonic in $ \\Omega \\subset \\mathbb{R}^n$ functions with the same zero set $Z$. We show that the ratio $f$ of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum principles. For $n=3$ we also prove the Harnack inequality and the gradient estimate for the ratios of harmonic functions, namely ${ \\sup\\limits_{K} |f| \\leq C \\inf\\limits_{K}| f| \\quad \\& \\quad \\sup\\limits_{K} |\\nabla f| \\leq C \\inf\\limits_{K}| f| }$ for any compact subset $K$ of $\\Omega$, where the constant $C$ depends on $K$, $Z$, $\\Omega$ only. In dimension ","authors_text":"Alexander Logunov, Eugenia Malinnikova","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-02-12T16:38:55Z","title":"On ratios of harmonic functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2888","kind":"arxiv","version":2},"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:acdfb643261e57063934f6a29a5e090a6b847bcae688a2e1b7e51a1dff5b095f","target":"record","created_at":"2026-05-18T02:25:26Z","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":"b41627d6a64bb536a6eaba3816b559a555e00b0ef3f95d0975da6b37f444cd68","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-02-12T16:38:55Z","title_canon_sha256":"5b42c578e3f583ee6e96d4528057124c349734268ed9b299f2cc7227a18a8be1"},"schema_version":"1.0","source":{"id":"1402.2888","kind":"arxiv","version":2}},"canonical_sha256":"56050d2c9055f1bee9c6e7d2d8f69ce5ed47b7a9e4ffab776bffdea26399cee1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56050d2c9055f1bee9c6e7d2d8f69ce5ed47b7a9e4ffab776bffdea26399cee1","first_computed_at":"2026-05-18T02:25:26.599114Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:26.599114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r58Sku++1Or37/ix4qfmItxa1V1CPW6V5/wL/i5+nDfOBiZn2eyz/9jT/AMjD8bxYxk6HBeGLJCTzU8hCC6oAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:26.599537Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.2888","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:acdfb643261e57063934f6a29a5e090a6b847bcae688a2e1b7e51a1dff5b095f","sha256:761550c78a89dc0c7686b8a231e5e840195e0c3f61a9e741f7b8c09acc5ee585"],"state_sha256":"8aeee673799ca9090aa67104ad973ebc8899d5d1c9814112a56e78ac585cf0f5"}