{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:RPGMSOEMICA5VXYF662RBC65TX","short_pith_number":"pith:RPGMSOEM","schema_version":"1.0","canonical_sha256":"8bccc9388c4081dadf05f7b5108bdd9dcdeee42d9ea5911bd15544372ac29ef0","source":{"kind":"arxiv","id":"1402.4232","version":1},"attestation_state":"computed","paper":{"title":"Harnack Estimates for Nonlinear Backward Heat Equations in Geometric Flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongxin Guo, Masashi Ishida","submitted_at":"2014-02-18T06:54:00Z","abstract_excerpt":"Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation \\begin{eqnarray*} \\frac{\\partial f}{\\partial t} = -{\\Delta}f + \\gamma f\\log f +aSf \\end{eqnarray*} where $a$ and $\\gamma$ are constants and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our abstract formulation provides a unified framework for some known results proved b"},"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":"1402.4232","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-02-18T06:54:00Z","cross_cats_sorted":[],"title_canon_sha256":"7b20ceeb8684c71016da9c7d8744f32818bbe9a2b47612aaca39e91114e21e83","abstract_canon_sha256":"687e518803136735acac183aa673987e2131ed16173c0481cc98d047be62dcdf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:44.433176Z","signature_b64":"kWTnI1rlpHtAbjdwG3h4YbxH9XSAiloJ/8i92fjr0dLrtHeXtBiWWLeMvMWwAFbj5pGmAYl8aKBgfMU09i38DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8bccc9388c4081dadf05f7b5108bdd9dcdeee42d9ea5911bd15544372ac29ef0","last_reissued_at":"2026-05-18T02:58:44.432536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:44.432536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harnack Estimates for Nonlinear Backward Heat Equations in Geometric Flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongxin Guo, Masashi Ishida","submitted_at":"2014-02-18T06:54:00Z","abstract_excerpt":"Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation \\begin{eqnarray*} \\frac{\\partial f}{\\partial t} = -{\\Delta}f + \\gamma f\\log f +aSf \\end{eqnarray*} where $a$ and $\\gamma$ are constants and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our abstract formulation provides a unified framework for some known results proved b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4232","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":"1402.4232","created_at":"2026-05-18T02:58:44.432645+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.4232v1","created_at":"2026-05-18T02:58:44.432645+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.4232","created_at":"2026-05-18T02:58:44.432645+00:00"},{"alias_kind":"pith_short_12","alias_value":"RPGMSOEMICA5","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"RPGMSOEMICA5VXYF","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"RPGMSOEM","created_at":"2026-05-18T12:28:46.137349+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/RPGMSOEMICA5VXYF662RBC65TX","json":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX.json","graph_json":"https://pith.science/api/pith-number/RPGMSOEMICA5VXYF662RBC65TX/graph.json","events_json":"https://pith.science/api/pith-number/RPGMSOEMICA5VXYF662RBC65TX/events.json","paper":"https://pith.science/paper/RPGMSOEM"},"agent_actions":{"view_html":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX","download_json":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX.json","view_paper":"https://pith.science/paper/RPGMSOEM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.4232&json=true","fetch_graph":"https://pith.science/api/pith-number/RPGMSOEMICA5VXYF662RBC65TX/graph.json","fetch_events":"https://pith.science/api/pith-number/RPGMSOEMICA5VXYF662RBC65TX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX/action/storage_attestation","attest_author":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX/action/author_attestation","sign_citation":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX/action/citation_signature","submit_replication":"https://pith.science/pith/RPGMSOEMICA5VXYF662RBC65TX/action/replication_record"}},"created_at":"2026-05-18T02:58:44.432645+00:00","updated_at":"2026-05-18T02:58:44.432645+00:00"}