{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:AVTLAEN7YTJYN6H2XYODHNDZKZ","short_pith_number":"pith:AVTLAEN7","schema_version":"1.0","canonical_sha256":"0566b011bfc4d386f8fabe1c33b479565f3a1d854c7e0137dc870dea1f4c3234","source":{"kind":"arxiv","id":"1411.5087","version":4},"attestation_state":"computed","paper":{"title":"Volume doubling, Poincar\\'e inequality and Guassian heat kernel estimate for nonnegative curvature graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.DG","authors_text":"Paul Horn, Shing-Tung Yau, Shuang Liu, Yong Lin","submitted_at":"2014-11-19T01:26:59Z","abstract_excerpt":"By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincar\\'e inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is "},"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":"1411.5087","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-11-19T01:26:59Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"1b00552931bea208c08feae3e83d302bb8ea5b777a587817d0154de8a7adbb6c","abstract_canon_sha256":"aaeaa78ca950b4a7b833adf5632b5c2fb1aac9dadac4526f93374001a37a7a8e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:16.065000Z","signature_b64":"K8igurHmrqdV/nadc3u7lP10u77EQAHQaQntnvVUAJuy8WRSBvl5Lc8llIaFCMqazTfYNn6JydJtlXm3VMOSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0566b011bfc4d386f8fabe1c33b479565f3a1d854c7e0137dc870dea1f4c3234","last_reissued_at":"2026-05-18T01:25:16.064548Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:16.064548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Volume doubling, Poincar\\'e inequality and Guassian heat kernel estimate for nonnegative curvature graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.DG","authors_text":"Paul Horn, Shing-Tung Yau, Shuang Liu, Yong Lin","submitted_at":"2014-11-19T01:26:59Z","abstract_excerpt":"By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincar\\'e inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5087","kind":"arxiv","version":4},"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":"1411.5087","created_at":"2026-05-18T01:25:16.064617+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.5087v4","created_at":"2026-05-18T01:25:16.064617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5087","created_at":"2026-05-18T01:25:16.064617+00:00"},{"alias_kind":"pith_short_12","alias_value":"AVTLAEN7YTJY","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"AVTLAEN7YTJYN6H2","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"AVTLAEN7","created_at":"2026-05-18T12:28:19.803747+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/AVTLAEN7YTJYN6H2XYODHNDZKZ","json":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ.json","graph_json":"https://pith.science/api/pith-number/AVTLAEN7YTJYN6H2XYODHNDZKZ/graph.json","events_json":"https://pith.science/api/pith-number/AVTLAEN7YTJYN6H2XYODHNDZKZ/events.json","paper":"https://pith.science/paper/AVTLAEN7"},"agent_actions":{"view_html":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ","download_json":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ.json","view_paper":"https://pith.science/paper/AVTLAEN7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.5087&json=true","fetch_graph":"https://pith.science/api/pith-number/AVTLAEN7YTJYN6H2XYODHNDZKZ/graph.json","fetch_events":"https://pith.science/api/pith-number/AVTLAEN7YTJYN6H2XYODHNDZKZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ/action/storage_attestation","attest_author":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ/action/author_attestation","sign_citation":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ/action/citation_signature","submit_replication":"https://pith.science/pith/AVTLAEN7YTJYN6H2XYODHNDZKZ/action/replication_record"}},"created_at":"2026-05-18T01:25:16.064617+00:00","updated_at":"2026-05-18T01:25:16.064617+00:00"}