{"as_of":"2026-08-19T12:58:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:a64865d2f2c2ae533525d43b39d1c79acd389f8266edb686e17d633451326ac3","coverage":[{"denominator":43,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":43,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-07T14:28:44.579343Z","state":"measured"},{"denominator":43,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":43,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-19T06:32:44.657259+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2505.19113/citation-record","integrity":"/paper/2505.19113/integrity","json":"/paper/2505.19113/citation-record.json","paper":"/paper/2505.19113"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:54.759109Z","title":"Bakry, M","venue":null,"work_id":"8fa801ab-4029-42a6-a7c1-f0cc8b17bee9","year":1983},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:39.893140Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:53fc653c8c67bcf7cb93bb5047444ac38edcfeb79e6b92cfc7a35f86382b4df4","observation_id":"af96b05f-19af-47a8-9639-e11a8209812f","resolution":{"observed_at":"2026-08-07T14:28:54.845720Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:54.471427Z","title":"Bakry, Z.-M","venue":null,"work_id":"f6df5c96-59bc-479f-9591-93d2dd83d5d3","year":2000},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.020798Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:26a1dafe75a702030af2bc4683c286c68ceb2f27d141abc03d2df0d167c24592","observation_id":"8e1b1160-ae3e-45e9-b806-933994e29bde","resolution":{"observed_at":"2026-08-07T14:28:54.592167Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:54.256189Z","title":null,"venue":null,"work_id":"d14bb354-1d2e-4eb9-853b-7dbd6439de71","year":2004},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.168304Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:e6f223f089e474398f9ecaec782867de18277204dd35361e06bcc3e41b1965b2","observation_id":"2cc948b6-624c-45a1-b468-96ffdcd33a76","resolution":{"observed_at":"2026-08-07T14:28:54.363592Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:54.016595Z","title":null,"venue":null,"work_id":"4de94544-57ea-4e6f-b56a-e1057f417a69","year":1989},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.292303Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:a6c3d449e5d546e6b58f9c55be80051bb93d4cecd1066d0dca38e045c238a70e","observation_id":"0d387a58-39da-42c5-a20c-791eee947bff","resolution":{"observed_at":"2026-08-07T14:28:54.111120Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:53.770611Z","title":"Fujitani, Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds withε- range,manuscripta math., vol","venue":null,"work_id":"2c3edd04-71a9-45bd-adf8-646a69665b66","year":2024},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.408281Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:bb536e32d635a355d87c876c0730178ab63f912cd11ca5ec24d8c4a7ce21cbed","observation_id":"58a76bec-928c-4cfb-b89a-10634b144ae5","resolution":{"observed_at":"2026-08-07T14:28:53.859378Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:53.563706Z","title":"Fujitani, Analysis of harmonic functions under lower bounds ofN−weighted Ricci curvature with ε−range,Journal of Mathematical Analysis and Applications, vol","venue":null,"work_id":"3e1a8a65-28ce-4dba-a2bc-786673942812","year":2025},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.574968Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:6a7cdd1af819ccaa9788faf94a143a5fee0160e7c7c6eb1f6fb24c5cbef46010","observation_id":"0f2158ed-f972-467a-9c02-5f5e4e620cb2","resolution":{"observed_at":"2026-08-07T14:28:53.672895Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:53.308267Z","title":"Grigor’yan, Heat Kernel and Analysis on Manifolds, AMS/IP Studies in Advanced Mathematics, vol","venue":null,"work_id":"695f710f-7aa3-4c66-9ac3-1d7ef88ea315","year":2009},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.685680Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:e69448dc7bb8605dfef58c568ec344666e9ce3ab5eb3daf2f48669277946afc7","observation_id":"ad19355e-0a1c-4f4a-a7d3-293a70fa8553","resolution":{"observed_at":"2026-08-07T14:28:53.451287Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:53.076024Z","title":null,"venue":null,"work_id":"988fb83f-557d-4765-9d51-e39286ead4e9","year":2021},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.815162Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:dd959b698583c5b3372f6a09d74b81d1657b46bb2b73832dcea1afc727916f01","observation_id":"843765f7-66ba-45d9-9347-d77e08bcf6a6","resolution":{"observed_at":"2026-08-07T14:28:53.197609Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:52.835837Z","title":"Hamilton, The formation of singularities in the Ricci flow, inSurveys in Differential Geometry, Vol","venue":null,"work_id":"792194dc-4928-49c1-bb9b-e75ac2c34d55","year":1993},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:40.967378Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:abcdcdbb65b0fd9457c0256de5a82dfc804982aec2a680ac409a3a0a7bd69a48","observation_id":"d693fe9b-99df-4c0c-b30f-c8ea2f10d134","resolution":{"observed_at":"2026-08-07T14:28:52.926583Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:52.581704Z","title":"Karp and P","venue":null,"work_id":"41fbbe97-6e68-437f-8b65-4b2cd85cb476","year":1982},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.107552Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:2e1976589fb0071407171a7c0086576a67ad67bf769c9b7d8786ef4ec8c3dfa3","observation_id":"9f692fb2-4a9b-42ff-864c-1db5f1dd4ade","resolution":{"observed_at":"2026-08-07T14:28:52.723096Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:52.352513Z","title":"Kuwae and X.-D","venue":null,"work_id":"38e00222-a5fd-4988-a2c9-ef350b7d8259","year":2022},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.234674Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:42a1ca5b5d6114a2cc470ff78e6e1fc4cbc6a9f0ca7c5ddc81c40d3582f8e8e9","observation_id":"ec258d86-c9af-4afd-aa1f-15c89972c7d4","resolution":{"observed_at":"2026-08-07T14:28:52.471333Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:52.144200Z","title":"Kuwae and Y","venue":null,"work_id":"43ac1972-92d5-4b8f-a214-1dbcde62c62d","year":2021},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.362897Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:acdb2cba00b0d141291d5bc0c7b0989ddd68a9a9ac4aeeee72f0a56347a0e366","observation_id":"514cdab4-f18b-4b37-89d3-e77d1fc61f32","resolution":{"observed_at":"2026-08-07T14:28:52.254103Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:51.909962Z","title":"Kuwae and Y","venue":null,"work_id":"82fe25bd-bc1a-4581-8f5d-a13added5182","year":2022},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.431282Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:0b0765e2eeb0830424bcdaa3c33be1449d329f3b65a16a841d620d0b3aeb5cfe","observation_id":"cc3c6bb0-aa61-43d1-8906-027588145197","resolution":{"observed_at":"2026-08-07T14:28:52.027829Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:51.666745Z","title":"Kuwae and Y","venue":null,"work_id":"85c04849-b991-4557-b187-64670d475bf6","year":2023},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.520987Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:c9a015f373d75116c7e41ac0416ca6c1d6563d027e16309fbc04056fc8bceee1","observation_id":"06b0f683-4864-4e52-96ab-3ca630cdba74","resolution":{"observed_at":"2026-08-07T14:28:51.784558Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:51.340380Z","title":null,"venue":null,"work_id":"4f24989a-5ae7-47c4-a9cc-459c311d3176","year":2009},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.617712Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:c375925e284c5dee18f82543e939fe1107866b0e11863ff20f575dea7adde223","observation_id":"346fbd78-34d9-4287-8269-78d644a41d82","resolution":{"observed_at":"2026-08-07T14:28:51.478654Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:51.081537Z","title":"Li, Geometric analysis","venue":null,"work_id":"d8cbe71c-3cc4-47c7-8651-33f44a969107","year":2012},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.748294Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:904ee7bc47a2b9a4996fc0c7119f772a7c5e6f481d970e50a25c5b6767cd0162","observation_id":"2b57d25e-c7b6-4033-9892-f64c185ef109","resolution":{"observed_at":"2026-08-07T14:28:51.203898Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:50.852348Z","title":"Li, Uniqueness ofL 1 solutions for the Laplace equation and the heat equation on Riemannian manifolds,Journal of Differential Geometry, vol","venue":null,"work_id":"98737676-702d-4fcf-b92b-ecb5900a91d1","year":1984},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.843985Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:935332c5cf996f8ebb840cda4397b69079bfa8080f75bf39a056f90a6ba25486","observation_id":"4f9837df-9087-4f3b-b59b-816d6b453b10","resolution":{"observed_at":"2026-08-07T14:28:50.924221Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:50.586808Z","title":null,"venue":null,"work_id":"2b89f34f-abd3-4bac-a43f-13db48cf4ca7","year":1984},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:41.935702Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:2f0057a38420767266313591a75538fa57541cf159a96f36d19b73d7773115b8","observation_id":"c5f3f62c-7cd3-40d7-aa7d-cf4c97a8b835","resolution":{"observed_at":"2026-08-07T14:28:50.687186Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:50.262532Z","title":"Li, S.-T","venue":null,"work_id":"47a417c8-db72-4d42-97e5-6edcaba91b98","year":1986},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.041313Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:bcb686eb3fda0df0b512fef07f43fa302290081907de4069e8fe484742424514","observation_id":"8db549e6-e83a-4d43-aba0-fc457d73e351","resolution":{"observed_at":"2026-08-07T14:28:50.408180Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:50.039279Z","title":"Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, Journal of Mathematics Pure and Applied, vol","venue":null,"work_id":"8b09bf91-9c1a-4238-80ce-86a7866ade44","year":2005},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.131772Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:6fb34151e37a25b1f180d83481f2ab66ae507d35e31aeba59af8b3df7140efdf","observation_id":"88da2ab9-2569-457e-a3df-d742c624a44a","resolution":{"observed_at":"2026-08-07T14:28:50.122027Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:49.792842Z","title":"Li, Li-Yau-Hamilton estimates and Bakry-Émery-Ricci curvature,Nonlinear Anal.113 (2015), 1–32, Elsevier, Amsterdam","venue":null,"work_id":"98b297c9-a0f9-4b12-b601-14aedf4623fc","year":2015},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.253978Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:5e0394a824d7eff2ff63f4c438237df0c37bf3825f1161d4a1e952a3ce1af59b","observation_id":"086a18bc-825e-4691-b42f-59389833525b","resolution":{"observed_at":"2026-08-07T14:28:49.896120Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:49.394973Z","title":null,"venue":null,"work_id":"b9f17a35-d5c0-437a-9836-dd9f1757add6","year":2021},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.346848Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:6f3bda2a65461249bc5136c85bbc567588f9f75da892fd9d38c3fab6a92bc580","observation_id":"ee567f3b-3643-4e01-9f48-a82fcda1fbe1","resolution":{"observed_at":"2026-08-07T14:28:49.534577Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:49.080362Z","title":null,"venue":null,"work_id":"23acfc45-b59a-4d1e-a6bf-d3b0029e7ac5","year":2022},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.447575Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:2f69233ff054e8e631057cfb8768282c3eea1b4a15c1ecb5646c3cc1c7bd0cd5","observation_id":"f70a1650-06e7-48e5-8420-b1983421bbb7","resolution":{"observed_at":"2026-08-07T14:28:49.212364Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:48.859900Z","title":"Munteanu and J","venue":null,"work_id":"730a8a77-2b4c-43ad-9e80-d2272c5d8727","year":2014},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.535781Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:91e566297b94c7d2a86258c01d7e1615209d113046d2432d3fd54f970a1fc31d","observation_id":"4c6f992f-2427-443c-95e6-4d386a6a24c8","resolution":{"observed_at":"2026-08-07T14:28:48.945067Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:48.630422Z","title":"Ohta,(K,N)-convexity and the curvature-dimension condition for negative N, The Journal of Geometric Analysis26, no","venue":null,"work_id":"0e670bc6-49d0-45af-ae6b-1d08e7976155","year":2016},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.665058Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:193753e552c390b4b783ce8866c132ca54462e99b1e074e2c2860d8797c77397","observation_id":"148837bf-dd63-45e4-89a6-7b4bbca1a959","resolution":{"observed_at":"2026-08-07T14:28:48.742503Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:48.374871Z","title":"Ohta,Comparison Finsler geometry, Springer Monographs in Mathematics, Springer, Cham, [2021] ©2021","venue":null,"work_id":"1d65795e-ff81-4d93-8671-ce5d9a915722","year":2021},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.751258Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:fab74562fb9da48a1e6fa274016385e661514bb84256dc88f79f64638f952adb","observation_id":"d2e1df53-98fb-42e9-8ac5-0a5126b98f08","resolution":{"observed_at":"2026-08-07T14:28:48.476426Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:48.104166Z","title":null,"venue":null,"work_id":"bc0e2104-f24b-4a2e-a7ee-de24360688b6","year":2025},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.857924Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:f05a335f65eae0815303b39727073cbeb69bad3845b7ee2379f89b78b48a1dee","observation_id":"ad6acfb7-b078-4ab0-9a36-e83c82654b7f","resolution":{"observed_at":"2026-08-07T14:28:48.261799Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:42.982237Z","title":"Perelmann, The entropy formula for the Ricci flow and its geometric applications","venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:42.982237Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:a1e81098dd4c79fb5524bc20d4cd73ab2fa090cbbc85133c1c417a792a4005d4","observation_id":"a75f7247-7dd3-45d9-b37b-43c3428f2440","resolution":{"observed_at":"2026-08-07T14:28:42.982237Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:47.832965Z","title":"Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds.J","venue":null,"work_id":"02e95ea8-3fef-4c7c-92a2-e245905b9e3e","year":1992},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.114295Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:9b932845f0692aa89925168f0a0392ec79b8015b53fe45e43c422d5cbc50efbd","observation_id":"4847ebf9-d24a-4c5e-a99f-ef3d91e26ed8","resolution":{"observed_at":"2026-08-07T14:28:47.970461Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:47.587717Z","title":"Saloff-Coste, Aspects of Sobolev-Type Inequalities","venue":null,"work_id":"7e8e7f9a-eec9-400d-8500-41b8405945cb","year":2002},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.218165Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:de17ed62ff443fe2c1efd79bdb6686a9965f48ceea7d930859bb4fc7088a4127","observation_id":"db03c5d3-9a6a-4671-8c59-48a5134d58c7","resolution":{"observed_at":"2026-08-07T14:28:47.721205Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:47.312752Z","title":"Sturm, On the geometry of metric measure spaces","venue":null,"work_id":"85448e4b-0555-4c1e-8698-5ad85752e2e6","year":2006},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.301711Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:8f214089afc1c11351c707cb56d8331a034d461ab81304c9f2c0ed67deef5964","observation_id":"082d0f08-aead-4e00-b890-cb90c5f3be46","resolution":{"observed_at":"2026-08-07T14:28:47.452608Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:47.075097Z","title":"Sturm, On the geometry of metric measure spaces","venue":null,"work_id":"59cd8620-6bb8-43b8-a9ec-21f320c9ad19","year":2006},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.385323Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:8aaa1e21c059ef928ce0feed73ad22039ad76232f811165f0f667064036cd90e","observation_id":"e22e1081-6907-41f6-b220-482db18c1ec3","resolution":{"observed_at":"2026-08-07T14:28:47.187175Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:46.827892Z","title":"Pigola, M","venue":null,"work_id":"cf006092-489e-4a98-84e9-ce740a33f67f","year":2005},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.497752Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:0e000b3e1f13072ebb9970f904b375a7e6961d8cc6c0b29d0b5e0a0299527f38","observation_id":"a987821c-1b8a-408c-a0dc-b8a7a2882f8d","resolution":{"observed_at":"2026-08-07T14:28:46.937133Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:46.627041Z","title":null,"venue":null,"work_id":"5602cb78-19e1-4683-b504-be569654845c","year":2023},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.599717Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:046e965bde8223feaf051aa889010513acc1fe58638ddbf9a4658da653443651","observation_id":"ab3c7182-a22e-4264-90d6-8f8613e8e75f","resolution":{"observed_at":"2026-08-07T14:28:46.725416Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:46.393516Z","title":null,"venue":null,"work_id":"2583579d-aadd-4c8c-a683-c2eff88cfe9a","year":2024},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.721221Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:c68dc32867287662cbba78dab1e7c4a0051c942836f607960f0a51cfbe812c80","observation_id":"b79c230a-994e-4ea1-a7bf-5ed3a98e78e8","resolution":{"observed_at":"2026-08-07T14:28:46.491899Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:46.179820Z","title":null,"venue":null,"work_id":"3225a8ad-47ec-49d6-90ee-7997762d2f4a","year":2009},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.814164Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:0b58cdfcab459d7ab0a417a8e0ba7ba6713dc233b4ff45b1325a939016e33b8f","observation_id":"792e0bad-cb1f-4412-9d4b-ba57015d14f7","resolution":{"observed_at":"2026-08-07T14:28:46.274011Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:45.931070Z","title":"Wu,L p-Liouville theorems on complete smooth metric measure spaces,”Bulletin des Sciences Mathématiques, vol","venue":null,"work_id":"dec943ef-d3fc-49b2-80e3-9e4892293e5c","year":2014},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:43.914916Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:3feb1fdf2c825325913418a55bba177fe9efd0f33544a8743ef125dcacae94b9","observation_id":"746ca29a-bdcc-4cf5-ac70-83ea35800f37","resolution":{"observed_at":"2026-08-07T14:28:46.041793Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:45.700129Z","title":"Wu and P","venue":null,"work_id":"b50a66b4-0aa0-4ca7-89e2-7bcaba40341e","year":2015},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:44.020880Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:fa90e6eafb169bd0c83243f0f2a2b237e9c3b89a1d0ad4d218efad5769251be4","observation_id":"0c4d03b1-695c-48c5-84eb-f18fcd288551","resolution":{"observed_at":"2026-08-07T14:28:45.816548Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:45.487700Z","title":"Wu and P","venue":null,"work_id":"151271ed-d326-4c6e-8298-be408c5a4bdb","year":2016},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:44.160464Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:0979318fc8ac374e0da58369ad93afda8079152aa5419ae22bf24224210c4dca","observation_id":"fd3f4d2c-d685-4363-bed3-61c46ff81d04","resolution":{"observed_at":"2026-08-07T14:28:45.572639Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1602.08000","last_updated":"2016-02-25T17:20:02Z","snapshot_observed_at":"2026-08-18T20:20:07.449463Z","submitted_at":"2016-02-25T17:20:02Z","title":"On the geometry of Riemannian manifolds with density","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1602.08000","snapshot_observed_at":"2026-08-07T14:28:44.263898Z","title":"Wylie and D","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:44.263898Z"},"links":{"cited_paper":"/paper/1602.08000","citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:adea1b6e8966c46ac692116f45db94d14f5cbc9d3cd0c8046fd08b7e3c0d4d3c","observation_id":"d2ee9650-a80d-4272-8c82-d082187dcd27","resolution":{"observed_at":"2026-08-07T14:28:44.263898Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:45.299204Z","title":null,"venue":null,"work_id":"d526faed-9c4b-4673-abb2-ddd50d571abc","year":2019},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:44.342216Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:356b1d992bbf1438147ffc1140922a2ac65d4aa050bffc9d6392257995a54a2c","observation_id":"3ba35629-ea98-4b73-a903-65f09244919c","resolution":{"observed_at":"2026-08-07T14:28:45.391682Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:45.084306Z","title":"Zhang, M","venue":null,"work_id":"5bf68b68-57a2-4214-be1d-fc8f93780135","year":2018},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:44.448747Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:dd8949bfd71c7b571cee374cfc0b67463a80215bed37285524d96185d4a56956","observation_id":"46020f38-96fe-4cb5-92a3-cc98aedb2ac3","resolution":{"observed_at":"2026-08-07T14:28:45.175569Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T14:28:44.896017Z","title":null,"venue":null,"work_id":"018258a7-7c34-4010-b538-b583dc1227a6","year":2019},"citing_paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-07T14:28:44.579343Z"},"links":{"citing_paper":"/paper/2505.19113"},"observation_digest":"sha256:663cc57f9bc4e6768a42c79ed7d00a9890357c28d8b9a80a0edb31eb447c5387","observation_id":"18dd68af-a0a9-4155-b5db-febdb694e747","resolution":{"observed_at":"2026-08-07T14:28:44.981090Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2505.19113","last_updated":"2025-05-25T12:13:52Z","latest_version":1,"primary_category":"math.DG","snapshot_observed_at":"2026-08-18T07:21:00.214753Z","submitted_at":"2025-05-25T12:13:52Z","title":"Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\\epsilon$-range and it's application"},"reference_resolution":{"displayed":43,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":15,"verified_exact":0,"verified_fuzzy":28},"total_outbound_references":43},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"thesis":"As of 19 August 2026, this Paper Citation Record lists 43 of 43 outbound references and 0 inbound Pith citation observations for arXiv:2505.19113."}