{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:2I3MQVF7MA3Q2543M57SJYF6M3","short_pith_number":"pith:2I3MQVF7","schema_version":"1.0","canonical_sha256":"d236c854bf60370d779b677f24e0be66fa1967162161cc2b39490459aeadbd66","source":{"kind":"arxiv","id":"1801.09209","version":1},"attestation_state":"computed","paper":{"title":"Nash inequality for Diffusion Processes Associated with Dirichlet Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang, Weiwei Zhang","submitted_at":"2018-01-28T11:13:00Z","abstract_excerpt":"For any $N\\ge 2$ and $\\alpha=(\\alpha_1,\\cdots, \\alpha_{N+1})\\in (0,\\infty)^{N+1}$, let $\\mu^{(N)}_{\\alpha}$ be the Dirichlet distribution with parameter $\\alpha$ on the set $\\Delta^{ (N)}:= \\{ x \\in [0,1]^N:\\ \\sum_{1\\le i\\le N}x_i \\le 1 \\}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form\n  $${\\scr E}_\\alpha^{(N)}(f,f):= \\sum_{n=1}^N \\int_{ \\Delta^{(N)}} \\bigg(1-\\sum_{1\\le i\\le N}x_i\\bigg) x_n(\\partial_n f)^2(x)\\,\\mu^{(N)}_\\alpha(d x)$$ with Domain ${\\scr D}({\\scr E}_\\alpha^{(N)})$ being the closure of $C^1(\\Delta^{(N)})$. We prove the Nash inequality\n  $$\\mu_\\alpha^"},"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":"1801.09209","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-28T11:13:00Z","cross_cats_sorted":[],"title_canon_sha256":"3dabf3c09f5bfb00415e3a16e2883b13165910a8184ccae9795f934664a3b765","abstract_canon_sha256":"d82f436e3930fd2ccd46d06a9654e30b202956ab0a44b736284029560db160b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:02.496373Z","signature_b64":"cTr3SqEGKS0sJqhPVMy6b4H07uduxCM2gYagQPHSY4cgoUB7P4wsZrn0jkWk6HIaT2qm3bD6dN0dUDLM9+NXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d236c854bf60370d779b677f24e0be66fa1967162161cc2b39490459aeadbd66","last_reissued_at":"2026-05-18T00:19:02.495653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:02.495653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nash inequality for Diffusion Processes Associated with Dirichlet Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang, Weiwei Zhang","submitted_at":"2018-01-28T11:13:00Z","abstract_excerpt":"For any $N\\ge 2$ and $\\alpha=(\\alpha_1,\\cdots, \\alpha_{N+1})\\in (0,\\infty)^{N+1}$, let $\\mu^{(N)}_{\\alpha}$ be the Dirichlet distribution with parameter $\\alpha$ on the set $\\Delta^{ (N)}:= \\{ x \\in [0,1]^N:\\ \\sum_{1\\le i\\le N}x_i \\le 1 \\}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form\n  $${\\scr E}_\\alpha^{(N)}(f,f):= \\sum_{n=1}^N \\int_{ \\Delta^{(N)}} \\bigg(1-\\sum_{1\\le i\\le N}x_i\\bigg) x_n(\\partial_n f)^2(x)\\,\\mu^{(N)}_\\alpha(d x)$$ with Domain ${\\scr D}({\\scr E}_\\alpha^{(N)})$ being the closure of $C^1(\\Delta^{(N)})$. We prove the Nash inequality\n  $$\\mu_\\alpha^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09209","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":"1801.09209","created_at":"2026-05-18T00:19:02.495765+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.09209v1","created_at":"2026-05-18T00:19:02.495765+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.09209","created_at":"2026-05-18T00:19:02.495765+00:00"},{"alias_kind":"pith_short_12","alias_value":"2I3MQVF7MA3Q","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_16","alias_value":"2I3MQVF7MA3Q2543","created_at":"2026-05-18T12:32:02.567920+00:00"},{"alias_kind":"pith_short_8","alias_value":"2I3MQVF7","created_at":"2026-05-18T12:32:02.567920+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/2I3MQVF7MA3Q2543M57SJYF6M3","json":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3.json","graph_json":"https://pith.science/api/pith-number/2I3MQVF7MA3Q2543M57SJYF6M3/graph.json","events_json":"https://pith.science/api/pith-number/2I3MQVF7MA3Q2543M57SJYF6M3/events.json","paper":"https://pith.science/paper/2I3MQVF7"},"agent_actions":{"view_html":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3","download_json":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3.json","view_paper":"https://pith.science/paper/2I3MQVF7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.09209&json=true","fetch_graph":"https://pith.science/api/pith-number/2I3MQVF7MA3Q2543M57SJYF6M3/graph.json","fetch_events":"https://pith.science/api/pith-number/2I3MQVF7MA3Q2543M57SJYF6M3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3/action/storage_attestation","attest_author":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3/action/author_attestation","sign_citation":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3/action/citation_signature","submit_replication":"https://pith.science/pith/2I3MQVF7MA3Q2543M57SJYF6M3/action/replication_record"}},"created_at":"2026-05-18T00:19:02.495765+00:00","updated_at":"2026-05-18T00:19:02.495765+00:00"}