{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:VGTB2BVC6BK3IG3QNVC3MNDKO2","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":"75697056a2282607a1d6bd4b6a67e35e489472de2205631d0dd2bad699019564","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-13T17:55:01Z","title_canon_sha256":"36251ef781ee965cc977782733c7b552479dcb13089be13d2cbba6872b1fe523"},"schema_version":"1.0","source":{"id":"1705.04864","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.04864","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"arxiv_version","alias_value":"1705.04864v2","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04864","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"pith_short_12","alias_value":"VGTB2BVC6BK3","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VGTB2BVC6BK3IG3Q","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VGTB2BVC","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:366974c2f3c339525a0dda83ff1128db4ac9efd8852fcb6a6270c75fa4202536","target":"graph","created_at":"2026-05-18T00:40:23Z","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":"This paper proves that given a doubling weight $w$ on the unit sphere $\\mathbb{S}^{d-1}$ of $\\mathbb{R}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\\geq \\max_{x\\in \\mathbb{S}^{d-1}} \\frac {K_w} {w(B(x, n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1,\\cdots, z_N$ on $\\mathbb{S}^{d-1}$ which admits a strict Chebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x) d\\sigma_d(x)$, $$ \\frac 1{w(\\mathbb{S}^{d-1})} \\int_{\\mathbb{S}^{d-1}} f(x) w(x)\\, d\\sigma_d(x)=\\frac 1N \\sum_{j=1}^N f(z_j),\\ \\ \\forall f\\in\\Pi_n^d, $$","authors_text":"Feng Dai, Han Feng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-13T17:55:01Z","title":"Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04864","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":"sha256:0f635a2d816ef99f96e3140b032b5ea7606bdc5eca4bc8677d99d9da8c61a271","target":"record","created_at":"2026-05-18T00:40:23Z","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":"75697056a2282607a1d6bd4b6a67e35e489472de2205631d0dd2bad699019564","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-13T17:55:01Z","title_canon_sha256":"36251ef781ee965cc977782733c7b552479dcb13089be13d2cbba6872b1fe523"},"schema_version":"1.0","source":{"id":"1705.04864","kind":"arxiv","version":2}},"canonical_sha256":"a9a61d06a2f055b41b706d45b6346a7680a0fe6173f09ea7fda4f3450fb066b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9a61d06a2f055b41b706d45b6346a7680a0fe6173f09ea7fda4f3450fb066b6","first_computed_at":"2026-05-18T00:40:23.712800Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:23.712800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xMFAnco1VDWAGX28r9nWEHYLjfoR2eiLF8ij/FY06jbPhVexNx9eHGl2K8tSORTd2+CN+7X1G01MLxOlSePyBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:23.713326Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.04864","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f635a2d816ef99f96e3140b032b5ea7606bdc5eca4bc8677d99d9da8c61a271","sha256:366974c2f3c339525a0dda83ff1128db4ac9efd8852fcb6a6270c75fa4202536"],"state_sha256":"563da1a4a6c99b7fdacb2dc2127198d03a613c5480d69363782c788ffde65053"}