{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HE6I26PNRRPDKQ2GFUPJVYZTXP","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":"287d46cc9b79726cdffae133657b9fe36c65cc2514c2b9fa626ded4e3b539c47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-04-06T15:49:30Z","title_canon_sha256":"83ca25058b10945644ea162e5e302ab760ee50db0679db300970893ae0c3acde"},"schema_version":"1.0","source":{"id":"1804.02327","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.02327","created_at":"2026-05-17T23:49:43Z"},{"alias_kind":"arxiv_version","alias_value":"1804.02327v2","created_at":"2026-05-17T23:49:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02327","created_at":"2026-05-17T23:49:43Z"},{"alias_kind":"pith_short_12","alias_value":"HE6I26PNRRPD","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HE6I26PNRRPDKQ2G","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HE6I26PN","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:631239cd1e942e5727857a111397d37b3596db22248360792473fc30ee72bfe3","target":"graph","created_at":"2026-05-17T23:49:43Z","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":"We discuss the classical problem of how to pick $N$ weighted points on a $d-$dimensional manifold so as to obtain a reasonable quadrature rule $$ \\frac{1}{|M|}\\int_{M}{f(x) dx} \\simeq \\frac{1}{N} \\sum_{n=1}^{N}{a_i f(x_i)}.$$ This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional $$ \\sum_{i,j =1}^{N}{ a_i a_j \\exp\\left(-\\frac{d(x_i,x_j)^2}{4t}\\right) } \\rightarrow \\min, \\quad \\mbox{where}~t \\sim N^{-2/d},$$ $d(x,y)$ is the geodesic distance and $d$ is the dimension of the manifold. This yields poi","authors_text":"Jianfeng Lu, Matthias Sachs, Stefan Steinerberger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-04-06T15:49:30Z","title":"Quadrature Points via Heat Kernel Repulsion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02327","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:e83077a1cf526520b0b565dc4c0ed921bfe5ba5e88b7204f8e385baad16bc36a","target":"record","created_at":"2026-05-17T23:49:43Z","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":"287d46cc9b79726cdffae133657b9fe36c65cc2514c2b9fa626ded4e3b539c47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-04-06T15:49:30Z","title_canon_sha256":"83ca25058b10945644ea162e5e302ab760ee50db0679db300970893ae0c3acde"},"schema_version":"1.0","source":{"id":"1804.02327","kind":"arxiv","version":2}},"canonical_sha256":"393c8d79ed8c5e3543462d1e9ae333bbffc53304dcd2bf09c1cf7e638fd6215f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"393c8d79ed8c5e3543462d1e9ae333bbffc53304dcd2bf09c1cf7e638fd6215f","first_computed_at":"2026-05-17T23:49:43.927995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:43.927995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Gt112Ix9bT01W9hojQQOreN5kWgcfd2WkPAhqsTKARhDfcisZhURw2ofXjwyJYRXgxi/XXlKrrooUKZysD8zBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:43.928648Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.02327","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e83077a1cf526520b0b565dc4c0ed921bfe5ba5e88b7204f8e385baad16bc36a","sha256:631239cd1e942e5727857a111397d37b3596db22248360792473fc30ee72bfe3"],"state_sha256":"c5b0b0c38397dbc485373f2fad9dd69c58dcba217429f0bbd721b3c5c503441a"}