{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GQSPB5MEAIIMCLI56MJUWRA2BT","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":"1cd4bd1bafe544baa8124ea15bca0055e8f24e19f30a6a5a213d6ac83b64fe55","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-04-09T01:18:05Z","title_canon_sha256":"d93f9c5284c7db4d246efcaf82261aed13308185922a2625c89d13cf37a0e509"},"schema_version":"1.0","source":{"id":"1204.1779","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.1779","created_at":"2026-05-18T03:58:15Z"},{"alias_kind":"arxiv_version","alias_value":"1204.1779v1","created_at":"2026-05-18T03:58:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.1779","created_at":"2026-05-18T03:58:15Z"},{"alias_kind":"pith_short_12","alias_value":"GQSPB5MEAIIM","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GQSPB5MEAIIMCLI5","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GQSPB5ME","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:4b0fb3689f67e27272c76d8cfe7e0d15e126cef1ed9a6af6bc4d7b474466a365","target":"graph","created_at":"2026-05-18T03:58:15Z","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":"Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoir's method with one more combinatorial object, called regular t-wise balanced designs. Many cubature of small indices with few points are provided, which are used to update Shatalov's table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflecti","authors_text":"Hiroshi Nozaki, Masanori Sawa","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-04-09T01:18:05Z","title":"Remarks on Hilbert identities, isometric embeddings, and invariant cubature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1779","kind":"arxiv","version":1},"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":"sha256:0fb649509ea42ad2f31461bc569e74b9cc5300a70fb9e19c7b72912f1e379bb2","target":"record","created_at":"2026-05-18T03:58:15Z","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":"1cd4bd1bafe544baa8124ea15bca0055e8f24e19f30a6a5a213d6ac83b64fe55","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-04-09T01:18:05Z","title_canon_sha256":"d93f9c5284c7db4d246efcaf82261aed13308185922a2625c89d13cf37a0e509"},"schema_version":"1.0","source":{"id":"1204.1779","kind":"arxiv","version":1}},"canonical_sha256":"3424f0f5840210c12d1df3134b441a0cdf815190086fa3ffed98b1c290b4d6cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3424f0f5840210c12d1df3134b441a0cdf815190086fa3ffed98b1c290b4d6cf","first_computed_at":"2026-05-18T03:58:15.210335Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:15.210335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XtqE60ycidEuxPszLQ0mKWXto8v3Kj7OTU3QbyrPQxb8jAiSNe6Ai6Wy/B5Lqd2xuLOSosU65loErLLuelZmAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:15.211082Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.1779","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0fb649509ea42ad2f31461bc569e74b9cc5300a70fb9e19c7b72912f1e379bb2","sha256:4b0fb3689f67e27272c76d8cfe7e0d15e126cef1ed9a6af6bc4d7b474466a365"],"state_sha256":"b7ee8585f95b1e7d38dc3d4cc4ab4a3e4dd765d5e3823449b54ab126e150c7fe"}