{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QQNFSU365N4TAXHPKSN5CVQDSW","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":"7663659518ed98f72b35dd20fddebf92277a8a3449c78f4467dacc23d6e21fe2","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-11-13T08:19:57Z","title_canon_sha256":"234d062df5df4cbc92bf0daf3361d8fb1653e6e06fc97f45d9236ea76e60cace"},"schema_version":"1.0","source":{"id":"1511.04199","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.04199","created_at":"2026-05-18T01:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1511.04199v1","created_at":"2026-05-18T01:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04199","created_at":"2026-05-18T01:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"QQNFSU365N4T","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QQNFSU365N4TAXHP","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QQNFSU36","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:27a560ba58407e36c0d1350f96ce40e13552e6ffdbf6bbbe7d45e5302a1bfbec","target":"graph","created_at":"2026-05-18T01:26:59Z","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":"In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\\mathbb{R}/\\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality (Courant-sharp situation). Following the strategy of {\\AA}. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains on the torus, deduced as, in the work of P. B\\'erard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Perez","authors_text":"Corentin L\\'ena","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-11-13T08:19:57Z","title":"Courant-sharp eigenvalues of the three-dimensional square torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04199","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":"sha256:57b9a2c4000b6aedcf5323de341be988f9aaddb52bdc4ebeb65c018f27ff0cbc","target":"record","created_at":"2026-05-18T01:26:59Z","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":"7663659518ed98f72b35dd20fddebf92277a8a3449c78f4467dacc23d6e21fe2","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-11-13T08:19:57Z","title_canon_sha256":"234d062df5df4cbc92bf0daf3361d8fb1653e6e06fc97f45d9236ea76e60cace"},"schema_version":"1.0","source":{"id":"1511.04199","kind":"arxiv","version":1}},"canonical_sha256":"841a59537eeb79305cef549bd1560395bc156dad12032daadf35ba59f617ae9b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"841a59537eeb79305cef549bd1560395bc156dad12032daadf35ba59f617ae9b","first_computed_at":"2026-05-18T01:26:59.419720Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:59.419720Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W3AGwSGY4cnOpkDe0R5LGrqmToDVcG2CBTzX/vlZtqgeTieNqmIH1cmGD2qT1KcjFyakyxyTmOcGlohZn14UAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:59.420431Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.04199","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57b9a2c4000b6aedcf5323de341be988f9aaddb52bdc4ebeb65c018f27ff0cbc","sha256:27a560ba58407e36c0d1350f96ce40e13552e6ffdbf6bbbe7d45e5302a1bfbec"],"state_sha256":"5a71b05d44da8e50c51ed7ee513f1eff724916f9e5938bb7416b72b5c39cc3ca"}