{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:255UTKAEZO5K4WLOBTXYTDAQWF","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":"ef2cbe32b7c1148bc797ba1886717644f8372efd2a626a62c64fced2b69cab8a","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-05-12T11:32:35Z","title_canon_sha256":"8e8328c5e4f917e562af623b1393e494e8fd17b82a7b1fee467a3bc5cf0dfd64"},"schema_version":"1.0","source":{"id":"1105.2431","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.2431","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"arxiv_version","alias_value":"1105.2431v2","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.2431","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"pith_short_12","alias_value":"255UTKAEZO5K","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"255UTKAEZO5K4WLO","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"255UTKAE","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:bb1e34da8e0d048b01180de211831dd05b2de4bd26da2a693b0023f26cfe4094","target":"graph","created_at":"2026-05-18T04:09:54Z","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":"It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\\in\\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-\\Delta_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals $(\\a_j,\\b_j)\\subset[0,\\infty)$, $j=1,...,m$ ($m\\in\\mathbb{N}$), for an arbitrarily small $\\delta>0$ and for an arbitrarily la","authors_text":"Andrii Khrabustovskyi","cross_cats":["math-ph","math.AP","math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-05-12T11:32:35Z","title":"Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2431","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":"sha256:829542168f208c0b2ffb67e2437851046e5dbe4faa44c45b22215ace9838b1c3","target":"record","created_at":"2026-05-18T04:09:54Z","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":"ef2cbe32b7c1148bc797ba1886717644f8372efd2a626a62c64fced2b69cab8a","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-05-12T11:32:35Z","title_canon_sha256":"8e8328c5e4f917e562af623b1393e494e8fd17b82a7b1fee467a3bc5cf0dfd64"},"schema_version":"1.0","source":{"id":"1105.2431","kind":"arxiv","version":2}},"canonical_sha256":"d77b49a804cbbaae596e0cef898c10b14331c6612cfc496b8f36750d83f67e06","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d77b49a804cbbaae596e0cef898c10b14331c6612cfc496b8f36750d83f67e06","first_computed_at":"2026-05-18T04:09:54.533200Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:54.533200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KXlSYBiL0W16aJT3Rp21K2rDLLywy7tg9ksnxWLIAlIIKvroqbsKwR9WaW5xCeVeCinzsLA5+xYSWbdVtLkIAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:54.533830Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.2431","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:829542168f208c0b2ffb67e2437851046e5dbe4faa44c45b22215ace9838b1c3","sha256:bb1e34da8e0d048b01180de211831dd05b2de4bd26da2a693b0023f26cfe4094"],"state_sha256":"aa794a9be0c11c283ba501ac83ea6baf2116aa6cb9928608e1a7d08312ff123c"}