{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","short_pith_number":"pith:ZY5DZ4ZZ","canonical_record":{"source":{"id":"1201.3729","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-01-18T09:24:49Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"5e6bd706fb421c6a671185760a4c8c7f7cca7b765bcb141ecf802f798f5c264e","abstract_canon_sha256":"18087797690bd1e07137084568d21e2b21c023aa07cfed882e42d96356661507"},"schema_version":"1.0"},"canonical_sha256":"ce3a3cf339ce157e32514ca6dde7c0124ff91902f0ffa2753b4d623feb5a1b74","source":{"kind":"arxiv","id":"1201.3729","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3729","created_at":"2026-05-18T04:03:08Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3729v2","created_at":"2026-05-18T04:03:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3729","created_at":"2026-05-18T04:03:08Z"},{"alias_kind":"pith_short_12","alias_value":"ZY5DZ4ZZZYKX","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZY5DZ4ZZZYKX4MSR","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZY5DZ4ZZ","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3729","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-01-18T09:24:49Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"5e6bd706fb421c6a671185760a4c8c7f7cca7b765bcb141ecf802f798f5c264e","abstract_canon_sha256":"18087797690bd1e07137084568d21e2b21c023aa07cfed882e42d96356661507"},"schema_version":"1.0"},"canonical_sha256":"ce3a3cf339ce157e32514ca6dde7c0124ff91902f0ffa2753b4d623feb5a1b74","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:08.726308Z","signature_b64":"kZlLSgQWGvc3Igoc3DAlXjNKbS/zPvllGgFM5a0dw2A2SDPR9o1uVaR7Yh9R1tgbw7V6j2pKSUUI4Qe/8FHXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce3a3cf339ce157e32514ca6dde7c0124ff91902f0ffa2753b4d623feb5a1b74","last_reissued_at":"2026-05-18T04:03:08.725444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:08.725444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3729","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:03:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q93AemVxjm4srCwuynnaz6rzsC7kJtUQ5bmucLOZPrRd0seAfpHyq0Mqk12+W24yDYi/89ZOhylwulqJpKdqDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:25:45.990992Z"},"content_sha256":"837e14573ee535ecac195a3fe45787c9c104f3ac44f97af64eaa82e706e15423","schema_version":"1.0","event_id":"sha256:837e14573ee535ecac195a3fe45787c9c104f3ac44f97af64eaa82e706e15423"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Periodic elliptic operators with asymptotically preassigned spectrum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.SP","authors_text":"Andrii Khrabustovskyi","submitted_at":"2012-01-18T09:24:49Z","abstract_excerpt":"We deal with operators in $\\mathbb{R}^n$ of the form $$\\mathbf{A}=-{1\\over \\mathbf{b}(x)}\\sum\\limits_{k=1}^n\\ds{\\partial\\over\\partial x_k}(\\mathbf{a}(x){\\partial \\over\\partial x_k})$$ where $\\mathbf{a}(x),\\mathbf{b}(x)$ are positive, bounded and periodic functions. We denote by $\\mathbf{L}_{\\mathrm{per}}$ the set of such operators. The main result of this work is as follows: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(\\alpha_j,\\beta_j)\\subset[0,L]$, $j=1,...,m$ ($m\\in\\mathbb{N}$) we construct the family of operators $\\{\\mathbf{A}^\\varepsilon\\in \\mathbf{L}_{\\mathrm{pe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3729","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:03:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fshxPw2qjroggsPQFPXV7+6NRZ83KEyOdUJ9mgUl1+7CPa1AuVjWbrWmvVsTu7FYUiUbfeMelAU/Dfmvzs1CCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:25:45.991358Z"},"content_sha256":"21b1057cdbd1f92720b13f00ec18cac0d6a553bcfca2555f3f0528aaa57c647d","schema_version":"1.0","event_id":"sha256:21b1057cdbd1f92720b13f00ec18cac0d6a553bcfca2555f3f0528aaa57c647d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/bundle.json","state_url":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T01:25:45Z","links":{"resolver":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","bundle":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/bundle.json","state":"https://pith.science/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZY5DZ4ZZZYKX4MSRJSTN3Z6ACJ","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":"18087797690bd1e07137084568d21e2b21c023aa07cfed882e42d96356661507","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-01-18T09:24:49Z","title_canon_sha256":"5e6bd706fb421c6a671185760a4c8c7f7cca7b765bcb141ecf802f798f5c264e"},"schema_version":"1.0","source":{"id":"1201.3729","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3729","created_at":"2026-05-18T04:03:08Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3729v2","created_at":"2026-05-18T04:03:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3729","created_at":"2026-05-18T04:03:08Z"},{"alias_kind":"pith_short_12","alias_value":"ZY5DZ4ZZZYKX","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZY5DZ4ZZZYKX4MSR","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZY5DZ4ZZ","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:21b1057cdbd1f92720b13f00ec18cac0d6a553bcfca2555f3f0528aaa57c647d","target":"graph","created_at":"2026-05-18T04:03:08Z","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 deal with operators in $\\mathbb{R}^n$ of the form $$\\mathbf{A}=-{1\\over \\mathbf{b}(x)}\\sum\\limits_{k=1}^n\\ds{\\partial\\over\\partial x_k}(\\mathbf{a}(x){\\partial \\over\\partial x_k})$$ where $\\mathbf{a}(x),\\mathbf{b}(x)$ are positive, bounded and periodic functions. We denote by $\\mathbf{L}_{\\mathrm{per}}$ the set of such operators. The main result of this work is as follows: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(\\alpha_j,\\beta_j)\\subset[0,L]$, $j=1,...,m$ ($m\\in\\mathbb{N}$) we construct the family of operators $\\{\\mathbf{A}^\\varepsilon\\in \\mathbf{L}_{\\mathrm{pe","authors_text":"Andrii Khrabustovskyi","cross_cats":["math-ph","math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-01-18T09:24:49Z","title":"Periodic elliptic operators with asymptotically preassigned spectrum"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3729","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":"sha256:837e14573ee535ecac195a3fe45787c9c104f3ac44f97af64eaa82e706e15423","target":"record","created_at":"2026-05-18T04:03:08Z","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":"18087797690bd1e07137084568d21e2b21c023aa07cfed882e42d96356661507","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-01-18T09:24:49Z","title_canon_sha256":"5e6bd706fb421c6a671185760a4c8c7f7cca7b765bcb141ecf802f798f5c264e"},"schema_version":"1.0","source":{"id":"1201.3729","kind":"arxiv","version":2}},"canonical_sha256":"ce3a3cf339ce157e32514ca6dde7c0124ff91902f0ffa2753b4d623feb5a1b74","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce3a3cf339ce157e32514ca6dde7c0124ff91902f0ffa2753b4d623feb5a1b74","first_computed_at":"2026-05-18T04:03:08.725444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:08.725444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kZlLSgQWGvc3Igoc3DAlXjNKbS/zPvllGgFM5a0dw2A2SDPR9o1uVaR7Yh9R1tgbw7V6j2pKSUUI4Qe/8FHXDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:08.726308Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3729","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:837e14573ee535ecac195a3fe45787c9c104f3ac44f97af64eaa82e706e15423","sha256:21b1057cdbd1f92720b13f00ec18cac0d6a553bcfca2555f3f0528aaa57c647d"],"state_sha256":"661d16728773662dc7f0d41177b4afa6b7edf4c8b381fd351e2563de58aaed66"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iDEqvigNECBXEMBJBfhMapKwzud5SiTnvqsFBvqh2XFv5mtEFJ16chsCmdIDr4rl1tc4TSN0dsHrgtjV3iWHDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T01:25:45.993327Z","bundle_sha256":"9630587e35839e34fa5f0e16ee27f173dce8176643f58cd714497987d535456e"}}