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Then we take $(q_{11},\\dots, q_{nn})\\in ]0,+\\infty[^n$ and $p \\in Q\\equiv \\prod_{j=1}^{n}]0,q_{jj}[$. If $\\epsilon$ is a small positive number, then we define the periodically perforated domain $\\mathbb{S}[\\Omega_{p,\\epsilon}]^{-} \\equiv \\mathbb{R}^n\\setminus \\cup_{z \\in \\mathbb{Z}^n}\\mathrm{cl}\\bigl(p+\\epsilon \\Omega +\\sum_{j=1}^n (q_{jj}z_j)e_j\\bigr)$, where $\\{e_1,\\dots,e_n\\}$ is the canonical basis of $\\mathbb{R}^n"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1307.1612","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-05T13:35:50Z","cross_cats_sorted":[],"title_canon_sha256":"9336a10d58c3a3b5f812fdc4304a252a6a2ad6866ded2f4a2b70f02693849041","abstract_canon_sha256":"239afb7847b288ee5c390cb5f60c360c6645320a2ba978a1884d73c56a2e5091"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:12.403495Z","signature_b64":"w33hRRe2SsOptSdK2/Vl3VwzxwTdbKIjasNP50+oXH1ZDaLGfP7yffRdG4VC0Ws2sba14VYJ8SY9QxWmbEOoCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae214cdf46946e5aeba4f8c0d71ae7e4a3a088cb6b62998313164156e1d6a11a","last_reissued_at":"2026-05-18T03:19:12.402981Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:12.402981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. 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If $\\epsilon$ is a small positive number, then we define the periodically perforated domain $\\mathbb{S}[\\Omega_{p,\\epsilon}]^{-} \\equiv \\mathbb{R}^n\\setminus \\cup_{z \\in \\mathbb{Z}^n}\\mathrm{cl}\\bigl(p+\\epsilon \\Omega +\\sum_{j=1}^n (q_{jj}z_j)e_j\\bigr)$, where $\\{e_1,\\dots,e_n\\}$ is the canonical basis of $\\mathbb{R}^n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1612","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1307.1612","created_at":"2026-05-18T03:19:12.403065+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.1612v1","created_at":"2026-05-18T03:19:12.403065+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1612","created_at":"2026-05-18T03:19:12.403065+00:00"},{"alias_kind":"pith_short_12","alias_value":"VYQUZX2GSRXF","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"VYQUZX2GSRXFV25E","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"VYQUZX2G","created_at":"2026-05-18T12:28:04.890932+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S","json":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S.json","graph_json":"https://pith.science/api/pith-number/VYQUZX2GSRXFV25E7DANOGXH4S/graph.json","events_json":"https://pith.science/api/pith-number/VYQUZX2GSRXFV25E7DANOGXH4S/events.json","paper":"https://pith.science/paper/VYQUZX2G"},"agent_actions":{"view_html":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S","download_json":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S.json","view_paper":"https://pith.science/paper/VYQUZX2G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.1612&json=true","fetch_graph":"https://pith.science/api/pith-number/VYQUZX2GSRXFV25E7DANOGXH4S/graph.json","fetch_events":"https://pith.science/api/pith-number/VYQUZX2GSRXFV25E7DANOGXH4S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S/action/storage_attestation","attest_author":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S/action/author_attestation","sign_citation":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S/action/citation_signature","submit_replication":"https://pith.science/pith/VYQUZX2GSRXFV25E7DANOGXH4S/action/replication_record"}},"created_at":"2026-05-18T03:19:12.403065+00:00","updated_at":"2026-05-18T03:19:12.403065+00:00"}