{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:DME3L3GBATMGLXXFUGJU2W7AHA","short_pith_number":"pith:DME3L3GB","schema_version":"1.0","canonical_sha256":"1b09b5ecc104d865dee5a1934d5be0381ee60c2ea09883c7ce3688135c061a71","source":{"kind":"arxiv","id":"1709.05109","version":1},"attestation_state":"computed","paper":{"title":"Boundary optimization for rough sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konrad Engel, Tran Dan Thu","submitted_at":"2017-09-15T08:52:34Z","abstract_excerpt":"Let $n > m\\ge 2$ be integers and let $\\mathcal{A}=\\{A_1,\\dots,A_m\\}$ be a partition of $[n]=\\{1,\\dots,n\\}$. For $X \\subseteq [n]$, its $\\mathcal{A}$-boundary region $\\mathcal{A}(X)$ is defined to be the union of those blocks $A_i$ of $\\mathcal{A}$ for which $A_i\\cap X\\neq \\emptyset$ and $A_i\\cap ([n] \\setminus X)\\neq \\emptyset$. For three different probability distributions on the power set of $[n]$, partitions $\\mathcal{A}$ of $[n]$ are determined such that the expected cardinality of the $\\mathcal{A}$-boundary region of a randomly chosen subset of $[n]$ is minimal and maximal, respectively. "},"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":"1709.05109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-15T08:52:34Z","cross_cats_sorted":[],"title_canon_sha256":"3954891e9e60c193a96d340e322869c4f7381ae8b4b288cdefebd3f0daeb79a3","abstract_canon_sha256":"085c86a4b415c355e9c6ddcdcada49181f2b980a6ff7096b1cd95e2108e9a032"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:49.921280Z","signature_b64":"bAVswUVxbiKkggg5ArGcL+1vOznYnJ+uKOt0q1rzB9/7qAxHGrhs75zEY+FM0b1Z820dhL0gbFTHaIlofY3WBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b09b5ecc104d865dee5a1934d5be0381ee60c2ea09883c7ce3688135c061a71","last_reissued_at":"2026-05-17T23:42:49.920813Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:49.920813Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundary optimization for rough sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konrad Engel, Tran Dan Thu","submitted_at":"2017-09-15T08:52:34Z","abstract_excerpt":"Let $n > m\\ge 2$ be integers and let $\\mathcal{A}=\\{A_1,\\dots,A_m\\}$ be a partition of $[n]=\\{1,\\dots,n\\}$. For $X \\subseteq [n]$, its $\\mathcal{A}$-boundary region $\\mathcal{A}(X)$ is defined to be the union of those blocks $A_i$ of $\\mathcal{A}$ for which $A_i\\cap X\\neq \\emptyset$ and $A_i\\cap ([n] \\setminus X)\\neq \\emptyset$. For three different probability distributions on the power set of $[n]$, partitions $\\mathcal{A}$ of $[n]$ are determined such that the expected cardinality of the $\\mathcal{A}$-boundary region of a randomly chosen subset of $[n]$ is minimal and maximal, respectively. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05109","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":"1709.05109","created_at":"2026-05-17T23:42:49.920879+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.05109v1","created_at":"2026-05-17T23:42:49.920879+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05109","created_at":"2026-05-17T23:42:49.920879+00:00"},{"alias_kind":"pith_short_12","alias_value":"DME3L3GBATMG","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"DME3L3GBATMGLXXF","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"DME3L3GB","created_at":"2026-05-18T12:31:10.602751+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/DME3L3GBATMGLXXFUGJU2W7AHA","json":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA.json","graph_json":"https://pith.science/api/pith-number/DME3L3GBATMGLXXFUGJU2W7AHA/graph.json","events_json":"https://pith.science/api/pith-number/DME3L3GBATMGLXXFUGJU2W7AHA/events.json","paper":"https://pith.science/paper/DME3L3GB"},"agent_actions":{"view_html":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA","download_json":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA.json","view_paper":"https://pith.science/paper/DME3L3GB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.05109&json=true","fetch_graph":"https://pith.science/api/pith-number/DME3L3GBATMGLXXFUGJU2W7AHA/graph.json","fetch_events":"https://pith.science/api/pith-number/DME3L3GBATMGLXXFUGJU2W7AHA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA/action/storage_attestation","attest_author":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA/action/author_attestation","sign_citation":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA/action/citation_signature","submit_replication":"https://pith.science/pith/DME3L3GBATMGLXXFUGJU2W7AHA/action/replication_record"}},"created_at":"2026-05-17T23:42:49.920879+00:00","updated_at":"2026-05-17T23:42:49.920879+00:00"}