{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:YGE5E2OCNOWOSM5LOH26EEG4NT","short_pith_number":"pith:YGE5E2OC","schema_version":"1.0","canonical_sha256":"c189d269c26bace933ab71f5e210dc6ceab20abb5a205110f21176c0a8c49990","source":{"kind":"arxiv","id":"1806.00992","version":2},"attestation_state":"computed","paper":{"title":"Integrality of Subgradients and Biconjugates of Integrally Convex Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Akihisa Tamura, Kazuo Murota","submitted_at":"2018-06-04T07:43:10Z","abstract_excerpt":"Integrally convex functions constitute a fundamental function class in discrete convex analysis. This paper shows that an integer-valued integrally convex function admits an integral subgradient and that the integral biconjugate of an integer-valued integrally convex function coincides with itself. The proof is based on the Fourier-Motzkin elimination. The latter result provides a unified proof of integral biconjugacy for various classes of integer-valued discrete convex functions, including L-convex, M-convex, L$_{2}$-convex, M$_{2}$-convex, BS-convex, and UJ-convex functions as well as multi"},"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":"1806.00992","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-04T07:43:10Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"5711a2b2869e0e31fd0aa16c029c62836af299882d231a9e4f5be2065ce979ce","abstract_canon_sha256":"859fd1a2516d3fadf872e0e7b64878756b2ea7fabb9fa11bba19ca51a20fa9ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:16.010217Z","signature_b64":"Bq5UiHe50bRkwd/RlmcM8ogf/j9cmXsudxOWKlWO3G4JNkZNAO5A0O6hfmU9NK77EhOGOmmGpSaq+oYhmSI7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c189d269c26bace933ab71f5e210dc6ceab20abb5a205110f21176c0a8c49990","last_reissued_at":"2026-05-18T00:06:16.009526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:16.009526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integrality of Subgradients and Biconjugates of Integrally Convex Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Akihisa Tamura, Kazuo Murota","submitted_at":"2018-06-04T07:43:10Z","abstract_excerpt":"Integrally convex functions constitute a fundamental function class in discrete convex analysis. This paper shows that an integer-valued integrally convex function admits an integral subgradient and that the integral biconjugate of an integer-valued integrally convex function coincides with itself. The proof is based on the Fourier-Motzkin elimination. The latter result provides a unified proof of integral biconjugacy for various classes of integer-valued discrete convex functions, including L-convex, M-convex, L$_{2}$-convex, M$_{2}$-convex, BS-convex, and UJ-convex functions as well as multi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00992","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1806.00992","created_at":"2026-05-18T00:06:16.009659+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.00992v2","created_at":"2026-05-18T00:06:16.009659+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00992","created_at":"2026-05-18T00:06:16.009659+00:00"},{"alias_kind":"pith_short_12","alias_value":"YGE5E2OCNOWO","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"YGE5E2OCNOWOSM5L","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"YGE5E2OC","created_at":"2026-05-18T12:33:04.347982+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/YGE5E2OCNOWOSM5LOH26EEG4NT","json":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT.json","graph_json":"https://pith.science/api/pith-number/YGE5E2OCNOWOSM5LOH26EEG4NT/graph.json","events_json":"https://pith.science/api/pith-number/YGE5E2OCNOWOSM5LOH26EEG4NT/events.json","paper":"https://pith.science/paper/YGE5E2OC"},"agent_actions":{"view_html":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT","download_json":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT.json","view_paper":"https://pith.science/paper/YGE5E2OC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.00992&json=true","fetch_graph":"https://pith.science/api/pith-number/YGE5E2OCNOWOSM5LOH26EEG4NT/graph.json","fetch_events":"https://pith.science/api/pith-number/YGE5E2OCNOWOSM5LOH26EEG4NT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT/action/storage_attestation","attest_author":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT/action/author_attestation","sign_citation":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT/action/citation_signature","submit_replication":"https://pith.science/pith/YGE5E2OCNOWOSM5LOH26EEG4NT/action/replication_record"}},"created_at":"2026-05-18T00:06:16.009659+00:00","updated_at":"2026-05-18T00:06:16.009659+00:00"}