{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:R7AZJZNPOCGI4H2NXNZ47KZUPA","short_pith_number":"pith:R7AZJZNP","schema_version":"1.0","canonical_sha256":"8fc194e5af708c8e1f4dbb73cfab347801924147c4d05620e4537856b32eb0b2","source":{"kind":"arxiv","id":"1211.0747","version":2},"attestation_state":"computed","paper":{"title":"Conditional Analysis on R^d","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Michael Kupper, Nicolas Vogelpoth, Patrick Cheridito","submitted_at":"2012-11-05T02:34:07Z","abstract_excerpt":"This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring $L^0$ of measurable functions on a $\\sigma$-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of $L^0$-affine sets, $L^0$-convex sets, $L^0$-convex cones, $L^0$-hyperplanes, $L^0$-half-spaces and $L^0$-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study $L^0$-linear, $L^0$-affine, "},"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":"1211.0747","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-05T02:34:07Z","cross_cats_sorted":[],"title_canon_sha256":"51acfc1ce1cb716ac64fae09b9a13f9e5cacaa58f2f0528036e2a84bdbcdfcd2","abstract_canon_sha256":"ff0cc414804a1a53f2eff1fa994f268413bf97c2ae9cd0cdbeed1efedea9f5c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:25.806996Z","signature_b64":"Tr4SMNcvsUEJtLCqAe428RctxiuunGoe149KdPstLTvrIBGl+oejTya0Aehwd3UOI5BA9+HO970xGU28PfjfAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8fc194e5af708c8e1f4dbb73cfab347801924147c4d05620e4537856b32eb0b2","last_reissued_at":"2026-05-18T02:39:25.806418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:25.806418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Conditional Analysis on R^d","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Michael Kupper, Nicolas Vogelpoth, Patrick Cheridito","submitted_at":"2012-11-05T02:34:07Z","abstract_excerpt":"This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring $L^0$ of measurable functions on a $\\sigma$-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of $L^0$-affine sets, $L^0$-convex sets, $L^0$-convex cones, $L^0$-hyperplanes, $L^0$-half-spaces and $L^0$-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study $L^0$-linear, $L^0$-affine, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.0747","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":"1211.0747","created_at":"2026-05-18T02:39:25.806513+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.0747v2","created_at":"2026-05-18T02:39:25.806513+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.0747","created_at":"2026-05-18T02:39:25.806513+00:00"},{"alias_kind":"pith_short_12","alias_value":"R7AZJZNPOCGI","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"R7AZJZNPOCGI4H2N","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"R7AZJZNP","created_at":"2026-05-18T12:27:20.899486+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/R7AZJZNPOCGI4H2NXNZ47KZUPA","json":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA.json","graph_json":"https://pith.science/api/pith-number/R7AZJZNPOCGI4H2NXNZ47KZUPA/graph.json","events_json":"https://pith.science/api/pith-number/R7AZJZNPOCGI4H2NXNZ47KZUPA/events.json","paper":"https://pith.science/paper/R7AZJZNP"},"agent_actions":{"view_html":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA","download_json":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA.json","view_paper":"https://pith.science/paper/R7AZJZNP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.0747&json=true","fetch_graph":"https://pith.science/api/pith-number/R7AZJZNPOCGI4H2NXNZ47KZUPA/graph.json","fetch_events":"https://pith.science/api/pith-number/R7AZJZNPOCGI4H2NXNZ47KZUPA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA/action/storage_attestation","attest_author":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA/action/author_attestation","sign_citation":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA/action/citation_signature","submit_replication":"https://pith.science/pith/R7AZJZNPOCGI4H2NXNZ47KZUPA/action/replication_record"}},"created_at":"2026-05-18T02:39:25.806513+00:00","updated_at":"2026-05-18T02:39:25.806513+00:00"}