{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:Z6NVGVMNSYG552LHNT3HLVBSOG","short_pith_number":"pith:Z6NVGVMN","schema_version":"1.0","canonical_sha256":"cf9b53558d960ddee9676cf675d4327182083a5ac6493d111fc022017dd0cff7","source":{"kind":"arxiv","id":"1804.08289","version":2},"attestation_state":"computed","paper":{"title":"$C^1$ mappings in $\\mathbb{R}^5$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Pawe{\\l} Goldstein, Piotr Haj{\\l}asz","submitted_at":"2018-04-23T08:47:25Z","abstract_excerpt":"We find a counterexample to a conjecture of Ga{\\l}\\k{e}ski by constructing for some positive integers $m<n$ a mapping $f\\in C^1(\\mathbb{R}^n,\\mathbb{R}^n)$ satisfying $\\mathrm{rank}\\, Df\\leq m$ that, even locally, cannot be uniformly approximated by $C^2$ mappings $f_\\varepsilon$ satisfying the same rank constraint $\\mathrm{rank}\\, Df_\\varepsilon\\leq m$."},"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":"1804.08289","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-23T08:47:25Z","cross_cats_sorted":[],"title_canon_sha256":"aa5a30d7c93bdee05b571379152a11549464a0cc23118c1eb14d53f9f90ec6f2","abstract_canon_sha256":"620cbf181711e0205d67b13a6e409c5c35a87d1cae84d4490fb8c109d1d2eacf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:21.415265Z","signature_b64":"ilTZOLvGWHo5pltojrFplt/sysVH+HaEEQwVfYGMrYO4VIIR9PssXTwjGUNkxEbgjqiJp++VkUCFc0GXTE/vBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf9b53558d960ddee9676cf675d4327182083a5ac6493d111fc022017dd0cff7","last_reissued_at":"2026-05-18T00:14:21.414608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:21.414608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$C^1$ mappings in $\\mathbb{R}^5$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Pawe{\\l} Goldstein, Piotr Haj{\\l}asz","submitted_at":"2018-04-23T08:47:25Z","abstract_excerpt":"We find a counterexample to a conjecture of Ga{\\l}\\k{e}ski by constructing for some positive integers $m<n$ a mapping $f\\in C^1(\\mathbb{R}^n,\\mathbb{R}^n)$ satisfying $\\mathrm{rank}\\, Df\\leq m$ that, even locally, cannot be uniformly approximated by $C^2$ mappings $f_\\varepsilon$ satisfying the same rank constraint $\\mathrm{rank}\\, Df_\\varepsilon\\leq m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08289","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":"1804.08289","created_at":"2026-05-18T00:14:21.414716+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.08289v2","created_at":"2026-05-18T00:14:21.414716+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.08289","created_at":"2026-05-18T00:14:21.414716+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z6NVGVMNSYG5","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z6NVGVMNSYG552LH","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z6NVGVMN","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/Z6NVGVMNSYG552LHNT3HLVBSOG","json":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG.json","graph_json":"https://pith.science/api/pith-number/Z6NVGVMNSYG552LHNT3HLVBSOG/graph.json","events_json":"https://pith.science/api/pith-number/Z6NVGVMNSYG552LHNT3HLVBSOG/events.json","paper":"https://pith.science/paper/Z6NVGVMN"},"agent_actions":{"view_html":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG","download_json":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG.json","view_paper":"https://pith.science/paper/Z6NVGVMN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.08289&json=true","fetch_graph":"https://pith.science/api/pith-number/Z6NVGVMNSYG552LHNT3HLVBSOG/graph.json","fetch_events":"https://pith.science/api/pith-number/Z6NVGVMNSYG552LHNT3HLVBSOG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG/action/storage_attestation","attest_author":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG/action/author_attestation","sign_citation":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG/action/citation_signature","submit_replication":"https://pith.science/pith/Z6NVGVMNSYG552LHNT3HLVBSOG/action/replication_record"}},"created_at":"2026-05-18T00:14:21.414716+00:00","updated_at":"2026-05-18T00:14:21.414716+00:00"}