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We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\\psi(x) = (x, y_1(x)-y_2(x)), x \\in [0,\\alpha]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\\alpha]$. In other words: singularities propagate along arcs with finite turn."},"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":"1002.2911","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-02-15T16:43:45Z","cross_cats_sorted":[],"title_canon_sha256":"81e4f3991b48bdff29c8b82022414398d397e562589002e47d859f37ae4b338f","abstract_canon_sha256":"e7d7ab97170c9c21dcefdfaa2d8b924d7a6b05ccd721cb38b6b185ee749829bc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:21:47.781646Z","signature_b64":"RGUi+4wWFlL3BN4LY1Ddf9Al6dU4A4rBlzmehEpvSlq/UPcDyzXqwbHfeTCdIfMXYm8Ybs5+H/nVo6RjLMGlAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f82910fd9d8b42f4f11fce135ba889320e449cc3f64a3faf8791ad657a6e7e6","last_reissued_at":"2026-07-04T17:21:47.781211Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:21:47.781211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on propagation of singularities of semiconcave functions of two variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ludek Zajicek","submitted_at":"2010-02-15T16:43:45Z","abstract_excerpt":"P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in $\\R^n$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\\psi(x) = (x, y_1(x)-y_2(x)), x \\in [0,\\alpha]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\\alpha]$. In other words: singularities propagate along arcs with finite turn."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.2911","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1002.2911/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1002.2911","created_at":"2026-07-04T17:21:47.781277+00:00"},{"alias_kind":"arxiv_version","alias_value":"1002.2911v1","created_at":"2026-07-04T17:21:47.781277+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.2911","created_at":"2026-07-04T17:21:47.781277+00:00"},{"alias_kind":"pith_short_12","alias_value":"R6BJCD6Z3C2C","created_at":"2026-07-04T17:21:47.781277+00:00"},{"alias_kind":"pith_short_16","alias_value":"R6BJCD6Z3C2C6TYR","created_at":"2026-07-04T17:21:47.781277+00:00"},{"alias_kind":"pith_short_8","alias_value":"R6BJCD6Z","created_at":"2026-07-04T17:21:47.781277+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/R6BJCD6Z3C2C6TYR7TQTLOUISM","json":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM.json","graph_json":"https://pith.science/api/pith-number/R6BJCD6Z3C2C6TYR7TQTLOUISM/graph.json","events_json":"https://pith.science/api/pith-number/R6BJCD6Z3C2C6TYR7TQTLOUISM/events.json","paper":"https://pith.science/paper/R6BJCD6Z"},"agent_actions":{"view_html":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM","download_json":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM.json","view_paper":"https://pith.science/paper/R6BJCD6Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1002.2911&json=true","fetch_graph":"https://pith.science/api/pith-number/R6BJCD6Z3C2C6TYR7TQTLOUISM/graph.json","fetch_events":"https://pith.science/api/pith-number/R6BJCD6Z3C2C6TYR7TQTLOUISM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM/action/storage_attestation","attest_author":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM/action/author_attestation","sign_citation":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM/action/citation_signature","submit_replication":"https://pith.science/pith/R6BJCD6Z3C2C6TYR7TQTLOUISM/action/replication_record"}},"created_at":"2026-07-04T17:21:47.781277+00:00","updated_at":"2026-07-04T17:21:47.781277+00:00"}