{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WODCGC7EMZ5KT5ZBTCE5NJBKER","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3763e6608da6d312d479da14098f6e94e2350e4934c16f0a0c4397a115bd84cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-06T15:35:42Z","title_canon_sha256":"6bf5d93ffbd885c7d58eb562f32ccad460c89560357fab6871c8592c1d452441"},"schema_version":"1.0","source":{"id":"1508.01436","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.01436","created_at":"2026-05-18T01:35:42Z"},{"alias_kind":"arxiv_version","alias_value":"1508.01436v1","created_at":"2026-05-18T01:35:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.01436","created_at":"2026-05-18T01:35:42Z"},{"alias_kind":"pith_short_12","alias_value":"WODCGC7EMZ5K","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WODCGC7EMZ5KT5ZB","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WODCGC7E","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:b873d9a791913ed9c558db92d9fd1efb78db4cb0bdf8e2c4d4ebef56d2a85bd2","target":"graph","created_at":"2026-05-18T01:35:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"According to the Ambrosetti-Prodi theorem, the map $F(u)= - \\Delta u - f(u)$ between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function $f$ is required. We show in two different ways that, under mild conditions, convexity is indeed necessary. If $f$ is not convex, there is a point with at least four preimages under $F$. More, $F$ generically admits cusps among its critical points. We present a larger class of nonlinearities $f$ for which the critical set of $F$ has cusps. The results are true for a class of boundary conditions.","authors_text":"Andr\\'e Zaccur, Carlos Tomei, Marta Calanchi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-06T15:35:42Z","title":"Abundance of cusps and a converse to the Ambrosetti-Prodi theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01436","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b842d3115c9a94170a647fa615eda3bd9010be64063a61b7c1cff115628f9b20","target":"record","created_at":"2026-05-18T01:35:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3763e6608da6d312d479da14098f6e94e2350e4934c16f0a0c4397a115bd84cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-06T15:35:42Z","title_canon_sha256":"6bf5d93ffbd885c7d58eb562f32ccad460c89560357fab6871c8592c1d452441"},"schema_version":"1.0","source":{"id":"1508.01436","kind":"arxiv","version":1}},"canonical_sha256":"b386230be4667aa9f7219889d6a42a247183a80f892ca8fe4547f0c3277741b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b386230be4667aa9f7219889d6a42a247183a80f892ca8fe4547f0c3277741b7","first_computed_at":"2026-05-18T01:35:42.628700Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:42.628700Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gbWQSamPKyVD4LaE04uKk+t2KD7BbNqGluOWS9vPkP1D6oEakuw6QF+5PISoaLMiydejo9oBXo7QjyLAwyAjDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:42.629368Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.01436","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b842d3115c9a94170a647fa615eda3bd9010be64063a61b7c1cff115628f9b20","sha256:b873d9a791913ed9c558db92d9fd1efb78db4cb0bdf8e2c4d4ebef56d2a85bd2"],"state_sha256":"718ffa8e7b3e673b77d9428253ff4caf1885e166215c04615007a58986cf904f"}