{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:AAL6E3MI2P57MBPTMVQBGJ5IQU","short_pith_number":"pith:AAL6E3MI","canonical_record":{"source":{"id":"2201.10689","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2022-01-26T00:29:52Z","cross_cats_sorted":[],"title_canon_sha256":"edb37819802fd59c071c2bb622ce01773de7b75f14bbfc8746d6a0e21af0236c","abstract_canon_sha256":"2d92563836d2ec6baecd85b8a48e0544ad8c227d2099556ee567826389083f44"},"schema_version":"1.0"},"canonical_sha256":"0017e26d88d3fbf605f365601327a8850c4187ea043f4cc0350e828dc9e5bc68","source":{"kind":"arxiv","id":"2201.10689","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.10689","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"arxiv_version","alias_value":"2201.10689v2","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.10689","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"pith_short_12","alias_value":"AAL6E3MI2P57","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"pith_short_16","alias_value":"AAL6E3MI2P57MBPT","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"pith_short_8","alias_value":"AAL6E3MI","created_at":"2026-07-05T05:58:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:AAL6E3MI2P57MBPTMVQBGJ5IQU","target":"record","payload":{"canonical_record":{"source":{"id":"2201.10689","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2022-01-26T00:29:52Z","cross_cats_sorted":[],"title_canon_sha256":"edb37819802fd59c071c2bb622ce01773de7b75f14bbfc8746d6a0e21af0236c","abstract_canon_sha256":"2d92563836d2ec6baecd85b8a48e0544ad8c227d2099556ee567826389083f44"},"schema_version":"1.0"},"canonical_sha256":"0017e26d88d3fbf605f365601327a8850c4187ea043f4cc0350e828dc9e5bc68","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:58:14.974079Z","signature_b64":"ZTugnf6oMFOy4sjqAf3ProiTIs9hzNMUwEZEyKZccwnRKYfsF+t+NUl2AOXfVB2v4e2MuQsUiEJeqhGI/+wrDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0017e26d88d3fbf605f365601327a8850c4187ea043f4cc0350e828dc9e5bc68","last_reissued_at":"2026-07-05T05:58:14.973736Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:58:14.973736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2201.10689","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:58:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/eG0nhgiADJgM8eqCrzVOjnQOUPaBnSBR/70HGf8rO8A3XQ3hsA6srw6vYP0eHJacEPSEptAgzPikpsNbzsWDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T09:43:42.698238Z"},"content_sha256":"ec4072050dcd263474bb7da51132c3f839cb58b8efae8b5ecbf6bf328113ed79","schema_version":"1.0","event_id":"sha256:ec4072050dcd263474bb7da51132c3f839cb58b8efae8b5ecbf6bf328113ed79"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:AAL6E3MI2P57MBPTMVQBGJ5IQU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Boris Mordukhovich, Dang Van Cuong, Gary Sandine, Nguyen Mau Nam","submitted_at":"2022-01-26T00:29:52Z","abstract_excerpt":"In this paper we revisit a theorem by Rockafellar on representing the relative interior of the graph of a convex set-valued mapping in terms of the relative interior of its domain and function values. Then we apply this theorem to provide a simple way to prove many calculus rules of generalized differentiation of set-valued mappings and nonsmooth functions in finite dimensions. These results improve upon those in [14] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.10689","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.10689/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:58:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1vLIWfmeq7kzgBcPsHYK1P3sWuuimviL9jLSIU4THnLKlotxvUjkbWtq31WNpCa0tXETjx7WFIx9n1iNVfGqAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T09:43:42.698724Z"},"content_sha256":"082a5f42fba2442e410e258d230260a414bfa33675492fc69a171eafe8df1add","schema_version":"1.0","event_id":"sha256:082a5f42fba2442e410e258d230260a414bfa33675492fc69a171eafe8df1add"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU/bundle.json","state_url":"https://pith.science/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-04T09:43:42Z","links":{"resolver":"https://pith.science/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU","bundle":"https://pith.science/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU/bundle.json","state":"https://pith.science/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AAL6E3MI2P57MBPTMVQBGJ5IQU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:AAL6E3MI2P57MBPTMVQBGJ5IQU","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":"2d92563836d2ec6baecd85b8a48e0544ad8c227d2099556ee567826389083f44","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2022-01-26T00:29:52Z","title_canon_sha256":"edb37819802fd59c071c2bb622ce01773de7b75f14bbfc8746d6a0e21af0236c"},"schema_version":"1.0","source":{"id":"2201.10689","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.10689","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"arxiv_version","alias_value":"2201.10689v2","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.10689","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"pith_short_12","alias_value":"AAL6E3MI2P57","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"pith_short_16","alias_value":"AAL6E3MI2P57MBPT","created_at":"2026-07-05T05:58:14Z"},{"alias_kind":"pith_short_8","alias_value":"AAL6E3MI","created_at":"2026-07-05T05:58:14Z"}],"graph_snapshots":[{"event_id":"sha256:082a5f42fba2442e410e258d230260a414bfa33675492fc69a171eafe8df1add","target":"graph","created_at":"2026-07-05T05:58:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2201.10689/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we revisit a theorem by Rockafellar on representing the relative interior of the graph of a convex set-valued mapping in terms of the relative interior of its domain and function values. Then we apply this theorem to provide a simple way to prove many calculus rules of generalized differentiation of set-valued mappings and nonsmooth functions in finite dimensions. These results improve upon those in [14] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges.","authors_text":"Boris Mordukhovich, Dang Van Cuong, Gary Sandine, Nguyen Mau Nam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2022-01-26T00:29:52Z","title":"Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.10689","kind":"arxiv","version":2},"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:ec4072050dcd263474bb7da51132c3f839cb58b8efae8b5ecbf6bf328113ed79","target":"record","created_at":"2026-07-05T05:58:14Z","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":"2d92563836d2ec6baecd85b8a48e0544ad8c227d2099556ee567826389083f44","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2022-01-26T00:29:52Z","title_canon_sha256":"edb37819802fd59c071c2bb622ce01773de7b75f14bbfc8746d6a0e21af0236c"},"schema_version":"1.0","source":{"id":"2201.10689","kind":"arxiv","version":2}},"canonical_sha256":"0017e26d88d3fbf605f365601327a8850c4187ea043f4cc0350e828dc9e5bc68","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0017e26d88d3fbf605f365601327a8850c4187ea043f4cc0350e828dc9e5bc68","first_computed_at":"2026-07-05T05:58:14.973736Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:58:14.973736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZTugnf6oMFOy4sjqAf3ProiTIs9hzNMUwEZEyKZccwnRKYfsF+t+NUl2AOXfVB2v4e2MuQsUiEJeqhGI/+wrDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:58:14.974079Z","signed_message":"canonical_sha256_bytes"},"source_id":"2201.10689","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec4072050dcd263474bb7da51132c3f839cb58b8efae8b5ecbf6bf328113ed79","sha256:082a5f42fba2442e410e258d230260a414bfa33675492fc69a171eafe8df1add"],"state_sha256":"d9c4d9fc63c1928ffe19a636a7835f3ef9baace5e65052dd9430e717465b65ec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TgSEc4+6XVSVuuLjajdHWdfTB4C4NoBn9I2O7HQBV/c9q0I6XY5d1yqpHAYrNu0vtTU41NI9VzBOpC2WA1M4DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T09:43:42.702654Z","bundle_sha256":"62c86374dbc48f3769e482c5ef00326366f361d9d5c815214ba82430c9d2ac92"}}