{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:TJVDFNUSX3IRDWDPHFP4VWKAPI","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":"b58da8e21af00eeadf627bf98e4535f768d1aac4725d0afa1c1b4ab74cd9ce24","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-10-08T13:40:06Z","title_canon_sha256":"1908155429909cd0fb3c60c469cbc6a604139f6ace6a387fb6012fb1bf83ce97"},"schema_version":"1.0","source":{"id":"0810.1433","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0810.1433","created_at":"2026-07-04T15:14:58Z"},{"alias_kind":"arxiv_version","alias_value":"0810.1433v1","created_at":"2026-07-04T15:14:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.1433","created_at":"2026-07-04T15:14:58Z"},{"alias_kind":"pith_short_12","alias_value":"TJVDFNUSX3IR","created_at":"2026-07-04T15:14:58Z"},{"alias_kind":"pith_short_16","alias_value":"TJVDFNUSX3IRDWDP","created_at":"2026-07-04T15:14:58Z"},{"alias_kind":"pith_short_8","alias_value":"TJVDFNUS","created_at":"2026-07-04T15:14:58Z"}],"graph_snapshots":[{"event_id":"sha256:9fb5b76490b98b0308a614719bb1b0691ae73a7fbfb2c63aaed7004e078de0a6","target":"graph","created_at":"2026-07-04T15:14:58Z","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/0810.1433/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems, concerning extendability of continuous convex functions from a closed subspace of a normed linear space, complement recent results of J.Borwein, V.Montesinos and J.Vanderwerff.","authors_text":"Libor Vesely, Ludek Zajicek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-10-08T13:40:06Z","title":"On extensions of d.c. functions and convex functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.1433","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":"sha256:5017692f6d1115330a1339bb288d546df03e640f30f24d312b8b568d19724271","target":"record","created_at":"2026-07-04T15:14:58Z","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":"b58da8e21af00eeadf627bf98e4535f768d1aac4725d0afa1c1b4ab74cd9ce24","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-10-08T13:40:06Z","title_canon_sha256":"1908155429909cd0fb3c60c469cbc6a604139f6ace6a387fb6012fb1bf83ce97"},"schema_version":"1.0","source":{"id":"0810.1433","kind":"arxiv","version":1}},"canonical_sha256":"9a6a32b692bed111d86f395fcad9407a264e10bcc25bd08668a541bdeaea7e49","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a6a32b692bed111d86f395fcad9407a264e10bcc25bd08668a541bdeaea7e49","first_computed_at":"2026-07-04T15:14:58.557030Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:14:58.557030Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q24oFehpdO+3cJxrg65E1Ocq1NxuKMQCDpjlQwWhnTOizJX9XudxNspJYQTI5RpVZiIKjwkS4zMQxG97448mBw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:14:58.557437Z","signed_message":"canonical_sha256_bytes"},"source_id":"0810.1433","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5017692f6d1115330a1339bb288d546df03e640f30f24d312b8b568d19724271","sha256:9fb5b76490b98b0308a614719bb1b0691ae73a7fbfb2c63aaed7004e078de0a6"],"state_sha256":"60efdc7c41368552914a27184c01814a6ebc23b8fe0b56516ac85fefc7f81da9"}