{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:Q6EMGZYI2ESHV4IXQJT3D6CNL5","short_pith_number":"pith:Q6EMGZYI","canonical_record":{"source":{"id":"1212.0887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-04T21:58:50Z","cross_cats_sorted":[],"title_canon_sha256":"e54761a7d6c3a80a607d7efd82a900b46abe92f5cf57ed1498a4a430e62af064","abstract_canon_sha256":"27ead70e0cc7d6bad1398547f6621ff38cf723700a76a5577030299bcb876d8e"},"schema_version":"1.0"},"canonical_sha256":"8788c36708d1247af1178267b1f84d5f70b22c6fc71a1965eadc459e1dec9f76","source":{"kind":"arxiv","id":"1212.0887","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.0887","created_at":"2026-05-18T03:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"1212.0887v1","created_at":"2026-05-18T03:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0887","created_at":"2026-05-18T03:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"Q6EMGZYI2ESH","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"Q6EMGZYI2ESHV4IX","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"Q6EMGZYI","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:Q6EMGZYI2ESHV4IXQJT3D6CNL5","target":"record","payload":{"canonical_record":{"source":{"id":"1212.0887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-04T21:58:50Z","cross_cats_sorted":[],"title_canon_sha256":"e54761a7d6c3a80a607d7efd82a900b46abe92f5cf57ed1498a4a430e62af064","abstract_canon_sha256":"27ead70e0cc7d6bad1398547f6621ff38cf723700a76a5577030299bcb876d8e"},"schema_version":"1.0"},"canonical_sha256":"8788c36708d1247af1178267b1f84d5f70b22c6fc71a1965eadc459e1dec9f76","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:07.055801Z","signature_b64":"I1RG5rVMGkxt/SLffJo7r5jcy3QCC+7m5t/jXScA0FXZTV8jOdVH/KBCK/gk9u5I/HJ7vyO615QfRwh0grBPAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8788c36708d1247af1178267b1f84d5f70b22c6fc71a1965eadc459e1dec9f76","last_reissued_at":"2026-05-18T03:39:07.055141Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:07.055141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.0887","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QqdCeckl8g43iPuSYUTg6/uIHTlqn+1cySXZFWXkFTcVmu+qabnmrgFVVb6E96F/qPcc2nTGrWZTsfCi31b0Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T03:54:57.637788Z"},"content_sha256":"aaf1d16e39ca2ade2046409caba9f99ae7b07b0a8a3590c64d1f8c8d4b527a34","schema_version":"1.0","event_id":"sha256:aaf1d16e39ca2ade2046409caba9f99ae7b07b0a8a3590c64d1f8c8d4b527a34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:Q6EMGZYI2ESHV4IXQJT3D6CNL5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On strong $(\\alpha,\\F)$-convexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Judit Mak\\'o, Kazimierz Nikodem, Zsolt P\\'ales","submitted_at":"2012-12-04T21:58:50Z","abstract_excerpt":"In this paper, strongly $(\\alpha,T)$-convex functions, i.e., functions $f:D\\to \\R$ satisfying the functional inequality $$ f(tx+(1-t)y)\\leq tf(x)+(1-t)f(y)-t\\alpha\\big((1-t)(x-y)\\big)-(1-t)\\alpha\\big(t(y-x)\\big)$$ for $x,y\\in D$ and $t\\in T\\cap[0,1]$ are investigated. Here $D$ is a convex set in a linear space, $\\alpha$ is a nonnegative function on $D-D$, and $T\\subseteq\\R$ is a nonempty set. The main results provide various characterizations of strong $(\\alpha,T)$-convexity in the case when $T$ is a subfield of $\\R$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0887","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":""},"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":"2026-05-18T03:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JEe+kj/0K0zdE5dr8JBayqJ6YLOobl4Ef5pjFbFZ0zaqFFh8E/hze3CxmFOpdOhoygeBB/IBsQZgT3vLTKBrAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T03:54:57.638438Z"},"content_sha256":"dcfe2f3e2ec31d4e326c437274a59f9b9b0cb90605b60b81bb3a0482e34d95cc","schema_version":"1.0","event_id":"sha256:dcfe2f3e2ec31d4e326c437274a59f9b9b0cb90605b60b81bb3a0482e34d95cc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5/bundle.json","state_url":"https://pith.science/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5/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":"2026-06-06T03:54:57Z","links":{"resolver":"https://pith.science/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5","bundle":"https://pith.science/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5/bundle.json","state":"https://pith.science/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q6EMGZYI2ESHV4IXQJT3D6CNL5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:Q6EMGZYI2ESHV4IXQJT3D6CNL5","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":"27ead70e0cc7d6bad1398547f6621ff38cf723700a76a5577030299bcb876d8e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-04T21:58:50Z","title_canon_sha256":"e54761a7d6c3a80a607d7efd82a900b46abe92f5cf57ed1498a4a430e62af064"},"schema_version":"1.0","source":{"id":"1212.0887","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.0887","created_at":"2026-05-18T03:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"1212.0887v1","created_at":"2026-05-18T03:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0887","created_at":"2026-05-18T03:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"Q6EMGZYI2ESH","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"Q6EMGZYI2ESHV4IX","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"Q6EMGZYI","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:dcfe2f3e2ec31d4e326c437274a59f9b9b0cb90605b60b81bb3a0482e34d95cc","target":"graph","created_at":"2026-05-18T03:39:07Z","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":"In this paper, strongly $(\\alpha,T)$-convex functions, i.e., functions $f:D\\to \\R$ satisfying the functional inequality $$ f(tx+(1-t)y)\\leq tf(x)+(1-t)f(y)-t\\alpha\\big((1-t)(x-y)\\big)-(1-t)\\alpha\\big(t(y-x)\\big)$$ for $x,y\\in D$ and $t\\in T\\cap[0,1]$ are investigated. Here $D$ is a convex set in a linear space, $\\alpha$ is a nonnegative function on $D-D$, and $T\\subseteq\\R$ is a nonempty set. The main results provide various characterizations of strong $(\\alpha,T)$-convexity in the case when $T$ is a subfield of $\\R$.","authors_text":"Judit Mak\\'o, Kazimierz Nikodem, Zsolt P\\'ales","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-04T21:58:50Z","title":"On strong $(\\alpha,\\F)$-convexity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0887","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:aaf1d16e39ca2ade2046409caba9f99ae7b07b0a8a3590c64d1f8c8d4b527a34","target":"record","created_at":"2026-05-18T03:39:07Z","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":"27ead70e0cc7d6bad1398547f6621ff38cf723700a76a5577030299bcb876d8e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-04T21:58:50Z","title_canon_sha256":"e54761a7d6c3a80a607d7efd82a900b46abe92f5cf57ed1498a4a430e62af064"},"schema_version":"1.0","source":{"id":"1212.0887","kind":"arxiv","version":1}},"canonical_sha256":"8788c36708d1247af1178267b1f84d5f70b22c6fc71a1965eadc459e1dec9f76","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8788c36708d1247af1178267b1f84d5f70b22c6fc71a1965eadc459e1dec9f76","first_computed_at":"2026-05-18T03:39:07.055141Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:07.055141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I1RG5rVMGkxt/SLffJo7r5jcy3QCC+7m5t/jXScA0FXZTV8jOdVH/KBCK/gk9u5I/HJ7vyO615QfRwh0grBPAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:07.055801Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.0887","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aaf1d16e39ca2ade2046409caba9f99ae7b07b0a8a3590c64d1f8c8d4b527a34","sha256:dcfe2f3e2ec31d4e326c437274a59f9b9b0cb90605b60b81bb3a0482e34d95cc"],"state_sha256":"702eb986d5707c9de833658ed13b4a3f9540124a16ec55e1e91ae917bd13a090"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eZjeJhrbluziAAHZCt8h8Jb4L7UO9vFgriWtLFkCxiPpMMFGm7LfQP/P9NATivA62V54m3ikYskXCW+QMmgXAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T03:54:57.642783Z","bundle_sha256":"a7cf7d1d75fcf1fdc5f08617adfa7483c87ba72fc851e90fc6c769da21254b6f"}}