{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:E5IPHO3F67PZCCG57Z2P4BGW3A","short_pith_number":"pith:E5IPHO3F","canonical_record":{"source":{"id":"1902.01244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-02-04T15:17:54Z","cross_cats_sorted":[],"title_canon_sha256":"511cf1121da1a43c2542ebbe3c15d651d7a173a7a97eb6fbc1dac3765425f4c4","abstract_canon_sha256":"88d241f1dfb102716ce4b3c35024274b0afb6a8b3319eb0e51b28f3d7b984a95"},"schema_version":"1.0"},"canonical_sha256":"2750f3bb65f7df9108ddfe74fe04d6d804be84d6166ac8aeab733a6957f2bdf8","source":{"kind":"arxiv","id":"1902.01244","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.01244","created_at":"2026-05-17T23:54:49Z"},{"alias_kind":"arxiv_version","alias_value":"1902.01244v1","created_at":"2026-05-17T23:54:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.01244","created_at":"2026-05-17T23:54:49Z"},{"alias_kind":"pith_short_12","alias_value":"E5IPHO3F67PZ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"E5IPHO3F67PZCCG5","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"E5IPHO3F","created_at":"2026-05-18T12:33:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:E5IPHO3F67PZCCG57Z2P4BGW3A","target":"record","payload":{"canonical_record":{"source":{"id":"1902.01244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-02-04T15:17:54Z","cross_cats_sorted":[],"title_canon_sha256":"511cf1121da1a43c2542ebbe3c15d651d7a173a7a97eb6fbc1dac3765425f4c4","abstract_canon_sha256":"88d241f1dfb102716ce4b3c35024274b0afb6a8b3319eb0e51b28f3d7b984a95"},"schema_version":"1.0"},"canonical_sha256":"2750f3bb65f7df9108ddfe74fe04d6d804be84d6166ac8aeab733a6957f2bdf8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:49.329772Z","signature_b64":"sJwQ2oIz+hxqStS8mVrmtd3ByEOENU2gsoXNUQts3cUh3pgdwOX7Dno+M7JMSAbBBePNPUgo4iGFOZbHKTLZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2750f3bb65f7df9108ddfe74fe04d6d804be84d6166ac8aeab733a6957f2bdf8","last_reissued_at":"2026-05-17T23:54:49.329253Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:49.329253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1902.01244","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":"2026-05-17T23:54:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ycQndTfgMOO3yIr4KBWB/2d440LeE/I18CJy1YyVpkdkt+gkztdGzlrSOT81Z1dXKszM+J5dID58HAjAu8JuCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T20:36:52.502377Z"},"content_sha256":"6d35c86b0a21c0c683012885d637498782a640ce73de66778a38f02597154f05","schema_version":"1.0","event_id":"sha256:6d35c86b0a21c0c683012885d637498782a640ce73de66778a38f02597154f05"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:E5IPHO3F67PZCCG57Z2P4BGW3A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Martingale-like sequences in Banach lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Melnikov, Haile Gessesse","submitted_at":"2019-02-04T15:17:54Z","abstract_excerpt":"Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437--456]. In these frameworks, a collection of bounded $X$-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\\mathcal{E}$-martingales is a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.01244","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":"2026-05-17T23:54:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DTd6M/i+Jj7hJ5e1JQh6dU7BGYXCjV8AGhE9yHQGVtos1sh62NWsF0Li6/LabCh6njLOcXeZ+1VwdKgW1/kWAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T20:36:52.502730Z"},"content_sha256":"e47b87030856082b17468fd85ab35b6f0da19c670d6070d5a85ef668b2b7771c","schema_version":"1.0","event_id":"sha256:e47b87030856082b17468fd85ab35b6f0da19c670d6070d5a85ef668b2b7771c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E5IPHO3F67PZCCG57Z2P4BGW3A/bundle.json","state_url":"https://pith.science/pith/E5IPHO3F67PZCCG57Z2P4BGW3A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E5IPHO3F67PZCCG57Z2P4BGW3A/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":"2026-05-18T20:36:52Z","links":{"resolver":"https://pith.science/pith/E5IPHO3F67PZCCG57Z2P4BGW3A","bundle":"https://pith.science/pith/E5IPHO3F67PZCCG57Z2P4BGW3A/bundle.json","state":"https://pith.science/pith/E5IPHO3F67PZCCG57Z2P4BGW3A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E5IPHO3F67PZCCG57Z2P4BGW3A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:E5IPHO3F67PZCCG57Z2P4BGW3A","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":"88d241f1dfb102716ce4b3c35024274b0afb6a8b3319eb0e51b28f3d7b984a95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-02-04T15:17:54Z","title_canon_sha256":"511cf1121da1a43c2542ebbe3c15d651d7a173a7a97eb6fbc1dac3765425f4c4"},"schema_version":"1.0","source":{"id":"1902.01244","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.01244","created_at":"2026-05-17T23:54:49Z"},{"alias_kind":"arxiv_version","alias_value":"1902.01244v1","created_at":"2026-05-17T23:54:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.01244","created_at":"2026-05-17T23:54:49Z"},{"alias_kind":"pith_short_12","alias_value":"E5IPHO3F67PZ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"E5IPHO3F67PZCCG5","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"E5IPHO3F","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:e47b87030856082b17468fd85ab35b6f0da19c670d6070d5a85ef668b2b7771c","target":"graph","created_at":"2026-05-17T23:54:49Z","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":"Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437--456]. In these frameworks, a collection of bounded $X$-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\\mathcal{E}$-martingales is a","authors_text":"Alexander Melnikov, Haile Gessesse","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-02-04T15:17:54Z","title":"Martingale-like sequences in Banach lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.01244","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":"sha256:6d35c86b0a21c0c683012885d637498782a640ce73de66778a38f02597154f05","target":"record","created_at":"2026-05-17T23:54:49Z","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":"88d241f1dfb102716ce4b3c35024274b0afb6a8b3319eb0e51b28f3d7b984a95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-02-04T15:17:54Z","title_canon_sha256":"511cf1121da1a43c2542ebbe3c15d651d7a173a7a97eb6fbc1dac3765425f4c4"},"schema_version":"1.0","source":{"id":"1902.01244","kind":"arxiv","version":1}},"canonical_sha256":"2750f3bb65f7df9108ddfe74fe04d6d804be84d6166ac8aeab733a6957f2bdf8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2750f3bb65f7df9108ddfe74fe04d6d804be84d6166ac8aeab733a6957f2bdf8","first_computed_at":"2026-05-17T23:54:49.329253Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:49.329253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sJwQ2oIz+hxqStS8mVrmtd3ByEOENU2gsoXNUQts3cUh3pgdwOX7Dno+M7JMSAbBBePNPUgo4iGFOZbHKTLZAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:49.329772Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.01244","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d35c86b0a21c0c683012885d637498782a640ce73de66778a38f02597154f05","sha256:e47b87030856082b17468fd85ab35b6f0da19c670d6070d5a85ef668b2b7771c"],"state_sha256":"8cb4a94f9b6c9c5546a441cd27ebd31bbeb7a49c1035e56bd5d52a331e7ef822"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+gG9DF067AsQIop4PZzNUjMG1VXa+E2Gfd9arLUaAema2KJ6vAHL91hCSB7cXfQ8f4zuGgrmlKOvUsHWacLvCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-18T20:36:52.503922Z","bundle_sha256":"b2acf673a4d2e2f0296cb7ff986e14e92460e58bc175ce173bb1f50167e2f73f"}}