{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ESMSHTFNYIMYR62RE7V347N4MW","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":"6923ecbfc3ccceddc0b4477f0ac819db88ef9003e5895aade68e2b9e7a3935e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-29T16:49:57Z","title_canon_sha256":"5211eb88173833a808dec8f6910d61c618e7bd2fd35f66171d7be6e5cdf26002"},"schema_version":"1.0","source":{"id":"1609.09423","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.09423","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"arxiv_version","alias_value":"1609.09423v2","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.09423","created_at":"2026-05-18T00:53:02Z"},{"alias_kind":"pith_short_12","alias_value":"ESMSHTFNYIMY","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"ESMSHTFNYIMYR62R","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"ESMSHTFN","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:dedaab33dfaa73e56fc4844499208b6e9b369d44eeb2ccce501abcfec8b77e83","target":"graph","created_at":"2026-05-18T00:53:02Z","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":"After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metri","authors_text":"Ben Berckmoes, Jan Van Casteren, Mark Sioen, Tim Hellemans","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-29T16:49:57Z","title":"An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.09423","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:e8444b6e8e742aed0e353c8d60dcdffe375b65a3aa71bd27cd636f1b6d412e5f","target":"record","created_at":"2026-05-18T00:53:02Z","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":"6923ecbfc3ccceddc0b4477f0ac819db88ef9003e5895aade68e2b9e7a3935e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-29T16:49:57Z","title_canon_sha256":"5211eb88173833a808dec8f6910d61c618e7bd2fd35f66171d7be6e5cdf26002"},"schema_version":"1.0","source":{"id":"1609.09423","kind":"arxiv","version":2}},"canonical_sha256":"249923ccadc21988fb5127ebbe7dbc658f9de8bdd72ab68abb102bffdf9ae2ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"249923ccadc21988fb5127ebbe7dbc658f9de8bdd72ab68abb102bffdf9ae2ac","first_computed_at":"2026-05-18T00:53:02.247288Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:02.247288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vknvpxrXvVXo9WdsQRh46z1kymj+tIDApDsQRWmTKvpXpga0ShDBSNl9teBBDfYoPwSVtNHvCtHua2q0mefQBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:02.248053Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.09423","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8444b6e8e742aed0e353c8d60dcdffe375b65a3aa71bd27cd636f1b6d412e5f","sha256:dedaab33dfaa73e56fc4844499208b6e9b369d44eeb2ccce501abcfec8b77e83"],"state_sha256":"aac9f58cbdeb1be81037362f15a6e3c2f072a3edda7dc235a2f5266d7772dc95"}