{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PP6MRAUNRVP5XR4GJR4XCOVAU5","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":"9dc03c38c4e913863e8a539be67ad2034dd9aec76267fd43c87d62a687979788","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-11-27T11:09:13Z","title_canon_sha256":"f4c6bcdd4c57c2b77445dab1e0275180be97e4d8acd8621bfc68c5b6981f7135"},"schema_version":"1.0","source":{"id":"1811.10912","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.10912","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"arxiv_version","alias_value":"1811.10912v1","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.10912","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"pith_short_12","alias_value":"PP6MRAUNRVP5","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PP6MRAUNRVP5XR4G","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PP6MRAUN","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:c868bb6d8542133f05ab17cd7f528bcbcb5f14fd08a6056b94ace6209f347092","target":"graph","created_at":"2026-05-17T23:59:45Z","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":"Let $G$ be a metric group and let $\\sA ut(G)$ denote the automorphism group of $G$. If $\\sA$ and $\\sB$ are groups of $G$-valued maps defined on the sets $X$ and $Y$, respectively, we say that $\\sA$ and $\\sB$ are \\emph{equivalent} if there is a group isomorphism $H\\colon\\sA\\to\\sB$ such that there is a bijective map $h\\colon Y\\to X$ and a map $w\\colon Y\\to \\sA ut (G)$ satisfying $Hf(y)=w[y](f(h(y)))$ for all $y\\in Y$ and $f\\in \\sA$. In this case, we say that $H$ is represented as a \\emph{weighted composition operator}. A group isomorphism $H$ defined between $\\sA$ and $\\sB$ is called \\emph{separ","authors_text":"Margarita Gary, Marita Ferrer, Salvador Hern\\'andez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-11-27T11:09:13Z","title":"Representation of Group Isomorphisms I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10912","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":"sha256:7a26682b3699e2d57037492aecc78bb52bb8c5c792e38ccdaf0d83feba8cbddf","target":"record","created_at":"2026-05-17T23:59:45Z","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":"9dc03c38c4e913863e8a539be67ad2034dd9aec76267fd43c87d62a687979788","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-11-27T11:09:13Z","title_canon_sha256":"f4c6bcdd4c57c2b77445dab1e0275180be97e4d8acd8621bfc68c5b6981f7135"},"schema_version":"1.0","source":{"id":"1811.10912","kind":"arxiv","version":1}},"canonical_sha256":"7bfcc8828d8d5fdbc7864c79713aa0a750d53277029e7bd460c25f737eb170ec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7bfcc8828d8d5fdbc7864c79713aa0a750d53277029e7bd460c25f737eb170ec","first_computed_at":"2026-05-17T23:59:45.678101Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:45.678101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ep2L9qva7k4id0CITGsyZA/TFBGPcjVQNm9YXDlotjA+H2b6Ip3svc/o6H1RzKhVvlV68onkGbJbzAR8PWL/Bg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:45.678531Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.10912","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7a26682b3699e2d57037492aecc78bb52bb8c5c792e38ccdaf0d83feba8cbddf","sha256:c868bb6d8542133f05ab17cd7f528bcbcb5f14fd08a6056b94ace6209f347092"],"state_sha256":"a6047d15b4b01dabef2ec1e863f0950b2d1d9a513173a2b93cd4573d9d77f136"}