{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PT7RGZAM2K5TA6QHSPHPMO6XAZ","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":"6674b81849556834aaf8a600f34304de39d96cba7668ef638632415481ed4d9c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-05-24T15:46:05Z","title_canon_sha256":"b2f90f4ca9cc5a162227c2feafa396cd1bc7ab35562ea37fbb89b9b02940eb2c"},"schema_version":"1.0","source":{"id":"1205.5489","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.5489","created_at":"2026-05-18T03:55:00Z"},{"alias_kind":"arxiv_version","alias_value":"1205.5489v1","created_at":"2026-05-18T03:55:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5489","created_at":"2026-05-18T03:55:00Z"},{"alias_kind":"pith_short_12","alias_value":"PT7RGZAM2K5T","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PT7RGZAM2K5TA6QH","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PT7RGZAM","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:3461a2fb0bdef5697696d749063a136ba0308d2ef0f70f4bb68e684f32017d94","target":"graph","created_at":"2026-05-18T03:55:00Z","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 $p,q,n$ be natural numbers such that $p+q=n$. Let $\\FF$ be either $\\CC$, the complex numbers field, or $\\HH$, the quaternionic division algebra. We consider the Heisenberg group $N(p,q,\\FF)$ defined as $N(p,q,\\FF)=\\FF^{n}\\times \\mathfrak{Im}\\FF$, with group law given by $$(v,\\zeta)(v',\\zeta')=(v+v', \\zeta+\\zeta'-{1/2} \\mathfrak{Im} B(v,v')),$$ where $B(v,w)=\\sum_{j=1}^{p} v_{j}\\bar{w_{j}} - \\sum_{j=p+1}^{n} v_{j}\\bar{w_{j}}$. Let $U(p,q,\\FF)$ be the group of $n\\times n$ matrices with coefficients in $\\FF$ that leave invariant the form $B$. In this work we compute explicit fundamental solut","authors_text":"Isolda Cardoso, Linda Saal","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-05-24T15:46:05Z","title":"Explicit fundamental solutions of some second order differential operators on Heisenberg groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5489","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":"sha256:1cde5ad792059e47677079e40ee4e348c6d4a1be9ac91b0cde49ce34527f3103","target":"record","created_at":"2026-05-18T03:55:00Z","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":"6674b81849556834aaf8a600f34304de39d96cba7668ef638632415481ed4d9c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-05-24T15:46:05Z","title_canon_sha256":"b2f90f4ca9cc5a162227c2feafa396cd1bc7ab35562ea37fbb89b9b02940eb2c"},"schema_version":"1.0","source":{"id":"1205.5489","kind":"arxiv","version":1}},"canonical_sha256":"7cff13640cd2bb307a0793cef63bd7065794a6c909b36140a31fd70b6a5dfccc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7cff13640cd2bb307a0793cef63bd7065794a6c909b36140a31fd70b6a5dfccc","first_computed_at":"2026-05-18T03:55:00.956339Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:00.956339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"72QCVfgNb8Z9L15fINF9vvrsLvoQh8PWLY62Cbqg+Mv8kXaFSh/yqJO5OMJerVtcKgEUPEpyOGf6r8YtEXjiBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:00.956894Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.5489","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1cde5ad792059e47677079e40ee4e348c6d4a1be9ac91b0cde49ce34527f3103","sha256:3461a2fb0bdef5697696d749063a136ba0308d2ef0f70f4bb68e684f32017d94"],"state_sha256":"b9c6580f065ad351d192d81ea2ecd6e013f1b7a9a2d38cd2f4c00c153b9a5596"}