{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:46QMPCLKNBX7KP54SL72SUXXUU","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":"7a95e3e8ad09f951b6d658eceb88a515003e051be6eaa5b0a92b5b3ca68e4524","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-04T22:26:48Z","title_canon_sha256":"77dc90339695615b78ef12be36cf8149e4ecd8004e41ff94620c8365013d3baf"},"schema_version":"1.0","source":{"id":"1409.1626","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1626","created_at":"2026-05-18T02:42:47Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1626v2","created_at":"2026-05-18T02:42:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1626","created_at":"2026-05-18T02:42:47Z"},{"alias_kind":"pith_short_12","alias_value":"46QMPCLKNBX7","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"46QMPCLKNBX7KP54","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"46QMPCLK","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:75b47cc110764179ebebc666261492da997310e8513a590f0012ec6287f6cd61","target":"graph","created_at":"2026-05-18T02:42:47Z","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":"We study Fuglede's $p$-module of systems of measures in condensers in Euclidean spaces and on polarizable Carnot groups. We apply and generalize a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in $\\mathbb R^2$, to a variety of settings. In the planar case, we apply Rodin's method to obtain estimates for the conformal module of a parallelogram and of a ring domain using directional dilatations. In $\\mathbb R^n,$ we identify the extremal function and compute the $p$-module of images of families of connecting c","authors_text":"Alexander Vasil'ev, Irina Markina, Melkana Brakalova","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-04T22:26:48Z","title":"Extremal functions for modules of systems of measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1626","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":"sha256:516208cc8c03e2a6074d22b3827d6be50618dc0416af6ad1e6278bae86e098e4","target":"record","created_at":"2026-05-18T02:42:47Z","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":"7a95e3e8ad09f951b6d658eceb88a515003e051be6eaa5b0a92b5b3ca68e4524","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-09-04T22:26:48Z","title_canon_sha256":"77dc90339695615b78ef12be36cf8149e4ecd8004e41ff94620c8365013d3baf"},"schema_version":"1.0","source":{"id":"1409.1626","kind":"arxiv","version":2}},"canonical_sha256":"e7a0c7896a686ff53fbc92ffa952f7a51465e8cac9d30640ca2b4e6833afae9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e7a0c7896a686ff53fbc92ffa952f7a51465e8cac9d30640ca2b4e6833afae9e","first_computed_at":"2026-05-18T02:42:47.402856Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:47.402856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C8yl/N6BI7ol28E+l6S6YypWBAZBYPsfy0au7SV2gyvdzNB6g8Px/st2CENG21gXQtHe+Ad/aH/P8PHTi7uWCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:47.403556Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.1626","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:516208cc8c03e2a6074d22b3827d6be50618dc0416af6ad1e6278bae86e098e4","sha256:75b47cc110764179ebebc666261492da997310e8513a590f0012ec6287f6cd61"],"state_sha256":"694bce4df200229453023598d535e5b098782a3706390dcf3dd13fdf3d926d14"}