{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MPU6MYE7DKCDWNK3MFYBOFD66G","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":"d5f38e831feffbc656f3dd0a07bed8e2f3605fdbf6bfdb0bad4ee4d57cb452ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-06-01T15:38:25Z","title_canon_sha256":"9c897e0a92e3e351741c7551a1804859796599d215e32831d37011e2800babc0"},"schema_version":"1.0","source":{"id":"1506.00542","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.00542","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"arxiv_version","alias_value":"1506.00542v1","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.00542","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"pith_short_12","alias_value":"MPU6MYE7DKCD","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MPU6MYE7DKCDWNK3","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MPU6MYE7","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:68628ca844a55529633ed1c6fed3df2fd27c1197f3a73ce12fcbcec2fe54d693","target":"graph","created_at":"2026-05-18T01:59:54Z","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":"In the article the authors consider the class ${\\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\\mathcal{P}_H^0(\\alpha)$ and $\\mathcal{G}_H^0(\\beta)$ of functions from ${\\mathcal H}_0$ and show that if $f\\in \\mathcal{P}_H^0(\\alpha)$ and $F\\in\\mathcal{G}_H^0(\\beta)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided ","authors_text":"Liulan Li, Saminathan Ponnusamy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-06-01T15:38:25Z","title":"Injectivity of sections of convex harmonic mappings and convolution theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00542","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:b11dfb196c04169f05732a0b211de164625bcadbdae8c48fd29ff392a7ab0f2b","target":"record","created_at":"2026-05-18T01:59:54Z","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":"d5f38e831feffbc656f3dd0a07bed8e2f3605fdbf6bfdb0bad4ee4d57cb452ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-06-01T15:38:25Z","title_canon_sha256":"9c897e0a92e3e351741c7551a1804859796599d215e32831d37011e2800babc0"},"schema_version":"1.0","source":{"id":"1506.00542","kind":"arxiv","version":1}},"canonical_sha256":"63e9e6609f1a843b355b617017147ef1aefd751d16a8608283772fb4fa32b86b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63e9e6609f1a843b355b617017147ef1aefd751d16a8608283772fb4fa32b86b","first_computed_at":"2026-05-18T01:59:54.295728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:54.295728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wbdaL+hIO8THktD1rxL2WRSc5WHLdfKYWyNa/H2cM0wyVmW12LE/IyCCNl/A2jqb4YWnZRA3gYczR9xKcIDcAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:54.296209Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.00542","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b11dfb196c04169f05732a0b211de164625bcadbdae8c48fd29ff392a7ab0f2b","sha256:68628ca844a55529633ed1c6fed3df2fd27c1197f3a73ce12fcbcec2fe54d693"],"state_sha256":"d77e8d2215c2d0263b1a38ded1c4df64cb7df7927d9770e15b707f2f3658376c"}