{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:3KFLQ4HHRKFXSJRZCAIRM7RAKE","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":"247560c8517dca44a4bee15b79d165cb1c310807aa5620c8faf9badf4bfa5186","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T18:13:55Z","title_canon_sha256":"d26f9acff6ba3f03d9924ed2982b6522b008ccb8e9085aa4202a3ecaceb798f8"},"schema_version":"1.0","source":{"id":"1410.1831","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.1831","created_at":"2026-05-18T02:40:54Z"},{"alias_kind":"arxiv_version","alias_value":"1410.1831v1","created_at":"2026-05-18T02:40:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1831","created_at":"2026-05-18T02:40:54Z"},{"alias_kind":"pith_short_12","alias_value":"3KFLQ4HHRKFX","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3KFLQ4HHRKFXSJRZ","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3KFLQ4HH","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:cc64f01ab8e09a2c2c3bd1ac080b2d9a1cae6b3a37c858ce03d61830a245b520","target":"graph","created_at":"2026-05-18T02:40:54Z","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":"We study stable solutions of the following nonlinear system $$ -\\Delta u = H(u) \\quad \\text{in} \\ \\ \\Omega$$ where $u:\\mathbb R^n\\to \\mathbb R^m$, $H:\\mathbb R^m\\to \\mathbb R^m$ and $\\Omega$ is a domain in $\\mathbb R^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $\\Omega=\\mathbb R^n$, and regularity results, when $\\Omega=B_1$, for stable solutions of the above system for a general nonlinearity $H \\in C^1(\\math","authors_text":"Mostafa Fazly","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T18:13:55Z","title":"Rigidity results for stable solutions of symmetric systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1831","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:a8367c722e19f03d3b5a7aa5ad09508528dd96f74d02678eaca9a8f6685c8c65","target":"record","created_at":"2026-05-18T02:40:54Z","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":"247560c8517dca44a4bee15b79d165cb1c310807aa5620c8faf9badf4bfa5186","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T18:13:55Z","title_canon_sha256":"d26f9acff6ba3f03d9924ed2982b6522b008ccb8e9085aa4202a3ecaceb798f8"},"schema_version":"1.0","source":{"id":"1410.1831","kind":"arxiv","version":1}},"canonical_sha256":"da8ab870e78a8b7926391011167e2051137e378924dd871067f658ebee104a86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da8ab870e78a8b7926391011167e2051137e378924dd871067f658ebee104a86","first_computed_at":"2026-05-18T02:40:54.650509Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:40:54.650509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0na1FZRCwdGb35rdm5nTzRVTTpAzPSzJVuEf43xYZ/31AaZwSoSrzDBE7Z+Vk4BdOZdtv9x9M6TC8TXKV6JHAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:40:54.651027Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.1831","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a8367c722e19f03d3b5a7aa5ad09508528dd96f74d02678eaca9a8f6685c8c65","sha256:cc64f01ab8e09a2c2c3bd1ac080b2d9a1cae6b3a37c858ce03d61830a245b520"],"state_sha256":"000363398411424398d8651467aed7e2c9aa5e660ad153b1390974e9a10fccb3"}