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The interesting fact is that the decay assumptions on the weight function $\\gamma(\\mathbf x') $ play the fundame"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1311.6084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-11-24T05:53:42Z","cross_cats_sorted":[],"title_canon_sha256":"670b5fb12c12a4e44772ca702370cf6961af0c69bc98303dbb19880be61d0c53","abstract_canon_sha256":"de80dea274b5315a7a5942b83d882bad844a985fb52695c48efd3a1796903529"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:20.609135Z","signature_b64":"2XMCmFHltpgJUIoSwW8BB7dpxamnk1yR/96LwPY3SRGvLtb4mRX11OWyKCTstQTTNTzQ0WJeuOFRgz7UqeCzCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77f6daa6ba7d29e1d353ad6aec7a5fe9090d1fcef40eac3a742b81e99ef7ed19","last_reissued_at":"2026-05-18T03:06:20.608292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:20.608292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher-dimensional solutions for a nonuniformly elliptic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2013-11-24T05:53:42Z","abstract_excerpt":"We prove $m$-dimensional symmetry results, that we call $m$-Liouville theorems, for stable and monotone solutions of the following nonuniformly elliptic equation \\begin{eqnarray*}\\label{mainequ} - div(\\gamma(\\mathbf x') \\nabla u(\\mathbf x)) =\\lambda (\\mathbf x' ) f(u(\\mathbf x)) \\ \\ \\text{for}\\ \\ \\mathbf x=(\\mathbf x',\\mathbf x'')\\in\\mathbf{R}^d\\times\\mathbf{R}^{s}=\\mathbf{R}^n,\n  \\end{eqnarray*} where $0\\le m<n$ and $0<\\lambda,\\gamma$ are smooth functions and $f\\in C^1(\\mathbf R)$. The interesting fact is that the decay assumptions on the weight function $\\gamma(\\mathbf x') $ play the fundame"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6084","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1311.6084","created_at":"2026-05-18T03:06:20.608423+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6084v1","created_at":"2026-05-18T03:06:20.608423+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6084","created_at":"2026-05-18T03:06:20.608423+00:00"},{"alias_kind":"pith_short_12","alias_value":"O73NVJV2PUU6","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"O73NVJV2PUU6DU2T","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"O73NVJV2","created_at":"2026-05-18T12:27:54.935989+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E","json":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E.json","graph_json":"https://pith.science/api/pith-number/O73NVJV2PUU6DU2TVVVOY6S75E/graph.json","events_json":"https://pith.science/api/pith-number/O73NVJV2PUU6DU2TVVVOY6S75E/events.json","paper":"https://pith.science/paper/O73NVJV2"},"agent_actions":{"view_html":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E","download_json":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E.json","view_paper":"https://pith.science/paper/O73NVJV2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6084&json=true","fetch_graph":"https://pith.science/api/pith-number/O73NVJV2PUU6DU2TVVVOY6S75E/graph.json","fetch_events":"https://pith.science/api/pith-number/O73NVJV2PUU6DU2TVVVOY6S75E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E/action/storage_attestation","attest_author":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E/action/author_attestation","sign_citation":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E/action/citation_signature","submit_replication":"https://pith.science/pith/O73NVJV2PUU6DU2TVVVOY6S75E/action/replication_record"}},"created_at":"2026-05-18T03:06:20.608423+00:00","updated_at":"2026-05-18T03:06:20.608423+00:00"}