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Under some natural assumptions on $f$ and $g$, by applying the \\emph{method of moving planes} directly to the system, we obtain symmetry on non-negative solutions without any decay assumption on the solutions at infinity."},"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":"1604.01465","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-06T02:15:38Z","cross_cats_sorted":[],"title_canon_sha256":"7bd297ce994ff77fb53aadcf086e7ed0da1e18c61c9582b28ddce9e315829f12","abstract_canon_sha256":"ec32c372229f53fe02291112790feb4901fd01b020e3678c9fe15a21f017e5f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:04.703412Z","signature_b64":"7OJLYikAOcrIYYCoiMviR9ZRBsBu6vNd+tJQGaNilrjv18bVGRSY6k5bIkm/TCSfmIXHwVVqQJ+xxvOeyie3AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4e5b2a80d09ca87a5b312b434ead6874c1902e491342cc353c4160376ba5e18","last_reissued_at":"2026-05-18T00:49:04.702752Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:04.702752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetry of Solutions for a Fractional System","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Pei Ma, Yan Li","submitted_at":"2016-04-06T02:15:38Z","abstract_excerpt":"We consider the following equations: \\begin{equation*} \\left\\{\\begin{array}{ll} (-\\triangle)^{\\alpha/2}u(x)=f(v(x)), \\\\ (-\\triangle)^{\\beta/2}v(x)=g(u(x)), &x \\in R^{n},\\\\ u,v\\geq 0, &x \\in R^{n}, \\end{array} \\right. \\end{equation*} for continuous $f, g$ and $\\alpha, \\beta \\in (0,2)$. Under some natural assumptions on $f$ and $g$, by applying the \\emph{method of moving planes} directly to the system, we obtain symmetry on non-negative solutions without any decay assumption on the solutions at infinity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01465","kind":"arxiv","version":2},"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":"1604.01465","created_at":"2026-05-18T00:49:04.702848+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.01465v2","created_at":"2026-05-18T00:49:04.702848+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.01465","created_at":"2026-05-18T00:49:04.702848+00:00"},{"alias_kind":"pith_short_12","alias_value":"2TS3FKANBHFI","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2TS3FKANBHFIPJNT","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2TS3FKAN","created_at":"2026-05-18T12:29:55.572404+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/2TS3FKANBHFIPJNTCK2DJ2WWQ5","json":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5.json","graph_json":"https://pith.science/api/pith-number/2TS3FKANBHFIPJNTCK2DJ2WWQ5/graph.json","events_json":"https://pith.science/api/pith-number/2TS3FKANBHFIPJNTCK2DJ2WWQ5/events.json","paper":"https://pith.science/paper/2TS3FKAN"},"agent_actions":{"view_html":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5","download_json":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5.json","view_paper":"https://pith.science/paper/2TS3FKAN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.01465&json=true","fetch_graph":"https://pith.science/api/pith-number/2TS3FKANBHFIPJNTCK2DJ2WWQ5/graph.json","fetch_events":"https://pith.science/api/pith-number/2TS3FKANBHFIPJNTCK2DJ2WWQ5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5/action/storage_attestation","attest_author":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5/action/author_attestation","sign_citation":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5/action/citation_signature","submit_replication":"https://pith.science/pith/2TS3FKANBHFIPJNTCK2DJ2WWQ5/action/replication_record"}},"created_at":"2026-05-18T00:49:04.702848+00:00","updated_at":"2026-05-18T00:49:04.702848+00:00"}