{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:EA6C7BVPSUP3KK5ZO4G4XZWNKI","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":"f479ded92055771ba04ef00b1fb219702388469fcb089938cf2bf37bfc16bfd9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-06-19T19:51:33Z","title_canon_sha256":"687ba55ac9c9a5b07aed0b1aea3a8a0dae912e746b32e1936c97050dd42e0803"},"schema_version":"1.0","source":{"id":"1406.5168","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5168","created_at":"2026-05-18T02:30:15Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5168v2","created_at":"2026-05-18T02:30:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5168","created_at":"2026-05-18T02:30:15Z"},{"alias_kind":"pith_short_12","alias_value":"EA6C7BVPSUP3","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"EA6C7BVPSUP3KK5Z","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"EA6C7BVP","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:61bdd134e59ae68bde0158a4d31cef8862712ce7871d151602a64a06b532d6a3","target":"graph","created_at":"2026-05-18T02:30:15Z","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":"This article carries out a qualitative analysis on a system of integral equations of the Hardy--Sobolev type. Namely, results concerning Liouville type properties and the fast and slow decay rates of positive solutions for the system are established. For a bounded and decaying positive solution, it is shown that it either decays with the slow rates or the fast rates depending on its integrability. Particularly, a criterion for distinguishing integrable solutions from other bounded and decaying solutions in terms of their asymptotic behavior is provided. Moreover, related results on the optimal","authors_text":"John Villavert","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-06-19T19:51:33Z","title":"Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5168","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:fd95b4aac731d4d98bdc5e5db386fb6e176e0b3f380c20b6de8668b4d3e18d5a","target":"record","created_at":"2026-05-18T02:30:15Z","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":"f479ded92055771ba04ef00b1fb219702388469fcb089938cf2bf37bfc16bfd9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-06-19T19:51:33Z","title_canon_sha256":"687ba55ac9c9a5b07aed0b1aea3a8a0dae912e746b32e1936c97050dd42e0803"},"schema_version":"1.0","source":{"id":"1406.5168","kind":"arxiv","version":2}},"canonical_sha256":"203c2f86af951fb52bb9770dcbe6cd522dd043e10b49c2dc1ac52c6a17fe3264","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"203c2f86af951fb52bb9770dcbe6cd522dd043e10b49c2dc1ac52c6a17fe3264","first_computed_at":"2026-05-18T02:30:15.520530Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:15.520530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C5opHRWJRoxto3S7NpLeM188qhKaAU4Y4Y28a9P6fgnVeRZw68araLquaBKuJHu2sYFJUHOfoZwn2obGDjXuBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:15.520990Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.5168","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fd95b4aac731d4d98bdc5e5db386fb6e176e0b3f380c20b6de8668b4d3e18d5a","sha256:61bdd134e59ae68bde0158a4d31cef8862712ce7871d151602a64a06b532d6a3"],"state_sha256":"4302cc08276434c79d4ebbbdccb947870f6c8accfc09f34eb74f4e0ec8b0d366"}