{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:SWDX73HSWIERQQ2I3FBTREJOS7","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":"373231b35e9ccd0ddc1e14c826d0e71b0f775f8805d191d143585b44cda31aca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-25T10:07:25Z","title_canon_sha256":"bd51c04051e328a5a4cea1506615d1e296243d4b6f46e492c3c2444f94d00344"},"schema_version":"1.0","source":{"id":"1211.5751","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5751","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5751v2","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5751","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"SWDX73HSWIER","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"SWDX73HSWIERQQ2I","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"SWDX73HS","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:2b6c461de53eb0bb21066b7418c8b9f95e16842965bc21b9b3da7d5aee9b2f3d","target":"graph","created_at":"2026-05-18T02:53:54Z","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 consider a class of semilinear elliptic system of the form $-\\Delta u(x,y)+\\nabla W(u(x,y))=0,\\quad (x,y)\\in\\R^{2}$ where $W:\\R^{2}\\to\\R$ is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system $-\\ddot q(x)+\\nabla W(q(x))=0,\\ x\\in\\R$, which connect the two minima of $W$ as $x\\to\\pm\\infty$ has a discrete structure, then the given system has infinitely many layered solutions.","authors_text":"Francesca Alessio","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-25T10:07:25Z","title":"Stationary layered solutions for a system of Allen-Cahn type equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5751","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":"sha256:90e51995af160dfcf7eaf55c6a69e2c0bbce456299b445c2bd593c3d4a3fb2c1","target":"record","created_at":"2026-05-18T02:53:54Z","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":"373231b35e9ccd0ddc1e14c826d0e71b0f775f8805d191d143585b44cda31aca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-25T10:07:25Z","title_canon_sha256":"bd51c04051e328a5a4cea1506615d1e296243d4b6f46e492c3c2444f94d00344"},"schema_version":"1.0","source":{"id":"1211.5751","kind":"arxiv","version":2}},"canonical_sha256":"95877fecf2b209184348d94338912e97e038afc8a695d7999f11fed02b07f5f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95877fecf2b209184348d94338912e97e038afc8a695d7999f11fed02b07f5f2","first_computed_at":"2026-05-18T02:53:54.032344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:54.032344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ia2HbxG7Uq1IFU35az5qmaMNbDkFQcq/sxuhvYZlipcuvySAHHDosJRLTeBiVnc1C4T15DkCYBfsT1U4MMl0DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:54.033129Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5751","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90e51995af160dfcf7eaf55c6a69e2c0bbce456299b445c2bd593c3d4a3fb2c1","sha256:2b6c461de53eb0bb21066b7418c8b9f95e16842965bc21b9b3da7d5aee9b2f3d"],"state_sha256":"2bd720b612f069dc5349d0e28d331d0eb467ab1ede97ec2fd0aeb7171567e4a1"}