{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:V3KOSICTWZL2UUJESJ5NLVNNYF","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":"59f2575b5e19caf5c0e3360a75c52dd75765d68e82aa8fe0ed190cbde4538e9b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-02-24T11:49:38Z","title_canon_sha256":"d43ccc270d824a60f8c67d546905bd9938e000fbba9cb693c843f6da228d7c34"},"schema_version":"1.0","source":{"id":"2302.12586","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2302.12586","created_at":"2026-07-05T08:42:20Z"},{"alias_kind":"arxiv_version","alias_value":"2302.12586v3","created_at":"2026-07-05T08:42:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.12586","created_at":"2026-07-05T08:42:20Z"},{"alias_kind":"pith_short_12","alias_value":"V3KOSICTWZL2","created_at":"2026-07-05T08:42:20Z"},{"alias_kind":"pith_short_16","alias_value":"V3KOSICTWZL2UUJE","created_at":"2026-07-05T08:42:20Z"},{"alias_kind":"pith_short_8","alias_value":"V3KOSICT","created_at":"2026-07-05T08:42:20Z"}],"graph_snapshots":[{"event_id":"sha256:59348885d62d0e18ce1648097fa1a7ab09d3ae2130d4ba2eecfc900b5b10126d","target":"graph","created_at":"2026-07-05T08:42:20Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2302.12586/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_\\alpha(\\rho):=\\frac{1}{q}\\int_{\\mathbb{R}^d}|\\rho(x)|^q dx+\\frac{1}{2}\\iint_{\\mathbb{R}^d\\times\\mathbb{R}^d}\\frac{\\rho(x)\\rho(y)}{|x-y|^{d-\\alpha}}dx dy-\\int_{\\mathbb{R}^d}V(x)\\rho(x)dx,$$ where $d\\ge 2$, $\\alpha\\in (0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non--trivial way on the c","authors_text":"Damiano Greco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-02-24T11:49:38Z","title":"Optimal decay and regularity for a Thomas--Fermi type variational problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.12586","kind":"arxiv","version":3},"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:c328e4fc993b582536cd55f4f48189e61befe35ec259c930601040cc865835bd","target":"record","created_at":"2026-07-05T08:42:20Z","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":"59f2575b5e19caf5c0e3360a75c52dd75765d68e82aa8fe0ed190cbde4538e9b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-02-24T11:49:38Z","title_canon_sha256":"d43ccc270d824a60f8c67d546905bd9938e000fbba9cb693c843f6da228d7c34"},"schema_version":"1.0","source":{"id":"2302.12586","kind":"arxiv","version":3}},"canonical_sha256":"aed4e92053b657aa5124927ad5d5adc1659a23b14659b553ce87f85f0d6f13d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aed4e92053b657aa5124927ad5d5adc1659a23b14659b553ce87f85f0d6f13d2","first_computed_at":"2026-07-05T08:42:20.644448Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:42:20.644448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F1nt9bF/L2Wbj5AatXqjwGrXD8y8nvH5hFJWUHl85qp3kn4a50pPVdsYgtvpXdo3IY82NWfPbJ/z5woVwl2pAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:42:20.644955Z","signed_message":"canonical_sha256_bytes"},"source_id":"2302.12586","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c328e4fc993b582536cd55f4f48189e61befe35ec259c930601040cc865835bd","sha256:59348885d62d0e18ce1648097fa1a7ab09d3ae2130d4ba2eecfc900b5b10126d"],"state_sha256":"bd124aa27ebe548ecf7f05d4f6e1ff92bfe209c8c49182b9aa2d83233fff93da"}