{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:S4CIBNB2KN2UHCLBKHGHXP7BUO","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":"9b987207cf7a46a139863636acacdf0e809a73efb66103ffc2b9bddc576923cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:17:14Z","title_canon_sha256":"5a7038ecdc4260a69bf06115a96c49944b3b0a6993e76cc8bcd0ee3c8e20e882"},"schema_version":"1.0","source":{"id":"1612.02935","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.02935","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"arxiv_version","alias_value":"1612.02935v2","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.02935","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"pith_short_12","alias_value":"S4CIBNB2KN2U","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"S4CIBNB2KN2UHCLB","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"S4CIBNB2","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:20d52c549b0166f1fa3c5f7b6e2fdc0937866f10c521039884c4506363423e7c","target":"graph","created_at":"2026-05-18T00:53:47Z","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":"In this note, we prove that the kernel of the linearized equation around a positive energy solution in $\\mathbb{R}^n$, $n\\geq 3$, to $-\\Delta W-\\gamma|x|^{-2}V=|x|^{-s}W^{2^\\star(s)-1}$ is one-dimensional when $s+\\gamma>0$. Here, $s\\in [0,2)$, $0\\leq\\gamma<(n-2)^2/4$ and $2^\\star(s)=2(n-s)/(n-2)$.","authors_text":"Fr\\'ed\\'eric Robert","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:17:14Z","title":"Nondegeneracy of positive solutions to nonlinear Hardy-Sobolev equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02935","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":"sha256:57106e16add33edc1813350f6b64d6b8a3cbdef9d8a44cf5e953c42af5f1dbad","target":"record","created_at":"2026-05-18T00:53:47Z","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":"9b987207cf7a46a139863636acacdf0e809a73efb66103ffc2b9bddc576923cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-09T08:17:14Z","title_canon_sha256":"5a7038ecdc4260a69bf06115a96c49944b3b0a6993e76cc8bcd0ee3c8e20e882"},"schema_version":"1.0","source":{"id":"1612.02935","kind":"arxiv","version":2}},"canonical_sha256":"970480b43a537543896151cc7bbfe1a3914c0f3f4987e92729d82d6f3146fd6a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"970480b43a537543896151cc7bbfe1a3914c0f3f4987e92729d82d6f3146fd6a","first_computed_at":"2026-05-18T00:53:47.498015Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:47.498015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PJbH7ebGQT39b7OhmTWVnOrEaFQgoxAeTnwhBaX0mKXXceSlQH0IDh0RV+kExXjPYeJv+xRwn7SMGqxWsRaEAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:47.498474Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.02935","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57106e16add33edc1813350f6b64d6b8a3cbdef9d8a44cf5e953c42af5f1dbad","sha256:20d52c549b0166f1fa3c5f7b6e2fdc0937866f10c521039884c4506363423e7c"],"state_sha256":"c3cbb84803ceac135e1ea373a80859393b79351bb74dd1e450525e72223e42a7"}