{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:Y2AJAI74RCGFXIKT7VUH26ROLR","short_pith_number":"pith:Y2AJAI74","canonical_record":{"source":{"id":"1610.06820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-21T15:17:55Z","cross_cats_sorted":[],"title_canon_sha256":"0676270aa193975a2da26412f4d93a5f483c054764aa5fbed22a38582d8afdeb","abstract_canon_sha256":"21d74c07a72d21aeabab3bfdc1bb719f139c86073a8dfc87edefbe3cb26f6da8"},"schema_version":"1.0"},"canonical_sha256":"c6809023fc888c5ba153fd687d7a2e5c6e33a960a91e6ad4b81bc9327df6004f","source":{"kind":"arxiv","id":"1610.06820","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06820","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06820v1","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06820","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"Y2AJAI74RCGF","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"Y2AJAI74RCGFXIKT","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"Y2AJAI74","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:Y2AJAI74RCGFXIKT7VUH26ROLR","target":"record","payload":{"canonical_record":{"source":{"id":"1610.06820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-21T15:17:55Z","cross_cats_sorted":[],"title_canon_sha256":"0676270aa193975a2da26412f4d93a5f483c054764aa5fbed22a38582d8afdeb","abstract_canon_sha256":"21d74c07a72d21aeabab3bfdc1bb719f139c86073a8dfc87edefbe3cb26f6da8"},"schema_version":"1.0"},"canonical_sha256":"c6809023fc888c5ba153fd687d7a2e5c6e33a960a91e6ad4b81bc9327df6004f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:41.628292Z","signature_b64":"2zw+zVZxpg4SCRnZ2cqmf0B8Sjo8afzbsypZtfc+x7OhKptrkjeapcxGCZ72Bw2+3xkWGEsn1AvXenHy3I63Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6809023fc888c5ba153fd687d7a2e5c6e33a960a91e6ad4b81bc9327df6004f","last_reissued_at":"2026-05-18T00:10:41.627484Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:41.627484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.06820","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yAJ1weYv7QFphuJj0yMAL0TlRUGv1pTZSa9SZyD6zL+wOFqt6o32b2hw5k3tEpgM18P82QwRXlrHjLoOQ0v7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:49:49.389488Z"},"content_sha256":"dadbfdb7638e951dd3a93e173a9e5acd124cc5b4966cbfa9f79d454ef5b8026b","schema_version":"1.0","event_id":"sha256:dadbfdb7638e951dd3a93e173a9e5acd124cc5b4966cbfa9f79d454ef5b8026b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:Y2AJAI74RCGFXIKT7VUH26ROLR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder, Luca Martinazzi, Stefano Iula","submitted_at":"2016-10-21T15:17:55Z","abstract_excerpt":"Let $m\\ge 2$ be an integer. For any open domain $\\Omega\\subset\\mathbb{R}^{2m}$, non-positive function $\\varphi\\in C^\\infty(\\Omega)$ such that $\\Delta^m \\varphi\\equiv 0$, and bounded sequence $(V_k)\\subset L^\\infty(\\Omega)$ we prove the existence of a sequence of functions $(u_k)\\subset C^{2m-1}(\\Omega)$ solving the Liouville equation of order $2m$ $$(-\\Delta)^m u_k = V_ke^{2mu_k}\\quad \\text{in }\\Omega, \\quad \\limsup_{k\\to\\infty} \\int_\\Omega e^{2mu_k}dx<\\infty,$$ and blowing up exactly on the set $S_{\\varphi}:=\\{x\\in \\Omega:\\varphi(x)=0\\}$, i.e. $$\\lim_{k\\to\\infty} u_k(x)=+\\infty \\text{ for }x\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06820","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5KdIR9YUXjJvMhW+j/aJUjebo8Irv+B0v3+8HdZswooEmQia8pTFtLSLcIkjUoJuqDdywOjClhTW7ka5VfRRBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:49:49.389836Z"},"content_sha256":"ad5c6cc7d50348d9cd3cf71c5a6132a13777a0b47144e71735801139c76936e4","schema_version":"1.0","event_id":"sha256:ad5c6cc7d50348d9cd3cf71c5a6132a13777a0b47144e71735801139c76936e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/bundle.json","state_url":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T06:49:49Z","links":{"resolver":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR","bundle":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/bundle.json","state":"https://pith.science/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y2AJAI74RCGFXIKT7VUH26ROLR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:Y2AJAI74RCGFXIKT7VUH26ROLR","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":"21d74c07a72d21aeabab3bfdc1bb719f139c86073a8dfc87edefbe3cb26f6da8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-21T15:17:55Z","title_canon_sha256":"0676270aa193975a2da26412f4d93a5f483c054764aa5fbed22a38582d8afdeb"},"schema_version":"1.0","source":{"id":"1610.06820","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06820","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06820v1","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06820","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"Y2AJAI74RCGF","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"Y2AJAI74RCGFXIKT","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"Y2AJAI74","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:ad5c6cc7d50348d9cd3cf71c5a6132a13777a0b47144e71735801139c76936e4","target":"graph","created_at":"2026-05-18T00:10:41Z","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":"Let $m\\ge 2$ be an integer. For any open domain $\\Omega\\subset\\mathbb{R}^{2m}$, non-positive function $\\varphi\\in C^\\infty(\\Omega)$ such that $\\Delta^m \\varphi\\equiv 0$, and bounded sequence $(V_k)\\subset L^\\infty(\\Omega)$ we prove the existence of a sequence of functions $(u_k)\\subset C^{2m-1}(\\Omega)$ solving the Liouville equation of order $2m$ $$(-\\Delta)^m u_k = V_ke^{2mu_k}\\quad \\text{in }\\Omega, \\quad \\limsup_{k\\to\\infty} \\int_\\Omega e^{2mu_k}dx<\\infty,$$ and blowing up exactly on the set $S_{\\varphi}:=\\{x\\in \\Omega:\\varphi(x)=0\\}$, i.e. $$\\lim_{k\\to\\infty} u_k(x)=+\\infty \\text{ for }x\\","authors_text":"Ali Hyder, Luca Martinazzi, Stefano Iula","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-21T15:17:55Z","title":"Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06820","kind":"arxiv","version":1},"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:dadbfdb7638e951dd3a93e173a9e5acd124cc5b4966cbfa9f79d454ef5b8026b","target":"record","created_at":"2026-05-18T00:10:41Z","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":"21d74c07a72d21aeabab3bfdc1bb719f139c86073a8dfc87edefbe3cb26f6da8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-10-21T15:17:55Z","title_canon_sha256":"0676270aa193975a2da26412f4d93a5f483c054764aa5fbed22a38582d8afdeb"},"schema_version":"1.0","source":{"id":"1610.06820","kind":"arxiv","version":1}},"canonical_sha256":"c6809023fc888c5ba153fd687d7a2e5c6e33a960a91e6ad4b81bc9327df6004f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6809023fc888c5ba153fd687d7a2e5c6e33a960a91e6ad4b81bc9327df6004f","first_computed_at":"2026-05-18T00:10:41.627484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:41.627484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2zw+zVZxpg4SCRnZ2cqmf0B8Sjo8afzbsypZtfc+x7OhKptrkjeapcxGCZ72Bw2+3xkWGEsn1AvXenHy3I63Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:41.628292Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.06820","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dadbfdb7638e951dd3a93e173a9e5acd124cc5b4966cbfa9f79d454ef5b8026b","sha256:ad5c6cc7d50348d9cd3cf71c5a6132a13777a0b47144e71735801139c76936e4"],"state_sha256":"1455c579143164d8664e7a2965a3b561de7dc14451234a9adbcac60b64f0db7f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OaFF+G+863Wwy4UYbIdcOtCxauuWCtEi/NsTSrSGH6F0enw3ItBMh+uQRVc3TWNzhlyWKkBAsUh2OvyF/C6EDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T06:49:49.391945Z","bundle_sha256":"c88d598045a89a1d8011b83883ff0c059563c5762e3d4997dd4fac6548776dc0"}}