{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:3GFNYEODOJNO52Q2HFMZXV4LJE","short_pith_number":"pith:3GFNYEOD","canonical_record":{"source":{"id":"1206.2000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-10T07:10:55Z","cross_cats_sorted":[],"title_canon_sha256":"f4ad107af168ce6782bf30592f3c2d245fbe505e9c87ffd200ebf8a7edf33328","abstract_canon_sha256":"903559dbc0dc02914d7d2f120fe47390fce557cf5f6e3b64f833c5cd3efb50b4"},"schema_version":"1.0"},"canonical_sha256":"d98adc11c3725aeeea1a39599bd78b492deff84dfd56da4939d0a2650f7d5fe6","source":{"kind":"arxiv","id":"1206.2000","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2000","created_at":"2026-05-18T03:39:06Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2000v1","created_at":"2026-05-18T03:39:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2000","created_at":"2026-05-18T03:39:06Z"},{"alias_kind":"pith_short_12","alias_value":"3GFNYEODOJNO","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3GFNYEODOJNO52Q2","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3GFNYEOD","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:3GFNYEODOJNO52Q2HFMZXV4LJE","target":"record","payload":{"canonical_record":{"source":{"id":"1206.2000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-10T07:10:55Z","cross_cats_sorted":[],"title_canon_sha256":"f4ad107af168ce6782bf30592f3c2d245fbe505e9c87ffd200ebf8a7edf33328","abstract_canon_sha256":"903559dbc0dc02914d7d2f120fe47390fce557cf5f6e3b64f833c5cd3efb50b4"},"schema_version":"1.0"},"canonical_sha256":"d98adc11c3725aeeea1a39599bd78b492deff84dfd56da4939d0a2650f7d5fe6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:06.272410Z","signature_b64":"1WLSSnJTXqPyWDurUldwq3hNmV1KGDSkSQNb7Q+pAp5hifuK0iD/yrVhm7+hCMWEH6UKif6ToDQtmSCXjmY+Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d98adc11c3725aeeea1a39599bd78b492deff84dfd56da4939d0a2650f7d5fe6","last_reissued_at":"2026-05-18T03:39:06.271927Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:06.271927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.2000","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":"2026-05-18T03:39:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K3JmHuwshW3uTBoXXTjDzcKqly38OKiKYr2Ksa/35VWQ4ajMSJH7/yCtbSINn2rKODSRr1qOBJmZ6MHizmbhAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T11:22:31.401749Z"},"content_sha256":"079d6bc9c6bd2be06d3f8b6a987ee8eb9c645bec0410a5bc4183564ee65f5722","schema_version":"1.0","event_id":"sha256:079d6bc9c6bd2be06d3f8b6a987ee8eb9c645bec0410a5bc4183564ee65f5722"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:3GFNYEODOJNO52Q2HFMZXV4LJE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A functional framework for the Keller-Segel system: logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"DIM), Jean Dolbeault (CEREMADE), Juan Campos Serrano (CEREMADE","submitted_at":"2012-06-10T07:10:55Z","abstract_excerpt":"This note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2000","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":"2026-05-18T03:39:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JIhac3nfOpcw9fsM/L0pcT3aeY+DGSehrV4WFkMhteInV5tQdBn2EJwXD1OyC+cZcApE44R6RfddqU2/P+AIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T11:22:31.402085Z"},"content_sha256":"57d0544d597b47f6a75710dbad2cb3a9966576f87e52f0f1323c499b76c01a0a","schema_version":"1.0","event_id":"sha256:57d0544d597b47f6a75710dbad2cb3a9966576f87e52f0f1323c499b76c01a0a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3GFNYEODOJNO52Q2HFMZXV4LJE/bundle.json","state_url":"https://pith.science/pith/3GFNYEODOJNO52Q2HFMZXV4LJE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3GFNYEODOJNO52Q2HFMZXV4LJE/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":"2026-06-08T11:22:31Z","links":{"resolver":"https://pith.science/pith/3GFNYEODOJNO52Q2HFMZXV4LJE","bundle":"https://pith.science/pith/3GFNYEODOJNO52Q2HFMZXV4LJE/bundle.json","state":"https://pith.science/pith/3GFNYEODOJNO52Q2HFMZXV4LJE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3GFNYEODOJNO52Q2HFMZXV4LJE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3GFNYEODOJNO52Q2HFMZXV4LJE","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":"903559dbc0dc02914d7d2f120fe47390fce557cf5f6e3b64f833c5cd3efb50b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-10T07:10:55Z","title_canon_sha256":"f4ad107af168ce6782bf30592f3c2d245fbe505e9c87ffd200ebf8a7edf33328"},"schema_version":"1.0","source":{"id":"1206.2000","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2000","created_at":"2026-05-18T03:39:06Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2000v1","created_at":"2026-05-18T03:39:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2000","created_at":"2026-05-18T03:39:06Z"},{"alias_kind":"pith_short_12","alias_value":"3GFNYEODOJNO","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"3GFNYEODOJNO52Q2","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"3GFNYEOD","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:57d0544d597b47f6a75710dbad2cb3a9966576f87e52f0f1323c499b76c01a0a","target":"graph","created_at":"2026-05-18T03:39:06Z","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 note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass.","authors_text":"DIM), Jean Dolbeault (CEREMADE), Juan Campos Serrano (CEREMADE","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-10T07:10:55Z","title":"A functional framework for the Keller-Segel system: logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2000","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":"sha256:079d6bc9c6bd2be06d3f8b6a987ee8eb9c645bec0410a5bc4183564ee65f5722","target":"record","created_at":"2026-05-18T03:39:06Z","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":"903559dbc0dc02914d7d2f120fe47390fce557cf5f6e3b64f833c5cd3efb50b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-10T07:10:55Z","title_canon_sha256":"f4ad107af168ce6782bf30592f3c2d245fbe505e9c87ffd200ebf8a7edf33328"},"schema_version":"1.0","source":{"id":"1206.2000","kind":"arxiv","version":1}},"canonical_sha256":"d98adc11c3725aeeea1a39599bd78b492deff84dfd56da4939d0a2650f7d5fe6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d98adc11c3725aeeea1a39599bd78b492deff84dfd56da4939d0a2650f7d5fe6","first_computed_at":"2026-05-18T03:39:06.271927Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:06.271927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1WLSSnJTXqPyWDurUldwq3hNmV1KGDSkSQNb7Q+pAp5hifuK0iD/yrVhm7+hCMWEH6UKif6ToDQtmSCXjmY+Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:06.272410Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.2000","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:079d6bc9c6bd2be06d3f8b6a987ee8eb9c645bec0410a5bc4183564ee65f5722","sha256:57d0544d597b47f6a75710dbad2cb3a9966576f87e52f0f1323c499b76c01a0a"],"state_sha256":"98413a820e0b5f671b432b4b685844f2f7b3b34e9944aa7ee04a331c7efa2da8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mk28dJFVqYQ1RRzQdb1E1VpQiOeZE1qD07T1SBvHHsFEKl70bx4F9K+Qo4BLw8OOiKHnckis89x08lzsxmP5Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T11:22:31.403960Z","bundle_sha256":"00c514cc406b8daa4cee71ab7e83a09af6b32befeddba4c057938eec5b7ec479"}}