{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4EPWCNFC23EIM4C7D4CLVMZKUU","short_pith_number":"pith:4EPWCNFC","canonical_record":{"source":{"id":"1602.06615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-22T00:28:55Z","cross_cats_sorted":[],"title_canon_sha256":"26716466b8a681a9fec8d339292d5a8477ee91c65f0451c8fadc15060ff4a784","abstract_canon_sha256":"e0c9dd570a81a3b4c20ec7452fdfaf391fff36a5c30f503339961ecab0cc323f"},"schema_version":"1.0"},"canonical_sha256":"e11f6134a2d6c886705f1f04bab32aa524cc7fca8c4515cb5e475f057e4903b5","source":{"kind":"arxiv","id":"1602.06615","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.06615","created_at":"2026-05-18T01:20:13Z"},{"alias_kind":"arxiv_version","alias_value":"1602.06615v1","created_at":"2026-05-18T01:20:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.06615","created_at":"2026-05-18T01:20:13Z"},{"alias_kind":"pith_short_12","alias_value":"4EPWCNFC23EI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4EPWCNFC23EIM4C7","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4EPWCNFC","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4EPWCNFC23EIM4C7D4CLVMZKUU","target":"record","payload":{"canonical_record":{"source":{"id":"1602.06615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-22T00:28:55Z","cross_cats_sorted":[],"title_canon_sha256":"26716466b8a681a9fec8d339292d5a8477ee91c65f0451c8fadc15060ff4a784","abstract_canon_sha256":"e0c9dd570a81a3b4c20ec7452fdfaf391fff36a5c30f503339961ecab0cc323f"},"schema_version":"1.0"},"canonical_sha256":"e11f6134a2d6c886705f1f04bab32aa524cc7fca8c4515cb5e475f057e4903b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:13.526773Z","signature_b64":"Bjs3volQqIFVibdfZiOMk9jXxzIlRXrmMyBeibsYlTiBzDsNPZanY0SKC6bTyW8y7VkuevVbdMzHfw88ZRCCCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e11f6134a2d6c886705f1f04bab32aa524cc7fca8c4515cb5e475f057e4903b5","last_reissued_at":"2026-05-18T01:20:13.526015Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:13.526015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.06615","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":"2026-05-18T01:20:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kWkIsHpIIKMMBDzvcPsygVp78gDn5o0NKq/ICCDRtnCrVzO6LcdUYoFIPJFk5nNvEua8oCkJLUrikeqxY0gcDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:53:40.094448Z"},"content_sha256":"54b162286e8bda7af460d0ecc0400b0771073f64acf04a28e394b1d771d2b3a4","schema_version":"1.0","event_id":"sha256:54b162286e8bda7af460d0ecc0400b0771073f64acf04a28e394b1d771d2b3a4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4EPWCNFC23EIM4C7D4CLVMZKUU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jos\\'e A. Carrillo, Yanghong Huang","submitted_at":"2016-02-22T00:28:55Z","abstract_excerpt":"Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernels are constructed by inverting Fredholm integral operators or by employing certain integral identities. These solutions are expected to be the global energy stable equilibria and to characterize the generic behaviors of stationary solutions for more general interactions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06615","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":"2026-05-18T01:20:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kgx6Qb/Tr4uVIGIqRfkimeXtWw/k6DwuEzelcS77vG/p9GpNEKopqWQqTbGcWq9D02ce/GMbCbDmxSJtw8z1CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:53:40.094795Z"},"content_sha256":"9a461a022e5b36bc716bd491c51cab90e0bc88a0776134de346e5020dfa84f71","schema_version":"1.0","event_id":"sha256:9a461a022e5b36bc716bd491c51cab90e0bc88a0776134de346e5020dfa84f71"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4EPWCNFC23EIM4C7D4CLVMZKUU/bundle.json","state_url":"https://pith.science/pith/4EPWCNFC23EIM4C7D4CLVMZKUU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4EPWCNFC23EIM4C7D4CLVMZKUU/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":"2026-06-23T16:53:40Z","links":{"resolver":"https://pith.science/pith/4EPWCNFC23EIM4C7D4CLVMZKUU","bundle":"https://pith.science/pith/4EPWCNFC23EIM4C7D4CLVMZKUU/bundle.json","state":"https://pith.science/pith/4EPWCNFC23EIM4C7D4CLVMZKUU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4EPWCNFC23EIM4C7D4CLVMZKUU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4EPWCNFC23EIM4C7D4CLVMZKUU","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":"e0c9dd570a81a3b4c20ec7452fdfaf391fff36a5c30f503339961ecab0cc323f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-22T00:28:55Z","title_canon_sha256":"26716466b8a681a9fec8d339292d5a8477ee91c65f0451c8fadc15060ff4a784"},"schema_version":"1.0","source":{"id":"1602.06615","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.06615","created_at":"2026-05-18T01:20:13Z"},{"alias_kind":"arxiv_version","alias_value":"1602.06615v1","created_at":"2026-05-18T01:20:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.06615","created_at":"2026-05-18T01:20:13Z"},{"alias_kind":"pith_short_12","alias_value":"4EPWCNFC23EI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4EPWCNFC23EIM4C7","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4EPWCNFC","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:9a461a022e5b36bc716bd491c51cab90e0bc88a0776134de346e5020dfa84f71","target":"graph","created_at":"2026-05-18T01:20:13Z","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":"Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernels are constructed by inverting Fredholm integral operators or by employing certain integral identities. These solutions are expected to be the global energy stable equilibria and to characterize the generic behaviors of stationary solutions for more general interactions.","authors_text":"Jos\\'e A. Carrillo, Yanghong Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-22T00:28:55Z","title":"Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06615","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":"sha256:54b162286e8bda7af460d0ecc0400b0771073f64acf04a28e394b1d771d2b3a4","target":"record","created_at":"2026-05-18T01:20:13Z","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":"e0c9dd570a81a3b4c20ec7452fdfaf391fff36a5c30f503339961ecab0cc323f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-22T00:28:55Z","title_canon_sha256":"26716466b8a681a9fec8d339292d5a8477ee91c65f0451c8fadc15060ff4a784"},"schema_version":"1.0","source":{"id":"1602.06615","kind":"arxiv","version":1}},"canonical_sha256":"e11f6134a2d6c886705f1f04bab32aa524cc7fca8c4515cb5e475f057e4903b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e11f6134a2d6c886705f1f04bab32aa524cc7fca8c4515cb5e475f057e4903b5","first_computed_at":"2026-05-18T01:20:13.526015Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:13.526015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bjs3volQqIFVibdfZiOMk9jXxzIlRXrmMyBeibsYlTiBzDsNPZanY0SKC6bTyW8y7VkuevVbdMzHfw88ZRCCCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:13.526773Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.06615","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:54b162286e8bda7af460d0ecc0400b0771073f64acf04a28e394b1d771d2b3a4","sha256:9a461a022e5b36bc716bd491c51cab90e0bc88a0776134de346e5020dfa84f71"],"state_sha256":"f1ef8bbbc9d76561de57c2ec348b4ccb8f8a54f737a4159a3953b478f53ac8d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2ZpngPkQdr97wYsTOd3Vf3ouanpuu3Z1ULimSjHVWL79QD/5ASIP+06X5XkpBalCU1TfX3InrO58L9INvyl6BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:53:40.096762Z","bundle_sha256":"7a70e2f0ae61309e54dcc961a9e9d1549776c618c54db645bd79dd4ac213de26"}}