{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:UX4I4YGZKTXLQNO2SY27MJBKBB","short_pith_number":"pith:UX4I4YGZ","canonical_record":{"source":{"id":"1603.02929","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-09T15:52:38Z","cross_cats_sorted":[],"title_canon_sha256":"89f41b6b6032babfdd7e044d1380a3341e0dce8161d30333fed2be30a69961d2","abstract_canon_sha256":"4dc99d02026774fad9dafc3fb4345ebe83b061ca7e6d28e2be27c8dc7396826c"},"schema_version":"1.0"},"canonical_sha256":"a5f88e60d954eeb835da9635f6242a0871b1959f399323d783a05702ea15e00f","source":{"kind":"arxiv","id":"1603.02929","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02929","created_at":"2026-05-18T01:09:32Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02929v2","created_at":"2026-05-18T01:09:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02929","created_at":"2026-05-18T01:09:32Z"},{"alias_kind":"pith_short_12","alias_value":"UX4I4YGZKTXL","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UX4I4YGZKTXLQNO2","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UX4I4YGZ","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:UX4I4YGZKTXLQNO2SY27MJBKBB","target":"record","payload":{"canonical_record":{"source":{"id":"1603.02929","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-09T15:52:38Z","cross_cats_sorted":[],"title_canon_sha256":"89f41b6b6032babfdd7e044d1380a3341e0dce8161d30333fed2be30a69961d2","abstract_canon_sha256":"4dc99d02026774fad9dafc3fb4345ebe83b061ca7e6d28e2be27c8dc7396826c"},"schema_version":"1.0"},"canonical_sha256":"a5f88e60d954eeb835da9635f6242a0871b1959f399323d783a05702ea15e00f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:32.074290Z","signature_b64":"p5W0dvYUdPYmRQx91oVHS2NZcOrcxJBMODFFTv54Q46nHwvvPT9bMEmlKpm5YlVH4zRyk4cafIHRkgmFntrpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5f88e60d954eeb835da9635f6242a0871b1959f399323d783a05702ea15e00f","last_reissued_at":"2026-05-18T01:09:32.073895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:32.073895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.02929","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:09:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z2ORPlGxzyyoFqY+m3yGjCrzVMG0HPfG9pbNt9R3K0oePrZXLRig5O+FgtG7Qo0iAYdyMfKlTKUSPbnWydfWDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:44:02.239988Z"},"content_sha256":"cfd89aa1fdf0c61d027c1e5a5a9eb5f39f4def5292b2afb45a5877725cbe3fc3","schema_version":"1.0","event_id":"sha256:cfd89aa1fdf0c61d027c1e5a5a9eb5f39f4def5292b2afb45a5877725cbe3fc3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:UX4I4YGZKTXLQNO2SY27MJBKBB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Barbara Niethammer, Juan J.L. Vel\\'azquez, Philippe Lauren\\c{c}ot","submitted_at":"2016-03-09T15:52:38Z","abstract_excerpt":"We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $\\gamma < 1$. Due to the property of the diagonal kernel, the value of a solution depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02929","kind":"arxiv","version":2},"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":"2026-05-18T01:09:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kpiyOWE2qIASEkEoRHV6tDkCWDunMcPUQchkY8RoZImJDrdOIKQO1tV7R7n9CIPtEJa2pnKPeGQwp44yFSLOCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:44:02.240715Z"},"content_sha256":"1084f34b709b71d0e0333dd856190823b839efec244c5a4c3580dd03e14fd607","schema_version":"1.0","event_id":"sha256:1084f34b709b71d0e0333dd856190823b839efec244c5a4c3580dd03e14fd607"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UX4I4YGZKTXLQNO2SY27MJBKBB/bundle.json","state_url":"https://pith.science/pith/UX4I4YGZKTXLQNO2SY27MJBKBB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UX4I4YGZKTXLQNO2SY27MJBKBB/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":"2026-06-08T15:44:02Z","links":{"resolver":"https://pith.science/pith/UX4I4YGZKTXLQNO2SY27MJBKBB","bundle":"https://pith.science/pith/UX4I4YGZKTXLQNO2SY27MJBKBB/bundle.json","state":"https://pith.science/pith/UX4I4YGZKTXLQNO2SY27MJBKBB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UX4I4YGZKTXLQNO2SY27MJBKBB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UX4I4YGZKTXLQNO2SY27MJBKBB","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":"4dc99d02026774fad9dafc3fb4345ebe83b061ca7e6d28e2be27c8dc7396826c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-09T15:52:38Z","title_canon_sha256":"89f41b6b6032babfdd7e044d1380a3341e0dce8161d30333fed2be30a69961d2"},"schema_version":"1.0","source":{"id":"1603.02929","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02929","created_at":"2026-05-18T01:09:32Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02929v2","created_at":"2026-05-18T01:09:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02929","created_at":"2026-05-18T01:09:32Z"},{"alias_kind":"pith_short_12","alias_value":"UX4I4YGZKTXL","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UX4I4YGZKTXLQNO2","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UX4I4YGZ","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:1084f34b709b71d0e0333dd856190823b839efec244c5a4c3580dd03e14fd607","target":"graph","created_at":"2026-05-18T01:09:32Z","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":"We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $\\gamma < 1$. Due to the property of the diagonal kernel, the value of a solution depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.","authors_text":"Barbara Niethammer, Juan J.L. Vel\\'azquez, Philippe Lauren\\c{c}ot","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-09T15:52:38Z","title":"Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02929","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:cfd89aa1fdf0c61d027c1e5a5a9eb5f39f4def5292b2afb45a5877725cbe3fc3","target":"record","created_at":"2026-05-18T01:09:32Z","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":"4dc99d02026774fad9dafc3fb4345ebe83b061ca7e6d28e2be27c8dc7396826c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-09T15:52:38Z","title_canon_sha256":"89f41b6b6032babfdd7e044d1380a3341e0dce8161d30333fed2be30a69961d2"},"schema_version":"1.0","source":{"id":"1603.02929","kind":"arxiv","version":2}},"canonical_sha256":"a5f88e60d954eeb835da9635f6242a0871b1959f399323d783a05702ea15e00f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5f88e60d954eeb835da9635f6242a0871b1959f399323d783a05702ea15e00f","first_computed_at":"2026-05-18T01:09:32.073895Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:32.073895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p5W0dvYUdPYmRQx91oVHS2NZcOrcxJBMODFFTv54Q46nHwvvPT9bMEmlKpm5YlVH4zRyk4cafIHRkgmFntrpCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:32.074290Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.02929","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfd89aa1fdf0c61d027c1e5a5a9eb5f39f4def5292b2afb45a5877725cbe3fc3","sha256:1084f34b709b71d0e0333dd856190823b839efec244c5a4c3580dd03e14fd607"],"state_sha256":"ae1e5820f77360aa02ce002a8184b075c51e08c8ca24240f559d2d313d935080"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PO9rA5xuymzWFISVMHJnUc0Ypzq9qhnzXuTiThUtVz32ED5gzLy7St8LY0MHQWjFIFByPowyq47gkc1FiGXtDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T15:44:02.244471Z","bundle_sha256":"fa69660bbdc37a426d59eae8e0b6d359e171084325123f358c13b67552da7337"}}