{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MV2GQ5K4JDHMAI4F5C5HGAVBFB","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":"4d4673c92ce734968cac02a104b1b440fd9fd7a14c34c1dea9fdfcc779520283","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-10-09T16:29:31Z","title_canon_sha256":"637fcf1aa6b91f8576f58bfa9b7e45ab319cff5d235f97cb3566bfeff56b1c2e"},"schema_version":"1.0","source":{"id":"1010.1857","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.1857","created_at":"2026-05-18T04:28:56Z"},{"alias_kind":"arxiv_version","alias_value":"1010.1857v2","created_at":"2026-05-18T04:28:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1857","created_at":"2026-05-18T04:28:56Z"},{"alias_kind":"pith_short_12","alias_value":"MV2GQ5K4JDHM","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MV2GQ5K4JDHMAI4F","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MV2GQ5K4","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:9144e5b40109002e576c8eee64952846767bd88be1fa3da39f0ead01a73ce938","target":"graph","created_at":"2026-05-18T04:28:56Z","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 consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with multiplicative kernel of homogeneity $2l\\lambda \\in (0,1)$. We establish rigorously that such solutions exhibit a singular behavior of the form $x^{-(1+2\\lambda)}$ as $x \\to 0$. This property had been conjectured, but only weaker results had been available up to now.","authors_text":"Barbara Niethammer, Juan J.L. Velazquez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-10-09T16:29:31Z","title":"Optimal bounds for self-similar solutions to coagulation equations with product kernel"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1857","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:e7c3947aef2f209b5047580c29ef44393ee92ce7669ba0edb5bc857810989826","target":"record","created_at":"2026-05-18T04:28:56Z","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":"4d4673c92ce734968cac02a104b1b440fd9fd7a14c34c1dea9fdfcc779520283","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-10-09T16:29:31Z","title_canon_sha256":"637fcf1aa6b91f8576f58bfa9b7e45ab319cff5d235f97cb3566bfeff56b1c2e"},"schema_version":"1.0","source":{"id":"1010.1857","kind":"arxiv","version":2}},"canonical_sha256":"657468755c48cec02385e8ba7302a128428e698274e658faa5a97ac6fb25b53e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"657468755c48cec02385e8ba7302a128428e698274e658faa5a97ac6fb25b53e","first_computed_at":"2026-05-18T04:28:56.661925Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:56.661925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TTfZYdp5IoqtDMeFjGMkuH8pWQoE9+gmNlR77sNB3cUZT4JgVuB9X4JeCSeJiN+nHU+dWU/J3qxLApHsZM3XBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:56.662507Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.1857","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e7c3947aef2f209b5047580c29ef44393ee92ce7669ba0edb5bc857810989826","sha256:9144e5b40109002e576c8eee64952846767bd88be1fa3da39f0ead01a73ce938"],"state_sha256":"43d5656d7b6164d8913f52c300586e45fdeebb48b9de9a9ae654199d302a0cbb"}