{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NC7ZUFPDV7HBQJ2MB4Q43QCJDL","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":"3d127042e1b26677649b6dd9846197266f1de243cc50354570b72cf8f554d748","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-01-09T18:54:52Z","title_canon_sha256":"68f500749bd764be1d0295e61fa065db3dc8c28400457a57fae2a3da2a3b41fe"},"schema_version":"1.0","source":{"id":"1801.03083","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.03083","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"arxiv_version","alias_value":"1801.03083v1","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.03083","created_at":"2026-05-18T00:26:21Z"},{"alias_kind":"pith_short_12","alias_value":"NC7ZUFPDV7HB","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NC7ZUFPDV7HBQJ2M","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NC7ZUFPD","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:cfb56f29310ab8a2960d743cf065204ba594f18bbbecc7182e049c00504dfcef","target":"graph","created_at":"2026-05-18T00:26:21Z","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":"In this article we study an extension of Smoluchowski's discrete coagulation equation, where particle in- and output takes place. This model is frequently used to describe aggregation processes in combination with sedimentation of clusters. More precisely, we show that the evolution equation is well-posed for a large class of coagulation kernels and output rates. Additionally, in the long-time limit we prove that solutions converge to a unique equilibrium with exponential rate under a suitable smallness condition on the coefficients.","authors_text":"Christian Kuehn, Sebastian Throm","cross_cats":["math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-01-09T18:54:52Z","title":"Smoluchowski's discrete coagulation equation with forcing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03083","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:f0b935704245829caac17966d686246019858a9ec1cefe2ba0393f74625dfbf9","target":"record","created_at":"2026-05-18T00:26:21Z","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":"3d127042e1b26677649b6dd9846197266f1de243cc50354570b72cf8f554d748","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-01-09T18:54:52Z","title_canon_sha256":"68f500749bd764be1d0295e61fa065db3dc8c28400457a57fae2a3da2a3b41fe"},"schema_version":"1.0","source":{"id":"1801.03083","kind":"arxiv","version":1}},"canonical_sha256":"68bf9a15e3afce18274c0f21cdc0491add6b5008484291ab266879539c238f9b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68bf9a15e3afce18274c0f21cdc0491add6b5008484291ab266879539c238f9b","first_computed_at":"2026-05-18T00:26:21.589671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:21.589671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Tm9Oc0PBIqdEgnMVH+h1Xfor01ot+CQpYKqUMR/qEpQ3fis9kyW5s5FliAj8RpncNPeOc5e823U88Kmxws6WAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:21.590215Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.03083","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0b935704245829caac17966d686246019858a9ec1cefe2ba0393f74625dfbf9","sha256:cfb56f29310ab8a2960d743cf065204ba594f18bbbecc7182e049c00504dfcef"],"state_sha256":"001a81c5e0c860a7783788a9d81747cd17ba2158fd31114a4ee180bb5abd7fe9"}