{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:FRSIWL7PX5RSLIMQVYDOUM6CXC","short_pith_number":"pith:FRSIWL7P","schema_version":"1.0","canonical_sha256":"2c648b2fefbf6325a190ae06ea33c2b8b7edb91f19da4354a5ee2bc23fd893b6","source":{"kind":"arxiv","id":"1612.06610","version":2},"attestation_state":"computed","paper":{"title":"Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one","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, Marco Bonacini","submitted_at":"2016-12-20T11:12:43Z","abstract_excerpt":"We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski's coagulation equation, for a class of kernels $K(x,y)$ which are homogeneous of degree one and satisfy $K(x,1)\\to k_0>0$ as $x\\to 0$. In particular, we establish the existence of a critical $\\rho_*>0$ with the property that for all $\\rho\\in(0,\\rho_*)$ there is a positive and differentiable self-similar solution with finite mass $M$ and decay $A(t)x^{-(2+\\rho)}$ as $x\\to\\infty$, with $A(t)=e^{M(1+\\rho)t}$. Furthermore, we show that (weak) self-similar solutions in the class of "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.06610","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-20T11:12:43Z","cross_cats_sorted":[],"title_canon_sha256":"04eac41b0d77c7061a9b5e2a0711fe8b5bdc6be338105a99db5e79192864bb0a","abstract_canon_sha256":"802745d8decc2e0f8f9f6bfa7584e195d783a82a8acbeb2c75a820dece1c0fdb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:23.375716Z","signature_b64":"IGwRPGNx1/Ka1CC0QreKTvsMNZttLhvMXzpXnTMejAZNffgxhVmifG2mKkS+6hWHqzviU7T1QtIoonXWquHiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c648b2fefbf6325a190ae06ea33c2b8b7edb91f19da4354a5ee2bc23fd893b6","last_reissued_at":"2026-05-17T23:58:23.375022Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:23.375022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one","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, Marco Bonacini","submitted_at":"2016-12-20T11:12:43Z","abstract_excerpt":"We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski's coagulation equation, for a class of kernels $K(x,y)$ which are homogeneous of degree one and satisfy $K(x,1)\\to k_0>0$ as $x\\to 0$. In particular, we establish the existence of a critical $\\rho_*>0$ with the property that for all $\\rho\\in(0,\\rho_*)$ there is a positive and differentiable self-similar solution with finite mass $M$ and decay $A(t)x^{-(2+\\rho)}$ as $x\\to\\infty$, with $A(t)=e^{M(1+\\rho)t}$. Furthermore, we show that (weak) self-similar solutions in the class of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06610","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.06610","created_at":"2026-05-17T23:58:23.375122+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.06610v2","created_at":"2026-05-17T23:58:23.375122+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06610","created_at":"2026-05-17T23:58:23.375122+00:00"},{"alias_kind":"pith_short_12","alias_value":"FRSIWL7PX5RS","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FRSIWL7PX5RSLIMQ","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FRSIWL7P","created_at":"2026-05-18T12:30:15.759754+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC","json":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC.json","graph_json":"https://pith.science/api/pith-number/FRSIWL7PX5RSLIMQVYDOUM6CXC/graph.json","events_json":"https://pith.science/api/pith-number/FRSIWL7PX5RSLIMQVYDOUM6CXC/events.json","paper":"https://pith.science/paper/FRSIWL7P"},"agent_actions":{"view_html":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC","download_json":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC.json","view_paper":"https://pith.science/paper/FRSIWL7P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.06610&json=true","fetch_graph":"https://pith.science/api/pith-number/FRSIWL7PX5RSLIMQVYDOUM6CXC/graph.json","fetch_events":"https://pith.science/api/pith-number/FRSIWL7PX5RSLIMQVYDOUM6CXC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/action/storage_attestation","attest_author":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/action/author_attestation","sign_citation":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/action/citation_signature","submit_replication":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/action/replication_record"}},"created_at":"2026-05-17T23:58:23.375122+00:00","updated_at":"2026-05-17T23:58:23.375122+00:00"}