{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FRSIWL7PX5RSLIMQVYDOUM6CXC","short_pith_number":"pith:FRSIWL7P","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"},"canonical_sha256":"2c648b2fefbf6325a190ae06ea33c2b8b7edb91f19da4354a5ee2bc23fd893b6","source":{"kind":"arxiv","id":"1612.06610","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.06610","created_at":"2026-05-17T23:58:23Z"},{"alias_kind":"arxiv_version","alias_value":"1612.06610v2","created_at":"2026-05-17T23:58:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06610","created_at":"2026-05-17T23:58:23Z"},{"alias_kind":"pith_short_12","alias_value":"FRSIWL7PX5RS","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FRSIWL7PX5RSLIMQ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FRSIWL7P","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FRSIWL7PX5RSLIMQVYDOUM6CXC","target":"record","payload":{"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"},"canonical_sha256":"2c648b2fefbf6325a190ae06ea33c2b8b7edb91f19da4354a5ee2bc23fd893b6","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"},"source_kind":"arxiv","source_id":"1612.06610","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":"2026-05-17T23:58:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yjcBSqv/R083rtvD33GA/r/5wY8kheada6Qa9XB6I1088YsQ96iwpDxbYTr4V0Jn0nL+m/eqAVSYhFE3MrgeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T06:30:35.095760Z"},"content_sha256":"d1978a7c5d9efdc52b5874509e969ea9914e5207d4c6429fb19a5534069a1a03","schema_version":"1.0","event_id":"sha256:d1978a7c5d9efdc52b5874509e969ea9914e5207d4c6429fb19a5534069a1a03"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FRSIWL7PX5RSLIMQVYDOUM6CXC","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:58:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kf3bnnTnRg6Jhqz5b95sVe4SAbTZnwaGc015sIVj8EcqZeRBUr9AhTf6SILZQbfv1N6SrrXm8JBdlgZDY5L6Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T06:30:35.096112Z"},"content_sha256":"e521408540afd5baf91a3c7e0d689fba26bb9340af97d7e2e07a8af0f8c63c98","schema_version":"1.0","event_id":"sha256:e521408540afd5baf91a3c7e0d689fba26bb9340af97d7e2e07a8af0f8c63c98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/bundle.json","state_url":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/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":"2026-06-20T06:30:35Z","links":{"resolver":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC","bundle":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/bundle.json","state":"https://pith.science/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FRSIWL7PX5RSLIMQVYDOUM6CXC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FRSIWL7PX5RSLIMQVYDOUM6CXC","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":"802745d8decc2e0f8f9f6bfa7584e195d783a82a8acbeb2c75a820dece1c0fdb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-20T11:12:43Z","title_canon_sha256":"04eac41b0d77c7061a9b5e2a0711fe8b5bdc6be338105a99db5e79192864bb0a"},"schema_version":"1.0","source":{"id":"1612.06610","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.06610","created_at":"2026-05-17T23:58:23Z"},{"alias_kind":"arxiv_version","alias_value":"1612.06610v2","created_at":"2026-05-17T23:58:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06610","created_at":"2026-05-17T23:58:23Z"},{"alias_kind":"pith_short_12","alias_value":"FRSIWL7PX5RS","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FRSIWL7PX5RSLIMQ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FRSIWL7P","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:e521408540afd5baf91a3c7e0d689fba26bb9340af97d7e2e07a8af0f8c63c98","target":"graph","created_at":"2026-05-17T23:58:23Z","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 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 ","authors_text":"Barbara Niethammer, Juan J.L. Vel\\'azquez, Marco Bonacini","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-20T11:12:43Z","title":"Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06610","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:d1978a7c5d9efdc52b5874509e969ea9914e5207d4c6429fb19a5534069a1a03","target":"record","created_at":"2026-05-17T23:58:23Z","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":"802745d8decc2e0f8f9f6bfa7584e195d783a82a8acbeb2c75a820dece1c0fdb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-20T11:12:43Z","title_canon_sha256":"04eac41b0d77c7061a9b5e2a0711fe8b5bdc6be338105a99db5e79192864bb0a"},"schema_version":"1.0","source":{"id":"1612.06610","kind":"arxiv","version":2}},"canonical_sha256":"2c648b2fefbf6325a190ae06ea33c2b8b7edb91f19da4354a5ee2bc23fd893b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c648b2fefbf6325a190ae06ea33c2b8b7edb91f19da4354a5ee2bc23fd893b6","first_computed_at":"2026-05-17T23:58:23.375022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:23.375022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IGwRPGNx1/Ka1CC0QreKTvsMNZttLhvMXzpXnTMejAZNffgxhVmifG2mKkS+6hWHqzviU7T1QtIoonXWquHiAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:23.375716Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.06610","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d1978a7c5d9efdc52b5874509e969ea9914e5207d4c6429fb19a5534069a1a03","sha256:e521408540afd5baf91a3c7e0d689fba26bb9340af97d7e2e07a8af0f8c63c98"],"state_sha256":"4aeaa6bea4b64a3cea95f1509ce037ca9649be05fec8ca0e1cb671b57193f915"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i2z0PPpNQbuRYeKweggb0bJeFoX+3ZWSGxb/Tdf6QRavJnYeN7GLmrR7J2YKusYbw/w2184Q74FCrYSQ/T6ADA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T06:30:35.098088Z","bundle_sha256":"bc6ad7d0dfc9e0d1d216a3dd56f2e172f6be2bb8a681f27b1ab40d5bd46e908a"}}