{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:J6HVLMICH46URBFTSSTU32UDLE","short_pith_number":"pith:J6HVLMIC","canonical_record":{"source":{"id":"0803.1462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-03-10T17:45:51Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2b38f0f2772e072d075f31f2ca1d7ab325e9b9dca7315f863d1d7625589fd823","abstract_canon_sha256":"e650a0534e3b566ce42e5f0d7be8d7243da081f4c67118b03659feefd3640824"},"schema_version":"1.0"},"canonical_sha256":"4f8f55b1023f3d4884b394a74dea835920d1d2b77e618f5f787faae26ddcf96d","source":{"kind":"arxiv","id":"0803.1462","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0803.1462","created_at":"2026-05-18T04:06:56Z"},{"alias_kind":"arxiv_version","alias_value":"0803.1462v2","created_at":"2026-05-18T04:06:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.1462","created_at":"2026-05-18T04:06:56Z"},{"alias_kind":"pith_short_12","alias_value":"J6HVLMICH46U","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"J6HVLMICH46URBFT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"J6HVLMIC","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:J6HVLMICH46URBFTSSTU32UDLE","target":"record","payload":{"canonical_record":{"source":{"id":"0803.1462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-03-10T17:45:51Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2b38f0f2772e072d075f31f2ca1d7ab325e9b9dca7315f863d1d7625589fd823","abstract_canon_sha256":"e650a0534e3b566ce42e5f0d7be8d7243da081f4c67118b03659feefd3640824"},"schema_version":"1.0"},"canonical_sha256":"4f8f55b1023f3d4884b394a74dea835920d1d2b77e618f5f787faae26ddcf96d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:56.980212Z","signature_b64":"zJzfCUXyvjEXnAwnua6+K5fhu9iAfHOgKADA0jilTdL/RiKS2XV8hx8BF+YE6rwahUgfn2thveTt4byXJTpsBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f8f55b1023f3d4884b394a74dea835920d1d2b77e618f5f787faae26ddcf96d","last_reissued_at":"2026-05-18T04:06:56.979675Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:56.979675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0803.1462","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":"2026-05-18T04:06:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wirEoZ7qTiHNJyLBCBfhOYIBw7emkQxXTjTb0sBGlMGQhvmPW3em3bZbfRfHAWE5epK+Jpwe6tZWROztfkQKBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T06:16:40.235728Z"},"content_sha256":"bae665653bbb3f0c4f8f39212d1ae08f4e9897c44b89b012e58acdf1dcc90178","schema_version":"1.0","event_id":"sha256:bae665653bbb3f0c4f8f39212d1ae08f4e9897c44b89b012e58acdf1dcc90178"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:J6HVLMICH46URBFTSSTU32UDLE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jos\\'e Alfredo Ca\\~nizo, St\\'ephane Mischler","submitted_at":"2008-03-10T17:45:51Z","abstract_excerpt":"We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\\alpha y ^\\beta + x^\\beta y^\\alpha$ with $-1 < \\alpha \\leq \\beta < 1$ and $\\lambda := \\alpha + \\beta \\in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\\alpha = 0$ we prove that two profiles with the same mass and moment of order $\\lambda$ are necessarily equal, while in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.1462","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":"2026-05-18T04:06:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g9V512CAvN7vuW6Zua+rX3ISgxP78WYYb++D9OZVkTG6GgboymYB8aRGCPDFsJau8VuldoZV95O+Z5cCW+FMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T06:16:40.236072Z"},"content_sha256":"c4778ccbda54925a4e3ff785d1e14568b78c70d171c237a5a0bae8e90349eb21","schema_version":"1.0","event_id":"sha256:c4778ccbda54925a4e3ff785d1e14568b78c70d171c237a5a0bae8e90349eb21"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/bundle.json","state_url":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J6HVLMICH46URBFTSSTU32UDLE/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":"2026-06-23T06:16:40Z","links":{"resolver":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE","bundle":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/bundle.json","state":"https://pith.science/pith/J6HVLMICH46URBFTSSTU32UDLE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J6HVLMICH46URBFTSSTU32UDLE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:J6HVLMICH46URBFTSSTU32UDLE","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":"e650a0534e3b566ce42e5f0d7be8d7243da081f4c67118b03659feefd3640824","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-03-10T17:45:51Z","title_canon_sha256":"2b38f0f2772e072d075f31f2ca1d7ab325e9b9dca7315f863d1d7625589fd823"},"schema_version":"1.0","source":{"id":"0803.1462","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0803.1462","created_at":"2026-05-18T04:06:56Z"},{"alias_kind":"arxiv_version","alias_value":"0803.1462v2","created_at":"2026-05-18T04:06:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.1462","created_at":"2026-05-18T04:06:56Z"},{"alias_kind":"pith_short_12","alias_value":"J6HVLMICH46U","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"J6HVLMICH46URBFT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"J6HVLMIC","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:c4778ccbda54925a4e3ff785d1e14568b78c70d171c237a5a0bae8e90349eb21","target":"graph","created_at":"2026-05-18T04:06:56Z","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 Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\\alpha y ^\\beta + x^\\beta y^\\alpha$ with $-1 < \\alpha \\leq \\beta < 1$ and $\\lambda := \\alpha + \\beta \\in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\\alpha = 0$ we prove that two profiles with the same mass and moment of order $\\lambda$ are necessarily equal, while in the ","authors_text":"Jos\\'e Alfredo Ca\\~nizo, St\\'ephane Mischler","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-03-10T17:45:51Z","title":"Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.1462","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:bae665653bbb3f0c4f8f39212d1ae08f4e9897c44b89b012e58acdf1dcc90178","target":"record","created_at":"2026-05-18T04:06:56Z","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":"e650a0534e3b566ce42e5f0d7be8d7243da081f4c67118b03659feefd3640824","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-03-10T17:45:51Z","title_canon_sha256":"2b38f0f2772e072d075f31f2ca1d7ab325e9b9dca7315f863d1d7625589fd823"},"schema_version":"1.0","source":{"id":"0803.1462","kind":"arxiv","version":2}},"canonical_sha256":"4f8f55b1023f3d4884b394a74dea835920d1d2b77e618f5f787faae26ddcf96d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4f8f55b1023f3d4884b394a74dea835920d1d2b77e618f5f787faae26ddcf96d","first_computed_at":"2026-05-18T04:06:56.979675Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:56.979675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zJzfCUXyvjEXnAwnua6+K5fhu9iAfHOgKADA0jilTdL/RiKS2XV8hx8BF+YE6rwahUgfn2thveTt4byXJTpsBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:56.980212Z","signed_message":"canonical_sha256_bytes"},"source_id":"0803.1462","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bae665653bbb3f0c4f8f39212d1ae08f4e9897c44b89b012e58acdf1dcc90178","sha256:c4778ccbda54925a4e3ff785d1e14568b78c70d171c237a5a0bae8e90349eb21"],"state_sha256":"ebf4fbfd953f0ecef7753e5f45b2502a1c5faa69191690c73bc06aa03024bbc3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cBLzT/sZNZlU373FYAWPhW8Q2Jly+Fz1O5M5ZBO04pRj9VYXYlZ7vkGyRmx3B3A4T/lTN3VSF3TC6xX7wROIBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T06:16:40.238045Z","bundle_sha256":"9ab0235015abe611c1834e52f3f0cbf44e8066514c868176b1c6f0c8f7cc742c"}}