{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JZNGQ5VTVFTF35HPKOAVE3KUHL","short_pith_number":"pith:JZNGQ5VT","canonical_record":{"source":{"id":"1501.02648","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-12T14:06:48Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"58a0b6e455add1c67bf1c5da18485ecc0b8520fbe9ca061fd7692c72a269bc25","abstract_canon_sha256":"4a70020b2b0d479ea5dbfea712bb231a8d8dd8e01069bfc5d8f3fe349fd9055e"},"schema_version":"1.0"},"canonical_sha256":"4e5a6876b3a9665df4ef5381526d543acf00d3e911a240e25da906428f14a713","source":{"kind":"arxiv","id":"1501.02648","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02648","created_at":"2026-05-18T02:29:37Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02648v1","created_at":"2026-05-18T02:29:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02648","created_at":"2026-05-18T02:29:37Z"},{"alias_kind":"pith_short_12","alias_value":"JZNGQ5VTVFTF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JZNGQ5VTVFTF35HP","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JZNGQ5VT","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JZNGQ5VTVFTF35HPKOAVE3KUHL","target":"record","payload":{"canonical_record":{"source":{"id":"1501.02648","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-12T14:06:48Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"58a0b6e455add1c67bf1c5da18485ecc0b8520fbe9ca061fd7692c72a269bc25","abstract_canon_sha256":"4a70020b2b0d479ea5dbfea712bb231a8d8dd8e01069bfc5d8f3fe349fd9055e"},"schema_version":"1.0"},"canonical_sha256":"4e5a6876b3a9665df4ef5381526d543acf00d3e911a240e25da906428f14a713","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:37.554328Z","signature_b64":"nbLki8P0MIAT/BslwTF7Yw3JbnatzN2e0IwDZRHY+EXiF9adpCi+vWTS+SWYAV4ECqYkjgb46mylsZI+E8F+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e5a6876b3a9665df4ef5381526d543acf00d3e911a240e25da906428f14a713","last_reissued_at":"2026-05-18T02:29:37.553891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:37.553891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.02648","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:29:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LD2kbCZ35Z104ajupC5nKMr2JC6Xp/CAswim5Dek3PaDrl4klo6PIxtwCFpNuXBFlOcmfNLL4Uvi/5EwTVEaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:12:40.380937Z"},"content_sha256":"083e791f9672a347209dedfafee520b0f874dc4e096369437f7cd7662d21ff04","schema_version":"1.0","event_id":"sha256:083e791f9672a347209dedfafee520b0f874dc4e096369437f7cd7662d21ff04"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JZNGQ5VTVFTF35HPKOAVE3KUHL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mean field dynamics of collisional processes with duplication, loss and copy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Federico Bassetti, Giuseppe Toscani","submitted_at":"2015-01-12T14:06:48Z","abstract_excerpt":"In this paper we introduce and discuss kinetic equations for the evolution of the probability distribution of the number of particles in a population subject to binary interactions. The microscopic binary law of interaction is assumed to be dependent on fixed-in-time random parameters which describe both birth and death of particles, and the migration rule. These assumptions lead to a Boltzmann-type equation that in the case in which the mean number of the population is preserved, can be fully studied, by obtaining in some case the analytic description of the steady profile. In all cases, howe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02648","kind":"arxiv","version":1},"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":"2026-05-18T02:29:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pk0018H15VuY3X/Xfu8I7D1crivLyhHI+T742IFe2eOS886vnpZVJvYpLMDHWPqo14n/8gDwOwDILFez0TWPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:12:40.381670Z"},"content_sha256":"dce0deb6c6b59a8135a3975c00ae7936bb90716e0ec65eab116268cfc61ee545","schema_version":"1.0","event_id":"sha256:dce0deb6c6b59a8135a3975c00ae7936bb90716e0ec65eab116268cfc61ee545"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL/bundle.json","state_url":"https://pith.science/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL/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":"2026-06-11T02:12:40Z","links":{"resolver":"https://pith.science/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL","bundle":"https://pith.science/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL/bundle.json","state":"https://pith.science/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JZNGQ5VTVFTF35HPKOAVE3KUHL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JZNGQ5VTVFTF35HPKOAVE3KUHL","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":"4a70020b2b0d479ea5dbfea712bb231a8d8dd8e01069bfc5d8f3fe349fd9055e","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-12T14:06:48Z","title_canon_sha256":"58a0b6e455add1c67bf1c5da18485ecc0b8520fbe9ca061fd7692c72a269bc25"},"schema_version":"1.0","source":{"id":"1501.02648","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02648","created_at":"2026-05-18T02:29:37Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02648v1","created_at":"2026-05-18T02:29:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02648","created_at":"2026-05-18T02:29:37Z"},{"alias_kind":"pith_short_12","alias_value":"JZNGQ5VTVFTF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JZNGQ5VTVFTF35HP","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JZNGQ5VT","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:dce0deb6c6b59a8135a3975c00ae7936bb90716e0ec65eab116268cfc61ee545","target":"graph","created_at":"2026-05-18T02:29:37Z","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 paper we introduce and discuss kinetic equations for the evolution of the probability distribution of the number of particles in a population subject to binary interactions. The microscopic binary law of interaction is assumed to be dependent on fixed-in-time random parameters which describe both birth and death of particles, and the migration rule. These assumptions lead to a Boltzmann-type equation that in the case in which the mean number of the population is preserved, can be fully studied, by obtaining in some case the analytic description of the steady profile. In all cases, howe","authors_text":"Federico Bassetti, Giuseppe Toscani","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-12T14:06:48Z","title":"Mean field dynamics of collisional processes with duplication, loss and copy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02648","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":"sha256:083e791f9672a347209dedfafee520b0f874dc4e096369437f7cd7662d21ff04","target":"record","created_at":"2026-05-18T02:29:37Z","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":"4a70020b2b0d479ea5dbfea712bb231a8d8dd8e01069bfc5d8f3fe349fd9055e","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-12T14:06:48Z","title_canon_sha256":"58a0b6e455add1c67bf1c5da18485ecc0b8520fbe9ca061fd7692c72a269bc25"},"schema_version":"1.0","source":{"id":"1501.02648","kind":"arxiv","version":1}},"canonical_sha256":"4e5a6876b3a9665df4ef5381526d543acf00d3e911a240e25da906428f14a713","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e5a6876b3a9665df4ef5381526d543acf00d3e911a240e25da906428f14a713","first_computed_at":"2026-05-18T02:29:37.553891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:37.553891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nbLki8P0MIAT/BslwTF7Yw3JbnatzN2e0IwDZRHY+EXiF9adpCi+vWTS+SWYAV4ECqYkjgb46mylsZI+E8F+CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:37.554328Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.02648","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:083e791f9672a347209dedfafee520b0f874dc4e096369437f7cd7662d21ff04","sha256:dce0deb6c6b59a8135a3975c00ae7936bb90716e0ec65eab116268cfc61ee545"],"state_sha256":"a38027217be9d25fc8780b97bbeebc5fe5a32a4dbf6ddb490446d8e60dff10c7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FcveM1VLtUheI1SE+MF6mFTtUOR/XU2eqqQfTXk0lwVBTiL48t0rVGWdy/YYh5uwT0j0yEUGOB7VVxFkdw9XCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T02:12:40.385580Z","bundle_sha256":"4833fc4b5f07d481a6776d5da7b832f4628f86e2e7f8507088b8d9fa4d193901"}}