{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3JKVJZLF6ZZOCZUYN3XQRT5CFU","short_pith_number":"pith:3JKVJZLF","canonical_record":{"source":{"id":"2601.01453","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.AP","submitted_at":"2026-01-04T09:39:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"133970bb49470ea15baab12af627e9b963efe1610421fc955db00bfa6e427fe3","abstract_canon_sha256":"433d7f0ea5d2560ed66a9a7124a446d0193f4c795edfaa75a0a250610232cc7c"},"schema_version":"1.0"},"canonical_sha256":"da5554e565f672e166986eef08cfa22d2d7763d3dcd86cee47671b9e61e19e17","source":{"kind":"arxiv","id":"2601.01453","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.01453","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"arxiv_version","alias_value":"2601.01453v2","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.01453","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"pith_short_12","alias_value":"3JKVJZLF6ZZO","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"pith_short_16","alias_value":"3JKVJZLF6ZZOCZUY","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"pith_short_8","alias_value":"3JKVJZLF","created_at":"2026-06-01T01:02:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3JKVJZLF6ZZOCZUYN3XQRT5CFU","target":"record","payload":{"canonical_record":{"source":{"id":"2601.01453","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.AP","submitted_at":"2026-01-04T09:39:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"133970bb49470ea15baab12af627e9b963efe1610421fc955db00bfa6e427fe3","abstract_canon_sha256":"433d7f0ea5d2560ed66a9a7124a446d0193f4c795edfaa75a0a250610232cc7c"},"schema_version":"1.0"},"canonical_sha256":"da5554e565f672e166986eef08cfa22d2d7763d3dcd86cee47671b9e61e19e17","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:02:28.315985Z","signature_b64":"5qwl1TGDUbWBcOf2vZcU/lsWbBenaYBnG0jmFpgWU2T0CiBl9TUDR56p3Ws6ty2SqWn77W9kezQTeKEWXCuUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da5554e565f672e166986eef08cfa22d2d7763d3dcd86cee47671b9e61e19e17","last_reissued_at":"2026-06-01T01:02:28.314860Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:02:28.314860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2601.01453","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":"2026-06-01T01:02:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qbuRcl36QoN1uBfzdcuFkzyVjZMJNqHuwIYY0lpK2YH1+xLnZCVAQLJ2mk561kAOFKPqPGU17C9wyQ8GvY77Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T09:18:44.579251Z"},"content_sha256":"dad6a7fae55876d2ecf48cc7e2309cd9274d486b53a0b23d888c281f8d54de2f","schema_version":"1.0","event_id":"sha256:dad6a7fae55876d2ecf48cc7e2309cd9274d486b53a0b23d888c281f8d54de2f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3JKVJZLF6ZZOCZUYN3XQRT5CFU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fragmentation-coagulation processes with advection or diffusion in space","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Jacek Banasiak, Nduduzo Majozi","submitted_at":"2026-01-04T09:39:36Z","abstract_excerpt":"In this paper, we consider a continuous fragmentation--coagulation model in which the reacting particles can be transported in physical space through either advection or diffusion. We prove new results on the generation of $C_0$-semigroups with parameter and use them to show that the Abstract Cauchy Problem associated with a more general version of the advection/diffusion--fragmentation problem generates a positive $C_0$-semigroup in spaces $L_1(\\mathbb R_+, X_x, (1+m^r)dm),$ where $m$ is the particle mass, $X_x$ is either the space of integrable or continuous functions with respect to the spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.01453","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.01453/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-01T01:02:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TwV7T7eseTde2lUhpdlE0PZ21KgmatS3yo2QyT1V7jBRRBrvkfVD2eFy+GnDYlEaaC2Z/2GXCf/DGiXFHHscAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T09:18:44.580000Z"},"content_sha256":"79887606344bdc57a998b0f88547b74312ca1569a70b48f46cb864aa03a8b05a","schema_version":"1.0","event_id":"sha256:79887606344bdc57a998b0f88547b74312ca1569a70b48f46cb864aa03a8b05a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU/bundle.json","state_url":"https://pith.science/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU/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":"2026-06-10T09:18:44Z","links":{"resolver":"https://pith.science/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU","bundle":"https://pith.science/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU/bundle.json","state":"https://pith.science/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3JKVJZLF6ZZOCZUYN3XQRT5CFU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3JKVJZLF6ZZOCZUYN3XQRT5CFU","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":"433d7f0ea5d2560ed66a9a7124a446d0193f4c795edfaa75a0a250610232cc7c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.AP","submitted_at":"2026-01-04T09:39:36Z","title_canon_sha256":"133970bb49470ea15baab12af627e9b963efe1610421fc955db00bfa6e427fe3"},"schema_version":"1.0","source":{"id":"2601.01453","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.01453","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"arxiv_version","alias_value":"2601.01453v2","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.01453","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"pith_short_12","alias_value":"3JKVJZLF6ZZO","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"pith_short_16","alias_value":"3JKVJZLF6ZZOCZUY","created_at":"2026-06-01T01:02:28Z"},{"alias_kind":"pith_short_8","alias_value":"3JKVJZLF","created_at":"2026-06-01T01:02:28Z"}],"graph_snapshots":[{"event_id":"sha256:79887606344bdc57a998b0f88547b74312ca1569a70b48f46cb864aa03a8b05a","target":"graph","created_at":"2026-06-01T01:02:28Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2601.01453/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we consider a continuous fragmentation--coagulation model in which the reacting particles can be transported in physical space through either advection or diffusion. We prove new results on the generation of $C_0$-semigroups with parameter and use them to show that the Abstract Cauchy Problem associated with a more general version of the advection/diffusion--fragmentation problem generates a positive $C_0$-semigroup in spaces $L_1(\\mathbb R_+, X_x, (1+m^r)dm),$ where $m$ is the particle mass, $X_x$ is either the space of integrable or continuous functions with respect to the spa","authors_text":"Jacek Banasiak, Nduduzo Majozi","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.AP","submitted_at":"2026-01-04T09:39:36Z","title":"Fragmentation-coagulation processes with advection or diffusion in space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.01453","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:dad6a7fae55876d2ecf48cc7e2309cd9274d486b53a0b23d888c281f8d54de2f","target":"record","created_at":"2026-06-01T01:02:28Z","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":"433d7f0ea5d2560ed66a9a7124a446d0193f4c795edfaa75a0a250610232cc7c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.AP","submitted_at":"2026-01-04T09:39:36Z","title_canon_sha256":"133970bb49470ea15baab12af627e9b963efe1610421fc955db00bfa6e427fe3"},"schema_version":"1.0","source":{"id":"2601.01453","kind":"arxiv","version":2}},"canonical_sha256":"da5554e565f672e166986eef08cfa22d2d7763d3dcd86cee47671b9e61e19e17","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da5554e565f672e166986eef08cfa22d2d7763d3dcd86cee47671b9e61e19e17","first_computed_at":"2026-06-01T01:02:28.314860Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T01:02:28.314860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5qwl1TGDUbWBcOf2vZcU/lsWbBenaYBnG0jmFpgWU2T0CiBl9TUDR56p3Ws6ty2SqWn77W9kezQTeKEWXCuUAw==","signature_status":"signed_v1","signed_at":"2026-06-01T01:02:28.315985Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.01453","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dad6a7fae55876d2ecf48cc7e2309cd9274d486b53a0b23d888c281f8d54de2f","sha256:79887606344bdc57a998b0f88547b74312ca1569a70b48f46cb864aa03a8b05a"],"state_sha256":"fdea3bc28ee9f2da79fccf677ce961559a35f42ba358ae306ba25455316be2d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U9RS8EEzoBtgcct1ma5lKIKZPpF/rnv8t5zOHufxCj900D4yBRKKXy68AN4x8TynFXr6X00bvrEuGEFknHwpBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T09:18:44.583491Z","bundle_sha256":"89de2e6fabaead814de0803c1d6dae7710158026f08563d1c271e579aae0dc45"}}