{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:X4WPR64JAEHXRMQFD2WIE4QTB5","short_pith_number":"pith:X4WPR64J","canonical_record":{"source":{"id":"2312.09692","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2023-12-15T11:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"37599845cf10538807a228c52a6b86f2e20b40fe3ee4d2f718add56ca8ae8815","abstract_canon_sha256":"f75ac0ec52e47a03427aa9208bff7de2529c5e2def3e1be49e2c87c366992bcc"},"schema_version":"1.0"},"canonical_sha256":"bf2cf8fb89010f78b2051eac8272130f49e3e9710b16bc9fb13a8923360eb707","source":{"kind":"arxiv","id":"2312.09692","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2312.09692","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"arxiv_version","alias_value":"2312.09692v2","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.09692","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"pith_short_12","alias_value":"X4WPR64JAEHX","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"pith_short_16","alias_value":"X4WPR64JAEHXRMQF","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"pith_short_8","alias_value":"X4WPR64J","created_at":"2026-07-05T08:31:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:X4WPR64JAEHXRMQFD2WIE4QTB5","target":"record","payload":{"canonical_record":{"source":{"id":"2312.09692","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2023-12-15T11:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"37599845cf10538807a228c52a6b86f2e20b40fe3ee4d2f718add56ca8ae8815","abstract_canon_sha256":"f75ac0ec52e47a03427aa9208bff7de2529c5e2def3e1be49e2c87c366992bcc"},"schema_version":"1.0"},"canonical_sha256":"bf2cf8fb89010f78b2051eac8272130f49e3e9710b16bc9fb13a8923360eb707","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:31:45.550615Z","signature_b64":"rIBF599OaBc8ERRyyt2VqhLxtSMYaW+hcW+2CDqg6wS6Ki3enVAaNL2c/1QAu397x3iV277onCOmMvH/wraGAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf2cf8fb89010f78b2051eac8272130f49e3e9710b16bc9fb13a8923360eb707","last_reissued_at":"2026-07-05T08:31:45.550199Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:31:45.550199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2312.09692","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":"2026-07-05T08:31:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cmpqeiX37Nxf/8Z9fB//wrq8syMdHUgHmtbwUaN55g02f2eLhvOakLfMpChfHxu5nVK0QBNtnzyCnTRz9wGPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T15:52:40.422452Z"},"content_sha256":"ab15c9cf57c94f87936b91567e2f9b2a988f8ed701ece716a9f5aeef4f20817f","schema_version":"1.0","event_id":"sha256:ab15c9cf57c94f87936b91567e2f9b2a988f8ed701ece716a9f5aeef4f20817f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:X4WPR64JAEHXRMQFD2WIE4QTB5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positivity and global existence for nonlocal advection-diffusion models of interacting populations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jonathan Potts, Mark Lewis, Thomas Hillen, Valeria Giunta","submitted_at":"2023-12-15T11:13:31Z","abstract_excerpt":"In this paper we study a broad class of non-local advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the non-local presence of others. Our model allows for different non-local interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of non-local advection-diffusion equation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.09692","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/2312.09692/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":"2026-07-05T08:31:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2BQWQf54H0gflNr1fBmls4NfDHgcLDzT6K5Iv+6Nu2vNv+3x55Ex0WGyhO1qJQXV/3wkhwiQCUPV2QCRXUotCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T15:52:40.423358Z"},"content_sha256":"b21fc4704faa67ac2105d322df17bd981d2230860da5a6e84696dc44544fdaec","schema_version":"1.0","event_id":"sha256:b21fc4704faa67ac2105d322df17bd981d2230860da5a6e84696dc44544fdaec"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X4WPR64JAEHXRMQFD2WIE4QTB5/bundle.json","state_url":"https://pith.science/pith/X4WPR64JAEHXRMQFD2WIE4QTB5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X4WPR64JAEHXRMQFD2WIE4QTB5/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":"2026-08-08T15:52:40Z","links":{"resolver":"https://pith.science/pith/X4WPR64JAEHXRMQFD2WIE4QTB5","bundle":"https://pith.science/pith/X4WPR64JAEHXRMQFD2WIE4QTB5/bundle.json","state":"https://pith.science/pith/X4WPR64JAEHXRMQFD2WIE4QTB5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X4WPR64JAEHXRMQFD2WIE4QTB5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:X4WPR64JAEHXRMQFD2WIE4QTB5","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":"f75ac0ec52e47a03427aa9208bff7de2529c5e2def3e1be49e2c87c366992bcc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2023-12-15T11:13:31Z","title_canon_sha256":"37599845cf10538807a228c52a6b86f2e20b40fe3ee4d2f718add56ca8ae8815"},"schema_version":"1.0","source":{"id":"2312.09692","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2312.09692","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"arxiv_version","alias_value":"2312.09692v2","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.09692","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"pith_short_12","alias_value":"X4WPR64JAEHX","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"pith_short_16","alias_value":"X4WPR64JAEHXRMQF","created_at":"2026-07-05T08:31:45Z"},{"alias_kind":"pith_short_8","alias_value":"X4WPR64J","created_at":"2026-07-05T08:31:45Z"}],"graph_snapshots":[{"event_id":"sha256:b21fc4704faa67ac2105d322df17bd981d2230860da5a6e84696dc44544fdaec","target":"graph","created_at":"2026-07-05T08:31:45Z","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/2312.09692/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we study a broad class of non-local advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the non-local presence of others. Our model allows for different non-local interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of non-local advection-diffusion equation","authors_text":"Jonathan Potts, Mark Lewis, Thomas Hillen, Valeria Giunta","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2023-12-15T11:13:31Z","title":"Positivity and global existence for nonlocal advection-diffusion models of interacting populations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.09692","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:ab15c9cf57c94f87936b91567e2f9b2a988f8ed701ece716a9f5aeef4f20817f","target":"record","created_at":"2026-07-05T08:31:45Z","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":"f75ac0ec52e47a03427aa9208bff7de2529c5e2def3e1be49e2c87c366992bcc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2023-12-15T11:13:31Z","title_canon_sha256":"37599845cf10538807a228c52a6b86f2e20b40fe3ee4d2f718add56ca8ae8815"},"schema_version":"1.0","source":{"id":"2312.09692","kind":"arxiv","version":2}},"canonical_sha256":"bf2cf8fb89010f78b2051eac8272130f49e3e9710b16bc9fb13a8923360eb707","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf2cf8fb89010f78b2051eac8272130f49e3e9710b16bc9fb13a8923360eb707","first_computed_at":"2026-07-05T08:31:45.550199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:31:45.550199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rIBF599OaBc8ERRyyt2VqhLxtSMYaW+hcW+2CDqg6wS6Ki3enVAaNL2c/1QAu397x3iV277onCOmMvH/wraGAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:31:45.550615Z","signed_message":"canonical_sha256_bytes"},"source_id":"2312.09692","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ab15c9cf57c94f87936b91567e2f9b2a988f8ed701ece716a9f5aeef4f20817f","sha256:b21fc4704faa67ac2105d322df17bd981d2230860da5a6e84696dc44544fdaec"],"state_sha256":"b3485f6b764b327990b4103fd508fc56f4b9f2808beb5a125ab220be3d885866"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"azha/0ajpnPn4pQjfjekfGcjUAdFXH3+0fdjsRC/HzHdRcsPhjmo6NiYVCyU2r9ToqZJWfIX7JCNOfJCc115Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T15:52:40.494001Z","bundle_sha256":"9f5712cee1f05c756c89c1b6ab5b9968be440d226310a41cc28b07af2d5de127"}}