{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:OP2DAMQJCE5V6CUQFY3KJOVRMG","short_pith_number":"pith:OP2DAMQJ","canonical_record":{"source":{"id":"1410.4681","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-17T10:26:28Z","cross_cats_sorted":[],"title_canon_sha256":"182d583c52d1c9221f52e013743b9480d6e9fc4d6793a193067515a8a35cc65f","abstract_canon_sha256":"341dce9ff95918f90d5f4f4a06c64c6105e9f9e518a3ea7fb5b0b97b031565c0"},"schema_version":"1.0"},"canonical_sha256":"73f4303209113b5f0a902e36a4bab161a5d2d230dde0f3284fa0a5fad7c6ede8","source":{"kind":"arxiv","id":"1410.4681","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.4681","created_at":"2026-05-18T01:37:28Z"},{"alias_kind":"arxiv_version","alias_value":"1410.4681v2","created_at":"2026-05-18T01:37:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.4681","created_at":"2026-05-18T01:37:28Z"},{"alias_kind":"pith_short_12","alias_value":"OP2DAMQJCE5V","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OP2DAMQJCE5V6CUQ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OP2DAMQJ","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:OP2DAMQJCE5V6CUQFY3KJOVRMG","target":"record","payload":{"canonical_record":{"source":{"id":"1410.4681","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-17T10:26:28Z","cross_cats_sorted":[],"title_canon_sha256":"182d583c52d1c9221f52e013743b9480d6e9fc4d6793a193067515a8a35cc65f","abstract_canon_sha256":"341dce9ff95918f90d5f4f4a06c64c6105e9f9e518a3ea7fb5b0b97b031565c0"},"schema_version":"1.0"},"canonical_sha256":"73f4303209113b5f0a902e36a4bab161a5d2d230dde0f3284fa0a5fad7c6ede8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:28.617244Z","signature_b64":"4aL4IO/vsB3TyaLP7gaSnX49dTa6hwK4cjVUyFs/98Kr/9GR/UMqE55B8Z6gPX3CX8bzrkPaOQDfYn+4INDPDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73f4303209113b5f0a902e36a4bab161a5d2d230dde0f3284fa0a5fad7c6ede8","last_reissued_at":"2026-05-18T01:37:28.616539Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:28.616539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.4681","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":"2026-05-18T01:37:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+OE3eJggN9+ir5+xS+ghZCPMAX/BU0d7x4r5TVjrxPNWIn/KkXS7QElPQUBwGtOxh79Q1aIGVZT5ui5QREw3DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:51:11.849455Z"},"content_sha256":"5f4a38bc6289d7d30cb2e60c0f84472ea55123a322bba4c4259a3ee2eb07e446","schema_version":"1.0","event_id":"sha256:5f4a38bc6289d7d30cb2e60c0f84472ea55123a322bba4c4259a3ee2eb07e446"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:OP2DAMQJCE5V6CUQFY3KJOVRMG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence and Uniqueness of Solution of a Continuous Flow Bioreactor Model with Two Species","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A.M Ramos, B. Ivorra, M. Crespo","submitted_at":"2014-10-17T10:26:28Z","abstract_excerpt":"In this work, we study the mathematical analysis of a coupled system of two reaction-diffusion-advection equations and Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacterias) and a diluted substrate (e.g., nitrate) in a continuous flow bioreactor. This type of bioreactor can be used, for instance, for water treatment. First, we prove the existence and uniqueness of solution, under the hypothesis of linear reaction by using classical results for linear parabolic boundary value problems. Next, we prove the existence and uniqueness of solution"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4681","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":"2026-05-18T01:37:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lbkSlXvcHNzJD8sVYrqoGNAH5MxZLHHFw7xgW5Obb1+JbOx5mK0t/KA5037hmh4th2hBueHsU04D3BfIbEKGDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:51:11.850135Z"},"content_sha256":"897fe1ff2e5a4e4f511cb15c685c3db6974a0ea5bf5825cf8f5a4ea73ebddbcf","schema_version":"1.0","event_id":"sha256:897fe1ff2e5a4e4f511cb15c685c3db6974a0ea5bf5825cf8f5a4ea73ebddbcf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG/bundle.json","state_url":"https://pith.science/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG/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":"2026-06-05T03:51:11Z","links":{"resolver":"https://pith.science/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG","bundle":"https://pith.science/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG/bundle.json","state":"https://pith.science/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OP2DAMQJCE5V6CUQFY3KJOVRMG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OP2DAMQJCE5V6CUQFY3KJOVRMG","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":"341dce9ff95918f90d5f4f4a06c64c6105e9f9e518a3ea7fb5b0b97b031565c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-17T10:26:28Z","title_canon_sha256":"182d583c52d1c9221f52e013743b9480d6e9fc4d6793a193067515a8a35cc65f"},"schema_version":"1.0","source":{"id":"1410.4681","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.4681","created_at":"2026-05-18T01:37:28Z"},{"alias_kind":"arxiv_version","alias_value":"1410.4681v2","created_at":"2026-05-18T01:37:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.4681","created_at":"2026-05-18T01:37:28Z"},{"alias_kind":"pith_short_12","alias_value":"OP2DAMQJCE5V","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OP2DAMQJCE5V6CUQ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OP2DAMQJ","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:897fe1ff2e5a4e4f511cb15c685c3db6974a0ea5bf5825cf8f5a4ea73ebddbcf","target":"graph","created_at":"2026-05-18T01:37:28Z","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"},"paper":{"abstract_excerpt":"In this work, we study the mathematical analysis of a coupled system of two reaction-diffusion-advection equations and Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacterias) and a diluted substrate (e.g., nitrate) in a continuous flow bioreactor. This type of bioreactor can be used, for instance, for water treatment. First, we prove the existence and uniqueness of solution, under the hypothesis of linear reaction by using classical results for linear parabolic boundary value problems. Next, we prove the existence and uniqueness of solution","authors_text":"A.M Ramos, B. Ivorra, M. Crespo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-17T10:26:28Z","title":"Existence and Uniqueness of Solution of a Continuous Flow Bioreactor Model with Two Species"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4681","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":"sha256:5f4a38bc6289d7d30cb2e60c0f84472ea55123a322bba4c4259a3ee2eb07e446","target":"record","created_at":"2026-05-18T01:37:28Z","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":"341dce9ff95918f90d5f4f4a06c64c6105e9f9e518a3ea7fb5b0b97b031565c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-17T10:26:28Z","title_canon_sha256":"182d583c52d1c9221f52e013743b9480d6e9fc4d6793a193067515a8a35cc65f"},"schema_version":"1.0","source":{"id":"1410.4681","kind":"arxiv","version":2}},"canonical_sha256":"73f4303209113b5f0a902e36a4bab161a5d2d230dde0f3284fa0a5fad7c6ede8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73f4303209113b5f0a902e36a4bab161a5d2d230dde0f3284fa0a5fad7c6ede8","first_computed_at":"2026-05-18T01:37:28.616539Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:28.616539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4aL4IO/vsB3TyaLP7gaSnX49dTa6hwK4cjVUyFs/98Kr/9GR/UMqE55B8Z6gPX3CX8bzrkPaOQDfYn+4INDPDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:28.617244Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.4681","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5f4a38bc6289d7d30cb2e60c0f84472ea55123a322bba4c4259a3ee2eb07e446","sha256:897fe1ff2e5a4e4f511cb15c685c3db6974a0ea5bf5825cf8f5a4ea73ebddbcf"],"state_sha256":"45c9b1af6674d98e88287a2107775b30431e38b5d3817c5eb1dc5c02acd162f9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HeyCMv7kn88TuIBhzY0+mfX6Uq+LA4nrhm6mDgyt3TiHwbAeLzUPZka0b24p+w34q9IIr0Iyw8HBP8QWVB8UCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T03:51:11.853682Z","bundle_sha256":"cb34eb8da0d20d07aad05f0c55a2e2aeb479df918a6ab7cd489b3cacd911ac4e"}}